springer series in light scattering: volume 2: light scattering, radiative transfer and remote

Springer Series in Light Scattering


Alexander Kokhanovsky Editor

Volume 2: Light Scattering, Radiative Transfer and Remote Sensing


Series editor

Alexander Kokhanovsky, Vitrociset Belgium, Darmstadt, Germany

Editorial Advisory Board

Thomas Henning, Max Planck Institute for Astronomy, Heidelberg, GermanyGeorge Kattawar, Texas A&M University, College Station, USAOleg Kopelevich, Shirshov Institute of Oceanology, Moscow, RussiaKuo-Nan Liou, University of California, Los Angeles, USAMichael Mishchenko, NASA Goddard Institute for Space Studies, New York, USALev Perelman, Harvard University, Cambridge, USAKnut Stamnes, Stevens Institute of Technology, Hoboken, USAGraeme Stephens, Jet Propulsion Laboratory, Los Angeles, USABart van Tiggelen, J. Fourier University, Grenoble, FranceClaudio Tomasi, Institute of Atmospheric Sciences and Climate, Bologna, Italy

The main purpose of new SPRINGER Series in Light Scattering is to presentrecent advances and progress in light scattering media optics. The topic is verybroad and incorporates such diverse areas as atmospheric optics, ocean optics,optics of close-packed media, radiative transfer, light scattering, absorption, andscattering by single scatterers and also by systems of particles, biomedical optics,optical properties of cosmic dust, remote sensing of atmosphere and ocean, etc.The topic is of importance for material science, environmental science, climatechange, and also for optical engineering. Although main developments in thesolutions of radiative transfer and light scattering problems have been achieved inthe 20th century by efforts of many scientists including V. Ambartsumian,S. Chandrasekhar, P. Debye, H. C. van de Hulst, G. Mie, and V. Sobolev, the lightscattering media optics still have many puzzles to be solved such as radiativeransfer in closely packed media, 3D radiative transfer as applied to the solution ofinverse problems, optics of terrestrial and planetary surfaces, etc. Also it has a broadrange of applications in many brunches of modern science and technology such asbiomedical optics, atmospheric and oceanic optics, and astrophysics, to name a few.It is planned that the Series will raise novel scientific questions, integrate dataanalysis, and offer new insights in optics of light scattering media.

More information about this series at http://www.springer.com/series/15365

http://www.springer.com/series/15365

Alexander KokhanovskyEditor

Springer Series in LightScatteringVolume 2: Light Scattering, RadiativeTransfer and Remote Sensing

123

EditorAlexander KokhanovskyVitrociset BelgiumDarmstadtGermany

ISSN 2509-2790 ISSN 2509-2804 (electronic)Springer Series in Light ScatteringISBN 978-3-319-70807-2 ISBN 978-3-319-70808-9 (eBook)https://doi.org/10.1007/978-3-319-70808-9

Library of Congress Control Number: 2017957680

© Springer International Publishing AG 2018This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

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Contents

1 Polarized Radiative Transfer in Optically Active Light ScatteringMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Margarita G. Kuzmina, Leonid P. Bass and Olga V. Nikolaeva

2 Advances in Spectro-Polarimetric Light-Scattering by ParticulateMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Romain Ceolato and Nicolas Riviere

3 Light Scattering by Large Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . 109Fabrice R. A. Onofri and Matthias P. L. Sentis

4 Volume Scattering Function of Seawater . . . . . . . . . . . . . . . . . . . . . 151Michael E. Lee and Elena N. Korchemkina

5 Remote Sensing of Crystal Shapes in Ice Clouds . . . . . . . . . . . . . . . 197Bastiaan van Diedenhoven

6 Light Scattering in Combustion: New Developments . . . . . . . . . . . . 251Alan R. Jones

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

v

Contributors

Leonid P. Bass Keldysh Institute of Applied Mathematics RAS, Moscow, Russia

Romain Ceolato Optronics Department, ONERA, The French Aerospace Lab,Toulouse, France

Bastiaan van Diedenhoven Center for Climate System Research, ColumbiaUniversity, New York, NY, USA

Alan R. Jones Department of Chemical Engineering, Imperial College, London,UK

Elena N. Korchemkina Marine Hydrophysical Institute of RAS, Sevastopol,Russia

Margarita G. Kuzmina Keldysh Institute of Applied Mathematics RAS,Moscow, Russia

Michael E. Lee Marine Hydrophysical Institute of RAS, Sevastopol, Russia

Olga V. Nikolaeva Keldysh Institute of Applied Mathematics RAS, Moscow,Russia

Fabrice R. A. Onofri IUSTI (UMR 7343, Aix-Marseille Université), NationalCenter for Scientific Research (CNRS), Marseille cedex 13, France

Nicolas Riviere Optronics Department, ONERA, The French Aerospace Lab,Toulouse, France

Matthias P. L. Sentis DEN/DMRC/SA2I/LGCI, Atomic Energy and AlternativeEnergies Commission (CEA), Bagnols-sur-Cèze, France

vii

Chapter 1Polarized Radiative Transfer in OpticallyActive Light Scattering Media

Margarita G. Kuzmina, Leonid P. Bass and Olga V. Nikolaeva

1.1 Introduction

The disperse media composed of non-spherical particles (say, dust aerosols layers,and ice crystal clouds) can appear both optically isotropic and optically anisotropic,depending on local optical characteristics of turbid medium in question and also onthe orientation of particles.

Chiral media belong to the type of optically anisotropic media that is charac-terized by circular birefringence and circular dichroism (different medium refractiveindex and different absorption of left-handed and right-handed circularly polarizedradiation). The media can be composed either of spherical particles consisting ofoptically active matter or of particles of special shape (non-spherical shape withbroken mirror symmetry). In the Earth atmosphere remote sensing problems theanisotropic media can be produced by ensembles of non-spherical aerosol particles,and ice crystals. The anisotropic media can also arise in the situations when aerosolcontains a mixed combination of organic and inorganic particles, and the biologicalaerosol component dominates.

Polarization characteristics of scattered radiation can provide a valuable infor-mation on medium optical properties and medium miscrostructure. The adequateinterpretation of optical and scattering characteristics of optically anisotropic mediacan be done on the solid ground of the polarized radiation transport theory inoptically anisotropic media (using the vector radiation transport equation, the VRTE,

M. G. Kuzmina (&) � L. P. Bass � O. V. NikolaevaKeldysh Institute of Applied Mathematics RAS, Miusskaya pl. 4, Moscow, Russiae-mail: [email protected]

L. P. Basse-mail: [email protected]

O. V. Nikolaevae-mail: [email protected]

© Springer International Publishing AG 2018A. Kokhanovsky (ed.), Springer Series in Light Scattering, Springer Seriesin Light Scattering, https://doi.org/10.1007/978-3-319-70808-9_1

1

for anisotropic media). For the retrieval algorithms, based on the inverse radiationtransfer problem solutions, the data of multi-angular and multi-spectral measure-ments of the Stokes vector of back-scattered solar radiation are used. Usually a kindof statistically optimized problem solution is applied (relied either on the usage oflook-up-tables or on direct radiative transfer calculations). An accurate accounting ofterrestrial underlying surface reflectance is also quite essential. Realistic models ofdisperse media play a significant role in the construction of the retrieval algorithms.

A detailed overview of recently developed aerosol retrieval algorithms, based onmeasurements of back-scattered polarized radiation, is given in (Kokhanovsky2015). The proper computational algorithms and codes for accurate VRTE solutionare necessary for realization of the developed retrieval procedures. Among a varietyof the developed RT-codes, the codes developed in (Katsev et al. 2009;Kokhanovsky et al. 2010; Cairns et al. 2010; Cheng et al. 2011; Dubovik et al.2011; Hasekamp 2011; Knobelspiesse 2011) should be marked. The TR-code,developed in (Nikolaeva et al. 2007; Bass et al. 2009, 2010) should be marked. TheTR-code, developed in (Nikolaeva et al. 2007; Bass et al. 2009, 2010) for theVRTE solution in 3D cylindrical geometry and successfully tested in a number ofatmosphere remote sensing problems, could be also mentioned as one of availablecodes for extension to radiation transfer problems for optically anisotropic media.

The overview of main topics considered in this review is given below. They arerelated to various aspects of polarized radiation transfer processes in opticallyanisotropic media.

In Sect. 1.2 the essential steps of vector transport equation deriving from thesystem of Maxwell equations for the problem of electromagnetic radiation multiplescattering by an ensemble of discrete isolated scatterers are outlined. The attention ispaid on the set of restrictions imposed on the system ensemble of scatterers—1953,radiation field in the process of transport equation deriving. Thework on the topic wasstarted long ago (Foldy 1945; Lax 1951; Watson 1953, 1969) and finally allowed toobtain the matrix and the vector transport equations for optically anisotropic media(Dolginov et al. 1970, 1995; Newton 1982; Kuzmina 1976, 1986a, b, 1987, 1989,1991; Zege and Chaikovskaya 1984; Kokhanovsky 1999a, b; Kokhanovsky 2000).Subsequently the strict and detailed way of the VRTE derivation, realized in(Mishchenko et al. 2002, 2006, 2007, 2011, 2016a, b; Mishchenko 2002, 2003,2008a, b, 2010, 2011, 2014a, b), allowed to additionally study the phenomenon ofmedium coherent backscattering (CB) (otherwise known as weak localization ofelectromagnetic waves) (Barabanenkov 1973; Barabanenkov et al. 1991, 1995;Mishchenko et al. 2002, 2006, 2007, 2011). Because the four-component vectortransport equation can be correctly used for radiation transport problems only in thecase ofweakly anisotropicmedia (Born et al. 1975;Kravtsov et al. 2007;Kravtsov andBieg 2010) some necessary information on quasi-isotropic approximation of geo-metrical optics for weakly anisotropic media is included as well (Sect. 1.2.3).

In Sect. 1.3 the peculiarities of radiation transfer processes in anisotropic opti-cally active media are discussed. The characteristic features of the vector transportequation for optically active media are the matrix extinction operator (that can beexpressed in terms of medium refraction indices), and the integral operator of

2 M. G. Kuzmina et al.

scattering, defined by the non-block-diagonal phase matrix of special type),(Sect. 1.3.1). The main properties of radiation transport problems for slabs ofoptically active media (including boundary conditions) are marked in Sects. 1.3.2and 1.3.3. The polarization characteristics of coherently scattered (refracted andattenuated) radiation propagating in slabs of optically active media, that can beobtained analytically, are presented in Sects. 1.3.4 and 1.3.5. The transport prob-lems for slabs with reflecting boundaries are discussed as well (Sect. 1.3.6).

In Sect. 1.4 the perturbation method developed for transport problems in slabs ofweakly anisotropic optically active media is presented. The method can be used forthe estimation of the total Stokes vector perturbation due to medium optical ani-sotropy (Sect. 1.4.1). The example of estimation of transport problem solutionperturbation for a slab of optically isotropic medium with scattering operator,specified by non-block-diagonal phase matrix, is given in Sect. 1.4.3. Similarly thepolarization characteristics perturbation due to utilizing of the transport equationwith scalar extinction operator (instead of the matrix one, valid for optically ani-sotropic medium) could be estimated. The situation has already been encounteredearlier in the study of multi-scattered polarized infrared radiation transport in ani-sotropic media formed by horizontally oriented ice crystals (Takano et al. 1993). Asit was pointed out (Mishchenko 1994a, b), the utilization of the VRTE with scalarextinction operator could provide a significant error in solution of the transportproblems. The comparison of the exact and the approximate solutions of similartransport problem for another type of anisotropic medium model (composed ofperfectly aligned prolate and oblate spheroids) has been fulfilled previously in(Tsang et al. 1991). And a significant discrepancy in solutions was demonstrated.

In Sect. 1.5 the results on radiative transfer problems in anisotropic media relatedto the Earth atmosphere remote sensing are presented. First of all these were theproblems for ice clouds (cirrus and cirrostratus), where disperse anisotropicmedia canbe formed by spatially oriented suspended tiny ice crystals. The well-known atmo-spheric optical phenomenon of halo is just created by light reflection from theseanisotropic media. Another familiar phenomenon is light pillars that is produced bylight reflection from anisotropic media formed by column-shaped ice crystals.Modeling of radiative transfer in turbid anisotropic media requires for construction ofthematrix extinction operator and the scattering phasematrix of theVRTE, governingradiation transport in anisotropicmedium. The Sect. 1.5.1 contains an overview of thepapers where various models of disperse anisotropic media were designed and theoperators of the VRTE were constructed. In particular, the disperse medium models,composed of chiral particles, were considered, and the extinction matrices for themedia were constructed (Ablitt et al. 2006; Liu et al. 2013). The multiply scatteredlight transfer in the chiral anisotropic medium was studied via Monte Carlo simula-tions, and the effects of medium chirality were elucidated (Ablitt et al. 2006). Themodels allowed to study the dependence of medium scattering macro-characteristicson the medium micro-structure parameters. For some medium models the backscat-tering efficiencies were estimated as well (Mishchenko et al. 1992; Gao et al. 2012).

The Monte-Carlo simulations of radiation transfer processes in various opticallyanisotropic media models of ice clouds were performed (including the simulations

1 Polarized Radiative Transfer in Optically … 3

of halos). Particularly, these results demonstrated, that the anisotropy of cloudmedium can strongly affect the optical properties of crystalline clouds (Prigarinet al. 2005, 2007, 2008).

Densely packed disperse media (where the assumption concerning scattererlocations in far zones of each other is violated) are shortly reviewed in Sect. 1.5.1(B). The ice and snow cover media, representing the interest for the Earth remotesensing problems, are usually modelled as random disperse media with denselypacked particles (Kokhanovsky 1998). Sometimes (mainly in the visible wave-length range) the snow layers may be also modelled as ice clouds consisting ofseparated fractal particles (Kokhanovsky 2003; Liou et al. 2011). The deriving ofthe VRTE for densely packed disperse media is not a simple task. For that reasonthe exact computer studies (in terms of the Maxwell equations) of multiply scatteredradiation transfer in densely packed media have been undertaken to clarify theconditions of the VRTE applicability. As it turned out, in a number of situations thequalitative agreement with the results of RT-calculations takes place (the corre-sponding references are included).

The features of radiative transfer phenomena in magnetoactive plasma (Ginzburget al. 1975) are shortly reflected in Sect. 1.5.2. The polarization states of the normalwaves in this kind of anisotropic media are not orthogonal, in general. But normalwaves are reduced to circular polarized waves (for lengthwise propagation) and tolinear polarized waves (for transverse propagation). In appropriate parametricdomain the magnetoactive plasma can possess a strong optical anisotropy.

Optically active media occurring in bio-medical field of research are touched inSect. 1.5.3. These media can be divided into two main classes—strongly scattering(turbid) and weakly scattering (transparent). The analysis of polarization charac-teristics of multiply scattered radiation in biological media is one the mostimportant instruments for estimation of internal structure of the media. Chiralmolecules are often enclosed in bio-tissues, and multi-scattered light depolarizationmeasurements are widely used for estimation the concentrations of optically activemolecules (such as glucose). So the design of adequate mathematical models ofdisperse bio-tissues is of importance. A closely related area of research concerns theapplication of radiation transport theory to the problems of non-invasive medicaldiagnostics of heterogeneities in biological tissues. The deterministic method ofnumerical transport equation solution for calculating the characteristics of multiplyscattered light in the 3D-regions (instead of Monte-Carlo simulations) has beenproposed in (Bass et al. 2009; Bass et al. 2010).

The Sects. 1.5.4 and 1.5.5 are devoted to some radiation transport problems foroptically anisotropic media that are encountered in rather less popular applicationareas, such as multi-scattered radiation transport in bio-medical anisotropic media,liquid crystals, layered anisotropic media, in two-dimensional periodic opticallyanisotropic structures known as photonic crystals. Interesting phenomena of reso-nant radiation interaction with the medium can arise in the layered structures. Theliquid crystals can demonstrate phase transitions of the second order and sponta-neous symmetry breaking. Besides, the studying of radiative transfer processes inthe multilayered anisotropic structures is of value in view of increasingly wide


applications of these anisotropic media. Thus, the short overview of some topics inthe research field can provide a glance at a new class of radiation transport problemswhere both theoretical and computational work are still at the very beginning.

1.2 Radiation Transfer Problems for Disperse OpticallyAnisotropic Media

1.2.1 The Radiation Transport Equation for SparseDisperse Media Derived from the Maxwell Equationsfor an Ensemble of Scatterers

As well known, the radiation transport equation is the basis for calculation ofvarious problems of radiation transfer in scattering media. Initially it was derivedphenomenologically via considerations of energy balance for volume element ofscattering and absorbing medium through which the radiation propagates (see, forinstance, Chandrasekhar 1960; Van de Hulst 1957, 1980; Rosenberg 1955). At thesame time it was clearly understood, that for correct and comprehensive derivationof transport equation describing electromagnetic (polarized) radiation transfer it isnecessary to study the underlying problem of classical statistical electrodynamics—the problem of multiple scattering of electromagnetic waves in disperse media,formed by ensembles of discrete isolated scatterers, the scattered radiation behaviorbeing described by the Maxwell equations. The approach allows to formulate thefull system of restrictions on the medium microstructure and the radiation fieldproperties, and thus allows to reveal the relation between classical statistical elec-trodynamics and phenomenological radiative transfer theory. The applicabilityconditions of the classical radiative transfer equation can be also clarified in theway. The versions of the program have been successfully fulfilled in a variety ofwell known papers and monographs (see, for example, Foldy 1945; Lax 1951;Watson 1969; Borovoi 1966a, b, 1967a, b, 1983, 2005, 2013; Barabanenkov 1973;Dolginov et al. 1970, 1975, 1995; Barabanenkov 1975; Kuzmina 1976; Rytov et al.1978; Ishimaru 1978; Apresyan et al. 1996; Tsang et al. 2001; Mishchenko 2002,2003, 2008a, b; Mishchenko et al. 2006; 2014, 2016a, b).

The problem of multiple scattering of classical electromagnetic radiation in asparse disperse medium (an ensemble of N;N � 1; sparsely randomly distributedisolated macroscopic scatterers) is considered as the base problem for deriving ofpolarized radiation transport equation (the VRTE). The following natural set ofrestrictions is often admitted: (1) quasi-monochromatic incident radiation field isconsidered, and radiation scattering is supposed to occur without frequency redis-tribution; (2) the inequalities k � l; ds � l are fulfilled, where l is the length offree radiation path between the acts of scattering, k being the radiation wavelength,ds being the average scatterer diameter; (2) each scatterer is located in the far-fieldzone of all the other scatterers, and so electromagnetic wave travelling from


scatterer to scatterer can be assumed quasi-spherical at the vicinity of each scatterer;(3) the scatterer velocities are small in comparison with phase velocity of incidentelectromagnetic wave (in continuous transparent medium into which all the scat-terers are embedded); (4) the scattering characteristics of all scatterers arestationary.

The system of Maxwell equations governing the process of multiple scattering ofclassical electromagnetic field by statistical ensemble of N macroscopic scatterershas been written down in various forms in many papers, listed above. If the scatterercenters are located at the spatial points specified by the radius-vectors r1; . . .rN , thestarting system of equation for the vector of electric field of the electromagneticwave at spatial point r the can be written as

EðrÞ ¼ EincðrÞþXNj¼1

Escðr; rjÞ; ð2:1:1Þ

where EincðrÞ is the vector of the electric field of incident wave, and Escðr; rjÞ is thevector of the electric field of electromagnetic wave, scattered by j-th scatterer. It canbe calculated via integration over the volume of space, occupied with the j-thscatterer, of the function UjðrÞEexcðr; rjÞ; where UjðrÞ is the “potential” function forthe j-th scatterer, defined by its refractive index, and Eexcðr; rjÞ is the vector ofelectric field, acting on the scatterer (the “exciting” field). The accurate calculationof Eexcðr; rjÞ requires the attraction of Lippmann–Schwinger integral equation.However, a simplified approach was developed in (Foldy 1945; Lax 1951; Watson1969) and was further used in (Dolginov et al. 1970, 1995; Barabanenkov et al.1991, 1995; Kuzmina 1976). Besides the natural restrictions on the ensemble ofscatterers and radiation field, listed above, two additional significant approxima-tions for radiation field were usually admitted:

• hEscðr; r0Þir0 � hEscðr; r0Þi (the ensemble averaged radiation field in the vicinityof the scatterer, located at the point r0; only slightly differs from the field whichwould exist there in the case if the scatterer is absent at the r0Þ;

• It is assumed, that the coherent scattering of electromagnetic waves in sparsedisperse media takes place only in the exactly forward-scattering direction (theinequality Ds � l is supposed to be satisfied, where Ds is the diameter of thewhole medium volume, and l is the value of the radiation free path in thedisperse medium).

These two approximations allow to obtain the system of equations for Eðr; rjÞ inthe form (2.1.1), where Escðr; rjÞ are expressed in terms of operators of scatteringamplitude, Aðr; s; s0Þ; and the free-space Green function operators Gð r� r0j jÞ:Performing the configuration ensemble averaging, analogous to that performed in(Lax 1951; Watson 1969), the VRTE can be derived for the Stokes vector Iðr; sÞ ¼½Iðr; sÞ;Qðr; sÞ;Uðr; sÞ;Vðr; sÞ�T in the form


ðs � rÞIðr; sÞþ rðr; sÞIðr; sÞ ¼ ðPIÞðr; sÞþFðr; sÞ; ð2:1:2Þ

where rðrÞ is the extinction matrix of medium volume element, ðPIÞðr; sÞ is theintegral operator of scattering, defined by scattering phase matrix of the mediumvolume element, and Fðr; sÞ is the Stokes vector of internal sources of radiation inthe medium. The extinction operator rðr; sÞ of the VRTE (2.1.2), can be expressedin terms of the refractive index operator, nðr; sÞ; corresponding to the effectivecontinuous transparent medium (in general, optically anisotropic), in whichcoherently scattered (i.e. refracted by the disperse medium) radiation propagates(see Sect. 1.3.1). The operator nðr; sÞ is expressed in terms of the ensemble aver-aged operator of scattering amplitude in forward-scattering direction, hAðr; s; sÞi[we use the notations from (Kuzmina 1976)]:

n2ðr; sÞ ¼ Iþ 4pðxcÞ�2hAðr; s; sÞi: ð2:1:3Þ

Since the effective medium, specifying the non-scattered radiation propagation,is usually turned out to be weakly optically anisotropic (due to the disperse mediumsparsity), the quasi-isotropic approximation of geometrical optics for opticallyanisotropic media is applicable (see Sect. 1.2.3). The main features of the VRTE foroptically active media are described in Sect. 1.3.

It is necessary to mention that the most consecutive, strict and detailed VRTEderivation has been later performed in the monograph (Mishchenko et al. 2006) (seealso (Mishchenko 2014a, b; Mishchenko et al. 2016a, b). In (Mishchenko et al.2006) the radiative transfer theory (RTT) is presented as a branch of classicalmacroscopic electromagnetics, and the detailed theory of multiple scattering ofelectromagnetic radiation in random discrete media composed of sparsely dis-tributed particles is developed. The Foldy–Lax equations for N-particle ensemble ofscatterers (which can be viewed as the basic equations of modern theory of multiplescattering in random particle ensembles) figure as a starting point for strict devel-opment of the phenomenological VRTE. The diagrammatic interpretation of theorder-of-scattering expansion for the scattered radiative field is exploited. TheTwersky approximation for the coherent radiation field (in the diagrammaticinterpretation the approximation means that all scattering paths going through aparticle two or more times are neglected) is used. Theoretically justified, ensembleaveraging procedures are used in the process of obtaining of statistically averaged(macroscopic) scattering and absorption characteristics of the disperse media.

As a result, the integral and the integro-differential versions of the VRTE areobtained in (Mishchenko et al. 2006). The coherency extinction matrix of theVRTE for the Stokes vector [that corresponds to rðr; sÞ in Eq. (2.1.2)] is expressedin terms of the ensemble averaged components of the amplitude scattering matrix(denoted by S

�in forward-scattering direction. The scattering phase matrix of the

VRTE is expressed in terms of sums of pairwise products of the S components. Theproperties of both coherent and diffuse (multiply scattered) radiation field are


extensively discussed. In particular, the attention is paid to ladder approximation ofdiagrammatic approach (consisting in keeping of only a certain class of diagrams)and its physical interpretation. The diagrams with crossing connectors and theircumulative contribution are analyzed as well. In addition the helpful discussion onmany problems concerning optical polarization measurements in the context ofphenomenological RTT approach is provided.

The coherent backscattering (CB) (weak localization of electromagnetic waves),one of the most remarkable effects caused by multiply scattered radiation in adisperse medium, has been studied in detail in (Mishchenko et al. 2006; Tishkovetset al. 2004) in the frames of theory of electromagnetic radiation multiple scatteringby an ensemble of particles. The CB belongs to radiation interference phenomena.Arising as a result of interference of scattered waves in the exactly backscatteringdirection, the CB reveals itself in a narrow interference peak of intensity and ischaracterized by a specific behavior of polarization. In fact, the CB should bequalified as a mesoscopic physical phenomenon emerging as a result of correlationof multi-particle scatterer groups of a disperse medium.

It is a significantly more difficult task to derive the VRTE for densely packeddisperse media composed of large scatterers, where the assumption concerningscatterer locations in wave zones of each other is violated. Among a variety ofapproaches to treatment of radiation transport problems in dense particulate mediawe would like to mention the three ones. In some parametrical domains (forinstance, in the case of media composed of moderate size scatterers studied inmicrowave spectral range) different procedures of replacement of the dispersedmedium by a continuous one with an effective refractive index can be used(Kokhanovsky 1999b, 2004; Tsang et al. 1991, 2001, 2011). A quasi-crystallineapproximation represents another approach developed for radiative transfer prob-lems in dense media (Lax 1951; Tsang et al. 2001). Being applied to problems ofremote sensing of snow in microwave spectral range, the quasi-crystallineapproximation provides taking into account the coherent interaction among thescatterers, located at the vicinity of each other, via weighted pair distributionfunction of particle positions. It permits to calculate the coherently transmittedradiation and radiation absorption in densely packed media composed of moderatesize non-spherical particles (Tsang et al. 2001).

An approach to analysis of macro-characteristics of densely packed dispersemedia composed of large non-spherical particles (including faceted particles imi-tating ice crystals of cirrus clouds) has been developed in a series of papers(Borovoi et al. 2002, 2006, 2007, 2010; Borovoi 2005, 2006; 2013). The approachis based on accurate estimation of scattered field in the near zone of single scatterervia introduction of the so-called shadow-forming field. The really existingshadow-forming field can be determined at any distance from the scattering particlein the near zone in the frames of physical optics, and there is a number ofadvantages to operate with it. For example, both Fresnel and Fraunhofer diffractioncan be taken into account by the method without tedious calculations (Borovoi2013). In the paper (Borovoi et al. 2010) a treatment of a disperse medium com-posed of large faceted particles has been fulfilled by the approach of


shadow-forming field analysis. The scattered field is succeded to present in the formof a set of plane parallel beams, each beam being a clearly defined as the physicalobject with finite transverse size, known shape and spatial location. The polariza-tion of each beam is described analytically (in terms of 2D electric field vector andthe Jones polarization matrix). The shadow-forming plane parallel beam is includedinto the superposition of scattered beams as the additional beam. It is taken intoaccount that all the beams undergo the Fraunhofer diffraction in the wave zone ofeach particle. The diffracted field is calculated via the vector Fraunhofer diffractionequation. As a result the analytical expression for the scattered field in the wavezone of the particle has been obtained in terms of shadow functions, containing allthe parameters of near-zone plane-parallel beams. Finally, the polarization char-acteristics of radiation, scattered by the particle, are expressed in terms of theMueller matrix (the interference of all the diffracted beams being taken intoaccount). In the case of statistical ensemble of scatterers having certain sizes,shapes, and spatial orientations the scattered field is naturally expressed through theensemble average of shadow functions. In addition, the method of calculation ofdiffraction contribution in near forward-scattering direction has been developed aswell.

To elucidate the issue to what extent the VRTE can be applied to densely packedmedia the numerical solutions of various problems of multiply scattered electro-magnetic radiation in densely packed ensembles of discrete scatterers have beenperformed. In a number of situations the qualitative agreement with the results ofRT calculations (with the CB accounting) takes place (Mishchenko et al. 2007,2008a, b, 2016a, b; Okada and Kokhanovsky 2009; Dlugach et al. 2011).

1.2.2 Optically Active Anisotropic Media

Optically active (gyrotropic, chiral) media are optically anisotropic media charac-terized by elliptical birefringence and elliptical dichroism. The more general type ofoptically anisotropic media, to which the optically active media belong—bian-isotropic media—can be specified by the following general form of constitutiveequations (constitutive relations, or material equations), reflecting themagneto-electric cross- coupling:

D ¼ eEþ nH; B ¼ lHþ gE; ð2:2:1Þ

whereE andH are applied electric andmagnetic fields,D andB are the correspondingvectors of electric and magnetic induction, e is the electric permittivity tensor, l is themagnetic permittivity tensor, n and g are tensors, defining the magneto-electriccross-coupling (Kong 1974, 1990; Landau and Lifshitz 1960). Being placed in anelectric or magnetic field bianisotropic media become both polarized andmagnetized.


The tensors n and g in the Eq. (2.2.1) are not independent tensors: the relation betweenthem should be obtained from the condition of energy conservation for electromag-netic field in the concrete bianisotropic medium. Magnetoelectrical materials weretheoretically predicted by L.E. Dzyaloshinsky (Dzyaloshinskii 1960) and observedexperimentally in 1960 by D.N. Astrov (Astrov 1960). Bianisotropic media can bedivided into the so-called reciprocal and nonreciprocal bianisotropic media, that areusually studied separately. For example, the reciprocal uniaxially anisotropic chiralmedia are characterized by the tensors (Kong 1990)

e ¼ e0diagfe; e; e1g; l ¼ l0diagfl; l;l1g; n ¼ c�1diagf0; 0;�in0g;g ¼ c�1diagf0; 0; in0g:

For nonreciprocal aniaxially anisotropic medium the tensors n and g are definedas

n ¼ c�1diagf0; 0; n0g; g ¼ c�1diagf0; 0; n0g:

Phenomenological theory of gyrotropic media, belonging to reciprocal bian-isotropic media, was developed by F.I. Fedorov (Fedorov 1976). It was shown thatfor gyrotropic media the Eq. (2.2.1) can be rewritten in the form (Fedorov 1976)

D ¼ eEþ icH; B ¼ lH� icTE; ð2:2:1�Þ

where the gyration tensor c (the c is a real-valued pseudo-tensor) defines mediumoptical activity (gyrotropy) (the symbol T denotes matrix transposition operation).Thus, gyrotripic media can be specified by a single gyration tensor c: For problemsrelated to propagation of quasi-monochromatic plane electromagnetic wavesEðrÞ ¼ E expð�ik � rÞ in non-magnetic gyrotropic media the first of the constitu-tive Eq. (2.2.1*) can be also written as

D ¼ eEþ cðr EÞ: ð2:2:1 � �Þ

In this case the gyration vector g can be defined as g ¼ fk; g ¼ ð0; 0; gÞ; where fis a pseudo-scalar (changing sign depending on the handedness of the coordinatesystem), and the Eq. (2.2.1**) can be represented in the k�dependent form

D ¼ eEþ ie0ðfk gÞE: ð2:2:2Þ

If the electromagnetic wave propagates in the z-direction, so that k ¼ð0; 0; kÞ; g ¼ ð0; 0; gÞ; one can present the Eq. (2.2.2) in the matrix form

D1

D2

D3

264

375 ¼ e0

n2 �ig 0ig n2 00 0 n2

24

35

E1

E2

E3

264

375; ð2:2:3Þ


where n2 ¼ e=e0: Obviously, the diagonal elements of the matrix in the Eq. (2.2.3)correspond to the phase velocity of electromagnetic wave in the isotropic mediumwith refractive index n; whereas the off-diagonal elements, proportional to g; definemedium optical activity. The normal modes of the chiral medium (right-hand andleft-hand circularly polarized electromagnetic waves that transmit through themedia without distortion at different phase velocities ncÞ are obtained as the eigenvectors of the matrix, entering to the Eq. (2.2.3), corresponding to the eigen valuesn,

n ¼ffiffiffiffiffiffiffiffiffiffiffiffiffin2 g

p: ð2:2:4Þ

The Faraday effect—a magneto-optical phenomenon of the polarization planerotation—is caused by the circular birefringence inherent to chiral medium (dif-ferent phase velocities for right-hand and left-hand circularly polarized waves). Therotatory power of the medium is proportional to ðn� � nþ Þ; the difference ofrefractive indices for the normal modes. Natural optical activity is inherent tomaterials with intrinsically helical microstructure. Examples include selenium,tellurium oxide ðTeO2Þ; quartz ða� SiO2Þ; and cinnabar ðHgSÞ:Many materials actas polarization rotators at the presence of acting magnetic field. Magnetoactiveplasma and liquid crystals present the other examples of gyrotropic media. At theabsence of external magnetic fields three types of weakly damping waves can existin an isotropic plasma: a transverse electromagnetic wave and two types of lon-gitudinal waves—a high-frequency plasma (Langmuir) wave and a low-frequencyion-sound wave.

The double refraction (birefringence) is one of the most significant phenomenainherent to all types of anisotropic media. In the simplest case of a dielectric(non-magnetic) anisotropic medium the birefringence is described in terms of theelectric permittivity tensor e: In general case the tensor e has six independentcomponents in an arbitrary coordinate system. For crystals of certain symmetriesthe tensor e possesses fewer independent components. The refractive indices for thenormal modes and their polarization states are traditionally determined via ananalysis of refractive index ellipsoid. In the simplest case of an uniaxial crystal therefractive index ellipsoid is a spheroid (the ellipsoid of rotation with n1 ¼ n2 ¼no; n3 ¼ neÞ: A monochromatic plane wave incident on the boundary between theisotropic and anisotropic media generates two refracted waves inside the anisotropicmedium with different directions of propagation and different polarizations. At theboundary between two isotropic media with different refractive indices the angles ofincidence h and refraction h1 are related by the Snell law

k0 sin h1 ¼ k sin h:


In the case of a uniaxial crystal two refracted waves arise inside the anisotropicmedium: an ordinary wave of orthogonal polarization (TE) at the an angle h ¼ hofor which

sin h1 ¼ no sin ho;

and an extraordinary wave of parallel polarization (TM) at an angle h ¼ he forwhich

sin h1 ¼ nðha þ heÞ sin he:

The values no and ne can be determined from the relation

1n2ðhÞ ¼

cos2ðhÞn2o

þ sin2ðhÞn2e

;

where na ¼ nðhÞ (see Fig. 1.1).Thus, if the incident wave carries two polarizations and the wave vector k is not

normal to the boundary between the isotropic and uniaxial anisotropic media, tworefracted waves emerge at the boundary (as shown in Fig. 1.1). If the wave vector kof the incident wave is normal to the boundary, in addition to the ordinary wavewith the wave vector ko, parallel to k; the refracted extraordinary wave at k ¼ ks isemerges as well. That is, the normal incidence on the boundary between isotropicand anisotropic media creates oblique refraction.

It should be stressed that the elliptic birefringence and elliptic dichroism areinherent to optically active media of general type (Fedorov 1976). But in the framesof present paper we concentrate attention mainly on radiative transfer problems(and the VRTE) for isotropic optically active media so far (see Sects. 1.3 and 1.4).For these media the circular birefringence and polarization plane rotation take placefor any direction in the medium. On the contrary, for gyrotropic crystals thepolarization plane rotation takes place only along the directions of optical axes

Fig. 1.1 Double refraction in an uniaxial crystal


(Fedorov 1976). Another feature of any optically anisotropic media (gyrotropicmedia including) is that the formulation of boundary conditions for radiativetransfer problems requires of special attention. Before consideration this peculiarityin more detail it is worth to shortly remind of the quasi-isotropic approximation ofgeometrical optics that is used in quasi-monochromatic radiation transfer problemsin weakly anisotropic media (Kravtsov and Orlov 1990, Kravtsov et al. 2007).

1.2.3 Quasi-Isotropic Approximation of Geometrical Optics

The coherent component of the radiation field in a scattering medium (non-scatteredradiation, refracted by the medium) can be considered as the radiation propagatingin the effective continuous transparent refractive medium which optical character-istics can be directly derived from the Maxwell equations (Kravtsov and Orlov1990). In a weakly anisotropic medium the monochromatic non-scattered radiationpropagates in the form of a transverse electromagnetic waves. The polarization ofthe transverse electromagnetic wave can be calculated in the basis fe1; e2g; so ase1 � e2 ¼ e1 � s ¼ e2 � s ¼ 0 where s ¼ _r (the unit vector tangent to the ray). Thecomplex-valued basis of circularly polarized waves can also be used:

e� ¼ ðe1 þ ie2Þ=ffiffiffi2

p; eþ ¼ ðe1 � ie2Þ=

ffiffiffi2

p: ð2:3:1Þ

Polarization evolution of a partially polarized electromagnetic wave propagatingin a weakly anisotropic inhomogeneous medium without scattering is governed bythe equation for the four-component Stokes vector

IðsÞ ¼ ½IðsÞ;QðsÞ;UðsÞ;VðsÞ�T � col½IðsÞ;QðsÞ;UðsÞ;VðsÞ� ð2:3:2Þ

which can be written as

_IðsÞ ¼ MIðsÞ; ð2:3:3Þ

where M is the differential Mueller matrix for the anisotropic medium. The matrixM can be expressed in terms of three-component vector G ¼ ðG1;G2;G3Þ(Kravtsov and Orlov 1990):

M ¼ImG0 ImG1 ImG2 ImG3

ImG1 ImG0 �ReG3 ReG2

ImG2 ReG3 ImG0 �ReG1

ImG3 �ReG2 ReG1 ImG0

2664

3775 ð2:3:4Þ


where the Mueller matrix can be presented as the sum of three terms (Azzam et al.1989). The first one, Ma ¼ ImG0diag½1; 1; 1; 1� describes attenuation common forall components of the Stokes vector. The second term, the dichroic one,

Md ¼0 ImG1 ImG2 ImG3

ImG1 0 0 0ImG2 0 0 0ImG3 0 0 0

2664

3775 ð2:3:5Þ

corresponds to the attenuation, responsible for dichroism (i.e., selective attenuationof the normal modes). Finally, the matrix

Mb ¼0 0 0 00 0 �ReG3 ReG2

0 ReG3 0 �ReG1

0 �ReG2 ReG1 0

2664

3775 ð2:3:6Þ

describes the birefringence.Thus, for multiply scattered polarized radiation transport problems in weakly

optically anisotropic chiral media the four-component VRTE with the matrixextinction operator (making sense the differential Mueller matrix) and the integraloperator of scattering, defined by the phase matrix of a non-block-diagonal form,can be adequately used.

1.3 Radiation Transport Problems for Optically ActiveMedia

1.3.1 The Radiation Transport Equation for IsotropicOptically Active Media

Recall that the Stokes parameters of a radiation beam, propagating in the directions;

IðsÞ ¼ ½IðsÞ;QðsÞ;UðsÞ;VðsÞ�T ð3:1:1Þ

are related to the time-averaged bilinear products of components of electric fieldvector E of plane quasi-monochromatic electromagnetic wave, E ¼E1e1 þE2e2; e1s ¼ e2s ¼ 0; accordingly to the expressions

I ¼ E1j j2 þ E2j j2D E

; Q ¼ E1j j2� E2j j2D E

; U ¼ 2Re E1E�2

� �� ;

V ¼ �2Im E1E�2

� �� :

ð3:1:2Þ


The Stokes parameters define the beam intensity I; the polarization degree p; theshape and orientation of the polarization ellipse in the basis fe1; e2g (see Fig. 1.2):

p ¼ ðQ2 þU2 þV2Þ1=2I

; v ¼ 12arctan

UQ; b ¼ 1

2arcsinð V

ðQ2 þU2 þV2Þ1=2Þ:

ð3:1:3Þ

The radiation transport equation governing polarized radiation transfer in ascattering and absorbing medium is usually written in terms of the Stokes vectorIðr; sÞ; defined by (3.1.1), (3.1.2) and depending on the coordinate of spatial point rin the medium and the unit vector s; defining the direction of radiation propagation.The matrix transport equation, written in terms of the coherence matrix q;

qðr; sÞ ¼ E1j j2 E1E�2

E1E�1 E1j j2

� �¼ 0:5

IþQ U � iVU � iV IþQ

� �; ð3:1:4Þ

is exploited as well (see, for example, Dolginov et al. 1970; Kokhanovsky 2000).We will further use the integro-differential vector transport equation in the form

ðs � rÞIðr; sÞþ rðrÞIðr; sÞ ¼ ðPIÞðr; sÞþFðr; sÞ; ð3:1:5Þ

where rðrÞ is the extinction operator, ðPIÞðr; sÞ is the integral scattering operator,defining multiple scattering of radiation in the medium, and Fðr; sÞ is the Stokesvector of internal sources of radiation in the medium. For weakly anisotropicoptically active media the extinction operator r can be expressed in terms of themedium refractive index operator n by the formula (Kuzmina 1976, 1986):

r ¼ �ixcTfn� I2 � I2 � n�gT�1; ð3:1:6Þ

where

Fig. 1.2 The polarization ellipse


n ¼ n� þ ij� 00 nþ þ ijþ

� �; 2

xcj ¼ rt ¼ rs þ ra ; T ¼

1 0 0 11 0 0 �10 1 1 00 �i i 0

2664

3775:

ð3:1:7Þ

Here n are the values of refractive indices for the radiation in the polarizationstates of the normal waves (i.e., the states of right and the left circular polarizationfor the chiral medium), j are the attenuation coefficients for the radiation in thesame polarization states, rs ; r

a ; r

t are the cross sections of scattering, absorption

and extinction for the radiation in the mentioned polarization states, I2 ¼diag½1; 1�; and � is the symbol of tensor product (see, for example, Dullemondand Peeters 1991–2010).

For isotropic optically active medium the extinction operator rðrÞ in the VRTE(3.1.5), is defined by the matrix (the notations from (Kuzmina 1986, 1989, 1991)are further used):

r ¼�rt 0 0 D�rt=20 �rt x

c Dn 00 � x

c Dn �rt 0D�rt=2 0 0 �rt

2664

3775 ð3:1:8Þ

where

�rt ¼ rþt þ r�t

2; Drt ¼ rþ

t � r�t ; Dn ¼ nþ � n�: ð3:1:9Þ

The integral operator of scattering in the VRTE (3.1.5) possesses the samestructure as that in the case of optically isotropic medium (due to the mediumgeometrical isotropy). It can be written as

ðPIÞðr; sÞ ¼ �rs4p

ZX

Lðs; s0ÞCðr; s � s0ÞLþ ðs0; sÞds0; ð3:1:10Þ

where Cðr; s � s0Þ is the scattering phase matrix, defining the law of scattering by themedium volume element and depending on variables s and s0 through the scalarproduct s � s0 � cos hs � c hsð being the angle of scattering), X is the unit sphere inthree-dimensional vector space R3. The matrix Lðs; s0Þ in Eq. (3.1.10) is the knownmatrix of the Stokes vector transformation at the transition from one polarizationbasis to another, and Lþ is the Hermitian conjugate to L: Geometrically isotropicoptically active media are characterized by the special type of phase matrices:


CðcÞ ¼a1ðcÞ b1ðcÞ c2ðcÞ b2ðcÞb1ðcÞ a2ðcÞ c3ðcÞ b3ðcÞ�c2ðcÞ �c3ðcÞ a3ðcÞ c1ðcÞb2ðcÞ b3ðcÞ �c1ðcÞ a4ðcÞ

2664

3775 ð3:1:11Þ

In addition to the usual normalization condition for the scattering phase function(the indicatrix of scattering) C11ðr; cÞ � a1ðr; cÞ;

12

Z1

�1

C11ðr; cÞdc ¼ 1;

there is another normalization condition for the element C14ðr; cÞ � b2ðr; cÞ; thatshould be fulfilled:

12

Z1

�1

C14ðr; cÞdc ¼ rþs � r�s

rþs þ r�s

¼ Drs2�rs

� Ds: ð3:1:12Þ

The relation (3.1.12) can be derived as the consequence of the energy conser-vation law for volume element of the medium in the situation when it is illuminatedby a mono-directed beam of circular polarized radiation. The value DsðrÞ deter-mines an essential macro-characteristics of scattering medium, that might be calledthe medium dichroism due to scattering. The condition DsðrÞ � 0 may be con-sidered as the necessary condition of the medium optical isotropy, whereas thecondition DsðrÞ 6¼ 0 can figure as the sufficient condition of the medium opticalanisotropy. Using the natural relation between the cross sections of extinction,scattering and absorption, one can obtain the relation between the full mediumdichroism DðrÞ ¼ ðrþ

t � r�t Þ=2�rt and the dichroism contributions DsðrÞ and DaðrÞdue to scattering and absorption, respectively:

DðrÞ ¼ kðrÞDsðrÞþ ð1� kðrÞÞDaðrÞ; ð3:1:13Þ

where

DsðrÞ ¼ ðrþs � r�s Þ=2�rs;DaðrÞ ¼ ðrþ

a � r�a Þ=2�ra; �kðrÞ ¼ �rs=�rt:

So, as one can see, instead of the two cross sections, rtðrÞ and rsðrÞ; that incombination with the scattering phase matrix Cðr; cÞ are sufficient to completelyspecify the VRTE for optically isotropic media, the six cross sectionsrt ðrÞ; rs ðrÞ; ra ðrÞ; and additionally nðrÞ; are needed to specify the VRTE foroptically active media. Surely, the equivalent collection of functions can also be used,for instance: �rtðrÞ; �kðrÞ ¼ �rsðrÞ=�rtðrÞ; �nðrÞ; DðrÞ ¼ ðrþ

t ðrÞ � r�t ðrÞÞ=2�rtðrÞ;DsðrÞ ¼ ðrþ

s ðrÞ � r�s ðrÞÞ=2�rsðrÞ; and dðrÞ ¼ ðx=cÞðnþ � n�Þ=�rt. We will use


further just the last collection offunctions and rewrite the extinction operator (3.1.8) inthe form

rextðrÞ ¼ �rtðrÞ1 0 0 DðrÞ0 1 dðrÞ 00 �dðrÞ 1 0

DðrÞ 0 0 1

2664

3775 ð3:1:8�Þ

where

�rt ¼ rþt þ r�t

2; D ¼ rþ

t � r�t2�rt

; d ¼ xc� n

þ � n�

�rt; ð3:1:14Þ

rt ðrÞ being the extinction cross sections, and n being the refractive indices forright and left circularly polarized radiation, respectively. The integral operator ofscattering (3.1.10) is specified by the phase matrix of scattering Cðr; cÞ; that isdefined accordingly Eq. (3.1.11). The additional normalization conditions, definedvia Eqs. (3.1.12) and (3.1.13), can be rewritten in the form

�rt ¼ �ra þ �rs ¼ �ra þ �rs2

Z1

�1

C11ðcÞdc; ð3:1:15Þ

D ¼ Da þDs ¼ Da þ 12

Z1

�1

C14ðcÞdc: ð3:1:16Þ

The relation (3.1.16) determines Ds, an essential macro-characteristics of thescattering medium (the medium dichroism due to scattering). The condition Ds � 0can figure as the necessary condition of optical isotropy of the medium, whereas thecondition Ds 6¼ 0 represents the sufficient condition of its optical anisotropy (i.e.,the optical activity).

1.3.2 Radiation Transport Problems for Slabs of IsotropicOptically Active Medium

The problem of polarized radiation transfer in a slab of isotropic optically activemedium is a boundary value problem for the VRTE. Let us consider the slab0 z H; z being the coordinate along the unit normal n to the plane z ¼ 0. Let sbe the unit vector of radiation transfer direction s 2 X;Xð being the unit sphere in


3D vector space R3�; and Iðr; sÞ be the four-component Stokes vector, defined in

(3.1.1), (3.1.2). Then we have the following boundary value problem for the VRTE:

l@Iðz; sÞ@z

þ rðzÞI ¼ ðPIÞðz; sÞþFðz; sÞ; ð3:2:1Þ

Iþ ð0; sÞ ¼ f þ0 ðsÞ; ð3:2:2Þ

I�ðH; sÞ ¼ f�HðsÞ: ð3:2:3Þ

Here l ¼ s � n ¼ cos h; while

Iþ ðz; sÞ ¼ Iðz; sÞ; l� 00; l\0

ð3:2:4Þ

I�ðz; sÞ ¼ 0; l� 0Iðz; sÞ; l\0:

ð3:2:5Þ

The functions f þ0 and f�H define the Stokes vectors of the external radiation at theboundaries z ¼ 0 and z ¼ H; while Fðz; sÞ defines the Stokes vector of internalvolume sources.

It should be noted that we do not concern here an interesting and noteworthyissue on the relation between vector space, affine space and point Euclidean space.For interested reader the monograph (Faure et al. 1964) could be recommended. Seealso (Rogovtsov 2015a; Rogovtsov et al. 2016).

The operator r ¼ rext for slabs is specified by the matrix (see 3.1.8*):

rðzÞ ¼ �rtðzÞ1 0 0 DðzÞ0 1 dðzÞ 00 �dðzÞ 1 0

DðzÞ 0 0 1

2664

3775 ð3:2:6Þ

where �rt; D and d are defined through rt and n accordingly to (3.1.14). Theintegral scattering operator ðPIÞðz; sÞ is specified by formulas (3.1.10), (3.1.11) withC ¼ Cðz; cÞ:

1.3.3 Boundary Conditions

The transport problem with the given mono-directed monochromatic radiationbeam, defined at the slab boundary, is a typical model problem in Earth remotesensing. In the case of oblique beam incidence at the boundary of a slab of opticallyanisotropic medium the two geometrically separated refracted beams arise inside


the slab due to the birefringence phenomenon. For fully polarized mono-directedbeam the amplitudes and the polarization states of these two refracted electro-magnetic waves can be exactly calculated (accordingly to the Fresnel formulageneralization (see, for instance, Fedorov 1976; Fedorov and Philippov 1976). Forweakly anisotropic media the angles of refraction of the two beams and theirpolarizations can be estimated in the approximation of weak medium anisotropy.

Let the incident non-polarized monochromatic mono-directed beam (planequasi-monochromatic electromagnetic wave) is defined by the Stokes vector

Iincð0; s0Þ ¼ ½I0; 0; 0; 0�T~dðs� s0Þ ¼ ½I0; 0; 0; 0�T~dðl� l0Þ~dðu� u0Þ; ð3:3:1Þ

where l0 ¼ s0 � n ¼ cos h0 and ~d is Dirac’s delta-function. Due to the geometricalmedium isotropy according to Snell’s law one has

sin hþ ¼ sin h0=nþ ; sin h� ¼ sin h0=n�;uþ ¼ u� ¼ u0: ð3:3:2Þ

Putting (for certainty) nþ [ n�ðDn[ 0Þ; we obtain

sin hþ ¼ sin h0ð1� Dn=2�nÞ; sin h� ¼ sin h0ð1þDn=2�nÞ: ð3:3:3Þ

The polarization states of two refracted beams can be easily calculated (Kuzmina1986a). In the case of mono-directed non-polarized monochromatic beam (3.3.1),incident on the slab boundary z ¼ 0 of homogeneous chiral mediumrt ¼ const; n ¼ const� �

; the refracted beam represents the superposition of twofully circularly polarized beams:

Irefrð0; sþ0 ; s�0 Þ ¼ �rt1þD2

I0½1; 0; 0; 1�T~dðl� lþ0 Þ~dðu� u0Þ

þ �rt1� D2

I0½1; 0; 0;�1�T~dðl� l�0 Þ~dðu� u0Þ;ð3:3:4Þ

where l0 ¼ s0 � n: So, the mono-directed beam of non-polarized external radiationof intensity I0, incident to the boundary of homogeneous chiral medium, is trans-formed inside the medium into superposition of two geometrically separated fullycircularly polarized beams of intensities I þ0 ¼ �rt 1þD

2 I0 (right-hand circularlypolarized beam) and I�0 ¼ �rt 1�D

2 I0 (left-hand circularly polarized beam).

The angles h0 are needed to be obtained.

1.3.4 Coherently Scattered Radiation in a Slab of ChiralMedium

The Stokes vector of non-scattered radiation in a slab, Icðz; sÞ ¼ Icðs; sÞ; can befound as the solution of the following boundary value problem:


l@Icðs; sÞ

@sþ rðsÞIc ¼ ~Fðs; sÞ; ð3:4:1Þ

Iþc ð0; sÞ ¼ I�c ðsH ; sÞ ¼ 0 ð3:4:2Þ

where l ¼ s � n ¼ cos h and

~Fðs; sÞ ¼ Fðs; sÞþ lf þ0 ðsÞ~dðsÞþ lj jf�HðsÞ~dðs� sHÞ; ð3:4:3Þ

s � sð0; zÞ ¼Zz

0

�rtðxÞdx; sH � sð0;HÞ: ð3:4:4Þ

The solution of the problem (3.4.1)—(3.4.4) may be written in the form

Icðs; sÞ ¼ l�1Zs

0

Gðs=l; s0=lÞ~Fþ ðs0; sÞds0 þ lj j�1ZsHs

Gðs0=l; s=lÞ~F�ðs0; sÞds0;

ð3:4:5Þ

Gðz; z0; lÞ � Gðs; s0; lÞ is the known Green function for a slab of non-scatteringgeometrically isotropic optically active medium (Kuzmina 1989, 1991;Kokhanovsky 1999a):

Gðz; z0; lÞ ¼ e��sðz;z0Þ

l

ch Dsðz;z0Þ2l 0 0 �sh Dsðz;z0Þ

2l

0 cos D/ðz;z0Þ

l � sin D/ðz;z0Þl 0

0 sin D/ðz;z0Þl cos D/ðz;z

0Þl 0

�sh Dsðz;z0Þ2l 0 0 ch Dsðz;z0Þ

2l

266664

377775; ð3:4:6Þ

�sðz; z0Þ ¼Zz

z0

�rtðxÞdx; ð3:4:7Þ

Dsðz; z0Þ ¼Zz

z0

½rþt ðxÞ � r�t ðxÞ�dx; D/ðz; z0Þ ¼

Zz

z0

½nþ ðxÞ � n�ðxÞ�dx: ð3:4:8Þ

If internal volume sources of radiation are absent, the coherently scattered radiationis obviously expresses by


Icðz; sÞ ¼ Gðz; 0; lÞf þ0 ðsÞþ GðH; z; lj jÞf�HðsÞ: ð3:4:9Þ

Further consideration easily shows, that the dynamical system (3.4.1) can berepresented in the form of four independent one-dimensional equations. To find theequations one should calculate the eigen values and eigen vectors of the extinctionoperator r of the Eq. (3.2.6):

k1 ¼ �rtðzÞ½1þDðzÞ� � rþt ðzÞ; Wð1Þ ¼ 1; 0; 0; 1½ �T ;

k2 ¼ �rtðzÞ½1� idðzÞ�; Wð2Þ ¼ 0; 1; �i; 0½ �T ;k3 ¼ �rtðzÞ½1þ idðzÞ�; Wð3Þ ¼ 0; 1; i; 0½ �T ;k4 ¼ �rtðzÞ½1� DðzÞ� � r�t ðzÞ; Wð4Þ ¼ 1; 0; 0; �1½ �T ;

ð3:4:10Þ

In the eigen basis fWð1Þ;Wð2Þ;Wð3Þ;Wð4Þg the extinction operator r is diagonal:

rð0Þ ¼ L�1rL; L ¼

1 0 0 1

0 1 1 0

0 �i i 0

1 0 0 �1

26664

37775 L�1 ¼ 1

2Lþ ;

rð0ÞðzÞ ¼ �rtðzÞdiagf1þDðzÞ; 1� idðzÞ; 1þ idðzÞ; 1� DðzÞg:

ð3:4:11Þ

Thus, if instead of the vector I ¼ I; Q; U; V½ �T we use the vector

U ¼ 12½IþV ; Qþ iU; Q� iU; I � V �T ; ð3:4:12Þ

we will have the system of independent equations for the components of the vectorU with the diagonal extinction operator defined by (3.4.11). To write the expressionfor the Green function Gðz; z0; lÞ; providing the solution to the problem fornon-scattered radiation transfer in terms of the vector U; it is convenient to intro-duce the optical thicknesses

skðz; z0Þ ¼Zz

z0

kkðxÞdx; ð3:4:13Þ

where kk are defined in (3.4.10). Finally we have


s1 ¼ sþ ðz; z0Þ ¼ Rzz0�rþt ðxÞdx ¼ �sþDs; s2 ¼ �sðz; z0Þ þ iD/ðz; z0Þ;

s3 ¼ �sðz; z0Þ � iD/ðz; z0Þ; s4 ¼ s�ðz; z0Þ ¼ Rzz0�r�t ðxÞdx ¼ �s� Ds;

ð3:4:14Þ

where

�sðz; z0Þ ¼Zz

z0

�rtðxÞdx ¼ 12½sþ ðz; z0Þ þ s�ðz; z0Þ�;

D/ðz; z0Þ ¼ /þ ðz; z0Þ � /�ðz; z0Þ; /ðz; z0Þ ¼ xc

Zz

z0

nðxÞdx:ð3:4:15Þ

So, for the Green function Gð0Þðz; z0; lÞ; providing the solution to the problem(3.4.1)–(3.4.2) in terms of U; we find the expression in the form of the diagonalmatrix:

Gð0Þðz; z0; lÞ ¼ diagfG1;G2;G3;G4g;Gkðz; z0; lÞ ¼ exp½�l�1skðz; z0Þ�; k ¼ 1; 2; 3; 4:

ð3:4:16Þ

It remains to note, that the optical thicknesses sþ and s� characterize spatialattenuation of radiation beams in the states of right and left circular polarizations,whereas the function D/ defines the phase incursion between the beams in samepolarization states.

1.3.5 The Equivalent System of Equations for Parametersof Polarization Ellipse

In order to study the behavior of polarization state of non-scattered radiation in aslab 0 z H of isotropic chiral medium we consider the simplest radiationtransport problem with homogeneous monodirected radiation beam incident to theboundary z ¼ 0 :

l@Iðz; sÞ@z

þ rðzÞI ¼ 0;

Iþ ð0; sÞ ¼ I0ðs0Þ ¼ I0~dðl� l0Þ~dðu� u0Þ;I�ðH; sÞ ¼ 0:

ð3:5:1Þ


It is convenient to introduce new variables

v ¼ 0:5 arctanðU=QÞ; Y ¼ V=I ¼ sinð2bÞ;Z ¼ 1� p2 ¼ I�2ðI2 � Q2 � U2 � V2Þ; ð3:5:2Þ

where ðv; bÞ are the parameters of polarization ellipse, p is the polarization degree(see 3.1.3). From the system of ordinary differential equations for the Stokesparameters the following system of the ODE equations for the variables (3.5.2) canbe easily derived (Kuzmina 1986b):

l@v@z

¼ xcDnðzÞ2

;

l@Y@z

¼ �DrtðzÞ2

ð1� Y2Þ;ð3:5:3Þ

l@Z@z

¼ DrtðzÞY � Z;

l@I@z

¼ ��rtðzÞI � DrtðzÞ2

Y � I;vð0Þ ¼ v0; Yð0Þ ¼ Y0; Zð0Þ

¼ 1� p20; Ið0Þ ¼ I0:

ð3:5:4Þ

As one can see, the equations for v and Y are independent on the other equationsof the system (3.5.3)–(3.5.4), and each of the equations can be exactly integrated:

vðz; lÞ ¼ v0 þ l�1 xc

Zz

0

Dnðz0Þ2

dz0; ð3:5:5Þ

Y ¼ tanhðar tanh Y0 � l�1 xc

Zz

0

Drtðz0Þ2

dz0Þ: ð3:5:6Þ

When Y has been obtained from (3.5.6), the function Z can be found from thethird equation of the system (3.5.4) (under Y known the equation is also exactlyintegrated). Finally we obtain the polarization degree p; dependent on Z accord-ingly to (3.5.2):

p2 ¼ 1� 1� p201� Y2

0sech2ðar tanh Y0 � DsðzÞ

2lÞ: ð3:5:7Þ

Now the intensity I can be found from the last equation of the system (3.5.3):


I ¼ 12I0ðe�sþ ðzÞ=l þ e�s�ðzÞ=lÞ½1� V0

I0tanhðDsðzÞ

2lÞ�: ð3:5:8Þ

In the case of homogeneousmedium DrtðzÞ � D ¼ const; DnðzÞ � d ¼ constð Þthe first two equations of system (3.5.3)–(3.5.4) are reduced to

2 _v ¼ d;

_Y ¼ �Dð1� Y2Þ: ð3:5:9Þ

As one can see from simple analysis of the system (3.5.9), at D 6¼ 0 the polar-ization state tends to the polarization eigenstate with the smaller value of absorption.At D ¼ 0 the rotation of polarization plane with constant speed d without change ofpolarization ellipse form takes place under �rtz=l increasing. Note, that in the casethe plane ðD; dÞ is the parametric space of dynamic system (3.5.9). In the case ofinhomogeneous medium all possible types of behavior of function Y ¼ YðDsðzÞ=2lÞat various relations between the sign of DðzÞ and the sign of initial value Y0 ¼ V0=I0are presented in the Fig. 1.3 (the solutions of the second equation of the system(3.5.9) at Dj j ¼ 0:1Þ (Kuzmina 1986b). Two types of function p2ðDsðzÞ=lÞ aredepicted in the Fig. 1.4: I : V0 [ 0;D\0; V0\0;D[ 0;II : V0 [ 0;D[ 0;V0\0;D\0; Although both the monotonic and the non-monotonic type of beha-viour with the distance are possible, the limit value is p ¼ 1 (full polarization). Thus,the optically active medium acts as a polarizer. In the case of chiral medium the anytype of radiation, propagating through the medium, is finally transformed into fullycircularly polarized radiation.

It is worth to note the feature of the parametrical domain ðD; dÞ of dynamicalsystem (3.5.9). The simple analysis shows, that the areas Gþ ¼ fD; d Dj [ 0; d 6¼0g and G� ¼ fD; d Dj \0; d 6¼ 0g are the areas of structural stability (robustness) ofthe system. The line D ¼ 0 is a bifurcation curve (it separates the parametric spaceinto two areas of structural stability, and the dynamical system itself becomes aconservative system on the line). The line d ¼ 0 is not a bifurcation curve, but thesystem (3.5.9) becomes a noncoarse dynamical system in the line (that is, the line

Fig. 1.3 The versions offunction Y Dj j ¼ 0:1ð Þ


d ¼ 0 represents a dangerous boundary in the parametric domain (Kuzmina 1986b;Bautin and Leontovich 1976).

Now it is worth to briefly summarize the main features of coherently scatteredradiation in the slabs of optically active media, that have been elucidated in theSects. 1.3.1–1.3.5.

1. The four-component vector transport Eq. (3.1.5) can be used in radiationtransport problems only in the case of weakly optically anisotropic media.

2. Geometrically isotropic optically active media are characterized by the specialtype of scattering phase matrices of the medium unit volume, defined by(3.1.11). There is a normalization condition on the phase matrix elementC14ðz; cÞ;

Ds � 12

Z1

�1

C14ðz; cÞdc;

permitting to find out whether the medium is optically isotropic (at Ds ¼ 0Þ; oroptically anisotropic (at Ds 6¼ 0Þ:3. For transport problems in optically active media it is necessary to use transport

equation with matrix differential operator of extinction, r; defined by the for-mula (3.1.8).

4. For transport problems with mono-directed external radiation beam, incident tothe slab boundary of optically anisotropic medium, it is necessary to calculatethe birefringent radiation in the slab.

Fig. 1.4 Two types offunctionp2 Dj j ¼ 0:1ð Þ


1.3.6 Radiation Transfer Problems for Slabs of ChiralMedia with Reflecting Boundaries

The multi-scattered radiation transport problem in a slab 0 z H is the solution ofthe boundary value problem for transport Eq. (3.2.1)–(3.2.3), where the extinctionoperator rðzÞ is defined by (3.2.6), and the integral operator PI is specified by theexpressions (3.1.10), (3.1.11). For analytical calculations the boundary conditionsat the slab boundaries are convenient to be putted zero, supposing that the addi-tional internal radiation sources to be localized at the slab boundaries z ¼ 0 andz ¼ H:

l@Iðz; sÞ@z

þ rðzÞI ¼ ðPIÞðz; sÞþ ~Fðz; sÞ; ð3:6:1Þ

Iþ ð0; sÞ ¼ 0; ð3:6:2Þ

I�ðH; sÞ ¼ 0; ð3:6:3Þ~Fðz; sÞ ¼ Fðz; sÞþ lf þ0 ðsÞ~dðzÞþ lj jf�HðsÞ~dðz� HÞ: ð3:6:4Þ

The solution of the problem (3.6.1)–(3.6.4) for coherently scattered radiation,Icðz; sÞ;

l@Icðz; sÞ

@zþ rðzÞIc � DIc ¼ Fðz; sÞ ð3:6:5Þ

Iþc ð0; sÞ ¼ 0; ð3:6:6Þ

I�c ðH; sÞ ¼ 0; ð3:6:7Þ

can be written in the form

Icðz; sÞ ¼ Aþ ~Fþ A�~F; ð3:6:8Þ

where

ðAþ ~FÞðz; sÞ ¼ l�1Rz0G z

l ;z0l

�~Fþ ðz0; sÞdz0; l� 0

0; l\0

8<: ð3:6:9Þ

ðA�~FÞðz; sÞ ¼0; l� 0

lj j�1RHzG z0

lj j ;zlj j

�~F�ðz0; sÞdz0; l\0

8<: : ð3:6:10Þ


The Green function Gðz=l; z0=lÞ ¼ Gðx� x0Þ in (3.6.9), (3.6.10), is known (see3.4.6) and can be written as

Gðx� x0Þ ¼ e�ðx�x0ÞchDÞðx� x0Þ 0 0 �shDðx� x0Þ

0 cos dðx� x0Þ � sin dðx� x0Þ 00 sin dðx� x0Þ cos dðx� x0Þ 0

�shDðx� x0Þ 0 0 chDðx� x0Þ

2664

3775;

ð3:6:11Þ

where x ¼ �rz=l:The solution of transport problem for non-scattered radiation for the slab

0 z H with reflecting boundary z ¼ H is often of special interest. The boundaryvalue problem can be written as

DIc ¼ Fðz; sÞ; ð3:6:12Þ

Iþc ð0; sÞ ¼ 0; ð3:6:13Þ

I�c ðH; sÞ ¼ ðRIþc ÞðH; sÞ; ð3:6:14Þ

where

ðRIþ ÞðH; sÞ ¼ZXþ

Rðs; s0ÞIþ ðH; s0Þds0; ð3:6:15Þ

Due to the linearity of the boundary value problem (3.6.12)–(3.6.15) it is pos-sible to find out the relation between the solution Icðz; sÞ of the problem for the slabwith nonreflecting boundaries and the corresponding solution ~Icðz; sÞ of the sameproblem with reflecting boundary z ¼ H:

~Icðz; sÞ ¼ Icðz; sÞþ ðA�RAþ ~FÞðz; sÞ: ð3:6:16Þ

It is natural to expect, that the relation between the solution Iðz; sÞ of problem(3.6.1)–(3.6.4) for multiply scattered radiation for slab with non-reflecting bound-aries and the solution ~Iðz; sÞ � IRðz; sÞ of the same problem for the slab withreflecting boundary z ¼ H can be found out. The relation can be written in the form(Kuzmina 1986b):

~IRðz; sÞ ¼ Iðz; sÞþ �SX1k¼1

ðR�SÞkRð�S~FÞðz; sÞ � S~Fþ �S½E � R�S��1R�S~F; ð3:6:17Þ

where


Iðz; sÞ ¼ S~F � D�1ð~Fþ PIÞ; �S~F ¼ S½f0~dðzÞþ fH~dðz� HÞ�;E ¼ diagf1; 1; 1; 1g:

ð3:6:18Þ

It should be noted, that the relation between the solution ~Iðz; sÞ � IRðz; sÞ ofmulti-scattered radiation transport problem for slab with reflecting boundary and thesolution Iðz; sÞof the same problem with non-reflecting boundary does not dependon concrete form of Green function, governing the behaviour of non-scatteredradiation in the slab. Similar relations are valid for transport problems for slabsof optically isotropic medium in analogous situations (Germogenova 1985;Germogenova et al. 1989).

1.4 The Estimation of Medium Weak AnisotropyInfluence by a Perturbation Method

1.4.1 The Reduction of Transport Problem for AnisotropicMedium to a Recurrently Solvable System of Problemsfor Isotropic Media

We return to the problem of multi-scattered radiation transport in a slab of geo-metrically isotropic optically active medium, that can be formulated as a boundaryvalue problem to the VRTE.

l@Iðs; sÞ

@zþ rðsÞI ¼ ðPIÞðs; sÞþFðs; sÞ; ð4:1:1Þ


I�ðsH ; sÞ ¼ f�HðsÞ; ð4:1:3Þ

where (see 3.1.8*, 3.1.14)

rðsÞ ¼ �rt

1 0 0 DðsÞ0 1 dðsÞ 00 �dðsÞ 1 0

DðsÞ 0 0 1

2664

3775; ð4:1:4Þ

P I is the integral operator of scattering specified by the expressions (3.1.10),(3.1.11), and


�rt ¼ 0:5ðrþt þ r�t Þ; s � sð0; zÞ ¼

Zz

0

�rtðxÞdx: ð4:1:5Þ

The functionsDs � Drs ¼ rþs � r�s ;Da � Dra ¼ rþ

a � r�a ; andDn ¼ nþ � n�

represent macro-characteristics of optically active medium. The optically anisotropicmedium may be considered as a weakly anisotropic one if Ds;Da and Dn (as thefunctions of variable zÞ are uniformly small for all z 2 ½0;H�: In the case a smallparameter can be introduced and a perturbation method may be developed. Introducethe values

�Ds ¼ supz2½0;H� DsðzÞj j; �Da ¼ supz2½0;H� DaðzÞj j;�d ¼ x

csupz2½0;H�

nþ ðzÞ � n�ðzÞj j�rtðzÞ ;

ð4:1:6Þ

where the symbol sup (supremum) denotes the least upper bound of the function(see, for example Rudin 1976). Then the value

e ¼ maxð�Ds; �Da; �dÞ ð4:1:7Þ

can figure as the mentioned small parameter.In terms of e the operators the transport Eq. (4.1.1) can be represented in the

following forms. The extinction operator can be written as

rðsÞ ¼ diagð1; 1; 1; 1Þþ e

0 0 0 ~DðsÞ0 0 ~dðsÞ 00 �~dðsÞ 0 0

~DðsÞ 0 0 0

2664

3775; ð4:1:8Þ

where

~D ¼ e�1DðsÞ; �d ¼ e�1dðsÞ; ~DðsÞ�� 1; ~dðsÞ�� 1; s 2 ½0; sH �: ð4:1:9Þ

Based on physical cosiderations (Zege and Chaikovskaya 1984; Zege et al.1991), the phase matrix Cðs; cÞ; defining the integral operator of scatteringaccordingly to (3.1.10) and (3.1.11), can be also presented in the form of the sum

Cðs; cÞ ¼ C0ðs; cÞþ e ~Cðs; cÞ; ð4:1:10Þ

where


C0ðs; cÞ ¼ �kðsÞ

a1 b1 0 0b1 a2 0 00 0 a3 c10 0 �c1 a4

2664

3775; ~Cðs; cÞ ¼

0 0 ~c2 ~b20 0 ~c3 ~b3

�~c2 �~c3 0 0~b2 ~b3 0 0

2664

3775

ð4:1:11Þ

the elements of ~Cðs; cÞ being of the Oð1Þ order.By using the decompositions (4.1.8) and (4.1.10) one may present the solution of

the boundary value problem (4.1.1)–(4.1.3) in the form of expansion into a series one powers:

Iðs; sÞ ¼X1n¼0

enIðnÞðs; sÞ; ð4:1:12Þ

l@Ið0Þðs; sÞ

@zþ Ið0Þ ¼ ðPð0ÞIð0ÞÞðs; sÞþFðs; sÞ; ð4:1:13Þ

Ið0Þþ ð0; sÞ ¼ f þ0 ðsÞ; ð4:1:14Þ

Ið0Þ�ðsH ; sÞ ¼ f�HðsÞ; ð4:1:15Þ

and

l@IðnÞðs; sÞ

@zþ IðnÞ ¼ ðPð0ÞIðnÞÞðs; sÞþ ðP� rÞIðn�1Þ; ð4:1:16Þ

IðnÞþ ð0; sÞ ¼ 0; IðnÞ�ðsH ; sÞ ¼ 0; n ¼ 1; 2; . . . ð4:1:17Þ

The convergence of the perturbation method has been analyzed for the case ofhomogeneous optically active medium DðsÞ ¼ const; dðsÞ ¼ constð Þ and externalbeam of non-polarized radiation, incident to the slab boundary z ¼ 0;

F � 0; f þ ð0; sÞ ¼ IincðsÞ½1; 0; 0; 0�T ; fðsH ; sÞ ¼ 0:

For the proof of the perturbation method convergence it was necessary to use theintegral vector transport equation instead of integro-differential transport Eq. (4.1.1)(Kuzmina 1991). It is worth mentioning that the properties of the scalar integraltransport equation and the integral characteristic equation, defining asymptoticcharacteristics of deep radiative regimes, were extensively analytically studied in(Maslennikov 1968, 1969). The effective methods of analytical and computationalstudies of the scalar integral characteristic equation were further proposed in thepublications (Rogovtsov et al. 2009, 2016; Rogovtsov 2015a, b). In particular, theapplication of general invariance principles to various scalar radiative transferproblems allowed to carry out a number of analytical results (such as analytical


representation of “surface” and “volume” Green functions, plane and sphericalalbedos and others) (Rogovtsov et al. 2016).

Finally, the estimate of the total Stokes vector perturbation has been derived inthe form

Iðs; sÞ � Ið0Þðs; sÞ�� Ce½1; 1; 1; 1�T ; ðs; sÞ 2 D ¼ ½0; sH � fXnfsjl ¼ 0gg:ð4:1:18Þ

[The Cartesian product of the sets is denoted by the symbol in Eq. (4.1.18)].The constant C in Eq. (4.1.18) depends on the essential parameters of the

transport problem. In the case of non-conservatively scattering medium ð�kðsÞ\1Þthe constant C can be estimated in terms of the following parameters

�k ¼ sups2½0;sH ��kðsÞ; Iincmax ¼ sups2X IincðsÞ

Cj j ¼ sups2½0;sH � supm;n C0mn

L2½�1;1�;

~C�� ¼ sups2½0;sH � supm;n ~Cmn

L2½�1;1�:

ð4:1:19Þ

As far as the estimated by the perturbation method Stokes vector

Iðs; sÞ ¼ Ið0Þðs; sÞj þCe½1; 1; 1; 1�T ; ð4:1:20Þ

has to satisfy the inequality I2 � Q2 � U2 � V2 � 0; the following constraint on thesmall parameter has been obtained (Kuzmina 1991):

e\e0ð1� �kÞð1� pmaxÞIð0Þminj

Cj jIðincÞmax

; ð4:1:21Þ

where

Ið0Þmin ¼ infðs;sÞ2D Ið0Þðs; sÞ;pmax ¼ sup ðs;sÞ2DðIð0ÞÞ�1½ðQð0ÞÞ2 þðUð0ÞÞ2 þðV ð0ÞÞ2�1=2;

ð4:1:22Þ

e0 being some constant, not depending on the parameters of transport problemfor slab of optically active medium, and the symbol inf (infimum) denotes thegreatest lower bound of the function (Rudin 1976). The constraint (4.1.21) for eshould imply that there are some transport problems for slabs of optically aniso-tropic media in which the perturbation of the solution due to medium anisotropymight be not small.


1.4.2 The Estimation of Weak Medium AnisotropyInfluence

The uniform estimate for total perturbation of multiple scattered radiation transportsolution for a slab of weakly anisotropic optically active medium as compared tocorresponding problem for optically isotropic medium (with mean optical charac-teristics) by a perturbation method has been obtained. However, some remarks onpossibility of estimation of weak medium anisotropy influence using the pertur-bation method series should be made.

The first note concerns the fact that the transport problem is defined in anon-compact domain D of variables ðs; sÞ : ðs; sÞ 2 D; D ¼ ½0; sH � fXnfs lj ¼0gg: Usually in such a case perturbation method expansions convergesnon-uniformly in D; especially in the situations when the equations for perturbedproblem and for unperturbed one differ qualitatively. It is just our case. Indeed, asthe results of qualitative analysis of the two-dimensional dynamical system (3.5.9)showed, the point D ¼ 0; d ¼ 0 in the parametric domain of the system (the pointjust corresponds to optically isotropic medium and e ¼ 0 in the perturbation methodexpansion) is a bifurcation point of the dynamical system (see Sect. 1.3.5).However, for the transport problem, formulated in terms of Stokes vector, thepolarization characteristics of radiation field represent the interest only in theregions where the radiation intensity does not vanish. So, the Stokes parameters arejust the adequate characteristics in the sense, and the global deviation of thetransport problem solution in the slab of optically active medium from that one inthe slab of corresponding isotropic medium still can be estimated.

The second note concerns the constraint (4.1.21) for e, obtained in the process ofthe perturbation method convergence proof. The constraint should imply that theradiation transport problems for optically active media actually should be bettertreated independently, as a special class of transport problems.

1.4.3 An Example: The Estimation of PolarizationCharacteristics Perturbation in a Slab of IsotropicMedium with Non-Block-Diagonal Scattering PhaseMatrix

Optically isotropic medium composed of chaotically distributed non-sphericalscatterers can be characterized by non-block-diagonal phase matrix of the type(3.1.11). Sometimes it can be of interest what is the effect of the phase matrixnon-block-diagonality on polarization characteristics of multiply scattered radiationin isotropic medium (in comparison with the same transport problem for themedium specified by the block-diagonal phase matrix). A qualitative answer can beobtained via application of the perturbation method.


Consider the radiation transport problem for the slab of 0 z H of opticallyisotropic medium with non-reflecting boundaries, defined by the equations (see3.2.1–3.2.5):

l@Iðz; sÞ@z

þ rðzÞI ¼ ðPIÞðz; sÞþFðz; sÞ; ð4:3:1Þ


I�ðH; sÞ ¼ f�HðsÞ: ð4:3:3Þ

where rt is the scalar operator, rt ¼ rtðzÞ � diagf1; 1; 1; 1g: Let the integral oper-ator of scattering P (see 3.1.10) is defined by the phase matrix

Cðz; cÞ ¼ Cð0Þðz; cÞþ eC

ð1Þðz; cÞ; ð4:3:4Þ

where

Cð0Þ ¼ �rs

a1 b1 0 0b1 a2 0 00 0 a3 c10 0 �c1 a4

2664

3775; Cð1Þ ¼ �rs

0 0 c2 b20 0 c3 b3

�c2 �c3 0 0b2 b3 0 0

2664

3775 ð4:3:5Þ

and e � 1 (see (4.1.10)–(4.1.11)). As it was marked in the Sect. 1.3.1, thecondition

12

Z1

�1

C14ðcÞdc � 12

Z1

�1

b2ðcÞdc ¼ 0 ð4:3:6Þ

should be fulfilled in the case of optically isotropic medium. The “influence” ofsmall non-block-diagonality of the phase matrix on the transport problem solutionfor a slab of optically isotropic medium can be estimated by the perturbationmethod (Kuzmina 1987).

It is convenient to schematically present the solution of multi-scattered radiationtransfer problem (4.3.1)–(4.3.5) in the compact form (the decomposition on suc-cessive orders of scattering)

Iðz; sÞ � S~F ¼X1k¼0

ðAPÞkA~Fðz; sÞ; ð4:3:7Þ

where ~F is defined by the formula


~Fðz; sÞ ¼ Fðz; sÞþ lf þ0 ðsÞ~dðzÞþ lj jf�HðsÞ~dðz� HÞ; ð4:3:8Þ

(See 3.4.3). The operator A in Eq. (4.3.7) is the operator of attenuation (extinction)of non-scattered radiation, defined by the expression

A~Fðz; sÞ ¼ Aþ ~Fþ A�~F; ð4:3:9Þ

where Aþ ~F; A�~F are defined accordingly to (3.6.9) and (3.6.10). For opticallyanisotropic medium the operator A is the matrix operator whereas for opticallyisotropic medium it is the scalar operator, defined by the scalar Green function

Gðz=l; z0=lÞ ¼ e��sðz;z0Þ

l diagf1; 1; 1; 1g;�sðz; z0Þ ¼Zz0

z

rtðxÞdx ð4:3:10Þ

Therefore, as far as radiation attenuation in optically isotropic medium does notchange the polarization characteristics of radiation AP ¼ PA

� �; the series (4.3.7)

can be rewritten in the form

IðisotrÞðz; sÞ � SðisotrÞ~F ¼X1k¼0

IðkÞ ¼X1k¼0

Akþ 1Pk~Fðz; sÞ: ð4:3:11Þ

As one can see, the k-the term of the expansion (4.3.11) may be presented in theform

IðkÞðz; sÞ ¼ A~PIðk�1Þ þ ASðisotrÞ �PIðk�1Þ; ð4:3:12Þ

where �P is the integral operator of scattering, defined by the phase matrix �C ¼Cð0ÞCð1Þ

(obviously, the matrix �C possesses the algebraic structure similar to that of

the matrix Cð1ÞÞ:

For the further analysis of the terms IðkÞðz; sÞ of the series (4.3.11) (expressedthrough PkI

�it is necessary to exploit the representation of I ðz; sÞ in the form of

Fourier series decomposition over the system of generalized spherical functionsfY l

msg. If the Stokes parameters are used as the vector Iðz; sÞ componentsIðz; sÞ ¼ ðI;Q;U;VÞT� �

; the Fourier series can be written in the form (Kuzmina1978);

Iðz; l;uÞ ¼X1l¼0

Xl

s¼�l

Y lsI

lsðz; l;uÞ; ð4:3:13Þ

where


Y ls ¼

Yl0s 0 0 0

0 12 ðYl

2s þ Yl�2sÞ i

2 ðYl2s � Yl

�2sÞ 0

0 i2 ðYl

2s � Yl�2sÞ 1

2 ðYl2s þ Yl

�2sÞ 0

0 0 0 Yl0s

26664

37775;

Ils ¼

Il0s12 ðQl

2s þQl�2sÞ � i

2 ðUl2s � Ul

�2sÞ12 ðUl

2s þUl�2sÞþ i

2 ðQl2s � Ql

�2sÞVl0s

26664

37775

ð4:3:14Þ

The functions ðPkIÞðz; l;uÞ; entering into the series (4.3.11), can be written interms of series over fY l

msg as

ðPkIÞðz; l;uÞ ¼X1l¼0

Xl

s¼�l

Y lsðC

ðlÞY ÞkIlsðs; sÞ; ð4:3:15Þ

where CðlÞY are the matrices, containing proper combinations of the decomposition

coefficients on fY lmsg of the phase matrix C (Kuzmina 1987).

The estimation of the solution perturbation caused by smallnon-block-diagonality of the phase matrix can be rather easily obtained for thesimplest transport problem—the axially symmetric transport problem for slab,illuminated by external mono-directed linearly polarized radiation beam, normallyincident to the boundary z ¼ 0 :

rt ¼ rð0Þt ðzÞ;Fðz; sÞ ¼ 0; fHðsÞ ¼ 0; f0ðsÞ � f�0 ¼ ðI0;Q0; 0; 0ÞTdðzÞ ð4:3:16Þ

Via using the expression (4.3.12) and the decomposition (4.3.15), the followingrelations can be obtained:

~P~F ¼ ~Pf�0dðzÞ ¼ I� � ð0; 0;U�;V�ÞT ; �P~F ¼ �I�;PðisotrÞI� ¼ I0 � ðI�;Q�; 0; 0ÞT ; ð4:3:17Þ

Using (4.3.7), one can finally obtain, that the Stokes vector Ið1Þðz; l;uÞ is of theform

Ið1Þðz; l;uÞ ¼ ð0; 0; U1ðz;l;uÞ; V1ðz; l;uÞÞT ; ð4:3:18Þ

where the functions U1ðz; l;uÞ and V1ðz; l;uÞ in (4.3.18) do not contain the zerol� harmonics. It is the consequence of the fact that due the medium isotropy

Ds ¼ 0ð ; see (3.1.13)) we have �Cð0Þ ¼ 0. Further qualitative analysis based on the

decomposition (4.3.15) allows to elucidate the general properties of the remainingfunctions IðkÞðz; sÞ; k[ 1 : the functions Ið2k�1Þðz; sÞ; defining the contributionsOðe2k�1Þ into the total solution, have the form


Ið2k�1Þðz; l;uÞ ¼ ð0; 0; U2k�1ðz; l;uÞ; V2k�1ðz;l;uÞÞT ; ð4:3:19Þ

whereas the functions Ið2kÞðz; sÞ; defining the contributions Oðe2kÞ have the form

Ið2kÞðz; l;uÞ ¼ ðI2kðz; l;uÞ; Q2kðz; l;uÞ; 0; 0ÞT : ð4:3:20Þ

The Fourier analysis allows to extract an essential qualitative information on theradiation transport problem solution for the slab of optically isotropic mediumspecified by non-block-diagonal phase matrix, defined by (4.3.4)–(4.3.5). Namely,we have the following result: (a) the multiply scattered light in the slab is weaklyelliptically polarized; (b) the deviations of both the radiation intensity and the linearpolarization degree from the corresponding characteristics of unperturbed transport

problem (specified with block-diagonal phase matrix Cð0ÞÞ are of Oðe2Þ (Kuzmina

1978).

1.5 An Outline of Some Results on Radiation TransferProblems in Anisotropic Media of Another Types

1.5.1 Anisotropic Media in the Earth Atmosphere RemoteSensing Problems

It is now well recognized that cirrus clouds have a major influence on theEarth-ocean-atmosphere energy balance. The macroscopic optical properties of adisperse media consisting of scattering particles, randomly oriented in the space, isultimately defined by particle microscopic characteristics (particle size, shape andthe refractive index) and the distribution function on the particle orientations. If theorientations of non-spherical particles of disperse medium are not totally random,the medium is proved to be optically anisotropic. The ice crystal clouds (cirrus andcirrostratus) provide the examples of optically anisotropic media, demonstrating thewell-known atmospheric optical phenomenon of halo. The crystals responsible forhalo may be horizontally oriented flat, hexagonal plates or oriented column-shapedcrystals. The ice crystals can be suspended near the ground, in which case they arereferred to as diamond dust. When the dust anisotropic medium is formed bycolumn-shaped crystals the known phenomenon of light pillars can be observed.

(A) Ice crystal clouds

Ice cloud disperse optically anisotropic media, formed by spatially orientedsuspended tiny ice crystals, belong to the general class of essentially opticallyanisotropic media, in which medium optical anisotropy is accompanied by itsgeometrical anisotropy. The well-known atmospheric optical phenomenon of halois just created by light reflection from these anisotropic media. Another familiar


phenomenon is light pillars that is produced by light reflection from anisotropicmedia formed by column-shaped ice crystals (see Figs. 1.5 and 1.6). Modeling ofpolarized radiative transfer in the anisotropic media requires construction of thematrix extinction operator and the scattering phase matrix of the vector transportequation, governing radiation transport in the anisotropic medium. Various modelsof disperse anisotropic media were designed and the operators of the VRTE wereconstructed. In particular, disperse medium models composed of chiral particleswere created (Ablitt et al. 2006; Liu et al. 2013). The radiative transfer in a chiralanisotropic medium was studied via Monte Carlo simulations, and the effects ofmedium chirality were elucidated (Ablitt et al. 2006). The optical properties ofscattering anisotropic medium models formed by ice crystals of cirrus clouds can beobtained based on geometrical and physical optics approaches (Borovoi et al. 2000,2006, 2007, 2010; Borovoi 2005, 2006, 2013).

Modeling of radiation transport processes in cirrus clouds is of importance forthe Earth atmosphere remote sensing problems (Takano et al. 1989, 1993;Mishchenko et al. 2000; Liou 2002; Mishchenko et al. 2002; Liou et al. 2011). Theattempt of radiative transfer problem analysis for optically anisotropic medium,formed by horizontally oriented ice cloud crystals, has been performed in (Takanoet al. 1993), based on vector transport equation with scalar extinction operator. As it

Fig. 1.5 The example of icehalo [www.ice-halo.net]

Fig. 1.6 The example oflight pillars [Sterlitamak,Russia, 19.12.2015,S. Lifanov]


http://www.ice-halo.net

was discussed in (Mishchenko 1994a, b), the approach could provide a significanterror in radiative transfer problem solutions. To estimate the error it would bedesirable to find the exactly solvable problem for the anisotropic medium forcomparison the exact and the approximate results. Such comparison was previouslyfulfilled for another type of anisotropic medium model (composed of perfectlyaligned prolate and oblate spheroids), and a significant discrepancy was demon-strated (Tsang et al. 1991; Ishimaru et al. 1984).

The Monte-Carlo simulations of radiation transfer in crystal cloud opticallyanisotropic media models have been performed (Grishin et al. 2004; Prigarin et al.2005). The medium models have been designed, and the functions entering into thevector transport equation have been calculated (Volkovitski et al. 1984; Takano andLiou 1989; Borovoi et al. 2000]; Grishin et al. 2004; Kokhanovsky 2005, 2006).

The Monte Carlo simulations of halos in crystal cloud models of opticallyanisotropic media have been performed. The computer simulation results demon-strated that the anisotropy of cloud medium can strongly affect the cloud opticalproperties. In particular, both halo patterns and angular distributions of the upwardand downward radiation are strongly dependent on optical anisotropy characteris-tics. Besides it was found, that the cloud optical anisotropy can result not only fromthe shape and spatial orientation of cloud particles, but, in addition, it can be aconsequence of non-poisson spatial particle distribution (Prigarin et al. 2005a, b,2007, 2008).

It should be added, that although many important studies of ice crystal cloudmedia, composed of non-randomly distributed particles, have been undertaken(Liou 2002; Kokhanovsky 2003, 2004, 2005a, b; Prigarin et al. 2005, 2007, 2008),further investigations of radiative transfer problems for optically anisotropic cloudmedia are still required.

(B) Densely packed disperse media

Radiation transport in dense scattering media is of interest both from theviewpoint of the Earth remote sensing problems and from the viewpoint of a varietyof other applications, including non-invasive medical investigation of biologicaltissues. Calculation of extinction matrix and scattering phase matrix for denselypacked media composed of non-spherical wavelength-sized scatterers demandstaking into account all the details of strongly inhomogeneous scattered radiationfield in the vicinity of any scatterer (Borovoi et al. 1983, 2005, 2013).

The necessity of studying of multi-scattered radiation transport processes in theEarth ice-snow cover follows from the fact that both ice and snow covers belong tothe class of strongly reflecting Earth surfaces (the reflectance of pure snow covercan achieve 90% in the visible wavelength band) (Kokhanovsky 1998; Farrell et al.2005; Kokhanovsky 2011). The importance of ice-snow cover monitoring is relatedto climatology problems: as it is established experimentally, the Earth surface,covered by ice and snow, is shrinking rather quickly over the past 25 years(Munneke 2009) (see Fig. 1.7).


In remote sensing problems both snow and clouds can be treated as dispersemedia consisting of mutually independent ice crystals. The snow layers can be alsomodelled as a random disperse media with densely packed particles ofnon-spherical shape (Kokhanovsky 1998). Sometimes the snow layer can be alsomodelled as an ice cloud consisting of fractal particles (in the visible wavelengthband) (Kokhanovsky 2003; Liou et al. 2011). Snow particle size, pollutant con-centrations, and the snow layer thickness represent the essential model parameters.The analytical approximation to radiation transport processes in snow cover layershas been developed (Kokhanovsky 2011). It provided the possibility to compare thecalculated snow cover characteristics (such as snow particle size, pollutant con-centration, snow cover albedo) with the results of satellite measurements(Kokhanovsky 2005a, 2011).

The results of accurate computer study of multiple electromagnetic radiationscattering by densely packed disperse medium models can be found in (Tse et al.2007; Tsang et al. 2007; Tseng 2008; Okada and Kokhanovsky 2009;Randrianalisoa 2010; Dlugach et al. 2011).

In another approach a two-layer model of radiation transfer in theatmosphere-snow system was designed, in which the lower layer was modelled as adisperse medium consisting of hexagonal ice crystals (Munneke 2009). It has beenpreviously hypothesized (Wiscombe and Warren 1980) that scattering andabsorptive properties of any ice crystal model can be approximated by the appro-priate disperse medium model composed of spherical particles, as long as thevolume-to-surface radio is conserved. Accurate computation of radiative transferproblem in the two-layer snow-atmosphere model has been performed. The mediummodel was further extended to that consisting of both snow and cloud layers. Theinfluence of cloud layer presence on snow surface albedo was demonstrated. Thus,it could be estimated as an additional evidence that clouds have a considerableimpact on the radiation balance of the atmosphere-snow system (Munneke 2009).In a whole, clouds increase the broadband clear-sky albedo of the snow cover. The

Fig. 1.7 The minimumsea-ice extent andconcentration in the ArcticOcean. http://www.ncidc.org,National Snow and Ice DataCenter, USA. [Munneke2009]


http://www.ncidc.org

concurrent observations were compared with model calculations, providing goodresults.

The major results, obtained for multiply cattered radiation transport problems insnow cover, were found under the assumption that the effective continuous med-ium, corresponding to coherently scattered radiation propagation, is an opticallyisotropic medium. However, in the case of densely packed ensembles of scatterersin a number of situations the effective medium may turned out to be opticallyanisotropic. There exist recent papers devoted to the studying of optical charac-teristics of these effective media (see, for example, Alonova et al. 2013). The modelof disperse medium of densely packed spheres, used in the problem of activemicrowave remote sensing of terrestrial snow, was treated in (Tsang et al. 2011),and a significant value of cross polarization was obtained. The results are consistentwith the experimental observations.

(C) The extinction and scattering phase matrices for models of disperseoptically anisotropic media

A great variery of disperse optically anisotropic medium models (formed byensembles of non-spherical particles with random and preferred types of particleorientations) have been designed with the aim of accurate calculation of extinctionoperators and scattering phase matrices of the VRTE governing the radiativetransfer in anisotropic media. It allowed to study the dependence of the mediumscattering macro-characteristics on the parameters of medium microstructure(Mishchenko et al. 1992; Alexandrov et al. 1993; Mishchenko 1994a, b; Bolgovet al. 1998; Roux et al. 2001; Mishchenko et al. 2007, 2016a, b; Xie et al. 2011;Shefer 2013, 2016; Gao et al. 2012, 2013; Liu et al. 2013; Marinyuk et al. 1992).For some models the medium backscattering efficiencies have been also estimated.For example, it was done for a model of polydisperse medium consisting of dis-ordered randomly distributed infinite Mie cylinders with different refractive indices.Under medium illumination perpendicularly to the cylinder axes the albedo prob-lem for homogeneous half-space was analyzed. The coherent backscattering factorsfor several two-dimensional medium models were found as well (Mishchenko et al.1992). In the paper (Gao et al. 2012) the medium phase matrices were calculated(by the Discrete Dipole Approximation method) and the backscattering efficiencieswere estimated for the disperse medium composed of small layered plates. Severalice cloud models consisting of smooth, roughened, homogeneous and inhomoge-neous hexagonal ice crystals with various aspect ratios were designed and studiedwith the aim of application to the satellite-based retrieval of ice cloud properties(Xie et al. 2011). The extinction matrices were calculated for the medium modelcomposed of plates (both infinite-radius plates and finite-size particles) (Gao et al.2013). A medium model composed of chiral particles was designed, a helical liquidcrystal model of a capsule shape being used for modeling of single medium scat-terer (Liu et al. 2013), The distribution on particle orientations of twist type wasconstructed. The matrix extinction operator of the VRTE (with 16 elements),


providing the medium ability of differentiating left and right circularly polarizedlight, was obtained.

1.5.2 Magneto-Gyrotropic Media

The magnetoactive plasma—cold rarefied plasma in a permanent magnetic fieldH0—represents an example of non-absorbing optically anisotropic (gyrotropic)medium with elliptical birefringence (Zheleznyzkov 1977, 1996; Dolginov et al.1995; Ginzburg et al. 1975). In appropriate parametric domain the magnetoactiveplasma can possess strong optical anisotropy. To analyze the anisotropy it isconvenient to present the components of dielectric permittivity tensor e in terms ofparameters u and v; where u ¼ x2

H=x2; v ¼ x2

L=x2;xH� the cyclotron frequency

for electron, xL� the plasma frequency. If one uses the coordinate system wherethe wave vector k is directed along the z axis, and H0 is located in the plane ðx; yÞ; abeing the angle between H0 and z axis (that is, H0 ¼ ð0;H0

y ;H0z ÞÞ; then the tensor e

can be written in the form:

e ¼e0 � g

ffiffiffiu

pig cos a �ig sin a

�ig cos a e0 � gffiffiffiu

pcos2 a 0

ig sin a gffiffiffiu

pcos a sin a e0 � g

ffiffiffiu

psin2 a

24

35; ð5:2:1Þ

where

e0 ¼ eðisotrÞ ¼ 1� v; g ¼ vffiffiffiu

p1� u

:

The tensor e completely defines the refractive index squares n2o; n2e of the normal

wave modes in the medium, and their polarization states can be found throughcalculation of the eigenvalues and eigenvectors of the two-dimensional projectione? of the tensor (5.2.1) (Zheleznyzkov 1977, 1996):

e? ¼ e0 � gffiffiffiu

pig cos a

�ig cos a e0 � gffiffiffiu

pcos2 a

� �: ð5:2:2Þ

Not writing down the explicit formulas for n2o; n2e and the polarization states

(Zheleznyzkov 1996), we can mark here some limit cases and qualitative conse-quences. In general case (at a 6¼ 0; a 6¼ p=2Þ the normal waves are ellipticallypolarized and their polarization states are almost orthogonal. The n2o; n

2e are

expressed in the form of complicated functions of u; v;x; which are simplified inthe cases of lengthwise a ¼ 0ð Þ and transverse ða ¼ p=2Þ directions of propagation.The polarizations of the normal waves are reduced to circular (for lengthwisepropagation) and to linear (for transverse propagation). It also should be noted that


in general case the polarization states of the normal waves are not orthogonal, andin the situation the radiation intensity does not equal to the sum of intensities of thenormal waves. The fact should be taken into account in transport problems forstrongly optically anisotropic media.

1.5.3 Optically Active Media Occurred in Bio-Medical Fieldof Research

Biological tissues belong to optically inhomogeneous absorbing media with therefracting indices greater than the refractive index of the air. They can be dividedinto two main classes—strongly scattering (turbid) and weakly scattering (trans-parent). The analysis of polarization characteristics of multiply scattered radiation inbiological media is one the most important instruments for estimation the featuresof the internal media structure (via solution of the inverse problems of radiationtransport). A wide variety of biological tissues belong to optically anisotropicmedia, demonstrating the birefringence of various types. For example, opticalanisotropy of bio-tissues can be a consequence of the refractive index difference ofthe base matter and the collagen fibers. Chiral molecules are typically enclosed inbio-tissues, and so circular birefringence and optical activity are two commonphenomena for radiation transfer in the media. In a whole, the bio-tissues usuallybelong to four large classes of optically anisotropic media: optically isotropicmedia, uniaxial crystals, biaxial crystals and optically active (chiral) media. Themeasurement of bio-tissue refractive indices is one of the actual problems ofbio-tissue optics. On the other hand, the results of analysis of multiply scatteredradiation transport through optically inhomogeneous bio-tissues provide a valuableinformation about the features of their internal structure. So, the design of adequatemathematical models of disperse bio-tissues is of importance. The four-componentvector transport equation with matrix differential operator is necessary for modelingthe radiation transport processes in optically anisotropic bio-tissue media.

In traditional polarimetry the multiply scattered light depolarization measure-ments are widely used for determining the concentrations of optically activemolecules (such as glucose) in the scattering medium. The accurate modelling ofpolarized light propagation in turbid media, serving as templates for the biologicaltissues, and the comparison of the results of modelling with the measured data oftendemonstrates good agreement (Maruo et al. 2003; Larin et al. 2002). An example ofapplication of Monte Carlo modeling of multiply scattered polarized light transportin linearly birefringent and optically active media, figuring as the models of bio-logical tissues, was provided in (Wood et al. 2007). Measurements were also madeusing a Stokes polarimeter that detected the scattered light in different geometries.The comparison of the results of Monte Carlo simulations with the measurementsshowed a close agreement between both the results.


A closely related area of research concerns the application of radiation transporttheory approaches to the problems of non-invasive medical diagnostics ofnon-heterogeneities in biological tissues. In the papers (Bass et al. 2009, 2010) themethod of non-heterogeneities retrieval was based on the solution of direct problem—the obtaining of multiply scattered radiation field in the 3D spatial regions ofoptically isotropic turbid media (in arbitrary 3D region in (x, y, z)-geometry andaxially symmetrical cylindrical region) under the illumination by an anisotropic(collimated) laser radiation source. The deterministic method for calculating themultiply scattered radiation field in the 3D regions was applied (instead of MonteCarlo simulations). The simulation of ultra-short light pulse propagation in turbidmedia was additionally realized (Bass et al. 2010), and the parallel computationalalgorithms were applied. The methods developed in (Bass et al. 2009, 2010) mightbe easily generalized to the corresponding problems for optically active media.

1.5.4 Multilayered Anisotropic Media

Multilayered plane structures consisting of various optically anisotropic materialshave become increasingly widely used in modern optical systems such asnarrow-band birefringent filters and many other semiconductor devices. In practicethe optically anisotropic media are made of thin films, composites, artificialmaterials. The characteristics of these devices and their design are usually based onthe detailed understanding of electromagnetic radiation propagation through theseanisotropic layered media. The development of general theory of electromagneticradiation propagation in birefringent layered media began since 1970-ths (see, forinstance Yeh 1979; 1980). Analytical approaches to studying of radiative processesin multilayer optically anisotropic structures were also developed (Stammes et al.2001; Farrell et al. 2005; Kiasat et al. 2011).

Such new phenomena as the exchange Bragg scattering, optical surface waves,oscillatory evanescent waves were found in these media. Since the birefringentmultilayer waveguides are of great importance in the integrated optics, the phe-nomena of radiation reflectance and transmittance in layered anisotropic mediawere extensively studied. The reflectance and transmittance coefficients for multi-layered birefringent media can be expressed in terms of the overall transfer matrixcomponents. The behavior of the evanescent and the guided waves give rise tointeresting features of radiation transport in the birefringent layered structures. Incontrast to optically isotropic media, where the evanescent waves have a pureimaginary propagation constant, in birefringent layered media the evanescent wavecan decay exponentially with an oscillatory intensity distribution. Another inter-esting feature of periodic plane structures concerns the possibility of resonantradiation interaction with the medium (when radiation wave length is approximatelyequal to the layered structure period).

Multilayered structures, designed based on the porous silicon (PSi), play cur-rently an important role in various applications. These include microcavities,


photonic crystals, waveguide structures, photodetectors, sensors, etc. Besides,optically active materials can be designed based on porous silicon structures (forinstance, via infiltration appropriate electro- or thermo-optic media into the pores).Therefore, the porous silicon structures are turned out to be excellent candidates fortunable optical interconnects and switches. Novel layered anisotropic structures arealso applied in material science, electro-analytical chemistry, biological interfaces,tissue engineering, physics, and optics.

Two-dimensional periodic optically anisotropic structures are known as photoniccrystals (PCs). These periodic structures have been currently extensively studieddue to their wide abilities to control the light flows. Light transfer inside PCs can beanalyzed via modelling the processes governed by the Maxwell equations.A variety of photonic devices (such as polarization-independent waveguides,wavelength demultiplexers, beam deflectors, and routers) can de designed by uti-lizing the interesting PC features (Kiasat et al. 2011; Giden et al. 2014).

Usually the PC lattices possess translational, rotational or mirror symmetry.However, under some conditions the PC cell may convert into a chiral medium.Photonic quasi-crystals are also represent significant interest and have been inten-sively studied, the PCs with chiral optical properties being of special interest.A variety of new photonic devices is expected to be created (beam routers, splitters,deflectors) based on understanding the features of the PC optical anisotropy.

1.5.5 Liquid Crystals and Optical Fibers

As well known, in a normal liquid the molecule arrangement is equally disorderedin all directions. Liquid crystals are anisotropic: the molecules have some degree ofalignment, and the liquid crystal properties depend on the direction. In the nematicphase, the molecules are not layered and are free to rotate or slide. In the smecticphase the molecules maintain the general order of the nematic phase, but in additionaligned into layers. In the cholesteric phase, the molecules are directionally orientedand stacked in a helical pattern, each layer being rotated at a slight angle (seeFig. 1.8). Because of their anisotropic structures, liquid crystals exhibit unusualoptical and electrical properties that are exploited in a great variety of applications.With the rapid development of nanosciences, and the synthesis of many new ani-sotropic nanoparticles, the number of various types of liquid crystals is quicklyincreasing. Theoretical study of fluid crystal microstructure is a quite complicatedtask because of their high density, many-particle correlations and anisotropy ofparticle interactions (de Gennes 1974; Chandrasekhar 1977; Yariv and Yeh 1984).Chiral liquid crystal molecules usually give rise to chiral mesophases. A number ofunusual interference effects can be observed in the chiral mesophases, which arevery interesting for applications. For example, chiral liquid crystals can be used astunable filters in electrooptical devices (for hyperspectral imaging).

Liquid crystals can demonstrate phase transitions of second order, spontaneoussymmetry breaking, strong fluctuations, discontinuity (Arsenova 2009).


Liquid crystal technology is exploited in many areas of science and engineering,as well as in device technology. Promising applications of this special kind ofmaterial are possible, providing new effective solutions to a great variety ofproblems.

Optical fibers are flexible, transparent fibers made by drawing glass of a diameterslightly thicker than that of a human hair. They are used as a means to transmit lightbetween the two ends of the fiber and demonstrate successful utilization in the fieldof fiber-optic communications (since signals travel along them with extremely smalldissipation). In addition, the fibers are also characterized by providing an electro-magnetic interference. Specially designed fibers are successfully used in a variety ofapplications (including creation of fiber sensors). Fibers are also actively used inremote sensing. The fiber optics is the actively developed field of applied scienceand engineering. Optical fibers are widely used as sensors in measurements ofintensity, phase, polarization, wavelength.

An optical fiber can be modelled as a cylindrical dielectric waveguide thattransmits light along its axis with the help of the process of total internal reflection.The fiber consists of a core surrounded by a cladding layer. To confine the opticalsignal in the core, the refractive index of the core must be greater than that of thecladding. The boundary between the core and cladding may either be abrupt orgradual. In addition to internal diffuse light scattering, attenuation also occur due toselective absorption of specific wavelengths. The design of optical fibers requiresthe selection of materials based on knowledge of its properties and limitations.

Deeper understanding of light propagation in fiber-based materials is possiblevia modelling multiply scattered light transport problems in infinitely long cylin-drical fibers, the structural properties of the fiber being taken into account.Numerical solution of radiative transfer equation by Monte Carlo method was so farused. In the way the relations between light diffusion and fiber structure charac-teristics were partially elucidated (Linder 2014) The propagation of short pulses inbirefringent single-mode fibers was studied as well (Menyuk 1988).

The radiation transport problems for optically anisotropic media of fiber-likegeometry might be considered as a new class of radiation transport problems wherethe work is still at the very beginning.

Fig. 1.8 The moleculearrangement in nematic,smectic and cholesteric liquidcrystals


Acknowlegements The work has been supported by the Fund of Fundamental Research RAS, theDepartment of Mathematical Sciences, Project 1.3.2, the Program #3.Our heartfelt thanks to A.A Kokhanovsky for his kind suggestion to prepare this review. Our

great thanks also to three anonymous reviewers for their helpful remarks and comments on themanuscript.One of the authors (M.G.K) is greatly thankful for many remarkable researchers on radiation

transport theory for interesting and helpful discussions on various aspects of polarized radiationtransfer in due time. These are M.V. Maslennikov (who for a long time was the chief of theDepartment of kinetic equations of Keldysh Institute of Applied Mathematics, RAS (KIAM RAS),and also the organizer and the head of regularly running seminar on kinetic equations at KIAMRAS), T.A. Germogenova, N.V. Konovalov (KIAM RAS), G.V. Rosenberg, V.I. Tatarsky, Yu.N.Barabanenkov, Yu.N. Gnedin, N.A. Silant’ev, H. Domke, E.P. Zege, L.I. Chaikovskaya, and manyothers. It is also a great pleasure to thank T. Nishida, K. Asano and the participants of the seminaron nonlinear equations of the Department of Mathematics, Kyoto University, for interesting andhelpful discussion on the VRTE properties for optically anisotropic media (1989).

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Chapter 2Advances in Spectro-PolarimetricLight-Scattering by Particulate Media

Romain Ceolato and Nicolas Riviere

2.1 Introduction

Electromagnetic light-scattering refers to the general physical process where elec-tromagnetic radiations do not follow trajectories predicted by the law of reflection.This phenomenon is a general term used to describe the light scattered by a surface,a single particle, as well as complex systems of particles. Particulate media aresystems of randomly positioned particles with arbitrary shapes in absorbing ornon-absorbing host-media. The light scattered by such media produces distinctsignatures or fingerprints, which may depend on wavelength, polarization,time-propagation, and directions. These signatures have been a growing interest forcharacterization of particulate media in different scientific and industrial fields. Thepurpose of this chapter is to provide insights into spectro-polarimetriclight-scattering by particulate media. It also sets out to demonstrate the place ofspectro-polarimetric light-scattering as a robust and comprehensive optical diag-nostic method for particulate media composed of small particles found in severalfields of science and engineering.

In atmospheric science, meteorology, and remote-sensing, particulate mediarefers to aerosols (e.g. dust, sand, spores, or sea salts) or hydrometeors (e.g. clouds,fog, rain). These particles are classified as either natural (i.e. produced by nature) oranthropogenic aerosols (i.e. produced from human activities). Much attention hasbeen given lately on fine and ultrafine anthropogenic aerosols such as soot and sulfateparticles as they present a severe respiratory health hazard in urban environments.High concentrations of airborne pollutants in the atmosphere result in substantial

R. Ceolato (&) � N. RiviereOptronics Department, ONERA, The French Aerospace Lab,31055 Toulouse, Francee-mail: [email protected]

N. Rivieree-mail: [email protected]


55

radiative perturbations which impact the energy transfer between the sun andearth-ground: this phenomenon is referred as radiative forcing and plays a significantrole in climate change. Soot particles are the second most important human emissionsconcerning climate-forcing, and only carbon dioxide is estimated to have a greatereffect (Bond et al. 2013). The study of particulate media is also of great interest indiverse fields such as combustion science for public health monitoring (e.g. com-bustion particles from diesel engines) or defense and security (e.g. combustionparticles from aircraft engines, plumes, rockets, or jets). Various processes produceparticulate media in numerous industrial products. For instance, nanoparticles pro-duced by different techniques (e.g. milling, attrition, and nucleation) which attractmuch attention today because of their unique chemical and physical properties. Inbiomedical research, particulate media in liquids are found in several living organ-isms (e.g. cells, nuclei, or bacteria). Bio-aerosols are another important issue forpublic health as these aerosols may present serious threats for societies (e.g. bio-logical warfare agents). Finally, in astrophysics and cosmology, the study of regolithor cosmic and interplanetary dust is of particular importance in understanding phe-nomena such as star or solar system formations (Pilbratt et al. 2010).

A wide variety of techniques is available to characterize particulate media.Optical techniques are good candidates for probing particulate media with particlessize close to the radiation wavelength because particles effectively scatter light withwavelengths close to their size. The first category of techniques implies samplingand collecting the particles for physicochemical analysis. These methods are oftenused in chemical or industrial engineering for their relative simplicity althoughsampling of particles may modify the characteristics of the particles. A secondcategory refers to remote probing technique. They require higher technicity andsophisticated analysis to extract relevant information about the particles but can beundertaken in-line without causing any disturbance of the media. Light-scatteringtechniques belong to both categories as they can characterize statistically significantnumbers of particles simultaneously. They provide valuable information such asparticle size distributions, morphology, concentration, internal structure, or relativecomplex refractive index. Depending on the applications or particles of interest,different terrestrial, remote (i.e. airborne, space) or laboratory instruments measurelight-scattering data to probe particulate media. Well known LIDAR (LIghtDetection And Ranging) techniques, the optical counterpart of microwave RADAR(RAdio Detection And Ranging), are used to remotely profile aerosols in theatmosphere spanning from the ultraviolet (UV) to the infrared (IR) spectrum withpolarization analysis. They provide long-range aerosol profiles and associatedinverse methods allow the retrieval of different microphysical properties. For lab-oratory analysis, SLS (Static Light-Scattering) or DLS (Dynamic Light-Scattering)are other well-established light-scattering techniques used to probe small particles.Light-scattering remains a complex problem though it provides precious informa-tion on an extensive range of particles such as aerosols, blood cells, or colloidalsuspensions. As an example, the light scattered by particulate media is governed bythe microphysical parameters of the particles, including the size, the shape, and thecomplex refractive index. However, this dependence is complex, and measurements

56 R. Ceolato and N. Riviere

cannot directly retrieve these physical parameters due to the non-unique solution ofthe solved problem. Light-scattering can be considered as a direct and an inverselight-scattering problem. Solving the direct problem implies computing the scat-tered radiation for a given particulate medium and illumination. Solving the inverseproblem means to retrieve the properties of a particulate medium from the scatteredradiation for a given illumination. In practice, inversion methods for light-scatteringhave been developed to retrieve the microphysical properties of particles fromlight-scattering measurements. Errors often marred these methods (e.g. errors oninitial parameters, invalid assumptions). Several challenges related to the inverselight-scattering problem are given below:

The non-unique solution. A first major challenge comes from the existence ofsingularities in the inverse scattering problem, i.e. identical light-scattering patternscan be obtained from different microphysical parameters. This problem is usuallyreferred as the non-uniqueness problem of light-scattering. The inverselight-scattering problem is not a well-posed problem in the sense of Hadamard(1902) and is termed as an ill-posed problem. Most of these inversion methods areconditioned to prior knowledge about the particulate media of interest. For instance,particle size can be retrieved from angular data as long as the refractive index isknown. The morphology or relative complex refractive index of the particles ismost of the time assumed or known a priori to retrieve the Particle Size Distribution(PSD) of particulate media from light-scattering measurements (e.g. LIDAR orSLS/DLS measurements and their associated inverse techniques). In cases wherethe relative complex refractive index is unknown or when it changes upon exper-imental conditions (e.g. heating or chemical processes), severe uncertainties resultin the sizing of the particles. In such cases, one can determine the relative complexrefractive index as long as PSD is provided by external instruments such as aerosolimpactors or particles counter (Gramm et al. 1974a, b) which may result inuncertainties.

The single-scattering assumption. A second challenge comes from the use ofunsuitable and unverified physical assumptions. Many light-scattering inversemethods are based on the single-scattering approximation, especially in the field ofLIDAR (Klett 1981). They assume that light is scattered only once by particlesbefore detection. This assumption is suitable and gives good results for particulatemedia with a low concentration or a small optical path (e.g. weak aerosol plumes,diluted colloidal solutions). For instance, the inversion of LIDAR measurements isbased on the single-scattering LIDAR equation to retrieve aerosol PSD or opticalcoefficients (Scheffold and Cerbino 2007). This operation could change thechemical properties of the media and may result in additional uncertainties on thevalue of the concentration. Multiple-scattering is most of the time ignore while itoccurs for intermediate and highly concentrated particulate media. Knowing whe-ther particulate media are dense, or non-dense is hard to define as it depends on theparticulate medium itself (e.g. size, concentration, optical index) but also on thewavelength of light: a given particulate medium can be regarded as dense in the UVspectrum while can be considered non-dense in the far-IR spectrum. In other words,

2 Advances in Spectro-Polarimetric Light-Scattering … 57

the single-scattering approximation has to be used with a great care before invertinglight-scattering data.

Limited information. A third challenge comes from experimental limitations.Light-scattering by particulate media depends on all the properties of the incidentlight: angular (or directional) distribution, polarimetric state, wavelength, and eventime for pulsed light sources. Most experimental instruments are not able to mea-sure all these modifications at the same time. They must be restricted to limitedangles, polarimetric states, or wavelengths (e.g. single wavelength LIDAR, unpo-larized SLS). Such limitations restrict by nature the capacity of light-scatteringbased instruments.

Experimental errors. Besides, statistical errors may induce incapacity to measurelight-scattering for a given particulate medium accurately. The solution of thescattering problem is highly sensitive to errors in measurements and calculatedscattering properties.

Light-scattering signatures often result in large amounts of information, andbecause analytical inversion is not always feasible because of the above challenges,the retrieval of microphysical parameters is a complex task. One option to relaxthese difficulties is to fuse spectral and polarimetric properties of scattered light.The merge of spectral, polarimetric, and multi-angle information aboutlight-scattering is proposed as a comprehensive tool to characterize particulatemedia by relaxing different assumptions about the media. Recently, a growingamount of interest has been raising in spectro-polarimetric light-scattering in severalfields from remote-sensing to wider applications such as astrophysics (Kimura et al.2003), biomedical (Ghosh et al. 2011; Patskovsky et al. 2014), active (Manninenet al. 2014) and passive remote-sensing (Powers and Davis 2012; Diner et al. 2013;Zieger et al. 2007), particle analysis (Tang and Lin 2013; Sharma et al. 2013;Bendoula et al. 2015), defense, and security (Lambert-Girard et al. 2012).Spectro-polarimetric light-scattering techniques are expected to improve theunderstanding of scattering phenomena and identification of relevant microphysicalparameters of complex particulate media. Although numerous topics involvespectral and polarimetric light-scattering, including spectral or polarimetric laserimaging systems (Powers 2012), or spectro-polarimetric imaging systems (Dineret al. 2013), these questions fall out of the scope of this review. We restrict theanalysis to the characterization of particulate media by spectro-polarimetriclight-scattering.

The recent advances of spectral and polarimetric light-scattering by particulatemedia, including single and multiple scattering, are reviewed as follow: (i) a briefintroduction to light-scattering by particulate media, (ii) a state-of-the-art of thedifferent existing spectral—from multi-spectral to hyperspectral—light-scatteringmethods, (iii) a state-of-the-art of the different polarimetric light-scattering methods,and (iv) a comprehensive review of the recent advances and contributions of themerging of spectro-polarimetric information for light-scattering, including radiativetransfer simulations and measurements carried out at ONERA, The FrenchAerospace Lab.


2.2 Problem of Light-Scattering by Particulate Media

The interaction between particles and incident electromagnetic waves producesoscillations of electric charges, which results in scattering and absorption.Particulate media are defined as heterogeneous media composed of particles (e.g.aerosols, colloids) within a host medium (e.g. air, water). Such media can be termedas non-dense or dense particulate media, depending on the volume concentration ofparticles. Light-scattering by particulate media depends on (i) the size, (ii) theshape, and (iii) the complex refractive index m ¼ n� i � k of the particles where n isthe real refractive index and k the absorption coefficient. The index m refers to therelative index between the particle and host media (e.g. air or water).

The problem of light-scattering by a random distribution of particles has foundbroad interest in different scientific fields such as atmospheric science, particlecharacterization, biomedical and remote-sensing, specially studied in the LIDARcommunity. This section reviews the general principle of light-scattering in regardwith the single scattering (i.e. when the number of scattering events is equal tounity) and multiple scattering approximations (i.e. when the number of scatteringevents is higher than unity).

2.2.1 Single Light-Scattering

Single light-scattering refers to the scattering of electromagnetic waves by a singleparticle. This approximation remains valid for non-dense or weak particulate mediawith low particles concentrations such as dispersed aerosols in the atmosphere. Forsuch media, as represented in Fig. 2.1, a single scattering event occurs between theemission and detection of light.

Light-scattering by a single particle has been an object of interest for more than acentury from the historical Lorenz works (Lorenz 1890), who solved the scatteringproblem for transparent spheres, and Mie works (Mie 1908) who published acomplete theory describing the interactions between an electromagnetic wave and ahomogeneous spherical metallic particle. The Lorenz-Mie theory provides solutions

Detection system

Incidentlight

Fig. 2.1 Representation ofsingle light-scattering


of the scattering problem for colloidal suspensions in the form of an infinite seriesof vector spherical harmonic expansion, which satisfies Maxwell’s equations(Maxwell 1865). From Lorentz-Mie solutions, several radiative parameters such asefficiency, cross-sections, or differential cross-sections relative to scattering,absorption, and extinction are derived. The solutions of the Lorenz-Mie theory aredirectly related to two physical parameters:

(1) the relative complex refractive index m = n/nh where n ¼ np � ikp is thecomplex refractive index of the particle and nh is the real refractive index of thehost medium,

(2) and the size-wavelength parameter x defined as x ¼ 2 � p � r=k where r is theradius of the spherical particle and k the considered wavelength (Bohren andHuffman 1998).

This parameter was used by Lord Rayleigh (Young 1981) to definelight-scattering by small particles compared to the wavelength and was later usedby Van de Hulst (1981) to categorize light-scattering into different light-scatteringregimes. Light-phase variations, which are induced when light passes through aparticle and defined by d ¼ 2m � x, govern these light-scattering regimes. All thevarious scattering regimes are reported in a m� x diagram similar to the one givenin Fig. 2.2 where the x ranges from zero for ultrafine particles to infinity for largeparticles and, m ranges from zero for optically soft particles to infinity for opticallyhard particles.

For a small relative complex refractive index, the scattering regimes are eitherRayleigh-Debye-Gans regime (region 1) for large or intermediate size parameter, oranomalous diffraction regime (region 2) for small size parameters. Rayleigh regime(region 6) applies for small size parameters with intermediate relative refractiveindex, and geometrical optics regime (region 3) applies for large size parameterswith intermediate relative index. For a large relative refractive index, the scatteringregimes are either total reflection regime (region 4), for a large or intermediate sizeparameter, or optical resonance regime (region 5) for a small size parameter. Mieregime (central region) applies for intermediate size parameters or particle sizescomparable to the wavelength and intermediate relative refractive index, and no

x

)Re(m

Mie region

Fig. 2.2 Light-scatteringregimes categorized in thex-m diagram. Region 1:Rayleigh-Debye-Gansregime. Region 2: Anomalousdiffraction regime. Region 3:Geometrical optics regime.Region 4: Total reflectionregime. Region 5: Opticalresonance regime. Region 6:Rayleigh regime


approximation can be made as light-phase shifts cannot be neglected. Furthercomprehensive discussions on light-scattering regimes can be found in Van deHulst works (Van de Hulst 1981; Mishchenko et al. 2002).

Regularly-shaped particles, such as spheres, follow the Lorenz-Mie theory. Thetheory remains valid with no size limitation as the solutions converge to the limit ofgeometric optics for large particles as well as to the limit of the Rayleigh theory forvery small particles including molecules. This theory has been later extended to amore general theory named Generalized Separation of Variable (GSV) method totake into account different regularly-shaped particles different from spheres such ashomogeneous spheroidal particles (Asano 1976; Gouesbet 2011). Problemsinvolving complex regularly-shaped particles with large varieties of geometries(axisymmetric, non-axisymmetric, composite or layered and inhomogeneous)(Doicu and Wriedt 1999) have used extensively another category of methodsknown as the Surface-Integral Equation (SIE). Different numerical implementationsof these methods are available, such as the superposition T-matrix (Mackowski1996; Mishchenko et al. 2000; Liu et al. 2008). One advantage of these methods istheir short computation time, though they remain limited to regularly-shaped par-ticles with a specific range of size parameters. In addition, one must avoid usingthese methods to compute the radiative properties of particles with high-aspectratios as the problem becomes ill-conditioned for very large objects compared witha wavelength. We refer to the work of Mishchenko et al. (2002) for a detailedreview of the advantages and constraints of the different light-scattering numericalmethods.

Irregularly-shaped or realistic particles usually do not present regularly-shapedgeometries. The simplistic concept of equivalent-volume or equivalent-surfacesphere is commonly used to compute light-scattering properties of particles withouttaking into account shape effects (Kalashnikova and Sokolik 2004). Severalinvestigations have shown that most of these particles have non-spherical geome-tries (Okada et al. 2001; Munoz and Hovenier 2011) and that optical properties mayvary considerably depending on their shape (Yang et al. 2007; Yi et al. 2011).However, for the sake of simplicity, most studies assumed that particles have aspherical or spheroidal geometry despite the fact that the majority of particles foundin nature (e.g. dust, ashes) are not regularly-shaped. Accurate computation oflight-scattering properties of irregularly-shaped particles requires the use of morecomplex and time expensive methods. Another category of methods based on theVolume-Integral Equation (VIE) provide solutions which fully account the particlesgeometry since the Lorenz-Mie and SIE methods do not directly apply toirregularly-shaped particles. These methods compute light-scattering by particles bydiscretization of the particle into a distribution of discrete dipoles regarding themorphology of the particle. Dipole-dipole couplings model the interactions betweendipoles (Markel et al. 1991). The total electric field is considered as an incidentelectric field and a superposition of all the electric field from all the surroundingdipoles. Implementations are used depending on the discretization techniques suchas the Discrete Dipole Approximation (DDA) proposed by Purcell and Pennypacker(1973). With increasing dipole number, these methods can be time-consuming and


singular kernel problems may rise. In spite of these limitations, these methodsremain traditional to compute light-scattering properties of irregularly-shaped par-ticles thanks to their flexibility and physical approach (Draine and Flatau 1994).Other electromagnetic scattering methods address the problem of electromagneticscattering by irregularly-shaped particles. For instance, the Finite Difference TimeDomain (FDTD) methods are widely used in the field of RADAR (Yee 1966; Tangand Aydin 1995) in order to solve complex scattering problems, such as electro-magnetic scattering by rough surfaces (Hastings and Schneider 1995; Sun et al.2013a, b, c), or scattering by arbitrary shaped and inhomogeneous particles (Yangand Liou 1996; Yang et al. 2000; Sun et al. 2009, 2011). Either DDA or FDTDtechniques present different advantages and drawbacks concerning applications andperformances.

2.2.2 Multiple Light-Scattering

Particulate media are composed of multiple particles for intermediate or highconcentration that the single-scattering approximation does not hold. For particulatemedia such as clouds, dense plumes, jets, or foams, multiple scattering events mayoccur between the emission and detection of light. Thus, the widely usedsingle-scattering approximation is no longer valid (Ishimaru et al. 1984) (Fig. 2.3).

The Radiative Transfer Theory (RTT), which originates from Lommel (1889)and Chwolson (1889) independent works, models the propagation of electromag-netic waves in particulate media with multiple scattering. Schuster (1905) intro-duced the two-stream Radiative Transfer Equation (RTE) which provides analyticalsolutions to a single layer plane-parallel radiative transfer equation. Using the

Detection system

Incidentlight

Fig. 2.3 Representation ofmultiple light-scattering


invariance principle, Ambartsumian (1943, 1947) proposed a simple solution of theRTE to solve the problem of diffuse reflection of light by a scattering medium, andChandrasekhar (1950) solved the problem for a finite isotropic scattering mediumusing the discrete ordinate method. This method models the particulate medium as asemi-infinite plane–parallel multi-layered isotropic scattering medium to accountmultiple-scattering. This theory was later extended with great results for severalapplications in astrophysics, remote-sensing, biomedical and other scientific areas,including anisotropic particulate media (Ishimaru et al. 1984). Traditionally, theRTE is considered as a phenomenological equation based on heuristic principles ofclassical radiometry; yet it was derived from first-principles and Maxwell’s equa-tions using statistical electromagnetics (Mishchenko 2011). However, no electro-magnetic coherence effects are modeled and taken into account since the RTE isonly based on light intensity flux transfers. It restricts the use and validity of thisapproach to particulate media with moderate particle concentration, i.e. where thedistance between particles is larger than the wavelength. One essential condition forvalidity is that the scattering events should always occur in the far-field region ofeach particle in order to prevent the apparition of short-range inter-particle corre-lations effects such as dependent scattering (e.g. aggregates, closely-packed parti-cles). This is sometimes referred as the independent multiple-scatteringapproximation.

In the following section, an overview of spectral light-scattering by particulatemedia is presented for a broad range of applications. The advantages of studyinglight-scattering properties at different wavelengths or frequencies are highlighted aswell as the associated challenges for optical diagnostic using this method.

2.3 Spectral Light-Scattering

Wavelength, or frequency, is a fundamental property of electromagnetic waves.Standard light-scattering methods have studied the propagation of monochromaticplane waves through particulate media. Radiative parameters such as extinction,scattering, and absorption coefficients, often present distinct wavelength depen-dencies due to different physical phenomena (Van de Hulst 1981). The scope of thissection is to present a non-exhaustive state-of-the-art of spectral light-scatteringmethods. Techniques with single wavelength capabilities are not considered in thissection. Though several methods measure broadband electromagnetic scatteringproperties in the microwave and terahertz region, these are out of the scope of thisreview as the spectral domain of interest is limited to methods which cover widespectral domain in the UV, visible, and IR wavelengths.

Light-scattering by spherical particles, according to Lorenz-Mie theory, exhibitsresonances which depend upon the values of size parameter x and relative complexrefractive index m. Single-wavelength light-scattering methods can address a singlelight-scattering regime in the m� x diagram presented in Fig. 2.2. However, for agiven single and fixed-wavelength, the scattered intensity shows dependencies on


both particle size and relative refractive index. This non-uniqueness problem hasraised a fundamental limitation of single-wavelength light-scattering methods:equivalent light-scattering signatures can be obtained either by changing the par-ticles size or by varying the particles relative refractive index. Therefore, theinference of the microphysical properties based only on single-wavelengthlight-scattering measurements has led to highly questionable results, unless addi-tional information was known a priori (Janzen 1979). In the absence of suchadditional information, the uniqueness of particle size and relative refractive indexinferred by single-wavelength light-scattering methods cannot be assured.

Spectral or multi-wavelength methods have been proposed (Ceolato 2016) toovercome the non-uniqueness problem of light-scattering and to infer microphysicalproperties with no a priori knowledge of the media. The objective of these spectralmethods is to address a full series of light-scattering regimes in the m� x diagramby analyzing light-scattering over a broad spectrum. For non-dispersive particulatemedia, i.e. m does not depend on the wavelength; m is invariant for a series ofx values: the m� x series describes a straight line in the diagram, perpendicularly tothe m-axis. For dispersive particulate media, i.e. m depends on the wavelength;m varies for a series of x values: the m� x series describes a curved line in thediagram, where the curvature of is directly related to the variation of the relativecomplex refractive index. Thanks to this simple idea, methods based on spectrallight-scattering intensities are proposed for unambiguous inference of microphys-ical properties of particulate media.

In astrophysics research, spectral light-scattering measurements were first carriedout to study particulate planetary surfaces. Also known as color-ratio imaging, thesetechniques have been effective ways to study the chemical and mineral compositionof the Moon for instance (McCord 1969; McCord et al. 1972; Pieters 1999).Because the illuminating/observing geometry plays a secondary role in the spectralreflectance of planetary surfaces, multi-spectral measurements at different phaseangles were first compared, without accounting the polarization effects, to providevaluable information on the lunar surface composition (McCord 1969). In the fieldof remote-sensing and atmospheric studies, sensing and detecting of atmosphericparticulate matter (e.g. biomass burning particles, soot particles, sea salts) have usedlight-scattering to retrieve microphysical properties of interest (Holben et al. 1998).Among them, spectral light-scattering techniques remain a vast area of research toremotely sense and detect particles and are usually classified into two categories:passive and active optical techniques. Both techniques have proved to be conve-nient to monitor atmospheric aerosols and to evaluate their impact on climate. Bothmethods present certain advantages and deficiencies (Deirmendjian 1980).

Total (or angularly integrated) spectral light-scattering measurements, such asextinction measurements, have always been experimentally much simpler thanmulti-directional or angularly resolved light-scattering. Nevertheless, the inverseproblem associated with extinction present no unique solution for a single wave-length: a measured extinction coefficient may correspond to any combination ofparticle size and relative complex refractive index (Janzen 1979). Ångström firstsuggested in 1929 an empirical relationship between the particle size and the


wavelength dependence of the extinction coefficient (or optical depth) of light byaerosols. A spectral relation between the atmospheric visibility (defined from theextinction coefficient) and backscattering coefficient was later proposed the 1950sbased on long-range atmospheric transmission measurements (Curcio and Knestrick1958). The advantage of broadband over monochromatic light sources has beentheoretically demonstrated to infer aerosols particle size distribution from thespectral extinction-to-backscatter ratio (Twomey and Howell 1965; Foitzik 1965).First numerical inversion based on spectral extinction coefficient measurements wasperformed by Yamamoto (1969) to retrieve the particle size distribution. Othermethods were later deployed to determine number-size distributions of particulatemedia composed of regularly (Yamamoto and Tanaka 1969; Box and McKellar1978; Nilsson 1979; Klett 1984) and irregularly-shaped (Liu et al. 1999; Tang2013) particles. For instance, spectral extinction measurements, from broadbandlight-source in the visible and infrared, coupled with Lorenz-Mie theory basedinversion methods were developed to probe various hydrometeors such as fog andhaze (Lenham and Clay 1982). Several techniques have been developed to improvenumerical inversion (Grassl 1971; Shaw et al. 1973a, b; King et al. 1978). Amongthem, the constraint imposed by the Kramers-Kronig relation (Landau and Lifshitz1960) has been proposed to determine the real and complex part of the relativecomplex refractive index of particles, including aerosols (Milham et al. 1981),water clouds, and hazes (Deirmendjian 1964).

Angular (or angularly resolved) spectral light-scattering methods have beendeveloped to probe particulate media from multi-spectral nephelometry to diffuselight-scattering, most of the time in weak multiple scattering regimes. Differentremote-sensing techniques use a natural light source (e.g. Sun or Moon) to performspectral light-scattering measurements. Such measurements are carried out todetermine aerosol spectral properties of particulate media in polluted, haze, orcloudless days (Olsen et al. 1983; Vasilyev et al. 1995). Numerous instrumentshave been developed to measure optical properties of airborne particles fromcommercially available such as sky-photometers or albedometers, which measurethe ratio of scattering to extinction or Single Scatter Albedo (SSA) (Dial et al.2010). Spectral dependency of optical thickness was reported for different particlesfrom desert dust, soot, or biomass burning (Vasilyev et al. 1995; Bergstrom et al.2007). The retrieval of aerosol size distribution can be performed from spectraloptical depth (Shaw et al. 1973a, b; Moorthy et al. 1991), whereas the retrieval ofother aerosol parameters such as the complex refractive index cannot usually beperformed without prior knowledge (Yang and Wenig 2009). Other remote-sensingtechniques use an artificial light source (e.g. lamp or laser) to perform spectrallight-scattering measurements. Spectral nephelometry consists in measuring thespectral light-scattering at multiple angles to investigate complex and dense par-ticulate media (Elias and Cotte 2008). For instance, the relative complex refractiveindex of polystyrene microspheres was inferred from spectral light-scatteringmeasurements (Ma et al. 2003). New techniques in nephelometry have beeninvestigated using a SuperContinuum (SC) laser source with a series of filters


(Sharma et al. 2013), or highly sensitive techniques based on Cavity Ring-Down(CRD) spectroscopy (Lang-Yona et al. 2009; Zhao et al. 2014).

Range-resolved light-scattering techniques were developed to fill the gaps in theparticle characterization. These active techniques use either incoherent or coherentartificial light sources (e.g. laser) and are important parts of atmospheric research. Inoptics, these techniques follow the principle of RADAR and are referred as LIDAR.First LIDAR systems employed incoherent continuous lamp to sense the molecularand aerosol composition of the atmosphere (Johnson et al. 1939; Bureau 1946).From the 1960s, with the development of the laser, several systems have integratedthese new coherent light sources to become useful tools for measuring the verticalprofiles of the aerosol optical properties. A series of laser pulses propagate throughthe atmosphere, and a sensor detects a small amount of backscattered light. Theextent and properties of particulate media in the atmosphere can be retrieved fromthe magnitude of the backscattered light. In principle, the use of multi-wavelengthor spectral LIDAR allows one to investigate the different ranges of particle sizedistribution of aerosols, which is due to the dependence of scattering properties onthe wavelength of incident light. Relying on significant advances in novel lasersources, the concept of spectral LIDAR has gained notoriety based on the differ-ential light-scattering and absorption over a spectral domain of interest (Fig. 2.4).

The first option for spectral LIDAR is to use different laser sources which emit alimited number of wavelengths to probe the atmosphere. Early works showed thefeasibility of inferring additional information on the particulate media usingsimultaneous multi-wavelength LIDAR measurements (Heintzenberg et al. 1981;Feingold and Grund 1994). Wood (1984) theoretically studied backscatteringcoefficient as a function of wavelength, measurable by a multi-wavelength LIDAR,to identify the chemical composition of atmospheric aerosols. Muller and Quenzel(1985) numerically investigated the feasibility of determining the PSD from aerosolbackscattering and extinction coefficients at four LIDAR wavelengths using arandomized minimization search technique. They found that accurately determinedbackscatter and extinction coefficients at four wavelengths carry enough informa-tion for a relatively good retrieval of the PSD, if the relative complex refractiveindex is known a priori. Qing et al. (1989) evaluated the feasibility of deriving PSDfrom extinction and backscattering coefficients measurements with amulti-wavelength LIDAR and proposed a regularization method assuming that

Range

ParticulateMedia

Detection system

Pulsed Laser source

Fig. 2.4 General principle of a LIDAR system


these coefficients can be accurately measured from the LIDAR data. Sasano andBrowell (1989) demonstrated the potential of a multi-wavelength LIDAR for dis-criminating the different aerosol types, such as maritime, continental, stratospheric,and desert, from the wavelength dependence of the aerosol backscattering coeffi-cient. LIDAR In-space Technology Experiment (LITE) was the first LIDAR systemsent into space to demonstrate and explore the capabilities of space LIDAR foratmospheric research (Couch et al. 1991). In addition to being a pioneer spaceborneLIDAR, LITE used three-wavelength laser sources to perform multi-spectralbackscatter LIDAR measurements. Post et al. (1992) at NOAA observed ejectafrom the eruption of Mt. Pinatubo at three wavelengths from visible to infrared. DelGuasta et al. (1994) fitted the multi-wavelength LIDAR data to a monomodallog-normal PSD and retrieved its associated mode radius. Based onmulti-directional or of angularly dependent light intensity measurements,Yoshiyama et al. (1996) estimated PSD from bistatic multi-wavelength LIDARmeasurements using 5–8 wavelengths. Rajeev and Parameswaran (1998) developedinversion methods dedicated to retrieving aerosols from dual-wavelength tomulti-wavelength LIDAR measurements without any assumption on the form of thesize distribution. Ligon et al. (2000) examined the feasibility of using a generalizedstochastic inversion methodology to estimate PSD from spectral backscattering andextinction coefficients. Extensive works of Müller et al. (1999, 2000) andVeselovskii et al. (2002, 2005) at the Leibniz Institute for Tropospheric Research(TROPOS) thoroughly investigated the advantage of multi-wavelength LIDARmeasurements to provide a robust estimation of PSD and refractive index of par-ticulate media. Jagodnicka et al. (2009) retrieved broad bimodal PSD frommulti-wavelength LIDAR measurements from UV to near-infrared (NIR). Withthese different studies, the concept of spectral LIDAR has emerged as a robust toolto probe microphysical properties of particulate media (e.g. PSD) with lessassumption or need of a priori knowledge. All these spectral LIDAR techniques usethe advantages of multi-spectral information to enhance the retrieval of micro-physical properties of particulate media in the atmosphere. Long range measure-ments can be achieved thanks to high power laser sources. This approach has amajor drawback in that it uses fixed wavelengths and cannot be easily tuned tocover a broad field of applications.

The second option for spectral LIDAR is to use a single tunable laser sourceemitting light over a large spectral domain. Different solutions are under consider-ation. A first solution uses tunable laser. Early work of Mudd et al. (1982) investi-gated the spectral backscattering coefficients of aerosols in a chamber as a function ofwavelength in the IR with a tunable carbon dioxide laser. Recent solutions based ontunable Optical Parametric Oscillator (OPO) lasers have been proposed to probeparticulate media. For instance, a spectral short and middle wave infrared (SWIR/MWIR) LIDAR has been developed for standoff bio-agent cloud detection usingsimultaneous broadband DIfferential SCattering (DISC) (Lambert-Girard et al.2012). A 1064 nm laser coupled with a tunable OPO is employed to generate abroadband light between 1.5 and 3.9 µm emitted towards a particulate media and atelescope measures the backscattered light at short range. Although these novel


techniques cover a relatively narrow bandwidth, they have the advantage of rangingprofiles at long distances and being versatile for numerous applications.

The third option for spectral LIDAR is to use broadband light sources with lowerpower such as SC laser sources, which generate directional broadband light usingcascaded nonlinear optical interactions in an optical fiber framework (Alfano andShapiro 1970a, b). In spite of their limited peak power and beam quality comparedwith OPO laser sources, the feasibility of using SC laser sources for spectralLIDAR has been proposed. Spectral LIDAR signals have been measured in theNIR-SWIR domain up to 4 km in altitude using high peak-power byfemtosecond-terawatt-laser sources (Méjean et al. 2003). Spectral LIDAR has beenconsidered for identification of solid targets both indoors and in the field with arange up to 1.5 km from moderate peak-power SC sources (Manninen et al. 2014).The main limitation of this approach remains in the lack of available high power SClaser sources, which limits the maximum range.

In biomedical research, the propagation of light through biomaterials such astissues is a complex problem. Light can be scattered by cell organelles (e.g. nuclei,mitochondria) with relative complex refractive indices different from the hostmedium (e.g. cytoplasm). The cell nuclei are appreciably larger than the opticalwavelength (typically 5–10 m vs. 0.5 m) and scatter light in the forward direction,and there is significant scattering in the backward direction (Brunsting andMullaney 1974; Mourant et al. 1998). Light-scattering by tissues have beenextensively studied both experimentally and theoretically. Early work of Perelmanet al. (1998) reported multi-spectral unpolarized measurements in order to retrievethe size distribution and density of epithelial nuclei. The wavelength dependence ofthe intensity of the light elastically scattered by the tissue structure was found to besensitive to changes in tissue morphology, for instance for precancerous lesions. Itwas reported that specific features of malignant cells, such as increased nuclear sizeand nuclear/cytoplasmic ratio, could be retrieved from spectral light-scattering.

Spectral light-scattering, also named Light-Scattering Spectroscopy (LSS), bytissues have been proposed to extract and quantify morphological changes takingplace during a disease, even at early stages of cancer for instance. The advantage ofspectral light-scattering was shown on dysplastic human tissues using in the visiblespectrum (Ghosh et al. 2011). The spectral correlation matrices presented inFig. 2.5 were computed to probe scatterers of different sizes from small intracellularorganelles, which strongly influence in the backscattering, to large nuclei, whichinfluence in the forward scattering. From such analysis, normal and dysplastictissues could be distinguished. LSS techniques have also been used to analyze andretrieve the properties of particles and particulate media from single microorgan-isms to complex cellular structures and tissues. The analysis of the spectral scatteredlight was carried out using different methods to extract self-similarities at varyingscales associated with the structural changes (Soni et al. 2011).

For instance, spectral Diffuse Optical Tomography (DOT) is a derivative ofspectral light-scattering techniques, which consists in recovering the scattering and


absorption properties maps of tissues in function of time, space, and wavelength(Dehghani et al. 2003). With increasing number of wavelengths, modern systemshave proved the ability to recover more physiologically relevant parameters;specifically, concentrations of species including oxygenated and deoxygenatedhemoglobin, lipids, and water (Corlu et al. 2005, 2007). Because DOT requires thesolution of an ill-posed scattering inverse problem, the utility of hyperspectralinformation (up to hundreds of wavelengths) has been demonstrated for diffuseoptical tomography to recover concentration images of multiple chromophores(Larusson et al. 2011a, b). Another growing area of applications of spectrallight-scattering involves the use of metal nanoparticles to form plasmonic hot spotsfor applications in biology or medicine (Patskovsky et al. 2014; Ray 2016).

In a nutshell, spectral light-scattering techniques have demonstrated their abilityto improve optical characterization of particulate media. A significant amount ofinformation provided by these techniques has been useful to solve numerousscattering problems in various fields of interest. However, most of them rely onnumerous assumptions, such as the spherical or spheroidal approximations, toreproduce the light-scattering properties of dust particles for instance (Duboviket al. 2006). For LIDAR observations, these strong assumptions, in addition to apoor understanding of light-scattering properties by irregularly shaped particles,may lead to a lower estimation of backscattering coefficients (Mishchenko et al.1997). As a consequence, the LIDAR ratio (i.e. extinction-to-backscattering coef-ficient) may be considerably higher for irregularly shaped particles (e.g. mineraldust, aggregates) than for spherical particles. It may lead to significant errors for theretrieval of microphysical properties of irregularly shaped particles. The develop-ment of polarimetric light-scattering offers new ways to investigate particulatemedia composed of irregularly shaped particles.

Fig. 2.5 Correlation matrices in the visible domain for (a) normal and (b) dysplastic samples(Ghosh et al. 2011)


In the following section, an overview of polarimetric light-scattering by par-ticulate media is given. The advantages of studying light-scattering properties byaccounting polarization states of light are discussed.

2.4 Polarimetric Light-Scattering

Polarization is a fundamental property of electromagnetic waves, which is modifiedby scattering depending on the microphysics of the particulate media. In thatrespect, knowledge of both intensity and polarization state of the scattered light mayprovide new possibilities for probing particulate media. Polarimetric effects areusually classified as: (i) diattenuation (i.e. differential attenuation of orthogonalpolarization), (ii) retardance (i.e. de-phasing of orthogonal polarization), and(iii) depolarization (i.e. randomization of polarization state). Among these effects,much attention has been directed toward depolarization. For instance, multiplescattering induces a change of polarization state while the wave is scattered severaltimes before detection. This modification of the polarization state may provideuseful information about particulate media and could be used as an indicator of theparticles concentration for instance. Different formalisms have been introduced todescribe polarimetric light-scattering for optical and radar applications.

The Jones formalism was introduced (Jones 1941) to describe interactionsbetween fully polarized electromagnetic waves and a medium. It is based on 2�1Jones vectors J ¼ ðJ1; J2Þ and 2�2 complex-valued dimensionless Jones scatteringmatrices S which operate both on the amplitude and the phase of the electromag-netic field as:

Jsca ¼ S � Jinc ¼ S11 S12S21 S22

� �� Jinc ð2:1Þ

where Jinc and Jsca are respectively the incident and scattered Jones vectors and, Sijare the complex elements of the Jones scattering matrix.

The Stokes-Mueller formalism was later introduced (Mueller 1943; Stokes 1852)to account partially polarized electromagnetic waves, which may arise from inco-herent phenomena such as multiple scattering. Unlike Jones formalism, theStokes-Mueller formalism relies on intensities rather than fields. It is based on 4�1Stokes vectors S ¼ ðI;Q;U;VÞT (Stokes 1852) and 4�4 real-valued dimensionlessMueller matrices M (Mueller 1943) which both operate on intensities as:

Ssca ¼ M � Sinc ¼M11 M12 M13 M14

M21 M22 M23 M24

M31 M32 M33 M34

M41 M42 M43 M44

0BB@

1CCA � Sinc ð2:2Þ

where Sinc and Ssca are respectively the incident and scattered Stokes vector.


The Mueller matrix represents a complete polarimetric transfer function of amedium in its interactions with polarized light. An equivalent of the Muellerscattering matrix is defined as:

M ¼ k2

4 � p2 � D2 � F ¼ k2

4 � p2 � D2 �F11 F12 F13 F14

F21 F22 F23 F24

F31 F32 F33 F34

F41 F42 F43 F44

0BB@

1CCA ð1:3Þ

where F is the 4�4 real-valued dimensionless phase matrix strongly dependent onthe microphysical properties of the particles with diameter D. Generally, this matrixhas non-zero values. The matrix can be significantly simplified for particles with ahigh degree of symmetry. Several elements of the matrix turn out to be equal to zerofor randomly-oriented or axially symmetric particles (Van de Hulst 1981;Mishchenko et al. 2002). Different properties of the scattered Stokes vectors includethe Degree Of Polarization (DOP), Degree Of Linear Polarization (DOLP), andDegree Of Circular Polarization (DOCP) as:

DOP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2 þU2 þV2

pI

; DOLP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2 þU2

pI

; DOCP ¼ VI

ð2:4Þ

Polarimetric light-scattering is generally categorized into two different methods.The first category, here stated as polarimetric contrast-enhancing methods, relieson the gating ability of polarization to subtract diffuse waves so as to enhancecontrast. Thus, only single scattered light contribution can be extracted, yetmaintaining a high DOP. Numerous techniques based on this method have beenproposed to enhance the imaging quality of optical systems from simple techniquesto advanced ones that maximize polarimetric contrast with adapted and predeter-mined polarization states to show distinctive polarimetric features (Richert et al.2009; Upadhyay et al. 2011). The second category, here stated as polarimetricquantitative methods, actually exploits the comprehensive information from thecomplete Mueller matrix using different polarization sensitive sensors. Thepolarization-preserving but also the cross-polarized or co-polarized components canbe used to infer microphysical properties of particulate media such particle size orshape (Mishchenko et al. 1995; Tishkovets et al. 2004). The latter category is ofgreat interest for optical diagnostic of particulate media.

Inferring information from polarimetric light-scattering from particulate media isa complex and challenging task. First attempts have been undertaken by inter-preting polarimetric data without properly modeling multiple scattering. Solutionsprovided by the Lorenz-Mie theory take into account polarization but are restrictedto a single particle and effects such as depolarization induced by multiple scatteringare not modeled. Most of these attempts led to large errors until multiple scatteringwas actually taken into account. Lyot in 1929 has called attention to the fact that themultiple-scattering DOP (i.e. resulting as the sum of all orders of scattering) issmaller than the single scattering DOP in a proportional manner. Van de Hulst


(1981) suggested that all the scattering of orders larger than the 1st order could beconsidered unpolarized. This simplistic approximation was proven to lead to sig-nificant errors. Hovenier (1971) later proposed a second approximation that all thescattering of orders larger than the 2nd order is unpolarized. This approximationresults in taking into account only first and second order of scattering to modelmultiple-scattering. While this approximation results in better agreement withmeasurements, it still leads to significant errors. All the scattering orders must betaken into account to model polarimetric light-scattering accurately.

In astrophysics, early studies were conducted to understand the polarimetricsignature of the scattered light from planetary regolith such as the Moon (Dollfusand Bowell 1971). Major polarimetry studies date back to the 1840s in France(Arago 1842; Lyot 1929) and later in USSR (Barabashev 1926). From astronomicalobservations, an inverse correlation was noticed empirically in 1912 between thesurface albedo and the DOP of the light scattered by the lunar surface at large phaseangles (Umov 1912). This so-called albedo-polarization law has been later cor-rected with further polarimetric measurements on many particulate surfaces andregolith (Dollfus 1971; Dollfus 1957; Bowell et al. 1972; Shkuratov 1980).A brightness enhancement toward exact backscattering direction and a branch ofnegative polarization (i.e. decreasing DOP reaching a minimum negative value at asmall phase angle) were also measured for many particulate media. Continuingsimulations progress has been made to explain these phenomena with mechanisms(e.g. coherent backscattering or shadow hiding opposition effect) to interpret pho-tometric and polarimetric measurements of light-scattering by regolith and partic-ulate media (Wolff 1975; Shkuratov 1981; Shkuratov et al. 1994; Zhang and Voss2009).

Different techniques have been developed to compute polarimetriclight-scattering with no limitation in terms of scattering orders, although the originalLorenz-Mie theory has been extended to systems of regularly shaped particles. Forinstance, the well-established RTT (Lommel 1889; Schuster 1905; Van de Hulst1981; Chandrasekhar 1950; Ambartsumian 1957) has been extended from a scalarequation to a vector equation to include polarization (Evans and Stephen 1991). TheVector Radiative Transfer Equation (VRTE) fully models the transformation of theStokes vector of electromagnetic waves due to its propagation and scattering in amedium. It is used to evaluate the solution of scattering problems in a polarimetricframework and gave good agreement with experiments for particulate media withmultiple scattering (Riviere et al. 2013).

Techniques to measure polarimetric light-scattering have been developed since1960s. These techniques investigate how a collimated beam of light is scattered atdifferent angles by particulate media such as aerosols. Different categories ofinstruments have been considered. Polarization sensitive nephelometers orgoniometers have permitted to measure the complete or partial Mueller matrix ofsingle particles or particulate media (e.g. water, quartz, silica, dust) (Pritchard andElliott 1960; Holland and Gagne 1970; Weiss-Wrana 1983; Kuik et al. 1991;Muñoz et al. 2000). Other instruments consider limited angular range such asforward-scattering (Eiden 1966; Gramm et al. 1974a, b). Angular information from


light-scattering is mostly sensitive to the particles size and relative complexrefractive index, while polarization is very sensitive to particles shape and relativecomplex refractive index (Ward et al. 1973; Tanaka et al. 1982; Zhao et al. 1997).When compared with Lorenz-Mie’s solutions for homogenous spheres, goodagreement was reported at forward scattering angles. However, significant errors areidentified at backscattering angles, which may be reduced by increasing the parti-cles absorption coefficients (Holland and Gagne 1970). This example illustrateshow inferring particles relative complex refractive index can be difficult forsingle-wavelength polarimetric light-scattering, especially when only backscatteredlight is measured. The problem of estimating microphysical properties usingangular and polarimetric light-scattering often remains an ill-posed problem innumerous scientific fields.

Polarimetric light-scattering has been neglected in the past decades for numerousapplications in remote-sensing while it offers an interesting tool to probe the earth’satmosphere. The light becomes partially polarized as it is scattered by the moleculesand aerosols in the atmosphere, whereas the direct sunlight is unpolarized. Thepolarization of skylight follows the well-established theory of Rayleigh scattering ina molecular atmosphere, with anomalies measured in unclear atmosphere caused byaerosols. Long after the hypothetic use of a dichroic material by the Vikings fornavigation (Roslund and Beckman 1994), qualitative measurements of the skylightwere first achieved at the beginning of the nineteenth century by Arago (1858). Forseveral decades, most polarimetric measurements used unpolarized natural lightsources and remained passive remote-sensing techniques. Pioneer quantitativepolarimetric light-scattering measurements of the atmosphere, i.e. DOP of skylight,were carried out before the 1960s by visual polarimeters (Comu 1890;Pyaskovskaya-Fesenkova 1958) and later with increased accuracy using electronicdevices in clear and polluted atmospheric conditions (Tousey and Hulburt 1947;Sekera 1957; Hariharan and Sekera 1966; Sun et al. 2014a, b), including polari-metric twilight sky measurements (Fesenkov 1966; Rozenberg 1968). For cloudresearch, an important challenge is to discriminate liquid water from ice waterclouds. Polarimetric information has become a valuable mean to remotely sense theshape of aerosols. In that respect, numerous studies have proved that liquid waterdroplets and ice water particles exhibit different polarimetric light-scattering sig-natures, which depends on cloud microphysics and optical properties. The use ofpassive polarimetric remote sensing began in the 1970s including polarimetriclight-scattering measurements from high-altitude balloon flights to distinguish liq-uid water clouds from ice water ones (Bums 1975; Stowe 1977).

A Swedish-built UV spectropolarimeter was launched on a Soviet satellite in theIntercosmos series to record resonance-line polarization in the solar spectrumaround 130–150 nm (Stenflo et al. 1976). Further polarimetric measurements werecarried out around 1980s in the Pioneer Venus Orbiter (Travis 1979) and the SpaceShuttle (NASA) (Whitehead 1990). Later, the POLarization and Directionality ofthe Earth’s Reflectance (POLDER) instrument (CNES, France) was a spacebornepolarimetric sensor launched in 1996 to collect polarized and directional solarradiation from space (Deschamps et al. 2002). Simultaneous retrieval of the particle


size and shape was achieved from POLDER simulations and measurements(Goloub et al. 1994; Breon and Goloub 1998). For instance, polarimetric featuressuch as zero polarization (between 75° and 120° from the incoming direction) andpolarization maximum (around 140°) were used respectively to infer the effectiveradius of cloud droplets and type of clouds (either liquid or ice water phase) by Sunet al. (2015a, b).

Active polarimetric remote-sensing have emerged from the mid-1960s with earlywork of Rozenberg who has investigated polarimetric light-scattering at night-timeusing an artificial illumination of the atmosphere and polarization sensitive sensors(Rozenberg 1968). Although the invention of laser opened new possibilities tosense particulate media in the atmosphere, active remote-sensing techniques such asLIDAR did not exploit polarization at first. Polarimetric LIDAR was proposed toprofile and characterize aerosols in the atmosphere as polarimetric measurementsprovide information about the shape of particles (Schotland et al. 1971; McNeil andCarswell 1975; Sassen 1991; Murayama 1996). Polarimetric LIDAR has shown itsability to distinguish spherical particles from irregularly shaped particles fromdepolarization measurements (Sassen 1991; Mishchenko and Sassen 1998; Sassenet al. 2007). This category of LIDAR is also used to study clouds and, for instance,to distinguish ice clouds from water clouds (Ansmann et al. 2005) or stratosphericclouds (Reichardt et al. 2000). It has also been used to identify and discriminatedifferent types of aerosols such as volcanic ash (Hayashida 1984; Sassen et al.2007), dust (Gobbi et al. 2000), black carbon (Li et al. 2015), contrails(Freudenthaler et al. 1996), or aerosolized biological warfare agents (Richardsonet al. 2008) from other aerosols present in the atmosphere.

In the field of biomedical studies, optical techniques mostly rely on photometricmeasurements without taking into account polarization effects (Boas et al. 2011).Early works from 1980s have shown that polarization could be a sensitive tool toinvestigate the modification of tissues, bacterial suspensions, or blood cells (Johnstonet al. 1988; Van deMerwe et al. 1989; Gross et al. 1991). Polarimetric light-scatteringtechniques range from the simplest to the most comprehensive, including ellipsom-etry (Dreher et al. 1992), fluorescence (Mohanty et al. 2001), Mueller matrix imaging(Dreher et al. 1992), or full Mueller endoscopy (Dreher et al. 1992). Although mosttechniques usually use polarization gating capabilities as a way to reducemultiple-scattering effects and improve signal-to-noise ratio, a category of techniquesfully exploits polarimetric light-scattering. These methods are more comprehensiveand complex by nature as it results in thorough interpretations of polarimetric mea-surements in order to extract useful information. It has been demonstrated thatpolarization could sense structural changes in malignant tissues as compared to thehealthy tissues for diagnosing diseases (Bordier et al. 2008; Ghosh et al. 2009) orcancerous cells (Anastasiadou et al. 2008) for biomedical diagnosis. Despite polari-metric light-scattering gains interest in biomedical fields, several strong challengesand limitations remain. Simultaneous occurrences of polarization effects (e.g. diat-tenuation or retardance) is a serious challenge for polarimetric light-scattering as itmay alter the measured signal, which can lead to erroneous estimation of thepolarization-preserving signal (Ghosh 2010).


In a nutshell, polarimetric light-scattering techniques provide useful additionalinformation regarding particulate media, such as the shape, concentration, localorganization, or orientation of particles. Merging directional and polarimetriclight-scattering information has been useful to probe complex or dense particulatemedia such as clouds or tissues. However, most polarimetric light-scatteringtechniques often assume no spectral dispersion of the materials or change in particlesize distribution. Such assumptions may lead to erroneous results from inversemethods. The development of spectral and polarimetric light-scattering intend tooffer a more robust and comprehensive method, without prior knowledge such asPSD or relative complex refractive index, to investigate complex particulate media,including particles with wide size distribution, materials with spectral dispersion,irregularly shaped particles, and dense media.

2.5 Spectral Polarimetric Light-Scattering

While several spectral or polarimetric methods have been proposed to studycomplex systems of small particles, the retrieval of the intrinsic microphysicalparameters of particulate media remains challenging without prior knowledge of thePSD or relative complex refractive index. Spectral and polarimetric light-scatteringare both sensitive to different microphysical properties of particulate media. As aresult, methods have been proposed to merge spectral and polarimetric properties oflight-scattering (Thompson et al. 1980; Chen et al. 1988). The goal of thesemethods is to identify microphysical parameters of small particles and particulatemedia, without assumptions or a priori information. Besides, the merge of spectraland polarimetric light-scattering may yield complementary information to retrievemicrophysical parameters of particulate media (Huckaby et al. 1994). This merge isexpected to ease the retrieval of additional microphysical parameters of the par-ticulate media, such as porosity, structuration, or compactness. All these terms referto the way the particles are arranged within complex particles, aggregates or clustersfor instance. This section presents a review of different emerging methods based onthe fusion of spectral and polarimetric light-scattering, also referred as polarizedspectroscopy, spectral Mueller matrix analysis, or spectro-polarimetry.

By merging spectral and polarimetric information from light-scattering, one cancompute the spectral DOP for particulate media. The spectral DOP is retrieved fromperpendicular and parallel polarized intensities as:

DOPðk; hÞ ¼ IPðk; hÞ � ISðk; hÞIPðk; hÞþ ISðk; hÞ ð2:5Þ

where k refers to the wavelength and h to the scattering angle.


The spectral DOP is also a wavelength-dependent polarimetric contrast betweentwo p and s orthogonal states of polarization. The variation of the spectral DOPoriginates from complex electromagnetic phenomena between particles:

i. Light-absorption by the particles. Absorption is the predominant phenomenonto explain the wavelength-dependence of the spectral DOP. For instance, whenpolarized light is scattered at absorbing wavelengths, the light is stronglyabsorbed and the spectral DOP is significantly enhanced at these wavelengths.

ii. Multiple-scattering by particles. This is another important phenomenon thatexplains the wavelength-dependence of polarization. The general idea lies inthe fact that light-scattering by particles is conditioned by the range of elec-tromagnetic interactions. This interaction range is directly related to the x-parameter, i.e. the ratio between the particle size and wavelength. For a givenparticle size, the range is minimal for x-parameter close to unity, whereas therange increases for larger values of x. Depending on the number of neighbor’sparticles involved in the interaction range, multiple scattering events mayoccur. Spectro-polarimetric light-scattering results from the number of parti-cles covered by one wavelength, which causes multiple scattering and depo-larization. Hence for a broadband light source, the part of multiple scattering,and as a consequence depolarization, varies with the wavelength.

iii. Spectral dependence of polarimetric effects. Effects such as diattenuation orretardance may be used in combination with the wavelength-dependence ofdepolarization for comprehensive assessment of particulate media.

Consequently, the variation of the spectral DOP results in a spectral gradient ofdegree of polarization rDOPðk; hÞ. This gradient is directly computed from thespectral DOP when spectral data are available, for instance from multi-spectral orhyperspectral sensors. In a similar way with the spectral slope or spectral gradientfor reflectance, this spectral gradient of DOP is defined as:

rDOPðk; hÞ ¼ dPðk; hÞdk

ð2:6Þ

The spectral gradient of DOP, qualified as positive or negative regarding the signof the gradient with the wavelength, turns out to be a promising tool for sensingdistinctive features related to the microphysics of particulate media and studying theinner structure of complex clusters of particles. In regard to the implication of themerging spectral and polarimetric information from light-scattering, a broad spec-trum of applications has emerged from aerosols characterization (Farhoud 1999) topurity of nanoparticles by Barreda et al. (2015a, b).

Remote-sensing has recently and progressively merged spectral and polarimetriclight-scattering techniques to probe particulate media remotely. Farhoud (1999) hasmeasured from spectro-polarimetric light-scattering a wavelength-dependence of thespectral DOLP and revealed a minimum polarization for aerosols characterization. Inpassive remote-sensing, the spectral polarization of clear and hazy daytime sky wasretrieved from simultaneous spectral irradiances measurements at high spectral and


angular resolutions. Such analysis combined with spectro-polarimetric RTT hasrevealed the combined angular and spectral effects of light-scattering on DOP. Itcould provide useful quantitative insights into how polarimetric spectra depend onscattering and absorption by tropospheric haze droplets and other aerosols (Suzukiet al. 1997; Lee and Samudio 2012; Ejeta et al. 2012; Hollstein et al. 2009; Boescheet al. 2006). For instance, Sun et al. (2013a, b, c) investigated the role of air bubblesand brine pockets contained in sea-ice and lake-ice from spectral polarized reflectanceand DOLP. In active remote-sensing, most techniques require additional informationregarding the aerosols to inverse the well-known LIDAR equations, due to ill-posedLIDAR problem. Such additional informationmay come from prior knowledge of theaerosols or from measurements provided by passive (e.g. sunphotometer) instru-ments, in which particles are collected and analyzed to determine their size, com-position or morphology. The idea of merging spectral and polarimetriclight-scattering information appears promising for future LIDAR applications.

Few studies have used multi-wavelength and polarization data for estimating theaerosol properties (Groß et al. 2011). It has been showed that using measurements attwo wavelengths helps to distinguish different types of aerosol (Sugimoto and Lee2006), most LIDAR systems employ only one wavelength for the analysis of theparticle linear depolarization ratio of dust by Sassen (1991, 2007). From simulations,Sakai et al. (2007) studied the relationship between the wavelength-dependence ofthe backscattering coefficients and the depolarization ratio measured by LIDAR atmultiple wavelengths. Several light-scattering calculations have shown that thespectral dependence of depolarization depends on the size distribution ofnon-spherical particles and could be used by Mishchenko and Sassen (1998) andWiegner et al. (2009). The development of multi-wavelength polarizationRaman LIDAR in the mid-1990s was one important step in our ability to characterizemineral dust particles (Müller et al. 1998; Althausen et al. 2000; Veselovskii et al.2002). First, dual-wavelength aerosol polarimetric LIDAR measurements werecarried out by Sugimoto et al. (2002), followed by multi-wavelength measurementsof the linear depolarization ratio by Freudenthaler et al. (2009). The Cloud-AerosolLidar and Infrared Pathfinder Satellite Observations (CALIPSO, NASA-CNES)(Winker et al. 2003) is a satellite mission with a unique payload consisting in atwo-wavelength polarimetric spaceborne LIDAR (CALIOP). These measurementsprovide information on the vertical distributions of aerosols and clouds, identificationof water/ice cloud (from the depolarization ratio provided by the two orthogonalpolarization channels), and a classification of aerosol PSD (from thewavelength-dependence of the backscattering coefficient). These multispectralpolarimetric LIDAR measurements were exploited to improve inverse schemes andto retrieve microphysical properties of aerosols (Dubovik et al. 2006; Müller et al.2013). Most of these spectro-polarimetric light-scattering based remote-sensingtechniques are non-imaging techniques (Goldstein and Chenault 2002). Severalimaging techniques have been proposed to exploit the advantages of mergingpolarimetric and spectral reflectance to enhance the contrast of man-made surface in ascene embedded or not in turbid media (Johnson et al. 1999; Le Hors et al. 2000;Giakos 2006; Wang et al. 2007).


In astrophysics or cosmology, spectral DOP, also called polarimetric colors,were measured at different wavelengths and polarization states for a variety ofparticles in space and particulate media such as planetary regolith surfaces. In themid-1950s, first multispectral polarimetric measurements of the Moon noticed awavelength dependence of the polarization (Dollfus 1957). Further measurementslater corroborated these results on lunar regolith and solid lava fragments (Coffeen1964). Lunar regolith appears to exhibit both wavelength and phase dependency ofthe polarization, while no wavelength dependence was reported for solid lavafragments. These studies are considered among the first publications suggesting thepotential of merging spectral and polarimetric light-scattering to identify micro-physical properties of particulate media. Spectro-polarimetric light-scatteringproperties of cosmic dust are hard to measure and, in most studies, onlymulti-spectral data are available (Myers and Nordsieck 1984; Le Borgne et al. 1987;Chernova et al. 1993; Crovisier et al. 1997; Kiselev et al. 2000). When measure-ments at different wavelengths are available, the spectral gradient was found todiffer significantly for different particles such as comets, dust, or asteroids.Regarding spectral DOP, positive or negative colors refer to the sign of the spectralgradient: for instance, positive color refers to positive spectral gradient values.Asteroid regolith often exhibits negative color, which may result from their highcompactness while most comic dust exhibits positive polarimetric color due to theirhigh porosity. Kolokova modeled accurately electromagnetic interactions betweenthe small particles of cosmic dust aggregates with different particles arrangement orporosity (Kolokolova and Jockers 1997; Gustafson and Kolokolova 1999; Kimura2003; Kolokolova 2010, 2016). Their results have shown a high dependence of thespectro-polarimetric light-scattering properties on their porosity. This relationshiporiginates from electromagnetic interactions, which contribute directly to thepolarimetric color. Ceolato et al. (2013) simulated broadband spectral and polari-metric light-scattering by small particles aggregates and super-aggregates. It hasbeen reported significant dependence of the aggregate parameters upon thespectro-polarimetric light-scattering signatures. Other laboratory measurementshave confirmed singular differences in the spectral DOP for natural particulatesurfaces such as planetary regolith or artificial powders (Sun et al. 2014a, b). Asreported by these different studies, the number of particles per single wavelength, aswell as the ratio between the gyration radius and the wavelength, is significantparameters which influence the spectro-polarimetric light-scattering by aggregates.This ratio mainly governs the influence of the structure of the aggregate uponspectro-polarimetric light-scattering: for small values, the influence of the structureis negligible, whereas, for large values, the structure becomes a governing factor ofthe signature. Spectral dependence of the DOLP was measured as well as anincrease within absorption bands, where the reflectance decreases sharply.

Early studies in astrobiology have proposed that the wavelength-dependence ofpolarization of light could be a key to detecting life traces in the universe and dis-tinguishing them from non-biological species (Pospergelis 1969;Wolstencroft 1974).These broadband polarimetric features, resulting from the homochiral nature of theorganics, are proposed as signatures of life called bio-signatures (Sparks et al. 2009;


Sterzik and Palle 2012; Berdyugina et al. 2016). Circular polarization of light, inparticular, was found to be not only non-zero but to vary distinctly with the wave-length within high absorption bands, depending on the number and size of particlesforming an aggregate of chloroplasts (Nagdimunov 2013). These studies showed thatthe more compact the aggregates, the more polarized is the scattered light, in a similarway to what has been studied for comic dust.

Biomedical research has been intensively active in merging spectral andpolarimetric properties of light-scattering. Apart from techniques where polarizationis used to suppress multiple-scattering from spectral light-scattering (Sokolov1999), polarimetric spectra of biological tissues were found to be strongly influ-enced by cell nuclei morphology and constituents. Skin and tissues are subject tomorphological, functional state, and structure modifications by different patholo-gies. These spectro-polarimetric modifications may imply changes of birefringenceand structure, which can be accurately monitored by the spectral and polarimetriclight-scattering (Sokolov 1999; Gurjar et al. 2001; Mourant et al. 2002; Zimnyakovet al. 2005). Pioneer work of Backman in 1999 proposed to measure both spectraland polarimetric light-scattering for quantitative analysis of biological samples. Theproposed technique uses a broadband tungsten lamp and polarizers with multi-channel spectrometers to measure the wavelength dependence of the polarizedbackscattered light in order to determine the size and relative complex refractiveindex of cell nuclei (Backman et al. 1999). Merging spectral and polarizationinformation provided a robust method of removing the diffuse component of thescattered light. The residual spectrum is mainly composed of light from singlebackscattering and provides histological information about the epithelial cells andcan be used to extract the size distribution, population density, and relative complexrefractive index of the nuclei by comparison with Lorenz-Mie theory. This methodwas proposed to detect and diagnose precancerous changes in tissues anddemonstrated the potential of spectro-polarimetric light-scattering to provideaccurate quantitative estimates of the size distributions of cell nuclei and additionalinformation, such as relative complex refractive index of cell organelles, which arevery difficult to obtain using existing methods. In the recent years, the use ofspectro-polarimetric light scattering to probe the morphological and structuralchanges of cells has become an active field of research. Different theoretical modelswere proposed based on Lorenz-Mie theory to analyze the single backscatteringpolarimetric spectra. For instance, a two-layered model was used to retrieve mor-phological parameters related to the abnormality of biological tissues, including themean diameter, size distribution and relative complex refractive index (Zhao et al.2014). Several techniques from laboratory have investigated the dependence of thespectro-polarimetric light-scattering on these parameters to endoscopic measure-ments (Ding et al. 2007; Qi et al. 2012). Results indicate that the angle of maximumdepolarization shifts toward larger values as the wavelength decreases from IR toUV (Ding 2007). Even though the angular shift is relatively small, it is sufficient toexhibit a high correlation with wavelength. Similar results were later reported usinga broadband laser source for cancerous cells (Ceolato et al. 2013). A majoradvantage of merging spectral and polarimetric information is to be able to probe


complex biological systems. Spectro-polarimetric measurements were carried outon bacteria colonies under different environments, and inverse power-law depen-dence over a broad range of wavelengths was extracted from the fractal analysis(Banerjee et al. 2013).

From a medical diagnostic perspective, the potential use of spectral polarimetrywas proposed to detect different pathologies such as leukemia, in which red bloodcells decrease in numbers as a result of an abnormal proliferation of white bloodcells. Red blood cells are known to play a major role in the absorption and scat-tering properties of blood (Tuchin 1997). Their variation in numbers in the bloodmay infer changes in the spectral polarimetric light-scattering signatures of blood.Recent results showed a strong depolarization of light by blood at 400–500 nm,where high absorption of hemoglobin may cause significant randomization ofpolarization (Swami et al. 2010). Spectro-polarimetric measurements have reporteda minimum of depolarization of light which may be related to the number of redblood cells (Aziz et al. 2013). This minimum of depolarization has been proposedto be used for diagnosing acute lymphoblastic leukemia. Experimental results havesuggested the potentials of the techniques based on the fusion of spectral andpolarimetric light-scattering for skin pathological diagnosis and treatment evalua-tion. Ramella (2011) measured the full Stokes vectors of chicken muscle at severalwavelengths in the visible domain. The change of depolarization with wavelength isalso explained by a decrease of the blood hemoglobin absorption, which is theprimary source of the light absorption in the visible wavelength range. Thus, theincrease of light penetration depth enhances the scattering and leads to a largerdepolarization of backscattered light for both healthy and cancerous tissues in thered part of the spectrum. Spectral Mueller matrix images of ex vivo human colontissues revealed a contrast enhancement between healthy and cancerous zones ofcolon specimen compared to unpolarized intensity images (Novikova et al. 2014).Similar results were reported on mice melanoma using spectral polarimetriclight-scattering measurements (Ceolato et al. 2015).

These early and recent works have demonstrated the advantage of couplingspectral and polarimetric methods for optical diagnostic of complex particulatemedia. In order to retrieve microphysical properties of these media, a compre-hensive quantitative approach is required. In the next section, a complete experi-mental and numerical method is detailed to compute accurate spectro-polarimetriclight-scattering signatures for a given particulate media.

2.6 Quantitative Spectro-Polarimetric Light-Scattering

ONERA, The French Aerospace Lab, has developed different experimental facilitiesfor optical diagnostic of particulate media. A variety of systems, such as laser-basednephelometers or LIDAR, have been used for years for optical diagnostic of complexdense systems of particles (Hespel and Delfour 2000; D’Abzac et al. 2012;Ceolato et al. 2012). Based on recent developments in light-sources andoptical-sensors, original concepts of optical characterization, including full spectral


range, polarimetric, and high angular resolution, have been investigated to performmore robust, accurate, and comprehensive remote optical diagnostic. This sectionreports recent works dedicated to performing a quantitative analysis ofspectro-polarimetric light-scattering. A systematic study aimed at searching themicrophysical parameters characterizing particulate media has been engaged and arereviewed in this section.

Developments in nanostructured fiber optics and compact pulsed lasers haveresulted in the conception of supercontinuum (SC) laser sources (Alfano andShapiro 1970a, b). These sources are generated using a pulsed laser propagatingthrough a non-linear medium. It results in a spectral broadening (Dudley et al.2006) and the creation of “white” directional coherent light. SC lasers are direc-tional and coherent light sources based on the supercontinuum generation phe-nomenon. Discovered in the 1970s, this phenomenon is the result of thepropagation of intense narrow-band ultrashort laser pulses inside a material (e.g.optical fiber or bulk). This phenomenon broadens the spectrum of thesingle-frequency pulses (Alfano and Shapiro 1970a, b) and results in the generationof a broadband laser light. This spectral broadening is a typical nonlinear effect dueto strong laser-field confinement and can extend from the IR to the visible. Theintense pulse induces atomic or molecular modification of the material, which inturn varies the dispersion law by optical Kerr effects. The variation in relativerefractive index produces non-linear phenomena resulting in the frequency sweepwithin the pulse by phase shift or self-phase modulation. Other more complexphenomena, such as stimulated Raman scattering and four-wave mixing (Coen et al.2002) can also be involved. With the current progress in nanophotonics, newbroadband laser sources are available based on SC generation from UV to IR.Ultrashort pulsed laser sources (e.g. femto or picosecond) are combined withPhotonic Crystal Fibers (PCF), which are used as the propagating material.The PCF has a small core diameter and has a nonlinear coefficient that can beengineered to achieve a specific output spectrum (Ranka et al. 2000). The coherenceproperties of SC lasers have been investigated using spectral interferometry(Zeylikovich et al. 2005). Although they present a low temporal coherence, thesesources yet retain a high degree of spatial coherence (Alfano and Shapiro 1970a, b).

In parallel, spectral or hyperspectral sensors have been used intensively for years inthe remote-sensing community (Shaw et al. 1973a, b; Manolakis and Marden 2003).Sensing with a high spectral resolution consists in measuring the intensity of lightover a wide range of wavelengths. This is accomplished by spectrophotometers orhyperspectral sensors, which have been used intensively in the remote-sensingcommunity (Ientilucci 2009). Hyperspectral sensing refers to the measurement oflight intensity for more than thousands of spectral bands. The outputs of hyperspectralsensors are tensors, or also named hypercubes, where the dimension is related to thenumber of bands. The combination of SC laser sources with hyperspectral sensors isof rising interest for a broad range of applications (Johnson et al. 1999; Peng and Lu2008; Zakian et al. 2009; Larusson et al. 2011a, b; Hakala et al. 2012).

Rather than a single or multi-wavelength approach, a spectro-polarimetriclaser-based scatterometer was designed at ONERA to characterize particulate


media. One advantage of spectro-polarimetric light-scattering measurements per-formed by this scatterometer is to analyze the spectral dependence of variousscattering media from rough surfaces to particulate media. The originality of thesystem is to use a SC laser source coupled with hyperspectral sensors. Nomonochromator is needed and the full polarimetric spectrum is measured at once.Hyperspectral, polarimetric, and directional light-scattering measurements are car-ried out simultaneously. This innovative instrument carries out in-line, andnon-destructive remote optical measurements. This spectral polarimetric scat-terometer measures the light scattering in visible and IR ranges (VIS/NIR/SWIR)from 480 to 2500 nm with a spectral resolution lower than 1 nm in the visiblenear-infrared range, and lower than 5 nm in the short-wave infrared (Riviere et al.2012; Ceolato et al. 2012). A simplified schematic of the design of the MELOPEEscatterometer is presented in Fig. 2.6.

The incident lighting system consists of a SC laser combined with an achromaticcollimator. The laser source is initially fully unpolarized at full power and is coupledto wide-band polarizers to select incident polarization states. The variation of theincident power ismeasured to quantify the stability of the laser source. The polarizers’efficiency was found to be greater than 98% on the whole spectral range. The sensingsystem consists of a CCD/InGaAs-sensor based spectrophotometer mounted on agoniometric platform at 1 m from the target. The instrument measures the scattered

Fig. 2.6 Schematic of the VIS/NIR hyperspectral version of MELOPEE instrument developed atONERA. The incident lighting system is a SC laser coupled to wide-band polarizers. The sensingsystem is composed of a CCD-based spectrophotometer mounted on a goniometric platform


light with an azimuthal angular resolution of 0.1° and provides high angle-resolvedmeasurements. A 5-axis fully automated sample holder facilitates the measurementprocedure. For light-scattering measurements, the principal plane is defined as theplane that contains the normal of the sample and the incident beam. Both in and out ofprincipal plane configurations are available with this scatterometer. Our instrumenthas been calibrated using different Lambertian reference standards such as LabSphereSpectralon®. Figure 2.7 presents the spectral and angular light-scattering measure-ments on LabSphere Spectralon® SRS-99 and SRS-20. These reference standards arefound to behave closely as a Lambertian surface at normal angle of incidence from480 to 1000 nm. Along with accurate spectro-polarimetric scatterometer, it isimportant to develop dedicated light-scattering models.

Along with MELOPEE scatterometer, ONERA has developped several numer-ical models to compute spectro-polarimetric light-scattering by a wide range ofparticulate media.

For particulate media composed of axisymmetric particles (e.g. spherical,spheroidal), a spectral and polarimetric RTT model was developed to model thespectro-polarimetric of dense particulate media. One essential condition for usingthe RTT approach to model multiple light-scattering by particulate media is that thedistance d between particles must be larger than the wavelength k such as d > k. Insuch a way, the scattering events always occur in the far-field region of each particleand prevent coherent effects such as dependent scattering. This spectral andpolarimetric RTT model solves the VRTE for an incident broadband illuminationand spectral detection.

Let us first consider a particulate media as an ensemble of N particles in randomorientations and positions in a non-scattering host medium. The spectral VRTE isgiven for a polarized broadband directional light source as:

u � rSðr; u; kÞþ kextðr; kÞ � Sðr; u; kÞ ¼14p

�Z 2p

0

Z 1

0du0 � Pðr; u; u0; kÞ � kscaðr; u; u0; kÞ�Sðr; u0; kÞþ kabsðr; u; kÞ � S0ðTÞ

ð2:7Þ

Fig. 2.7 Light-scattering measurements for Lambertian materials: Spectralon® SRS-99 andSRS-20. The illumination is P-polarized and the detection unpolarized


where kextðr; kÞ and kabsðr; u; kÞ refers respectively the spectral extinction andabsorption vectors. kscaðr; u; u0; kÞ refers to the spectral scattering matrix. S0ðTÞ ¼n2 � r � T4=p is the total blackbody Stokes vector, T the temperature and r ¼567051 � 10�8W �m�2 � K�4 the Stefan-Boltzmann constant.

The spectral Stokes vectors Sðr; u; kÞ are the spectral VRTE solutions whichencompass the spectro-polarimetric signatures of a particulate medium of interest asa function of position r, direction u, and wavelength k. The phase matrix Pðr; kÞ ofan ensemble of particles is calculated by integrating the 4 � 4 real-valueddimensionless Mueller matrix elements Mijðrp; r; kÞ over the PSD nðrpÞ of anensemble (i and j range from 1 to 4):

Pðr; kÞ ¼P11 P12 P13 P14

P21 P22 P23 P24

P31 P32 P33 P34

P41 P42 P43 P44

0BB@

1CCAwithPijðrp; r; kÞ

¼ k

p � k2sca

Zdrp �Mijðrp; r; kÞ � nðrpÞ ð2:8Þ

Particulate media are modeled for numerical purposes as semi-infiniteplane-parallel layers with constant radiative parameters as shown in Fig. 2.8. Weconsider multilayered isotropic scattering media. For instance, a particulate media

Fig. 2.8 Global view of the particulate media geometry and notation used for the simulation


inside a quartz or glass slab is modeled as a series of vertically inhomogeneouslayers containing randomly oriented particles of various geometries. The generaldescription of atmospheric layers is substituted herein by optical material interfaces,such as quartz or glass, with given relative complex refractive index. The extensionof this model to spectro-polarimetric light-scattering requires several modificationsincluding spectral dispersion of the interfaces or solving Fresnel equations at everyinterface for every wavelength.

Our model based on the spectral VRTE solves numerically a system ofL equations, where L is the number of discrete wavelengths composing thebroadband light source (typically L = 2000). The spectral operation consists ofcalculating the scattered spectral Stokes vector corresponding at each wavelength inthe spectral range. Azimuthal angles are expressed as a Fourier series and polar-ization rotations are performed directly in azimuth space. Subsequently, Fouriertransform is performed to retrieve the scattering matrix for each Fourier azimuthmode (Evans and Stephen 1991; Deuzé et al. 1989; Ishimaru et al. 1984). Multiplereflections between layers are also taken into account in the spectro-polarimetricradiance balance at the top and bottom of each layer. An adding-doubling techniqueis then deployed for each Fourier mode to model multiple scattering. This techniqueis numerically stable and is used to model multiple light-scattering in an intuitive,efficient, and simple way. A good selection of the Fourier modes and the quadraticangles for every wavelength is crucial for the computation.

Validation of the model has been carried out following a three-step procedure(Riviere et al. 2013). Briefly, (i) an analytical validation was conducted forRayleigh scattering from Coulson’s Table (1960), (ii) a stochastic method was usedto validate the model for Mie scattering and, (iii) experiments were carried out onpolystyrene particles in aqueous solution. As stated above, the spectral VRTEmodel is valid only for intermediate volume fractions, in dispersed-phase, as a largevalue of this parameter refers to a large number of multiple scattering events andmultiple reflections inside a glass cuvette for instance.

Spectro-polarimetric light-scattering simulations are carried out using ourspectral VRTE model for an incident broadband p-polarized collimated illumina-tion. Typical radiative transfer inputs are optical thickness or albedo. However, forbroadband calculations, these radiative parameters cannot be employed because oftheir spectral dependency. For broadband calculations, particulate media must berather described in terms of microphysical parameters such as volume fraction orrelative complex refractive index (both for particles and host medium), which arenot dependent on the wavelength of illumination. Thus, a particulate media com-posed of an ensemble of random spherical particles is modeled with three principalmicrophysical parameters:

(1) Particle size distribution, which follows for instance a log-normal size distri-bution such as:


dnðrpÞdrp

¼ N0

rp � lnð10Þ � r � ffiffiffiffiffiffi2p

p � exp � logðrp=rmÞ2 � r2

� �ð2:9Þ

where n(rp) is the PSD [m−3�m−1], rp the particle radius [m], No is the totalnumber density of particles [m−3], rm is the mean particle radius [m], and r isthe standard deviation of distribution.The volume fraction (Jeffrey and Acrivos 1976) describes the fraction of par-ticles in the host media volume (e.g. air, liquid water). It is calculated from thetotal particle volume Vpart [m

3�m−3] for a given PSD by:

Vpart ¼Z1

�1

p6r3pnðrpÞdrp ð2:10Þ

(2) Relative complex refractive index, which accounts for the spectral dispersionand absorption of both particles and host media. Analytical refractive indexmodels can be employed to compute spectral relative complex refractive indexof materials.

In what follows, a particulate media composed of spherical silica particles isconsidered. The microphysical parameters for the simulation correspond to a meanradius rm of 100 nm, a relative complex refractive index varying over the spectralrange (e.g. m = 1.59 + i 0.0 at 532 nm), and a volume fraction of 1%.

The first and second elements of the spectral Stokes vectors are computed andpresented in Fig. 2.9 against l = cos(h), the cosine of the scattering angle h. Thescattered intensity, related to the first Stokes matrix element I/Iinc, decreases forlarge wavelengths. This spectral dependence is shown in the backscattering(l ¼ �1) and forward scattering (l = +1) directions. This dependence could beunderstood as the variation of the single-scattering efficiency over a broadbandspectrum. The single-scattering efficiency is known to vary with the size parameter(i.e. ratio of the particle diameter to wavelength) (Mishchenko et al. 2002). Thus, italso varies when spectral scattered-intensities are computed over a wide spectralrange for a fixed particle size (here, particle radius is fixed). The spectral DOLP,related to the second Stokes matrix element Q/I, also exhibits a clear spectraldependence especially for reflection angles (between l ¼ �1 and l = 0). Moreinterestingly, a noticeable feature is found around l ¼ �5. It corresponds to amaximum depolarization where the spectral DOLP reaches low values. Similarfeatures have been previously reported from experiments (Müller et al. 1998) andare further analyzed in the next section. A numerical artifact produces unrealisticvalues at l = 0.

The microphysical parameters of particles are expected to play a major role inthe spectro-polarimetric light-scattering signatures of particulate media.A microphysical analysis is carried out to determine their impact on spectralpolarimetric light-scattering signatures. Such analysis is a prior requirement in thedevelopment of specific inversion methods merging spectral and polarimetric


information. In the following, spectro-polarimetric light-scattering signatures arecomputed with different mean particle radius, relative complex refractive indexes,and volume fractions in the range of validity of the VRTE model. The mean particleradius ranges from 35 to 450 nm, the relative complex refractive index is constantover the spectral range with fixed values ranging from 1.5 + i0.0 to 2.05 + i0.0, andthe volume fraction varies from 0.1 to 5.0%.

Figure 2.10 presents the spectral first Stokes elements (I/Iinc) resulting frommultiple light-scattering by particulate media with varying microphysical parame-ters. They are plotted in log-scale for an incident broadband p-polarized collimatedillumination and different sets of parameters. For each line, a single parametervaries when others remain constant. When not specified, the mean particle radiusfor a log-normal size distribution is rm = 100 nm, the volume fraction is f = 1.0%,and the relative complex refractive index is m = 1.50 + i 0.0.

Figure 2.11 presents the spectral second Stokes elements (Q/I) resulting frommultiple light-scattering by particulate media with varying microphysical parameters.They are plotted for an incident broadband p-polarized collimated illumination anddifferent sets of parameters. For each line, a single parameter varies when othersremain constant, similarly to Fig. 2.10. In the figure, red color refers to polarizedscattered light whereas blue color refers to unpolarized light: depolarization mainlyappears in the backward scattering directions (from µ = −1 to 0). For a given par-ticulate media, depolarization is reported in the backward directions (from µ = −1 toµ = 0) and strongly depends on the wavelength. It shows a directional dependenceand a spectral shift of the depolarization (i.e. low DOLP values).

Light-scattering by particulate media depend on the microphysical parameters ofparticles such as the particle size, relative complex refractive index, or volumefraction. Some of these parameters depends on the wavelength and may vary

Fig. 2.9 Simulation of the spectral Stokes vectors (I/Iinc in log-scale and Q/I in linear-scale)scattered by a particulate medium for a broadband (480–1020 nm, with spectral resolution of5 nm) p-polarized illumination


significantly for broadband calculation. Usual monochromatic light-scatteringtechniques investigate a single scattering regime. One advantage of spectral orbroadband light-scattering computations is to address distinct scattering regimes fora given particulate media. From Fig. 2.10, a change of scattering regime is iden-tified when the mean particle size or relative complex refractive index increases.However, no change is noticed for varying volume fraction. This is explained as thegoverning parameters of scattering (i.e. size and relative complex refractive index)are kept constant for different volume fractions. These results expose the interactionbetween microphysical parameters of particles and the wavelength of the light.Spectro-polarimetric light-scattering clearly provides interesting physical insightsabout the nature of particles in particulate media. Due to the interplay between thesize and relative complex refractive index, using only the spectral scattering

Fig. 2.10 Spectral computations of the first Stokes element (I/Iinc) in log-scale for an incidentbroadband (480–1020 nm, with spectral resolution of 5 nm) p-polarized collimated illumination.Variations of the mean particle size (rm), relative complex refractive index (m), and volumefraction (f) of particles


intensity may result in serious inversion hardships. The merge of spectral andpolarimetric information is proposed by computing the spectral second Stokesvector element (Q/I) or the spectral DOLP. Depolarization from multiple scatteringis a directional phenomenon (i.e. depends on the scattered angle) and can besimulated by our model over a wide spectral range from visible to near-infrared. Asshown in Fig. 2.11, the incident polarization is conserved for transmission angles(including forward-scattering), whereas significant depolarization appears atreflection angles (including backward-scattering). Depolarization frommultiple-scattering is known to depend on the particle size or relative complexrefractive index (Kim and Moscoso 2001) and has been accurately computed by ourmodel. When performing the microphysical analysis, the mean particle size, relative

Fig. 2.11 Spectral computations of the second Stokes element (Q/I) for an incident broadband(480–1020 nm, with spectral resolution of 5 nm) p-polarized collimated illumination. Variationsof the mean particle size (rm), relative complex refractive index (m), and volume fraction (f) ofparticles


complex refractive index, and volume fraction are important factors that drive thespectral and directional depolarization at reflection angles.

For particulate media composed of non-axisymmetric particles (e.g. fractalaggregates), other numerical models must be considered. A Spectral DiscreteDipole Approximation (SDDA) model (Ceolato et al. 2013) was developed tocompute the spectro-polarimetric light-scattering of fractal aggregates simulated byDiffusion-Limited Cluster-Aggregation (DLCA). The spectro-polarimetricscattered-light intensity curves were generated for coherent incident light with awide spectrum ranging from the ultraviolet (UV) to the Short Wave Infrared(SWIR). In the DDA model, the aggregate is discretized on a cubic lattice, witheach lattice being assigned an electric dipole that responds to the incident wave andcouples to the other dipoles in the aggregate.

The accuracy of the SDDA model is then determined by the fineness of thisdiscretization lattice (Yurkin and Hoekstra 2007). Applications often requirecomputations for aggregates in random orientations. Consequently, we split ourmodel into two operations: one is random orientation or orientational-averaging(Mishchenko and Yurkin 2017), and the other repeats the spectral calculations overthe SWIR–UV wavelength range. The averaging operation consists of repeatedcalculations for the aggregate fixed in a set of equally probable random orientationsfollowing (Hage et al. 1991). This results in a statistical ensemble-average of asingle aggregate morphology (Pathria 2003). However, rotating the aggregateposition relative to the dipole lattice can produce so-called shape errors, which areorientation dependent (Mishchenko et al. 2006). It is preferable then to fix the DDAlattice to the aggregate and rotate the incident wave propagation and polarizationdirections while being careful to preserve the transverse nature of the wave. Otherorientational-averaging schemes are possible such as analytical methods (Khlebstov2001) and numerical evaluation using the T-Matrix that is retrieved from the DDA(Mackowski 2002). The spectral operation consists of calculating thescattered-intensities for wavelengths in the spectral range. To account for spectraldispersion, we use the Sellmeier dispersion equation (Sellmeier 1871) to model, inthis case, the spectral properties of silica. This method could be widely extended toother aerosols such as soot particles.

The SDDA simulations are performed in the horizontal scattering plane, which isthe plane perpendicular to the linear polarization of the incident wave.Spectro-polarimetric scattered intensity curves are calculated with 1° angular reso-lution and averaged over the 200–2000 nm spectrum with 5 nm spectral resolution.

Results from the SDDA model are presented for a standard aggregate and asuperaggregate in both fixed and random orientations. The spectral and angularscattered-light intensities curves are plotted in Fig. 2.12 as a function of scatteringangle and wavelength. Referring to Fig. 2.12a, c, which are for a fixed orientation,the scattered-intensities display differences in the forward and backward directionsas the wavelength increases. In particular, there are distinct differences from 90° to180° between the two aggregates. Figure 2.12b, d for averaged orientation showssimilar features and can be directly compared to light scattering experimentalmeasurements (Sorensen et al. 1992). The V-shaped appearance of the plots near


90° is related to the first minimum after the Guinier crossover (Berg 2012). Inaddition, this feature persists after orientational averaging. Thus, this feature servesas a fingerprint, of sorts, independent of the aggregate’s orientation. It may be thatsuch fingerprints contain useful information about the aggregate morphology thatwould survive orientational averaging, and as such could find application inexperimental characterization of aggregating particle-systems.

In a nutshell, spectro-polarimetric light-scattering by particulate media appearsto present features related to distinctive microphysical parameters, such as:

1. Mean particle size. A spectral dependence of the depolarization on the meanparticle size is reported. For constant relative complex refractive index andvolume fraction, minimum DOLP values (i.e. maximum depolarization) appeararound µ = −0.5 and strongly varies with the wavelength and particles size. Onecan identify a limited spectral domain where the depolarization reaches amaximum value. This spectral domain is found variable from small particles to

Fig. 2.12 Spectral angular-light-scattering for the silica aggregates. The scattered intensity is lognormalized. Plots (a) and (c) show a fixed orientation for the standard and superaggregate,respectively. The corresponding orientational averaged plots are shown in (b) and (d)


large particles. Thus, this spectral feature would permit to identify the meanparticle size of an unknown particulate media.

2. Relative complex refractive index. A directional dependence of the depolariza-tion on the relative complex refractive index is reported. For constant particlesize and volume fraction, a maximum depolarization appears around µ = −0.5and strongly varies with directions. The shape of the limited spectral domainwith maximum depolarization is remarkably different when the relative complexrefractive index varies. As relative complex refractive index increases, directionsof maximum depolarization also increase. For instance, at µ = −1 (in the perfectbackscattering), DOLP ranges from 0.5 (i.e. partial depolarization) form = 1.59 + i 0.0 to 0.1 (i.e. near complete depolarization) for m = 2.05 + i 0.0.This directional feature is proposed as a tool to probe the relative complexrefractive index of particles composing an unknown particulate media.

3. Volume fraction. The limited spectral domain with maximum depolarizationshows neither spectral nor directional dependence on volume fraction. However,a dependence of the minimum DOLP value on the volume fraction is reported.For a given particle size and relative complex refractive index, the minimumDOLP value varies from 0.2 to 0.5 within an increase of the volume fraction.The minimum value of DOLP would allow retrieving the volume fraction ofparticulate media.

4. Porosity or structure. The aggregates or clusters structure formed by differentmonomers particles play an important role in the spectro-polarimetriclight-scattering. Distinct angular light-scattering features are displayed in thebackscattering domain (90°–180°). These features vary between the twoaggregates and depend on the wavelength.

2.7 Conclusion

Merging spectral and polarimetric light-scattering information is a growing interestin a wide range of scientific fields. Recently, it has been demonstrated how spectralpolarimetric light-scattering could be employed to probe microphysical propertiesof particulate media, for instance mean particle size or relative complex refractiveindex. This review highlighted the potential use of spectro-polarimetriclight-scattering. Fundamental advantages of merging spectral and polarimetricinformation for light-scattering are to address distinct scattering regimes, probesystems of particles at different scales, and extract useful information about aparticulate media of interest. The use of these spectral and polarimetric features foridentification of microphysical properties of an unknown particulate media is stillrecent. Future developments of robust inverse methods remain challenging. Thechallenge of simultaneous retrieval of microphysical properties of particulatemedia, without a priori knowledge or use of simplistic assumptions, has been moregenerally addressed throughout this review.


A comprehensive quantitative analysis of spectro-polarimetric light-scattering byparticulate media is presented. In order to perform this quantitative analysis, abroadband hyperspectral and polarimetric scatterometer was developed to measurespectro-polarimetric light-scattering by a large variety of particulate media. For thefirst time, an experimental set-up has been developed to measure spectro-polarimetriclight-scattering by combining a SC laser with a series of hyperspectral and polari-metric sensors. In parallel, different numerical models were developed to compute thespectro-polarimetric light-scattering signatures of particulate media. By solving thespectral VRTE, we have demonstrated that spectral depolarization, or spectral DOP,exhibits significant spectral and polarimetric features relative to a given particulatemedium. In a nutshell, the mean particle size has been found to relate to spectraldepolarization, the relative complex refractive index to directional depolarization, andthe volume fraction to the minimum DOLP value. By solving the DDA for abroadband polarized light, spectral and angular features were proposed to serve as afingerprint, of sorts, independent of the aggregate’s orientation. It may be that suchfingerprints contain useful information about the aggregate morphology that wouldsurvive orientational averaging. This quantitative analysis, based on experimental andtheoretical results, is a first stone for novel optical techniques based onspectro-polarimetric light-scattering to study particulate media.

Acknowledgements The authors would like to thank Matthew Berg for helpful commentsregarding radiative transfer equation and for many debates in the field of light-scattering. We arealso grateful to Michael I. Mishchenko, Gordon Videen, Ludmilla Kolokolova, and Alex J. Yuffafor fruitful discussions regarding spectral polarimetric light-scattering.

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Chapter 3Light Scattering by Large Bubbles

Fabrice R. A. Onofri and Matthias P. L. Sentis

3.1 Introduction

In the common acceptation the term bubble refers to gas particles in liquids, theterm particle being used here as a general term for a dispersed phase. In the two-and multiphase flow communities, a bubble is defined as a particle with densitylower than that of the surrounding medium (Clift et al. 1978). Thus, a bubble can bea gas particle in gas, a liquid, or a solid, or a liquid particle in a liquid or a solid. Thelightness of these particles is what characterizes them the most, explaining why, inspite of the presence of an additional thin soap film, e.g., soap bubbles (Salkin et al.2016) that are so familiar to children or helium-filled soap bubbles used to char-acterize the dynamics of large scale structures (Bosbach et al. 2009), may be alsoconsidered as bubbles. However, in what follows, to avoid any confusion, the latterwill be referred to as “coated bubbles” (they are also called “dirty bubbles” in somefields, e.g., Jonasz and Fournier 2007b).

Gas bubbles are commonly observed in nature. In our day-to-day life, air bub-bles in water are responsible for the white foam observed at the foot of a waterfallor at the surface of a raging sea (e.g., Woolf 2001; Trefil 1984; Minnaert 1980;Latham 2017). In oceans, they are often coated with an organic film, and they areknown to have a non-negligible influence on the climate balance (via carbondioxide and oxygen exchange with the atmosphere, the enhancement of oceansradiation backscattering, e.g., Zhang et al. 1998; Dave 1969; Johnson and Cooke

F. R. A. Onofri (&)IUSTI (UMR 7343, Aix-Marseille Université), National Center for Scientific Research(CNRS), 5 rue Enrico Fermi, 13453 Marseille cedex 13, Francee-mail: [email protected]

M. P. L. SentisDEN/DMRC/SA2I/LGCI, Atomic Energy and Alternative Energies Commission (CEA),30207 Bagnols-sur-Cèze, Francee-mail: [email protected]


109

1981) and on the reliability of ocean color remote sensing techniques (Gordon andBoynton 1998; Yan et al. 2002). In industry, air bubbles may be considered asextremely damaging, in the light of the ravages engendered by cavitation of shippropellers (Brennen 1995), or as beneficial when, for instance, employed in directcontact heat exchangers or to help mixing the ingredients at high temperature inlarge scale glass furnaces (Shelby 2005; Onofri et al. 2007b).

Gas bubbles in solids are responsible for the white color of icebergs (or, in asense, their blue color when absent) or the lightness of pumice stones (Woolf 2010;Bohren and Clothiaux 2008). These bubbles are frequently considered as a defect,although they can also be generated intentionally, with a laser for instance, toproduce wonderful 3D etched glass sculptures (also called bubblegrams).

In nature, liquid-liquid bubbles come as vesicles (Kita-Tokarczyk et al. 2005),emulsions (Tadros 2013), or coated bubbles (Liping et al. 2006), while in industrythey are at the core of a tremendous number of processes (i.e. for water treatment,nuclear fuel recycling, etc.) (Silvestre et al. 2009; Lamadie et al. 2012; Reinhardet al. 2005).

When observed in flows, one major difference between bubbles and droplets isthat the former can undergo larger morphological transitions or shape transforma-tions. Under certain circumstances, bubble sizes can also reach a few tens ofcentimeters, as scuba divers are well aware, or as it is commonly observed in bubblecolumns operated in a slug flow regime (Clift et al. 1978). According to Grace(1973), see also (Clift et al. 1978), for immiscible fluids (e.g. gas-gas bubbles areexcluded), it is possible to prepare a generalized graphical correlation of the bubbleshapes in terms of the Eotvos number, Eo ¼ gD2 ql � qg

� �r�1; Morton number,

Mo ¼ ql � qg� �

gl4l r�3q�2

l ; Reynolds number Re ¼ qlVTDð Þl�1l ; and Weber

number, We ¼ qlV2TDr

�1. In the latter equations, ql, ll are respectively the sur-rounding fluid density and dynamic viscosity; r the surface tension; D, VT, and qgthe diameter, the terminal velocity, and the density of the bubble, respectively,whereas g stands for the gravitational constant; see Fig. 3.1. Although in thisfamous diagram the boundaries between the principal shape regimes are somewhatarbitrary, it tells us that bubbles (it works also for drops) are spherical at lowReynolds and Eotvos numbers, ellipsoidal for intermediate Reynolds and Eotvosnumbers, and so on. Bubble shape is influenced by other factors, like the degree offluid contamination by surfactants or the wakes produced by the other surroundingbubbles. All this, plus bubbles’ coalescence and break-up phenomena, make themodeling of bubbly flows and the related Computationally Fluid Dynamics(CFD) calculations particularly challenging (e.g., van Sint Annaland et al. 2005;Bunner and Tryggvason 1999; Prosperetti and Tryggvason 2007). The same holdstrue for the experimental characterization of these fragile objects.

Whether for a better understanding of the role bubbles play in light and colorobserved in nature, or their importance in industrial applications, the characteri-zation of their shape, size, composition, and number density clearly matter. In thisview, the modeling of their light scattering properties and the development of

110 F. R. A. Onofri and M. P. L. Sentis

reliable optical particle characterization techniques is of scientific and technologicalinterest. In optics, the term bubble may also be employed for particles with arefractive index that is smaller than the one of the surrounding medium, i.e. havinga relative refractive index below unity. In most cases, this definition matches withthe one used in fluid mechanics, since, in the visible range, the bubbles’ materialdensity and real part of the refractive index are closely related. To the best of ourknowledge, this is always the case for gas bubbles in liquids or solids, but this is notnecessarily the case for liquid-liquid bubbles, especially when water is involvedwith an organic liquid (i.e. the refractive index of water is lower than most organicliquids, while its density is higher). Going back to the bubble shape diagrampresented in Fig. 3.1 [or to more recent numerical and experimental works, e.g., vanSint Annaland et al. 2005; Celata et al. 2007)], it is rather clear that the modeling ofthe light scattering properties of bubbles is also a challenging task since it has todeal with complex shaped scatterers with sizes ranging from the nanoscale, e.g.ultrafine bubbles (Ushikubo et al. 2010), to the centimetric scale.

This review intends to highlight the different methods developed so far in theliterature to predict the light scattering properties of bubbles. It is restricted to largebubbles and dilute media. The underlying problematic is essentially linked to the

Fig. 3.1 Grace’s diagram showing shape regimes for bubbles and drops, reproduced after Grace(1973). Triangular symbols stand for data recorded for the experimental investigation of thenear-critical-angle scattering (Onofri et al. 2009, 2011)

3 Light Scattering by Large Bubbles 111

characterization of bubbly flows or, more generally, multiphase flows, where mostoptical diagnostics are based on the analysis of the scattering diagrams withinterferometric or diffractometric techniques (Tatsuya et al. 2002; Vetrano et al.2004; Celata et al. 2007; Onofri et al. 2009, 2011; Tian et al. 2010; Ushikubo et al.2010). After this introduction, Sect. 3.2 reviews the rigorous electromagnetic modelavailable to predict their light scattering properties. It is shown that the Lorenz-MieTheory (LMT) and its generalizations remain essential to predict the scatteringproperties of large bubbles, while essentially limited to spherical bubbles. For morecomplex shaped bubbles, there exist other electromagnetic methods. However,since their capability in terms of maximum bubble size is far below what is neededfor multiphase flows investigations, they are only briefly evoked. In the same way,and because small bubbles are essentially spherical, all approximations that areessentially limited to small and soft particles (e.g. Rayleigh, Rayleigh-Gans-Debye,Perelman, etc.) are not addressed. The reader interested in such approximations canfind many reference books on this topic (e.g., Sharma and Somerford 2006; Bohrenand Huffman 1998). Section 3.3 is devoted to approximations for large bubbles.The fact is that for large bubbles, there exist only a few Physical OpticsApproximations (POAs) and Geometrical Optics Approximations (GOAs). Forlarge bubbles, nearly all POAs or semi-classical approximations are focused onparticular wave effects. In this category, we find the POA of the critical-scatteringintroduced by Marston and co-workers (Marston 1979, 1992, 1999, 2015; Marstonand Kingsbury 1981; Kingsbury and Marston 1981; Marston et al. 1982; Arnott andMarston 1988), denoted by M-POA in what follows, and a zero order approxi-mation derived from the Complex-Angular Momentum theory(CAM) (Fiedler-Ferrari et al. 1991; Fiedler-Ferrari 1983; Nussenzveig 1992).Finally, the few attempts to develop a predictive GOA and to couple it with POAsin a unified model are reviewed. Their aim is to describe, in an accurate andcomputational effective way, all main and subtle features (e.g. specular reflection,tunneling and Goos–Hänchen effects, etc.) that come into play in the scattering oflarge spherical (Marston and Kingsbury 1981; Kingsbury and Marston 1981;Marston et al. 1982; Langley and Marston 1984; Onofri 1999; Lock 2003; Yu et al.2008; Onofri et al. 2009; Sentis et al. 2016) and spheroidal bubbles (He et al. 2012;Onofri et al. 2012). Section 3.4 is an overall conclusion.

3.2 Rigorous Approaches

3.2.1 Lorenz-Mie Theory and Its Generalizations

The electromagnetic-light-scattering community recently celebrated (e.g., Hergertand Wriedt 2012) the one hundredth anniversary of the publication by Mie (1908)of what is now referred to as the Mie theory (that we prefer to call Lorenz-MieTheory, LMT). The latter solves in an exact manner, with a separation variable


method, the problem of absorption and scattering by a spherical, homogeneous,isotropic, and non-magnetic particle (sometime called a “Mie scatterer”) whenilluminated by an incident electromagnetic plane wave. Although the LMT back-ground and details are extensively discussed in many reference books (Kerker1969; Stratton 1941; Barber and Hill 1990; Bohren and Huffman 1998), in the nextsection we summarize the basic steps and results of this theory, which remainsessential (when it is not only available) for the characterization of bubbles byoptical means.

3.2.1.1 Spherical Homogenous Bubble with a Plane Wave Illumination

We consider a spherical bubble of radius a centered in the laboratory frame Oxyzð Þ.It is illuminated by an incident (subscript inc) linearly polarized and harmonic planewave with wavelength in air equal to k0. The incident light propagates along the yaxis with a perpendicular (subscript v ¼ 1) or parallel (subscript v ¼ 2) polariza-tion. For this particular wavelength, the surrounding medium and bubble mediumrefractive indices are equal to m1 (real) and m2, respectively, with m ¼ m2=m1\1denoting the bubble relative refractive index; see Fig. 3.2. The size parameter ofthis scatterer is equal to a ¼ 2 p am1=k0.

To be a solution of Maxwell’s equations, an electromagnetic wave (with elec-trical E and magnetic H fields) propagating in an isotropic, non-magnetic, andnon-electrically charged medium, has to satisfy the following equations (Bohrenand Huffman 1998):

r2Eþ k2E ¼ 0r2Hþ k2H ¼ 0

�ð3:1Þ

Fig. 3.2 a Coordinate system of the scattering problem and b LMT calculation of the near-fieldpower inside and outside an air bubble in water (a = 50 µm, m = 0.75, parallel pol.,k0 = 0.6328 µm)


where k is the wave vector with k2 ¼ x2el, and with x; e; l being respectively thepulsation of the incident wave, the permittivity, and permeability of the medium.The aforementioned problem can be reduced to the solving of the scalar waveequation:

r2wþ k2w ¼ 0 ð3:2Þ

where w is a function connected to the spherical harmonics by M ¼ r� ðrwÞ andN ¼ r�Mð Þ=k. In the spherical coordinate system r; h;/ð Þ, the scalar waveequation reads as

1r2

@

@rr2@w@r

� �þ 1

r2 sin h@

@hsin h

@w@h

� �þ 1

r2 sin h@2w

@/2 þ kw2 ¼ 0 ð3:3Þ

In the LMT a Separation Variable Method (SVM) is used to solve the previousequation, with

w r; h;/ð Þ ¼ RðrÞHðhÞUð/Þ ð3:4Þ

This procedure allows obtaining three equations (where here m and n are sep-aration and expansion constants):

d2U

d/2 þm2U ¼ 0 að Þ1

sin hddh sin h dHdh

� �þ n nþ 1ð Þ � m2

sin2 h

h iH ¼ 0 bð Þ

ddr r2 dRdr

� �þ k2r2 � n nþ 1ð Þ

R ¼ 0 cð Þ

8>>><>>>:

ð3:5Þ

The solutions of Eq. (3.5a) are of the following type:

Ue ¼ cos m/ð ÞUo ¼ sin m/ð Þ

�ð3:6Þ

The solutions of Eq. (3.5b) are the associated Legendre’s functions of the firstkind of degree n and order m, Pmn cos hð Þ. The solutions of Eq. (3.5c) are obtainedby introducing the change of variable q ¼ kr and by introducing the functionZ ¼ R

ffiffiffiq

p, Eq. (3.5c) can then be written as

qddq

qdZdq

� �þ q2 � nþ 1

2

� �2" #

Z ¼ 0 ð3:7Þ

We are looking for linearly independent solutions that are combinations ofspherical Bessel’s functions jn; yn; k

ð1Þn ; kð2Þn (Abramowitz and Stegun 1965; Chang

et al. 1996). Thus, the solutions of Eq. (3.4) are of the following form:


wemn ¼ cosðm/ÞPmn cos hð Þzn qð Þwomn ¼ sinðm/ÞPmn cos hð Þzn qð Þ ð3:8Þ

The different spherical Bessel functions jn; yn; kð1Þn ; kð2Þn are not defined in all

points of space. For instance, we have yn ! 1 when r ! 0; thus the later functioncannot be used to describe the internal (subscript int) electrical and magnetic fields.Conversely, it can be used to describe the scattered fields with the right asymptoticbehavior as yn ! 1=r when r ! 1.

Similar considerations allow obtaining the following formula for the internal andthe scattered (subscript sca) fields (Bohren and Huffman 1998):

Eint ¼P1n¼1

En cnMð1Þo 1n � jdnN

ð1Þe 1n

� �Hint ¼ �k1

xl1

P1n¼1

En dnMð1Þe 1n � jcnN

ð1Þo 1n

� �8>><>>:

Esca ¼P1n¼1

En janNð3Þe 1n � bnM

ð3Þo 1n

� �Hsca ¼ k2

xl2

P1n¼1

En jbnNð3Þo 1n � anM

ð3Þe 1n

� �8>><>>:

ð3:9Þ

with En ¼ jnE0 2nþ 1ð Þ=n nþ 1ð Þ, where j2 ¼ �1 stands for the imaginary unit andE0 the amplitude of the incident electric field. In Eq. (3.9) the electric and magneticfields are described by a linear combination of an infinite number of sphericalharmonics with complex coefficients. The coefficients an; bn are named the “ex-ternal scattering coefficients” and cn; dn the “internal scattering coefficients”.

The tangential components of the electromagnetic fields have to fulfill boundaryconditions onto the particle surface r ¼ að Þ for the local outgoing normal N :

Eint að Þ � Einc að Þ½ � � N að Þ ¼ 0;Hint að Þ �Hinc að Þ½ � � N að Þ ¼ 0:

�ð3:10Þ

From Eqs. (3.9) and (3.10) we get the following relations for the externalscattering coefficients (recalling that m is the particle relative refractive index):

an ¼ mwn mað Þw0n að Þ � wn að Þw0

n mað Þmwn mað Þn0n að Þ � nn að Þw0

n mað Þ ;

bn ¼ wn mað Þw0n að Þ �mwn að Þw0

n mað Þwn mað Þn0n að Þ �mnn að Þw0

n mað Þ ;ð3:11Þ

where the Ricatti-Bessel and spherical Hankel functions are related by wn qð Þ ¼qjn að Þ; nn að Þ ¼ qhð1Þn að Þ (Abramowitz and Stegun 1965). Expressions for theinternal scattering coefficients may be found in (Bohren and Huffman 1998). Tocompute efficiently the an; bn coefficients, it is necessary to introduce the loga-rithmic derivatives of the Riccati-Bessel functions:


an ¼ Dð3Þn

mDð1Þn að Þ � Dð1Þ

n mað ÞmDð2Þ

n að Þ � Dð1Þn mað Þ ; bn ¼ Dð3Þ

nDð1Þ

n að Þ �mDð1Þn mað Þ

Dð2Þn að Þ �mDð1Þ

n mað Þ ; ð3:12Þ

where

Dð1Þn zð Þ ¼ w0

n Zð Þwn Zð Þ ; Dð2Þ

n zð Þ ¼ n0n Zð Þnn Zð Þ ; Dð3Þ

n zð Þ ¼ wn Zð Þnn Zð Þ : ð3:13Þ

where here Z is denoting a real or complex variable, Z � a or Z � ma:For spherical particles and in the far-field, the scattered field can be expressed as

a function of the amplitude of the two linear polarization components of the inci-dent wave and two amplitude functions Sv :

Esca;2

Esca;1

� �¼ eikðr�yÞ

�ikRS2 00 S1

� �Einc;2

Einc;1

� �ð3:14Þ

with

S1 ¼X1n¼1

ð2nþ 1Þnðnþ 1Þ anpn þ bnsnð Þ

S2 ¼X1n¼1

ð2nþ 1Þnðnþ 1Þ ansn þ bnpnð Þ

ð3:15Þ

pn ¼ P1nsin h

; sn ¼ dP1nsin h

ð3:16Þ

From the previous equations and using the Poynting vector (e.g., Bohren andHuffman 1998), it is easy to derive the relations for the time averaged scatteringintensities Iv with I1 / S1j j2 and I2 / S2j j2. With the subscripts abs and ext forabsorption and extinction, respectively, we get the following for the cross sectionsof the particles:

Csca ¼ 2p

k2X1n¼1

ð2nþ 1Þ anj j2 þ bnj j2� �

Cext ¼ 2p

k2X1n¼1

ð2nþ 1ÞRe an þ bnf g

Cabs ¼ Cext � Csca

ð3:17Þ

In practice, due to the external scattering coefficients, the infinite expansionseries in Eqs. (3.15) and (3.17) are truncated when the expansion term exceed acertain value, i.e. nstop � aþ 4a1=3 þ 2 (e.g. Wiscombe 1980; Barber and Hill1990; Bohren and Huffman 1998). According to the localized principle (van de


Hulst 1957; Gouesbet and Lock 1994), this truncation is operated at a distancenstop þ 1=2� �

k=2pð Þ from the particle’s center. Although the LMT applies to allparticle sizes and refractive indices, due to some numerical difficulties in the cal-culations of special functions (e.g., Abramowitz and Stegun 1965; Chang et al.1996), caution must be paid on the predictions of most LMT codes available (e.g.,Wriedt 2017) when the particles are large (as order of magnitudes, one millimeterfor spheres and a few tens of micrometers for other shapes).

The Debye theory (Debye 1909) sheds more light on the physical meaning of theinfinite series used by LMT to describe the internal or scattered fields [i.e.Eqs. (3.15), (3.17)] by rearranging them to highlight the contribution of wavespartially reflected and partially transmitted by the particle. This leads to the intro-duction, like with GOA, of reflection and transmission coefficients for these partialwaves. Such an approach has brought considerable insights into the scattering bylarge refracting particles (essentially spherical and ellipsoid droplets, cylinders, etc.,e.g. Hovenac and Lock 1992; Lock and Adler 1997; Xu et al. 2010), but very littlefor bubbles (e.g., Wu et al. 2007). In fact, and although it is not yet fully clearwhether it is for profound physical or numerical reasons, it seems that codes usingDebye series still fail to predict the near-critical-angle scattering.

To illustrate and conclude on this part, and since LMT is extensively used in therest of this review for comparison purposes, we end this section with Fig. 3.2b. Thelatter shows the calculation of the near-field power inside and outside an air bubblein water with relative refractive index m ¼ 0:75 and radius a ¼ 50 lm, for a parallelpolarized plane wave with wavelength in air of k0 ¼ 0:6328 lm:

3.2.1.2 Other Bubble and Incident Beam Shapes

Over the years, the LMT, as a SVM, has been generalized to account for thescattering of a plane wave by spherically coated, multilayered, and chiral scat-terers (e.g., Kerker 1969; Bhandari 1985; Wu and Wang 1991; Lock 2008),spherical particles with inclusions (Borghese et al. 1994), spheroids (Asano andYamamoto 1975; Cooray and Ciric 1993; Farafonov et al. 1996), homogeneous ormultilayered right-angle cylinders (e.g., Cooke and Kerker 1969; Onofri et al.2004), etc. In the same way, LMT has been generalized to account for the scat-tering of spherical or more complex shaped particles when illumined by complexshaped incident beams, like continuous or pulsed circular and elliptical Gaussianbeams, Bessel beams, etc. (Gouesbet et al. 1988; Onofri et al. 1995; Barton 1997;Ren et al. 1997; Mees et al. 2001; Mitri 2011). As an example, in the frameworkof the Generalized Lorenz-Mie Theory (GLMT, see Gouesbet and Gréhan 2011),the amplitude functions S1 and S2 of a multilayered sphere (the coated sphere beinga particular case) illuminated by an arbitrary beam take the following form, seeOnofri et al. 1995, 1996):


S1 ¼X1n¼1

Xm¼n

m¼�n

2nþ 1n nþ 1ð Þ mAm

n pmj jn cos hð Þþ jBm

n smj jn cos hð Þ

exp jm/½ �

S2 ¼X1n¼1

Xm¼n

m¼�n

2nþ 1n nþ 1ð Þ Am

n smj jn cos hð Þþ jmBm

n pmj jn cos hð Þ

exp jm/½ �ð3:18Þ

where p mj jn cos hð Þ and s mj j

n cos hð Þ stand for the generalized Legendre functions. Thegeneralized scattering coefficients Am

n ¼ gmn;TMAn and Bmn ¼ gmn;TEBn are the product

of the external scattering coefficients of a multilayered sphere An and Bn, as derivedfor a plane wave illumination (Bhandari 1985; Wu and Wang 1991) by the beamshape coefficients gmn;TM and gmn;TE (Gouesbet and Gréhan 2011), describing allproperties of the incident beam. TM and TE refer to the beam Transverse Magneticand Transverse Electrical modes, respectively. To illustrate the importance of beamshape effects, Fig. 3.3 shows the evolution of the scattering diagram of anair-bubble in water as it moves transversely (along the x-axis, with y = z = 0)within a Gaussian beam with a beam waist radius x0 much smaller that the bubbleradius, x0=a � 0:16 (Krzysiek 2009). These calculations show clearly that theincident beam can alternatively amplify or dump different scattering mechanisms.As an example, for x � þ 200 lm; the critical-scattering is significantly amplified.This has huge consequences for optical particle characterizing techniques, since the

Fig. 3.3 GLMT calculations: scattering diagrams of an air bubble in water illuminated by aparallel polarized and focused Gaussian beam (2x0 = 75 µm, k0 = 0.532 µm; a = 238 µm, sizeparameter � 3747)


measurements may become strongly dependent on the bubble trajectories (Gréhanet al. 1996; Onofri et al. 2011).

3.2.2 Scattering by an Optically Thin Bubble Cloud

For the optical characterization of bubbly flows, clouds of bubbles are usuallyassumed to be randomly distributed in space and in low concentration (e.g., Tatsuyaet al. 2002; Onofri et al. 2007a, 2009; Dehaeck et al. 2005; Jonasz and Fournier2007c; Tian et al. 2010; Sentis et al. 2017). In such conditions, the ensemblescattering in the far-field of all illuminated bubbles is approximated by

�Isca / CN

Zamax

amin

Isca h; a;m; k0ð Þfn að Þda; ð3:19Þ

where CN is the bubble number concentration and fn að Þ stands for the normalizedBubble Size Distribution (BSD) in number. The term “low concentration” isemployed here to stress the fact that Eq. (3.19) does not apply to situations wheremultiple scattering cannot be neglected (while the multiple scattering regimedepends upon many other parameters that will be not discussed in the present paper,e.g., Mishchenko et al. 2006; Kokhanovsky 2006; Sentis et al. 2015). The term“spatially randomly distributed” is used here to stress on the fact that some coherentscattering are neglected, as well as bubble trajectory effects (provided that thediameter of the bubbles remains much smaller than the beam waist of the illumi-nation beam, e.g., Onofri et al. 2011).

As an illustration of the ensemble scattering of cloud of bubbles, Fig. 3.4a–ccompare the intensity profiles calculated with Eq. (3.19) and the LMT, in the nearcritical-angle region, for various log-normal BSD of spherical bubbles: (a) clouds ofair bubbles in water m ¼ 0:75 with different mean diameters �D ¼ 25� 800 lm buta constant BSD relative width: rD=�D ¼ 0:05; (b) like in (a) but for different BSDwidths rD=�D ¼ 0:05� 0:5; a constant mean �D ¼ 100 lm, and compositionm ¼ 0:75; (c) clouds of different compositions m ¼ 0:66� 0:75 but with the sameBSD: �D ¼ 100 lm and rD=�D ¼ 0:25. Classical effects, like the strong dependenceof the scattered intensity on the bubble size and the dumping of the high frequencyripple structures for increasing BSD relative widths are easily observed; seeFig. 3.4. More interesting, it is also found that (a) the mean diameter controls theangular spreading of the near-critical angle scattering patterns; (b) the BSD widthacts mainly on their visibility (contrast); and (c) the key influence of the relativerefractive index is essentially in their global angular position. The latter trends are atthe basis of the development of the critical-angle refractometry and sizing techniqueused to characterize single bubbles (Langley and Marston 1984; Onofri 1999) orclouds of bubbles (Onofri et al. 2007a, 2009, 2011).


3.2.3 Complex Shaped but Small to Moderate Sized Bubbles

There exist several rigorous electromagnetic approaches allowing for the handlingof scatterers with a complex shape and/or internal structure. Among the mostpopular, one can cite the Discrete Dipole and MultiLevel Fast MultipoleApproximations (DDA, MLFMA), the Finite-Difference Time-Domain (FDTD), orthe null-field (e.g. T-Matrix) methods (e.g., Draine and Flatau 1994; Mishchenkoand Travis 1998; Lin and Wang 2005; Wriedt 2007; Yurkin and Kahnert 2013;Zhang et al. 2015; Yang et al. 2015). Unfortunately, all these methods can hardlymanage scatterers with size parameters exceeding 10–600. This limit is acceptablefor atmospheric or aerosols physics (e.g., Bohren and Clothiaux 2008), where a lotof particles of interest remain in the nano- to micro-scale range. But this upper limit

Fig. 3.4 Scattering diagrams in the critical-angle region of bubble clouds with a constantcomposition and BSD width but different mean diameters; b constant composition and meandiameter but different BSD widths; c constant BSD and different compositions. d Comparison ofLMT, CAM, and POA (0, 0 + 1) predictions for the evolution of the visibility V, the globalangular position h1, and angular spreading h2 � h1 of the near-critical-angle scattering patternsversus the bubble clouds’ mean diameter. Offset and magnification factors are used for drawingconsiderations in (a) and (d)


is totally prohibitive for bubbly flows where, as pointed out in the introduction, thebubbles’ non-sphericity becomes only a real problem when their size is approachingthe millimetric scale, e.g. a � 4000 for a free rising air bubble in water (e.g., Celataet al. 2007, Onofri et al. 2011). However, in some particular situations that are outof the scope of this review, like high shear-stress flows, open-cell foams, or surfacevicinity, interest in the previously mentioned methods increases sharply (e.g., Veraet al. 2001; Eremina et al. 2006; Bunkin et al. 2009).

3.3 Approximations for Large Bubbles

3.3.1 Spherical Bubbles

3.3.1.1 Geometrical Optics Approximation

GOA is certainly the most appealing approach for large particles because, as ahigh-frequency approximation, its accuracy increases asymptotically with the par-ticle size parameter. It is computationally efficient and is expected to be straight-forward to apply to complex shaped particles, and thus large bubbles. Conversely,the validity of this approach is restricted to particles with large size parameters andwith smooth variations in their properties (i.e. shape, in the reflection and refractioncoefficients and their derivatives, etc.) with respect to the wavelength scale. Asmentioned previously, the formation of the GOA can take various forms (e.g. vande Hulst 1957; Macke and Mishchenko 1996; Muinonen et al. 1996; Sharma andSomerford 2006; Stavroudis 2006; Xu et al. 2006b; Jonasz and Fournier 2007a; Yuet al. 2008, 2013; Yang and Liou 2009). In the same way a GOA can take intoaccount different contributions, including some physical optics contributions (likephase terms and Fresnel coefficients), although, from a rigorous point of view,GOA should only refer to scalar rays of constant intensity (van de Hulst 1957;Nussenzveig 1992).

It seems that Davis (1955) was the first to publish a detailed work on a GOA forthe scattering of large spherical bubbles. For this, he used a trigonometric formalismthat is rather easy to implement in the case of spherical particles (e.g. van de Hulst1957; Yu et al. 2008; Sentis et al. 2016) but difficult to extend to spheroids (e.g.,Lock 1996a; Macke and Mishchenko 1996; He et al. 2012) and, in our opinion,impossible to extend to arbitrary shaped particles without introducing drasticsimplifications. The fact is that, for complex shaped particles, the vectorial for-malism is thought to be more appropriate (e.g. Muinonen et al. 1996; Macke andMishchenko 1996; Ren et al. 2011). However, in the present section, we prefer tosummarize, discuss, and extend (Sentis et al. 2016) the GOA model introduced byvan de Hulst (1957) since it takes into account, in a comprehensive and elegantmanner, the focal line and focal points that have a crucial influence on the finestructure of the scattering diagrams of large bubbles.


In van de Hulst’s GOA (van de Hulst 1957), which is expected to be valid forboth bubbles m\1ð Þ and refracting particles m[ 1ð Þ, the incident plane wave isdecomposed into rays characterized by a scattering order p ¼ 0; 1; � � � ; þ1 and bytwo incident angles, i1 and i2. The corresponding complementary angles s1 and s2are related by the Snell-Descartes law:

cosðs1Þ ¼ m cosðs2Þ: ð3:20Þ

The ray of order p emerges (after p� 1 internal reflections) in the directiondefined by the scattering angle hp with

hp ¼ 2s1 � 2ps2 for s1 2 �p=2; p=2½ �: ð3:21Þ

Because it is more convenient to address rays emerging exclusively in the rangehp 2 0½ ; p�; one can take advantage of the symmetry of the problem to completeEq. (3.21) as follows:

hp ¼ 2jpþ q 2s1 � 2ps2½ � for s1 2 0; p=2½ �; ð3:22Þ

where j is an integer allowing compensation for the number of rotations (which canbe important for higher-order rays) and q ¼ �1 allows restricting the scatteringdomain to hp 2 ½0; p� (Sentis et al. 2016). Total external reflection occurs fors1 � sc ¼ p=2� sin�1 mð Þ. In that case the critical ray p ¼ 0 emerges with thenear-critical-angle scattering angle hc ¼ 2 cos�1 mð Þ:

The amplitude for rays of order p is calculated, in the planar assumption limit(Nussenzveig 1992), using the Fresnel reflection coefficients rv :

r1 ¼ sinðs1Þ �m sinðs2Þsinðs1Þþm sinðs2Þ ; r2 ¼

m sinðs1Þ � sinðs2Þm sinðs1Þþ sinðs2Þ : ð3:23Þ

Using Eqs. (3.23), van de Hulst (1957) gives the amplitude of the field asso-ciated with each ray:

ep;v ¼ rv; for p ¼ 0;

ep;v ¼ 1� r2vh i

�rv p�1

; for p 1:ð3:24Þ

Although a more general and flexible approach was proposed recently to accountfor the local curvature of complex shaped particles or the incident wavefront (seenext section), the divergence factor (noted here Hp) that was introduced by van deHulst is perfectly suitable for spherical particles:

Hp ¼ cos s1ð Þ sin s1ð Þsin hð Þ dhp=ds1

�� : ð3:25Þ


The global phase delay rp;v of the ray of order p is the sum of three phase terms:

rp;v ¼ wp;v þ np þup; ð3:26Þ

where the phase wp;v corresponds to the phase of the complex amplitudes intro-duced in Eq. (3.24) and the phase np is associated with the optical path differencerelative to a reference ray passing through the sphere center and emerging in thesame direction as the ray of interest:

np ¼4 p ak0

m1 sin s1ð Þ � pm2 sin s2ð Þð Þ: ð3:27Þ

In Eq. (3.26) the phase up accounts for the passage of the ray of interest throughfocal points and focal lines. The latter are connected to the astigmatism (e.g.,Deschamps 1972; Nye 1999; Ren et al. 2011; Sentis et al. 2017) of the localwavefront associated to ray p; with

up ¼p2

p� 2jþ 12s� 1

2q

� �; ð3:28Þ

where in Eq. (3.28) the term s ¼ �1 depends on the sign of dhp=ds1 and can bedetermined as follows:

s ¼ 2� 2p tan s1ð Þ= tan s2ð Þð Þ2� 2p tan s1ð Þ= tan s2ð Þð Þj j : ð3:29Þ

As a result, in the GOA introduced by van de Hulst (1957), the complexamplitude of a scattered ray is expressed as

SGOAp;v hð Þ ¼ SGOAp;v

�� ejrp;v ¼ affiffiffiffiffiffiHp

pep;v�� ejrp;v : ð3:30Þ

3.3.1.2 Coupling Geometrical and Physical Optics Approximations

Forward Diffraction

A common approach (e.g., Keller 1962; Xu 2001; Xu et al. 2006b; Yu et al. 2008)to account for the forward diffraction is to sum the contributions of the GOA withthose of Fraunhofer’s Physical Optics Approximation (denoted by F-POA).Because only rays with p ¼ 0 are concerned, the contribution in terms of intensity

of the F-POA can be written as SF�POA0 ðhÞ�� 2¼ a4 J1 a sin hð Þ=a sin hð Þ½ �2, where J1

stands for the Bessel function of the first kind. If we add incoherently the GOA andF-POA contributions, we get for the total intensity scattered in the direction h(Sentis et al. 2016):


IGPOA�1v ðhÞ ¼

P1p¼0

SGOAp;v hð Þ��

��2

þ SF�POA0 ðhÞ�� 2; if h 2 0; hF�POA

max

;

P1p¼0


��2

; if h 2 hF�POAmax ; p

:

8>>>>><>>>>>:

ð3:31Þ

This model is referenced further on as the Geometrical and Physical OpticsApproximation model n°1 (GPOA-1). In Eq. (3.31), the GOA contribution ispolarization-dependent, in contrast to the F-POA term derived from scalardiffraction theory. In the same way, the contribution of forward diffraction is addedincoherently, although there is some attempt to couple coherently this contributionto the GOA (Yuan et al. 2016). hF�POA

max represents an upper boundary for theapplication of the F-POA. In the literature, hF�POA

max ¼ p=2 is generally used (e.g.,Keller 1962; Xu et al. 2006b; Yu et al. 2008) under the justification that diffractedlight does not propagate in the backward direction. This sounds reasonable, butrather qualitative. Indeed, from a numerical study based on Debye theory, a limitsuch as hF�POA

max ¼ p=4 appears to be more reasonable for large bubbles (at least forall cases considered here). Figure 3.5a compares LMT results with those obtainedwith Eq. (3.31) for an air bubble in water with a = 50 µm, m = 0.75,k0 ¼ 0:6328 lm, v ¼ 2, and p ¼ 0; 1; � � � ; 10: We find that the agreement is nearlyperfect over the entire angular range except in the forward diffraction h � 0ð Þ, andin the critical-angle region h � 80ð Þ. In the critical-angle region the coarsestructures (the bright and dark fringes localized at h1; h2; h3. . .) are out of phasecompared to LMT predictions, with also a plateau at the vicinity of hc. The latterdiscrepancies are not surprising since the GOA cannot manage the discontinuity inthe medium properties at the bubble interface (which generates the forwarddiffraction) and the discontinuity of the derivative of the reflection coefficient at thecritical angle.

Near Critical-Angle Region and the Tunneling Phase

The shift of the coarse structures observed in the near-critical-angle scatteringregion comes from an improper definition of the Fresnel coefficients in this zone.For bubbles, refraction no longer occurs for rays satisfying s1\sc: Using thedefinition of sc with 1� cos2 s1ð Þm2 � 0, it can be shown (Kerker 1969; Sentis et al.2016) that

sinðs2Þ ¼ �j cos2ðs1Þ=m2 � 1� �1=2

: ð3:32Þ

Equation (3.32) has two solutions. Only the negative one is physical, i.e.

sinðs2Þ ¼ �j cos2 s1ð Þ=m2 � 1ð Þ1=2. This solution is related to the existence of an


Fig.3.5

Com

parisonof

LMTresults

with

aGPO

A-1

(a=50

µm)andbGPO

A-2

(a=30–50

0µm),with

forbo

thcasesm

=0.75

,k0=0.63

28µm,v

=2,

p=1–10


evanescent wave below the bubble surface, see Figs. 3.2b and 3.6a, with penetra-tion depth:

K ¼ kj j�1 cos2 s1ð Þ=m2 � 1� ��1=2

; ð3:33Þ

where k is the wave vector inside the bubble, kj j ¼ 2 pm2=k0. This phenomenonenables better understanding of the sharp internal field gradient observed inFig. 3.2b near the critical point.

Given the above, the Fresnel reflection coefficients for s� sc take the followingform:

r1 ¼ sin s1ð Þþ j cos2 s1ð Þ �m2ð Þ1=2sin s1ð Þ � j cos2 s1ð Þ �m2ð Þ1=2

;

r2 ¼ m2 sin s1ð Þþ j cos2 s1ð Þ �m2ð Þ1=2m2 sin s1ð Þ � j cos2 s1ð Þ �m2ð Þ1=2

: ð3:34Þ

Because in this case the Fresnel coefficients are unimodular (i.e., rv ¼ 1), it is

preferable to rewrite them as rv ¼ ejwv and to use the general definition of the phaseof an arbitrary complex number (denoted by aþ jb, where a and b are two realnumbers):

w ¼ 2 tan�1 baþ

ffiffiffiffiffiffiffiffiffiffiffiffia2 þb2

p� �

if a[ 0 and b 6¼ 0;

�1 otherwise:

8<: ð3:35Þ

Fig. 3.6 a Evolution for a circular interface of the tunneling phase and penetration depth of theevanescent wave, and b the Goos-Hänchen angular shift


It can be shown (Yu et al. 2008) that when s� sc, the phase w0;v of rays p ¼ 0 isof the following form:

w0;1 ¼ 2 tan�1 cos2ðs1Þ �m2ð Þ1=2sin s1ð Þ

!

w0;2 ¼ 2 tan�1 cos2 s1ð Þ �m2ð Þ1=2m2 sin s1ð Þ

! ð3:36Þ

Thus for rays with p ¼ 0 and when s� sc, Eq. (3.26) must be replaced by

r0;v;t ¼ w0;v þ n0 þu0 ð3:37Þ

where the subscript t (for tunneling) is added to differentiate this particular phaseterm to the general one given by Eq. (3.26). Thus a GPOA-2 model can be built asfollows (Sentis et al. 2016):

IGPOA�2v ðhÞ ¼

SGOA0;v;t hð Þþ P1p¼1


��2


max



��2

; if h 2 hF�POAmax ; hc

P1p¼0


��2

; if h 2 hc; p� �

8>>>>>>>>>><>>>>>>>>>>:

ð3:38Þ

The phase w0;v is sometimes called the “tunneling phase” and represents anadditional phase delay. The expressions derived here for the tunneling phase areformally identical to the ones used by Yu et al. (2008). They are also similar tothose used in (Artmann 1948; Lötsch 1971; Marston and Kingsbury 1981; Langleyand Marston 1984; Onofri et al. 2009; Berman 2012). As shown in Fig. 3.6a thisphase term is strongly dependent on the polarization state and the complementaryincident angles. Figure 3.5b illustrates the effects of this tunneling phase on thenear-critical-angle scattering pattern of bubbles having different radii, i.e. a = 30,100, 300, and 500 lm The coarse structures are now in phase with LMT predic-tions, and the ripple structure seems to be in at least qualitative agreement. The onlyremaining significant discrepancies are observed at the close neighborhood of hc. Inthis angular region, the GPOA-2 predicts a plateau, whereas the LMT predicts acoarse maximum.


Goos-Hänchen Angular Shift

The classical Goos-Hänchen effect (Goos and Hänchen 1947; Artmann 1948;Renard 1964; Lotsch 1968; Berman 2012; Puri and Birman 1986; Lock 2003; Tranet al. 1995; Merano et al. 2007; Aiello and Woerdman 2008; Jörg et al. 2013;Stockschläder et al. 2014) refers to the apparent lateral displacement dv experiencedby a tiny linear polarized beam when it reflects under total reflection conditionsonto a planar dielectric or metallic medium; see Fig. 3.7a. This effect is alsoobserved in the case of a curved interface (Tran et al. 1995; Stockschläder et al.2014); see Fig. 3.7b. A recent work by Jörg et al. (2013) suggests that this effectcould be the counterpart of the Fourier filter effect and the Imbert-Fedorov effect inthe case of circularly or elliptically polarized waves (e.g., Aiello and Woerdman2008; Jörg et al. 2013; Stockschläder et al. 2014).

Although a consensus has not yet been reached in the literature, severalapproaches have been proposed to explain and quantify this beam displacement inthe context of GPOA (Renard 1964; Puri and Birman 1986; Merano et al. 2007;Berman 2012). The general underlying idea is that the Goos-Hänchen effect is theresult of an interference process involving the incident and the reflected waves.Most studies use the Artmann formula (Artmann 1948; Puri and Birman 1986;Merano et al. 2007) to estimate this effective lateral displacement dv:

dv sð Þ ¼ � k2p

dds

w0;v

� �: ð3:39Þ

Using Eqs. (3.20) and (3.23), we obtain

Fig. 3.7 Illustration of the Goos-Hänchen effect on a planar and b convex interfaces


d1 ¼ kp

cos s1ð Þcos2 s1ð Þ �m2ð Þ1=2

;

d2 ¼ m2

cos2 s1ð Þ 1þm2ð Þ �m2 d1:

ð3:40Þ

Other formulae are proposed in the literature, including one derived by Renard(1964) under energy flux conservation considerations that leads to

d1 ¼ kp

cos s1ð Þ sin2 s1ð Þ1�m2ð Þ cos2 s1ð Þ �m2ð Þ1=2

;

d2 ¼ 1�m2ð Þm2

m4 sin2 s1ð Þþ cos2 s1ð Þ �m2d1:

ð3:41Þ

In the case of a spherical bubble, the interface is convex relative to the incidentwave and concave relative to the tunneling rays. Two consequences can be inferred

from this. First, the lateral beam displacement dv induces an angular shift d_

v of therays (compared to the Snell-Descartes reflection law). Second, or subsequently, thesmall radius of curvature of micrometer- to millimeter-sized bubbles greatlyamplifies the importance of the Goos-Hänchen effect (which is difficult to observefor planar interfaces, e.g., Tran et al. 1995). As a result, when total reflection occursfor rays p ¼ 0ð Þ, i.e., s� sc, Eq. (3.21) takes the following particular form:

h0;v ¼ 2s1 � d_

v: ð3:42Þ

Fiedler-Ferrari et al. (1991) have derived, from the CAM (see below), the fol-

lowing expressions for the angular rotation d_

v:

d_

1 ¼ � k2 p a

ffiffiffiffiffiffiffi2m

p

1�mð Þ3=2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihc � hð Þ=2p ;

d_

2 ¼ � k2 p a

ffiffiffiffiffiffiffi2m

p

m2 1�mð Þ3=2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihc � hð Þ=2p :

ð3:43Þ

When h ! hc; Eqs. (3.43) reduce to Artmann’s expressions [Eqs. (3.41)] as

dv � a d_

v

�� . Figure 3.6b illustrates the evolution of d_

v for the parameters of

Fig. 3.5a. It increases rapidly as sc is approached, with a maximum angular shift ofapproximately 0.5° for parallel polarization.

Accounting in GPOA-1 for the Goos-Hänchen angular shift of rays p ¼ 0, usingEqs. (3.42) and (3.43) when s� sc; we get what is referred as the GPOA-3 model:


IGPOA�3v ðhÞ ¼

SGOA0;v h0ð Þþ P1p¼1


��2


max

SGOA0;v h0ð Þþ P1p¼1


��2


P1p¼1


��2

; if h 2 hc; p� �

8>>>>>>>>>><>>>>>>>>>>:

ð3:44Þ

Figure 3.8a shows, for different bubble radii, the effect of the Goos-Hänchen

angular shift d_

v calculated with the GPOA-3 model. At first glance, the agreementbetween GPOA-3 and LMT predictions is comparable with that found for GPOA-2,except that a small angular shift can be observed away from the critical region. Thiscan be explained by the fact that Eqs. (3.43) were derived from a first-order Taylorexpansion centered on the critical-scattering angle. In fact, the tunneling phase andGoos-Hänchen angular shift are two aspects of the same phenomena; as a result, theGoos-Hänchen effect should not be taken into account in GOA.

Physical Approximation of the Near Critical-Angle Scattering

For the critical-scattering angle hc the Fresnel reflection coefficient is only oncedifferentiable. This singularity generates what may be referred to as a weak caustic(Fiedler-Ferrari et al. 1991; Nussenzveig 1992) that cannot be managed by a GOA.Marston (1979, 1999, 2015), and then Marston and colleagues (Kingsbury andMarston 1981; Marston and Kingsbury 1981; Langley and Marston 1984; Marstonet al. 1982), have proposed a POA to account for this singularity in the case of

Fig. 3.8 Comparison between LMT and a GPOA-3 or b GPOA-3 for the same parameters than inFig. 3.5b


spherical bubbles. In this section, we only consider the POA contribution of raysp = 0 (Marston 1979), while Marston and colleagues have also considered thecoupling of rays p = 0 and p = 1 (e.g., Marston and Kingsbury 1981). In thisapproximation, noted M-POA in what follows, the near-critical-scattering pattern ismodeled using a procedure reminiscent of Airy’s theory of the rainbow(Fiedler-Ferrari et al. 1991). This approximation is derived through three majorsteps. First, a GOA is used to derive simple analytical expressions for the directionof propagation and the local curvature of the virtual wavefront associated with thecritical rays p ¼ 0ð Þ. Second, the propagation in the far-field of this virtual wave-front is calculated in the framework of scalar diffraction theory. Third, energy fluxconsiderations are used to obtain the normalization required to express the absolutescattered intensity. As a result of some simplifications related to the stationaryphase method used to calculate the far-field, the M-POA applies only to largebubbles and to the scattering domain h� hc.

With the M-POA, the amplitude of rays p ¼ 0 scattered by large bubbles in theregion h� hc reads as

SM�POA0;v ¼ F0

2ffiffiffi2

p exp j r0;v � p=4� �

; ð3:45Þ

with F0 gð Þ ¼ a F gð Þ � Fð�1Þj j and F gð Þ ¼ C gð Þþ jS gð Þ, where C gð Þ and S gð Þstand for Fresnel’s cosine and sine integrals (Marston 1979):

FðgÞ ¼ CðgÞþ iSðgÞ ¼Z g

0cos

p z2

2

� �dzþ j

Z g

0sin

p z2

2

� �dz: ð3:46Þ

z is a variable of integration and g hð Þ ¼ sin hc � hð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia=kð Þ sin scð Þp

a parameterquantifying the deviation from the critical-scattering angle. This parameter appearsonly as the upper boundary of the integral (like as a diffraction phenomenon).

With energy balance considerations derived from the work of Davis (1955), thescattered intensity of the coarse fringes, at distance R [ [ a from the bubblecenter, can be predicted implicitly by the M-POA using Eq. (3.45):

Ip¼0ðhÞ ¼ I0aR

� �2F0ðgÞ�F0ðgÞ8

; ð3:47Þ

where I0 represents the intensity of the incident plane wave. With Fð�1Þ ! 1=2,the function F0ðgÞ�F0ðgÞ ¼ CðgÞþ 1=2½ �2 þ SðgÞþ 1=2½ �2 is similar in form to thewell-known Fresnel (near Field) diffraction by a straight edge (Fowles 1987). Notethat Eq. (3.45) is polarization dependent, but not Eq. (3.47).

Determining the angular position of the extrema of the critical-scattering patterns(Onofri 1999) is equivalent to looking for the zeros of the derivative of the functionF0ðgÞ�F0ðgÞ:


CðgÞþ 1=2ð Þ cos p g2

2

� �þ SðgÞþ 1=2ð Þ cos p g2

2

� �¼ 0: ð3:48Þ

The solutions, i.e. the zeros gn of Eq. (3.48), can be obtained with a simplenumerical iterative procedure g1 � 1:2172;g2 � 1:8725; � � �ð Þ. For these zeros, theodd and even values of n identify the local maxima and minima of F0ðgÞ�F0ðgÞ,respectively (i.e. bright and dark fringes). From the definition of g and sc; one canlocalize extrema of critical-scattering:

hn ¼ hc � sin�1 gn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimk=affiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 1

ps !

ð3:49Þ

From Eq. (3.49) we find that the M-POA (p = 0) predicts a dependency for theangular spreading of the near-critical-angle scattering patterns with bubble sizes inhc � hð Þ / a�1=2. The latter is stronger than the one predicted by Airy’s theory forthe first rainbow: h� hrð Þ / a�2=3 (where hr represents the rainbow angle predictedby Geometrical Optics, e.g., Adam 2002). One merit of the M-POA is that it canprovide simple and useful relations for bubbles characterization. By way of illus-tration, for a bubble of known relative refractive index, by measuring the angularposition h1 of the first extrema of its near-critical-angle scattering pattern, one cansimply estimate its radius :

a � g21k0m

sin2 hc � h1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�m2

p : ð3:50Þ

If now the bubble size and relative index are unknown, they can be estimatedsimultaneously with a reasonable accuracy using (Onofri 1999; Krzysiek 2009) bymeasuring the angular position of the first two extrema:

m � sin p� h1 � X12ð Þ=2½ �;

a � mk0g21 þg2

2 � 2g1g2 cosD12

sin2 D12ð Þ sin h1 � X12ð Þ=2½ � ;ð3:51Þ

where X12 ¼ tan�1 sin D12ð Þ= cos D12ð Þ � x2=x1ð Þ½ � and D12 ¼ h1 � h2:Let us now couple the POA with the GPOA-2 (already accounting for the van de

Huslt GOA, the tunneling phase, and the Fraunhofer approximation). To do so, it isnecessary to account in the GPOA-2 for the contribution of the weak causticoccurring at the critical angle with the Goos-Hänchen shift for rays p ¼ 0ð Þ and,finally, to calculate in the far-field the resulting interference with higher-order rays.The phase term r0;v is taken from Eq. (3.37) for u0 ¼ 0: Thus we can write thebasic equations of what is referred further on as the GPOA-4 model (Sentis et al.2016):


IGPOA�4v hð Þ ¼

SM�POA0;v hð Þþ P1

p¼1SGOAp;v hð Þ

��2

þ SF�POAðhÞ�� 2; if h 2 0; hF�POAmax

;

SM�POA0;v hð Þþ P1

p¼1SGOAp;v hð Þ

��2


;

P1p¼0


��2

; if h 2 hc; p� �:

8>>>>>>>>>><>>>>>>>>>>:

ð3:52Þ

Note that in the GPOA-4 model, the M-POA is applied down to h ¼ 0, as it was

done by Marston. Adding the M-POA to the previous model significantly improvesthe agreement with the LMT in the near-critical-angle scattering region; seeFig. 3.8b for the dependency on the diameter (and Sentis et al. 2016 for thedependency on the refractive index). The fine structures are now perfectly repro-duced in this region. However, the M-POA tends to overestimate the intensitydecay of the scattering pattern slightly below hc. A more serious problem is that thelatter feature appears to be more pronounced for the largest bubbles, which is quitedisturbing regarding POA hypotheses. Two reasons explain the discrepanciesidentified in Fig. 3.8b. First, the Goos-Hänchen angular shift is not taken intoaccount in the amplitude term of Eq. (3.32). This can be easily corrected bymodifying the definition of the angular parameter g as follows:

g ¼ sin hc � hþ d_

v

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia=kð Þ sin scð Þ

pð3:53Þ

Second, the range of validity of the M-POA appears to be more limited thanexpected. Since this POA is based on a local approximation of the virtual wavefrontproperties at the vicinity of hc, it appears more reasonable to limit the application ofthe POA to this region, and in particular to the angular range h2; hc½ �; see Eq. (3.49).Thus the GPOA-5 model, where the term SM�POA

0;v;d_

v

hð Þ emphasizes on the fact that

Eq. (3.53) is used in the M-POA, reads as (Sentis et al. 2016)


IGPOA�5v hð Þ ¼



��2

þ SF�POAðhÞ�� 2; if h 2 0; hF�POAmax

;



��2

; if h 2 hF�POAmax ; h2

;

SM�POA

0;v;d_

v

hð Þþ P1p¼1


��2

; if h 2 h2; hc� �;

P1p¼0


��2

, if h 2 hc; p� �:

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð3:54Þ

Figure 3.9a–d compares the predictions of this model with LMT for different(a–b) bubble sizes and (c–d) relative refractive indices. It is found that the GPOA-5strongly improves the description of the scattering patterns in the region h2; hc½ �,

Fig. 3.9 Comparison between LMT and GPOA-5 (GPOA-2 plus modified M-POA withGoos-Hänchen shift): a full diagram and b zoom for various sizes; c full diagram and d zoomfor various relative refractive indices. All the other parameters are like to those of Fig. 3.5b. Thediagrams in (c) and (d) are scaled for clarity reasons


except for a residual discontinuity in the immediate vicinity of hc. While not shown,the same level of agreement is found for the perpendicular polarization.

Conclusion

The GPOA-5 provides, over a large angular range and a large range or parameters,surprisingly accurate results when compared to LMT predictions. In this model, toaccount for forward diffraction, the predictions of the Fraunhofer’s approximationare simply added incoherently to the GOA up to a maximum angle hF�POA

max whoseprecise determination remains an open problem. The main merit of this simplesolution is to improve the energy balance and account for the main features of theforward scattering pattern of large bubbles. However, it provides a rather poordescription of the coarse and fine structures of the scattering pattern in thenear-forward region. In addition, the F-POA is known to be a poor approximationwhen m ! 1. This is highlighted in Fig. 3.9c, d, where the agreement with LMT isdecreasing when the relative refractive index is approaching unity. For such bub-bles, instead of the Fraunhofer theory, it would be preferable to use approximationsderived from the anomalous diffraction theory (van de Hulst 1957). Despite theimprovement achieved with the GPOA-5, a discontinuity is still observed at thecritical-scattering angle, but this problem is intrinsic to the M-POA (Fiedler-Ferrariet al. 1991). It can also be noticed in Fig. 3.9c, d that the agreement found in thenear-backward region is perfect for a relative refractive index close to unity (i.e.m = 0.95), but it tends to decrease for smaller relative refractive indices (e.g.m = 0.66). This discrepancy could be attributed to a “transmitted wave glory”(Arnott and Marston 1988) that is not yet included in this model.

3.3.1.3 Zero-Order Transitional Complex Angular MomentumApproximation

Fiedler-Ferrari et al., have derived, for the near-critical-angle scattering of largespherical bubbles, a zero-order approximation from the Complex AngularMomentum (CAM) theory (Fiedler-Ferrari 1983; Fiedler-Ferrari et al. 1991;Nussenzveig 1992). At the vicinity of the critical angle, this approximationaccounts for two diffraction phenomena related to the Partial Reflection (PR) andTotal Reflection (TR) of the rays of order p ¼ 0 : SCAMv � SCAMv;PR þ SCAMv;TR , with

SCAMv;PR � e�j p=4

2ffiffiffip

p a exp �2j a sinh2

� �Z X¼0

�1þ jð Þ1exp �ju2� �

�1þ e�jp=6 mev=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�m2

p� �c ln0 Ai e�2jp=3X

� �1� e�j p=6 mev=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�m2

p� �c ln0 Ai e�2jp=3Xð Þ

8<:

9=;du;

ð3:55Þ


SCAMv;TR � e�j p=4

2ffiffiffip

p a exp �2j a sinh2

� �Z X¼0

�1þ jð Þ1exp �ju2� �

� exp 2j tan�1 mev=ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�m2

p� �c ln0 Ai Xð Þ½ �

h in odu;

ð3:56Þ

where Ai stands for the Airy function, and when keeping as possible thenotations of Nussenzveig (1992), X ¼ c a cos h=2ð Þ �m½ � þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a sin h=2ð Þpu

� �,

c ¼ 2=mað Þ1=3, ev¼1 ¼ 1; ev¼2 ¼ m�2: The integral Eqs. (3.55) and (3.56) couldappear to be much simple than the infinite expansion series of complex functionsobtained with LMT. However, their evaluation requires non-trivial integrations inthe complex plan that are time consuming and not always successful. Figure 3.10shows comparisons obtained with a Fortran code developed by Wiscombe(Fiedler-Ferrari et al. 1991) for the CAM approximation (thick gray line) and LMTresults after the ripple structure is subtracted (full thin line) for clarity reasons. Fourcases are considered in Fig. 3.10. Two bubble sizes (a = 50, 500 µm) and tworelative refractive indices (air bubble in water: m = 0.7488 and silicon oil droplet inwater: m = 0.9500). The results are rather disappointing and more specifically for asmall bubble having a relative refractive index close to unit.

Fig. 3.10 Comparison between LMT results (when the ripple structure is remove, dashed line)with CAM approximation (thick gray line) for two bubble sizes (a = 50, 500 µm) and two relativerefractive indices (air bubble in water: m = 0.7488 and silicon oil droplet in water: m = 0.95)


3.3.2 Complex Shaped Bubbles

3.3.2.1 Geometrical Optics Approximations

As already mentioned, GOAs come in trigonometric or vectorial forms. Dependingon the authors or the problem considered, they can also include different effects, e.g.incident beam shape (Xu et al. 2006a, b; Jiang et al. 2013), complex shaped scat-terers (Hovenac 1991; Xu et al. 2006a; He et al. 2012), tunneling phase(Fiedler-Ferrari et al. 1991; Lock 1996a; Yu et al. 2008), etc. With the exception ofsimple ray tracing techniques, it seems that only two studies were focused, or at leastpresent some results, on the scattering of large non-spherical bubbles, with atrigonometric (He et al. 2012) and a vectorial (Ren et al. 2011) formalism. In whatfollows, we just briefly review the background of the latter for reasons that werealready given.

As its name suggests, the Vectorial Complex Ray Model (VCRM) introduced byRen et al. (2011) is a GOA using a vectorial formalism. It allows predicting the lightscattering diagrams of large and virtually arbitrary shaped particles with a smoothsurface. In VCRM a ray possesses not only the four classical properties: direction,polarization, amplitude, and phase as in classical ray models, i.e. van de Hulst(1957), but also a new one, the wavefront curvature; Fig. 3.11. In practice, a ray ismodeled as follows:

Sp;v ¼ Ap;ve�jUp kp; ð3:57Þ

where Ap;v, Up stand for its amplitude and total phase respectively, and kp anormalized wave vector indicating its direction of propagation. Like in van deHulst’s GOA (see Sect. 3.1.1), the amplitude term is expressed as a function of adivergence factor Hp and the Fresnel coefficients:

Ap;v ¼ ffiffiffiffiffiffiHp

pep;v�� : ð3:58Þ

Fig. 3.11 a Schematic of the incident and scattered local wavefronts as well as the particle localsurface curvature taken into account by VCRM and b application to an ellipsoid (semi axes, a, b, c;incident plane wave direction tilted of hi) observed in the scattering plane (plane of symmetry) xz


However, the way VCRM calculates the divergence (astigmatism) factor istotally different:

Hp ¼ q011q021

q11q22

q012q022

q13q23. . .

q01pq02p

Rþ q01p� �

Rþ q02p� � ; ð3:59Þ

where R is the distance between the particle surface emergent point and theobservation point of the considered ray; q1q;q2q, with q ¼ 1; . . .; p, represent thetwo principal radii of curvature of the local incident wavefront (i.e. associated to theconsidered ray), and q01q; q

02q; the two principal radii of curvature of the refracted

local wavefront. The quantities 1= q1qq2q� �

and 1= q01qq02q

� �are the Gaussian

curvatures of the incident wavefront (defined locally by a 2� 2 curvature matrix Q)and the particle surface (defined locally by a 2� 2 local curvature matrix C)respectively. The 2� 2 local curvature matrix Q0 of a refracted ray can be obtainedby solving the wavefront equation:

k0 � kð Þ � NC ¼ k0P0TQ0P0 � kPTQP; ð3:60Þ

where Tð Þ and �ð Þ stand for the matrix transpose operator and matrix dot productrespectively, N the local normal to the particle surface and P a projection matrixallowing to apply the vectorial form of the Snell-Descartes law. Equation (3.60) isan algebraic equation that relates the curvature properties of the scattered andincident (e.g., plane wave and Gaussian beam, see Jiang et al. 2013) wavefronts,and the local principal curvatures of the particle surface. In VCRM, the phase-shiftsdue to the focal lines are derived a posteriori from the analysis of the astigmatism ofthe local wavefronts calculated with Eq. (3.60). In a few words, the other amplitudeand phase shifts terms are essentially calculated with the Fresnel coefficients (in theplanar limit) that are derived from the vectorial components of wave vector and not,like in Eq. (3.23), with trigonometric relations. Finally, like in the van de Hulst’sGOA, the fine and coarse structures of the scattering patterns are obtained bysumming the contribution of all rays. At the moment, the only POA taken intoaccount in VCRM is the forward diffraction using the F-POA (e.g. Ren et al. 2011)or a stochastic method (e.g., Yuan et al. 2016). VCRM predictions were found to bein a fairly good agreement with electromagnetic calculations and experimentalresults (for refracting spheres, spheroids, and cylinders) (Ren et al. 2011; Jiang et al.2013; Yang et al. 2015; Onofri et al. 2015).

As an illustration of VCRM’s capabilities, Fig. 3.12 compares the scatteringdiagrams of air ellipsoidal air bubbles in water with the same equivalent radius involume, a = 500 µm, but (a) different aspect ratios, c ¼ a=b with c � b and c ¼0:25; 0:5; 1; 2; 4; (b) an oblate bubble with c ¼ 0:25 that is tilted by hi ¼ 0; 10 and45

: For all considered cases, the calculations took only a few tens milliseconds on

a desktop computer. As it can be seen in Fig. 3.12, if the aspect ratio has a clear


influence on the angular frequency of the coarse structures observed in the criticalangle region, the effects of the tilt angle are even more noticeable.

3.3.2.2 Physical-Optics Approximations

Forward Diffraction

In the literature, depending of the application considered (e.g. far-field ornear-field), approximations for the forward scattering pattern of large and complexshaped particles are based on scalar theories of diffraction (Goodman 1960; Keller1962; Xu 2001; Sentis et al. 2017). Most of them are based on Babinet’s principle;the case of bubbles is not considered as meaningful excepted when, in the frame ofthe anomalous diffraction theory, their relative refractive index tends toward unity(Sharma and Somerford 2006). Finally, very few consider or offer the possibility tocouple diffraction effects with GOA (Keller 1962; Felsen 1984; Lock 1996a, b;Yuan et al. 2016).

Near-Critical-Angle Scattering

The POA of the critical-scattering, introduced by Marston (1979) in the case oflarge spherical bubbles, has been generalized to large spheroids with on-axis illu-mination (Onofri et al. 2012). In what follows, we review the first and basic steps ofthis model, with its geometry outlined in Fig. 3.13a. In a Cartesian coordinate

Fig. 3.12 Comparison of the scattering diagrams calculated with VCRM of air bubbles in water withthe same equivalent radius in volume, a = 500 µm, but a different aspect ratios, c = 0.25, 0.50, 1.0,2.0 and 4.0; b case c = 0.25 when the incident wave angle is tilted by hi = 0, 10 and 45°. Planewave illumination with parallel polarization and k0 = 0.532 nm, for rays of order p = 0, 1, … ,8.For drawing considerations, the curves are shifted each other’s by x100 in (a) and x1000 in (b)


system Oxyzð Þ, the equation of a spheroid with rotational symmetry about the xaxis and semi-axes ax; ay ¼ az is x2=a2x þ y2 þ z2ð Þ=a2y ¼ 1: Note that the coordinatesystem has been rotated compared to previous sections to keep as possible thenotations of pioneer works (Marston 1979; Onofri et al. 2012). Depending on itsaspect ratio c ¼ ax=ay, the spheroid may be oblate c� 1ð Þ or prolate c 1ð Þ.Owing to the symmetry of the problem, it is better to start with the scattering of abubble with an elliptical cross section, x2=a2x þ y2=a2y ¼ 1.

In the region x� 0; y 0ð Þ, the outgoing normal unit vector to the ellipse hassquared modulus N2 ¼ x2=a4x þ y2=a4y and coordinates cos i ¼ �x= a2xN

� �and

sin i ¼ y= a2yN� �

, from which we obtain

cos 2ið Þ ¼ 2x2= a4xN2� �� 1;

sin 2ið Þ ¼ �2xy= N2a2xa2y

� �:

ð3:61Þ

To describe the virtual wavefront of the reflected rays p ¼ 0ð Þ, a secondCartesian coordinate system Ocxfyfð Þ centered on the critical impact point xc; ycð Þis introduced, where

xc ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia4x= a2x þ a2y tan

2 ic� �r

;

yc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia4y= a2y þ a2x= tan

2 ic� �r

:

ð3:62Þ

The xf and x axes are parallel, with h ¼ x� xc [ 0 and yf\0, xf ¼h 1þ cos 2ið Þð Þ and yf ¼ y� ycð Þ � h sin 2ið Þ. The wavefront of rays p ¼ 0 can beparameterized in terms of the incident angle i (denoted by / in Marston 1979), butfor bubbles with an elliptical cross section, it is preferable to use x (Onofri et al.2012). To describe this wavefront, a third Cartesian coordinate system Ocuvð Þ is

Fig. 3.13 a Schematic of the POA model for spheroids with on-axis illumination and b shape ofthe wavefronts of rays p = 0 in the near-critical-angle scattering direction


defined, where the v axis is along the ray originating from the critical impact point,Oc. The transformation matrix between Ocxfyfð Þ and Ocuvð Þ is

uv

� �¼ sin 2icð Þ cos 2icð Þ

� cos 2icð Þ sin 2icð Þ� �

xfyf

� �: ð3:63Þ

We seek a relation v ¼ v uð Þ that is valid in the vicinity of the critical anglei � ic; i.e., u � v � 0. Omitting cubic and higher order terms, the Taylor expansionof the equation of the wavefront at x ¼ xc is

v � 0þ dv=duð Þuþ d2v=du2� �

u2=2!þO uð Þ3: ð3:64Þ

Letting q � u0 and p � v0 where ðÞ0 and ðÞ00 denote the first d=dxð Þ and secondderivatives d2=dx2

� �with respect to x, we find that dv=du ¼ p=q,

d2v=du2 ¼ qp0 � pq0ð Þ=q3, and

pc ¼ � cos 2icð Þx0f þ sin 2icð Þy0f ;qc ¼ sin 2icð Þx0f þ cos 2icð Þy0f :

ð3:65Þ

To determine the first two a priori non-zero terms in Eq. (3.64), it is necessary toevaluate the derivatives at x ¼ xc :

x0f ¼ 2x2c= a4xN2c

� �;

y0f ¼a2yxc

a2xycN2c

1a2y

� x2ca2x

1a2x

þ 1a2y

!" #:

ð3:66Þ

It is straightforward to show that pc ¼ 0 and qc 6¼ 0. The first analytical term inEq. (3.64) is therefore zero, and the second derivative reduces to d2v=du2 ¼ q�2p0,where at x ¼ xc;

p0c ¼ � cos 2/cð Þx00f þ sin 2/cð Þy00f : ð3:67Þ

For the second derivative at x ¼ xc; we obtain

x00f ¼ 8xc= a4xa2yN

4c

� �;

y00f ¼4

a2xa2y

2x2cyca2xN

4c

1a2y

� 1a2x

!þ a2y

ycN2c

y2ca2y

� x2ca2x

!" #� a4ya2xy3c

:ð3:68Þ


Finally, the approximation of the virtual wavefront is

v ¼ aeu2 with ae ¼ p0c= 2q2c� �

: ð3:69Þ

Two important conclusions can be drawn from Eq. (3.69). First, the wavefront isquadratic with coefficient ae. This coefficient, which can be calculated usingEqs. (3.61), (3.62), and (3.65)–(3.67), reduces to that derived for a sphere (Marston1979) when ax ¼ ay ¼ a: ae � as ¼ � a cos icð Þ�1 (Onofri et al. 2012). Second, inits general form, the coefficient ae obtained for the ellipsoid is equal to the one of asphere whose radius is equal to the radius of curvature of the ellipsoidal bubble in

the xyð Þ plane at the critical impact point, i.e., ae � as a � ac;jj ¼ a2xa2yN

3c

� �. This is

illustrated numerically in Fig. 3.13b by comparing the shape of the virtual wave-fronts of the rays p = 0 in the case of various spherical and elliptical bubbles.

From the conclusions derived from Eq. (3.69), it can be readily shown that thefar-field contribution of the weak caustic that was derived for a sphere also appliesto a spheroid. There are, however, two important differences: one in the definitionof the coefficient ae (described previously) and the second in the writing of theenergy balance conservation. At large distances R [ [ Max ax; ay

� �from the

spheroid center, the energy balance between the incident and the correspondingscattered rays can be expressed as (Davis 1955)

I0dS0 h;/ð Þ ¼ I h;/ð ÞdS h;/ð Þ: ð3:70Þ

The elementary projected surface associated to an incident pencil of light isdS0 � l0dx� y� d/ð Þ cos icj j, with / ¼ 0; 2p½ �. Near the critical point, the differ-

ential arc length along the ellipse, l0dx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þðy0Þ2

qdx; reduces to dlc ¼ ac;jjdh.

Recalling that h ¼ p� 2i, for the scattered rays we obtaindS ¼ �2Rdi� R sin 2icð Þ � d/j j. After some calculations and remembering thatF0 gð Þ�F0 gð Þ ! 2 when h ! 0, we find that for large bubbles and hc � h 0, thenear-critical angle scattering intensity of an ellipsoid illuminated along its majoraxis is

Iv h;/ð Þ ¼ I0ac;jjac;?R2

� �F0 gð Þ�F0 gð Þ

8; ð3:71Þ

where ac;? ¼ yc= sin ic denotes the second radius of curvature of the ellipsoid at the

critical impact point and g hð Þ ¼ sin hc � hþ d_

v

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiac;jj=k� �

sin scð Þq

. For a spher-

ical bubble, Eq. (3.71) is equivalent to Eq. (3.47).The results obtained with Eq. (3.71) are compared in Fig. 3.14 to those obtained

with LMT in the case of spheres and spheroids having the same radius of curvatureat the critical impact point (e.g. a ¼ ac;jj in the plane / ¼ 0

). We found the same

agreement between LMT and POA (p = 0) predictions as the one we got for


spheres; see Fig. 3.4b (Marston 1979). The coupling of this new POA (p = 0)(Onofri et al. 2012) with other approximations (e.g. GOA, F-POA, etc.) remains tobe seen, as well as its extension to arbitrary shaped bubbles.

3.4 Conclusion

The Lorenz-Mie and its generalizations remain essential for predicting the scat-tering properties of large bubbles, while essentially limited to spherical bubbles(homogeneous or coated). For more complex shaped bubbles, there exist twocomplementary alternatives. On one hand, we find exact electromagnetic methodsthat are, in practice, still limited to rather small to moderate sized particles. On theother hand, we find approximations. A GPOA, like the GPOA-5, accounts forvarious effects in the frame of the GOA (Fresnel coefficients, focal lines, tunnelingphase) and POA (forward diffraction, Goos–Hänchen effect, and singularity linkedto the transition to the total reflection). Although its predictions are not perfect, thiscomputationally efficient model allows for the reproduction of most of the coarseand fine structure of the scattering diagram of large spherical bubbles (the calcu-lation of a scattering diagram takes a few ten milliseconds on a desktop computer).To extend such a model to non-spherical bubbles, some of the most important stepshave been recently achieved (vectorial formalism accounting for the astigmatism inGOA, POA for spheroids, etc.). But there is still a lot of work to do to couple all ofthe latter developments in a general model that should open up perspectives invarious fields.

Fig. 3.14 a Comparison of the near-critical angle scattering diagrams and b angular position ofthe first two-fringes of spherical (LMT) and spheroidal bubbles (POA, p = 0)


Acknowledgements The authors want to acknowledge Prof. K-F. Ren for providing them cal-culations with VCRM. This work was partially funded by the French National Research Agency(ANR) under grants AMO-COPS (ANR-13-BS09-0008-02), Labex MEC (ANR-11-LABX-0092),and A*MIDEX (ANR-11-IDEX-0001-0).

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Chapter 4Volume Scattering Function of Seawater

Michael E. Lee and Elena N. Korchemkina

4.1 Introduction

The emergence of instrumentation capable of making direct measurements ofinherent optical properties (IOPs) has enabled investigations of the fundamentalcharacterization of light attenuation in aquatic environments. Evaluation of therange, precision and accuracy of the available instrumentation is necessary to allowfor the comparison and interpretation of ocean field data as well as establishing thevalidity of deriving other parameters from the measured values. While measure-ments of the absorption and attenuation coefficients of natural waters have becomeroutine, measurements of the volume scattering function (VSF) have remainedelusive. But recent years has seen a renewed interest in the volume scatteringfunction, and instruments now exist to measure this missing quantity in situ rela-tively easily. Measurements of the VSF and other optical properties are shown for avariety of areas, from turbid rivers to large phytoplankton blooms. We show theshape of the VSF is surprisingly variable, depending on both particle types andsizes.

Interpretation of ocean satellite data and inversion of optical characteristics ofthe water column using remote sensing of ocean color require both information onthe spatial and temporal variability of optical properties and accurate measurementof sub-pixel bio-optical parameters for the upper layer of the ocean. Measurementsof the spectral volume scattering function and spectral reflectance are among themost essential small-scale hydro-optical properties required to interpret of remotesensing of ocean color (Lee et al. 2007). Knowledge of scattering properties is

M. E. Lee (&) � E. N. KorchemkinaMarine Hydrophysical Institute of RAS, Kapitanskaya st., bldg. 2,Sevastopol, Russia, 299011e-mail: [email protected]

E. N. Korchemkinae-mail: [email protected]


151

especially important for the development of realistic optical models of the upperocean to improve our interpretation of the remote sensing of the sea in the visiblebands.

4.2 A New Approach on Measurement of VolumeScattering Functions of Seawater

4.2.1 Traditional Methods of Measuring the Propertiesof Light Scattering in Seawater

The propagation of light in the ocean depends on the absorption and scatteringcharacteristics of the substances in the water. The measure of the directional lightscattering is the volume scattering function (VSF), which is important for under-standing ocean color, radiative transfer in the water, and optical system perfor-mance. The VSF and the absorption coefficient completely determine the opticalproperties of a medium. When these two parameters are coupled with the angularand spectral distribution of the incident radiance field just below the surface, the fullradiative flux balance of the ocean can be simulated. Such computations are ofcentral importance in diverse areas like the upper ocean heat balance, the photo-synthetic productivity of the ocean, and the chemical transformation ofphoto-reactive compounds. Spectral VSF measurements are also key to currentapplications concerning the diagnosis of upper ocean constituents from inversion ofthe spectral distribution of upwelling radiance measured from airborne and satelliteplatforms, and to estimate efficacy of laser ranging.

The volume scattering function (VSF or b(h)), is radiometrically defined as theradiant intensity I, deriving from a volume element in a given direction (h), per unitof incident irradiance E, and per unit volume V, i.e.

bðhÞ ¼ dIðhÞEdV

ðm�1sr�1Þ; ð1Þ

The volume scattering function is often normalized by its angular integral toyield the phase function:

�bðhÞ ¼ bðhÞ2p

Rp0bðhÞ sin hdh

¼ bðhÞb

ðsr�1Þ; ð2Þ

which provides information on the relative angular distribution of the scattering.The denominator of Eq. 2 is the total volume scattering coefficient b(m−1). Thebackscattering coefficient bb(m

−1) is computed as

152 M. E. Lee and E. N. Korchemkina

bb ¼ 2pZp

p=2

bðhÞ sin hdh;

and the backscattering ratio is defined as bb/b. A similar computation can be donefor the forward hemisphere.

Although the VSF is crucial to ocean optics, very few VSF data have beencollected due to the complexity of the measurements, so the range of variability ofthe VSF in the ocean is not well enough investigated (Lee and Lewis 2003). Currentradiative transfer models rely on a very limited set of coarsely resolved measure-ments of the angular distribution of scattering made more than 40 years ago (Tylerand Richardson 1958; Petzold 1972; Kullenberg 1974; Morel 1973), despitewidespread agreement on the importance of the variability in the phase function inforecasting the underwater radiance distribution (Morel 1973; Plass et al. 1985;Mobley et al. 1993, 2002; Gordon 1994). As well, the VSF can be inverted toobtain information on the nature of the particulate matter in the oceans, which is ofwide interest (Brown and Gordon 1973; Zaneveld and Pak 1973; Zhang et al.2002). Instruments historically used to measure the VSF across a range of angles(e.g., polar nephelometers; Tyler and Richardson 1958; Kullenberg 1974) arecomplex and have no capability to measure the VSF over the full angular rangenecessary to solve the general radiative transfer equation.

There have been three instrumentation approaches applied to the measurement ofb(h). Differences in the approaches relate directly to attempts to address the dif-ferent problems that arise in making VSF measurements across the entire 0°–180°range, in the forward (small, 0°–10°), general (10°–170°), and backward (large,170°–180°) angle regions. The small angle problem relates to the high level ofbackground light generated by the direct beam. The intensity of the direct beam isseveral orders of magnitude greater than that of the measured scattered light. Themain problem in measuring scattering in the general angle region is the largedynamic range in the intensity of the scattered radiance (>107) and a very low signalof scattered light at angles in the vicinity of 90°. The problem of insufficient (anddifficult to determine) scattering volume is added to difficulties of measurements ofthe low light level for angles in the range near 180°. Incorporating the requirementsfor making measurements across the entire angular range into the capabilities of asingle instrument has posed a tremendous challenge to electro-optic instrumentdesigners. Consequently, separate instruments have been developed to accommo-date the different approaches required for the three angular ranges.

The most difficult problem of small angle measurements is the contamination ofthe scattered signal by light reflected and scattered from parts of the optical unit.The so-called small-angle technique has been developed to avoid this problem. Thesmall-angle technique is based on the illumination of a very small volume ofscattering media by a narrow parallel beam and measurement of the scattered lightin the focal plane of a receiving objective. The problem consists in the detection ofa very weak signal at small angles near the beam center, which interferes with the

4 Volume Scattering Function of Seawater 153

focal plane due to the imperfect character of the direct beam. For example, the beamcan exceed the scattered signal by up to 105 in clean ocean waters (Bauer and Morel1967; Petzold 1972; Agrawal and Pottsmith 2000).

The typical instruments used to measure of the general angular VSF havecomplex mechanical designs because their angular deviation is provided by rotatinga bulky light source or photodetector unit around the axis of the scattering volume(Petzold 1972). Due to interference by stray light the minimal forward angle islimited to approximately 10°. Backscattered light measurements are limited toangles less than 170° owing to physical restrictions created by the dimensions of thelight source and photodetector unit. For such instruments, it is also difficult toprovide sufficient shading to exclude ambient light.

Since modern photodetectors measure radiant fluxes, it is more convenient towrite Eq. (1) in terms of the initial and scattered radiant fluxes. The scattered flux,F(h), can be expressed as a function of the optical assembly parameters of thescattering meter as

FðhÞ ¼ IðhÞ � X � e�cr; ð3Þ

where X is the viewing solid angle of the photodetector (sr), c is the beam atten-uation coefficient (m−1), and r is the distance between the center of the scatteringvolume and the photodetector (m). The irradiance E in Eq. (1) is determined by theflux, F0, from the radiant beam, which penetrates into the sea water and is atten-uated along the path r1 from the light source to the center of the scattering volume:

E ¼ F0

Se�cr1 ; ð4Þ

where S is the normal cross-sectional area of the light beam. Combining Eqs. (1),(3) and (4),

bðhÞ ¼ FðhÞSecðrþ r1Þ

F0XVðhÞ : ð5Þ

To determinate absolute values of the scattering coefficient in a given direction,additional measurements of F0 and c are necessary but not always possible.Therefore, the measurements of the angular distribution of scattered light areusually made in relative units. If the polar nephelometer has the additionalrequirement of measuring the direct beam attenuation for h = 0 by the same pho-todetector, then its calibration for absolute values of b(h) can be carried out(Kullenberg 1968). In this case, in accordance to Lambert-Bouguer’s law, the lightreceived by the photodetector will be determined by:

Fðh ¼ 0Þ ¼ F0e�cðrþ r1Þ: ð6Þ

From Eqs. (5) and (6) we derive:


bðhÞ ¼ FðhÞSFðh ¼ 0ÞXVðhÞ : ð7Þ

The simultaneous measurements of scattered and direct attenuated fluxes allowthe determination of the absolute values of b(h), because other variables in Eq. (7)are known from the geometrical parameters of the optical assembly. This equationis strictly true only for a perfectly collimated source and collimated field of view ofthe receiver, where V(h) is known, and where all light scattered into directionsoutside the very narrow detector aperture is rejected.

Different devices are available to measure scattering at a few fixed angles, orover a narrow angular range in the forward direction (e.g., Sequoia Scientific, Inc.,Laser In Situ Scattering and Transmissometer (LISST); (Agrawal and Pottsmith2000), or the total scattering b (integral of the VSF over all solid angles), fromwhich the VSF can be inferred using models that theoretically relate scattering at agiven angle to elements of the full angular range (Man’kovsky 1971; Morel 1973;Oishi 1990; Maffione and Dana 1997; Boss and Pegau 2001). Because of the lackof direct measurement, these links are unreliable (but see Boss and Pegau 2001),mostly because the nature of the particles responsible for scattering in the ocean(particularly backscattering) is unknown (Zhang et al. 1998, 2002). Here wedescribe an advanced approach to the measurement of the VSF in the ocean (Haltrinet al. 1996; Lee and Lewis 2003). We present the theoretical background, theinstrument design, and the first volume scattering observations made in the ocean inthe last 40 years.

4.2.2 New Principles of Measurements of VolumeScattering Function Over the Wide Rangeof Scattering Angles in Seawater

The scientific team of Marine Hydrophysical Institute in Sevastopol has investi-gated a unique approach to determinate of the volume scattering function in thewhole range of angles. Main and novel features of the schematic are the use of arotating specially designed periscope prism, special shadowing technique to controlthe light beam thickness and carefully designed light trap showed on Fig. 4.1. Theprism rotates around the photodetector assembly axis that extends through thecenter of the scattering volume. Since the light source and the photodetector unitsare fixed, increasing their dimensions (length) does not result in significant designcomplexity, and the scanning arrangement is significantly simplified and reduced insize. The special shape of the periscope prism and its precisely adjusted dimensionsprovide the capability to measure the scattered light across all three angular regionswith an angular resolution of 0.3°.

In the first version the instrument for VSF measurement used a tungsten halogenlamp with a very small-sized filament as a light source. It provides a 75 mW light


flux. This lamp, coupled with a 100-mm-focus lens, provides a collimated lightbeam with a divergence of less than 0.1°. This beam penetrates through anobjective-illuminator into the water (sample volume) contained in a black chamberand irradiates the scattering volume. We assume that fluctuations in lamp intensityover the measurement sequence do not appreciably contribute to the overalluncertainty of the photon budget. During measurements of the volume scatteringfunction, the scattering volume varies in a complex fashion. The precision of thevolume scattering function determination, especially at small angles and for thoseclose to 180°, depends greatly on the accurate estimation of the scattering volume.With accuracy sufficient for practical applications the scattering volume variationfor large angles can be written as

VðhÞ ¼ V 90�ð Þ=sinðhÞ; ð8Þ

where, V(90°) is the scattering volume in the 90° direction, specified by the opticalassembly parameters. The usage of this equation is restricted to the general angleregion.

The light source, the photodetector (a photomultiplier), and the stepping motorof the angular scanner are placed in three separate, hermetically sealed compart-ments, mutually oriented at 90° with respect to the working volume of theinstrument. The working volume is contained within a light trap assembly

Fig. 4.1 Measurement approach to determination of the volume scattering function


consisting of a cup of light absorptive material (Delrin) enclosing a volume ofapproximately 1.5 L providing free exchange of ambient fluid. The water is slowlystirred within the chamber by rotation of the periscope: complete rotation aroundthe axis is normally takes 1.5 min.

As the scanning motor rotates, the scattered light is continuously directed by theperiscope prism into the hermetic case of the photodetector assembly, where it isfocused by an objective into the center of the field stop, and then directed onto aphotomultiplier photocathode. In front of the photocathode light filters are placed.In the prototype instrument, hereafter denoted as VSF meter, a single filter at555 nm was used. In 2002 a rotating filter wheel with 8 color filters (10 and 20 nmFWHM bandwidths) was introduced, so the instrument was named Multi-SpectralVolume Scattering Meter (MVSM). The filter set can be changed if desired. Theacceptance angle of the photodetector, determined by both the objective focus andthe field stop, has 3 changeable values. The volume scattering function measure-ment is performed under continuous rotation of the periscope prism. The intensityof the directly attenuated flux (c, m−1) is measured at 0°.

Efforts were taken to extend the angular range of the VSF measuring prototypeto 0.5° in the forward direction, and to >178° in the backward direction. To do thisa new shadowing method for small angle scattering measurements was developed.For this approach the periscope prism was designed with a parallel shift, to allowthe prism boundary to align precisely with the boundary edge of the optical axis ofthe system (Fig. 4.2). In this case the direct beam does not go into the receivingobjective, and the light scattered at angles � 0.5° reaches the photomultiplierwithout interference. For angles near zero, the beam edge slides along the prismboundary, but no direct light is received. With this design, the amount of lightinterference from the direct beam is decreased by several orders of magnitude. Foran ideal parallel beam, this approach would fully avoid interference from back-ground light. Unfortunately, real collimated beam sources have small spatialside-lobes, which pass light to the receiving objective. This problem was solved bynarrowing the beam width to a very small size for small angle measurements. Tocreate this very narrow beam, a changeable aperture was introduced. The beamwidth apertures are on a rotating cylinder. The aperture changes the beam widthsynchronously with the prism rotation so that from 0°–30° and 160°–180° the beamwidth is minimal, and for the other angles it is maximal. Accurate measurements upto 0.5° have been successfully made.

Fig. 4.2 A schematic diagram of the shadow method for small angle scattering measurements


Figure 4.2 shows a top-view, looking down along the rotational axis of theprism. At zero angle, the light source ray will slide along the side edge.

The second significant problem involved the adequate conditions for scatteringdetermination at angles near 180°. For most designs, the dimensions of the lightsource and the photodetector unit restrict measurements in the near-backwarddirection. Furthermore, accurate determination of the appropriate scattering volumebecomes difficult, due to specular scattering of the forward beam from the instru-ment surface into the backward direction. This problem was solved by introducingthe light trap consisting of two polished plates of dark glass fixed at an angle of 45°to the primary optical axis (Fig. 4.1). The non-scattered portion of the collimatedbeam is fully absorbed by polished plates because of the negligible differencebetween the refractive index of glass and the refractive index of seawater. Only afew fractions of percent of the incident light are reflected. Due to many reflectionsthe amount of background light is decreased more than 109 times. Corrections forattenuation in the scattering volume (see Eq. 7) were made using both the instru-ment itself and beam attenuation meter AC-9 (Wetlabs Inc.).

Figure 4.3 shows a general view of the multispectral volume scattering meter formaking laboratory and in situ measurements of the volume scattering function ofseawater and other natural waters.

Recently the scientific group of Marine Hydrophysical Institute modified andsuccessfully tested the prototype shown in Fig. 4.3. The new instrument usesmodern hi-tech electronic components. The modification includes the use of 8ultra-bright LEDs made by LedEngin, Inc., ranging from near UV to red boundaryof visible spectrum (Lee et al. 2015a). LEDs provide higher light energies com-pared to halogen lamps, allowing to decrease significantly the photomultiplier noisewhen detecting low light fluxes at high scattering angles. It is also worth noticingthat the smaller size of the light-emitting spot of the LED when compared to the

Fig. 4.3 A general view ofthe instrument for measuringthe volume scattering function


lamp, allows to decrease the divergence of the light beam and to increase theangular resolution of VSF measurements.

The application of ultra-bright LEDs also makes possible to measure thefluorescence of seawater with the same instrument (Lee et al. 2015b). The modi-fication made for this purpose consists in introducing the set of color filters beforethe photodetector. The 365 nm LED with 460 nm filter allows to estimate thefluorescence of dissolved organic matter, and 432 nm LED with 480 nm filtershows the fluorescence of chlorophyll-a. In addition, measurements of fluorescenceof specially made solution can provide the information need to develop methods forthe precise determination of the scattering volume as a function of the angle.

In order to evaluate in detail the instrument capability and performance, anumber of experiments were performed in Canadian and US laboratories. Theexperimental data confirmed preliminary estimate of the instrument specificationsand functioning. Successively experimental studies of VSF were performed withartificial media to mimic seawater components affecting water color. Theseexperiments were conducted with the purpose of developing new methods to solvebio-optical inverse problems, i.e., to determine absorption and scattering by pig-ments of phytoplankton in the upper layer of the ocean.

4.3 Laboratory Testing of VSF Meter

4.3.1 Primary Experiments to Measure Angular ScatteringProperties in Clean Seawater, and in Waterwith Bubbles of Different Size

The first version of the instrument was tested at the Marine Optics Department ofthe Marine Hydrophysical Institute in Sevastopol in the end of November, 1999.Measurements in air and in turbid water confirmed the correctness of VSF meterprinciples. A more careful testing and calibration of the improved version of theinstrument and some preliminary experiments were carried out using facilities ofthe Department of Oceanography, Dalhousie University, Halifax, Nova Scotia,Canada (Zhang et al. 2002).

First set of experiments was made to estimate the level of noise of the instru-ment, its sensitivity, dynamical range and characteristic of glint-rejection.Measurements were performed in distilled fresh water and in clean seawater.The VSF meter was submerged in the tank and the water was pumped through theworking chamber. The distilled water in the tank was changed five times to assurethe purity of the water filling the chamber of the instrument. After that, the tank wasfilled by clean seawater (filtered with 0.2 µm pore size filters).

Measurements showed that the instrument could distinguish between distilledand clean water. The measured volume scattering function of clean seawater differsfrom the pure water theoretical curve. It corresponds better to the typical volume


scattering functions of oceanic waters measured by (Kullenberg 1968). The volumescattering function of the Case-2 Halifax Harbor surface water also was measured.The volume scattering function curves displayed in Fig. 4.4 show difference inscattering properties.

The crucial problem is the glint-rejection characteristic of the instrument forsmall angles and angles close to 180°. The importance of glints depends on theturbidity of the water. The comparison of raw records for air and the cleanedseawater show that the glint probably is significant in angles less than 10°(Fig. 4.5). For the turbid water of the Halifax Harbor the glint is significant inangles less than 2°. Analysis of data obtained on solutions with well-known opticalproperties shows that measurements have good quality up to 170°.

Measurements in the tank demand big amounts of clean water, limiting thepossibility of laboratory test and calibration experiments. To overcome this limit theVSF meter offers the possibility to inject the fluid into the chamber. This isapproximately 1 L in volume. A volume scattering function of clean water wasmeasured once more in the chamber to verify the quality of previous results. Resultsindicate high agreement between the two sampling method (Fig. 4.6). Three typesof water were used: distilled water, clean distilled water and clean seawater. Thesame difference as in the tank was observed between distilled water and cleanseawater. However, clean fresh and salt waters gave practically the same signal.

During the last clean water test scattering by bubbles of different size wasinvestigated. Bubbles were generated by the special pumping system in a smalltank. Bubbles were then asymmetrically pumped into the working chamber of the

Fig. 4.4 Volume scatteringfunctions of different waters


instrument to ensure stochastic oscillations of size distribution. However all VSFvaried near the same typical curve built by averaging several measurements.

Three bubble size distributions were reproduced with diameters in the ranges of1–100, 1–50 and 1–20 lm. The frequency distribution of bubble sizes was pre-liminarily measured by applying a photographic method. Results are shown inFig. 4.7 together with an analytic approximation.

Fig. 4.5 Volume scatteringfunctions of different media

Fig. 4.6 Volume scatteringfunctions in differentmeasurement conditions:comparison measurements intank and in chamber


The bubble sizes followed a normal distribution. It is thus believed that thedetermined phase function represents that of the natural bubble populations(>10 lm) found in the ocean (Zhang et al. 2002). Averaged curves of the VSFmeasurements were compared with theoretical estimates of scattering obtained byapplying the Mie theory to the frequency distribution given in Fig. 4.7. Deviationsof the measured distribution from the theoretical one are explained by the fact thatthe bubble solution is not completely stable. In addition, a significant increase inscattering in the 160° region is probably due to the deposition of bubbles on theelements of the light trap design.

The clean water measurements were used as the reference for comparison oftheoretical simulations and observations. An inter-comparison of theoretical andempirical curves is shown in Fig. 4.8. An additional experiment included theinjection of biological surfactants into the water with bubbles. The goal of theexperiment was to estimate effect of biological films on the bubble surface, on theshape of volume scattering function. The experiment was carried out in four stages.The first stage consisted in the reference clean bubbles measurement (curve 1 onFig. 4.9). At the next stage, biological surfactants were injected into the tubetransporting bubbles from the tank to the working chamber (curve 2 on Fig. 4.9).Curve 3 shows the volume scattering function of bubbles after the injection.Successively, surfactants were injected into the tank where bubbles were produced.The curve 4 in Fig. 4.9 shows the volume scattering function related to this specificcase.

Observations were strongly affected by stochastic oscillation of bubbles, so evenafter the averaging data, differences among volume scattering functions are difficultto interpret. Nevertheless, after injection of surfactants it is possible to mention adecrease of the signal at angles 30°–60° and an increase at angles greater than 140°.It was noted that the signal does not return to the previous level even stopping theinjection of surfactants.

Fig. 4.7 Example of bubblesize distribution (black) andcorresponding normaldistribution (red line)


The laboratory measurements confirmed the theoretical prediction that bubblesof sizes populating the surface ocean waters (>10 lm) show elevated scattering forangles between 60° and 80° and that organic coating on the bubble surfaceincreases the scattering in the backward hemisphere but little change the scatteringin the forward directions, including the critical angles (Zhang et al. 2002).Specific VSF measurements performed in coastal waters showed the potentialexistence of submicron bubbles coated with organic film. This bubble population

Fig. 4.8 Comparison of VSFmeasurements with Mietheory

Fig. 4.9 Volume scatteringfunctions showing the effectof biological films


with negligible contribution to the total scattering (5%) accounts for 40% of thetotal backscattering observed in situ. The inclusion of bubble size distributiontowards smaller sizes would alter the shape of derived phase function in general.This would result in rather small changes in the backscattering ratio (<20%) as longas the slope of the size distribution is small, because most of the changes are in theforward (<10°) direction. However, the prominent peak in the VSF at the criticalangle observed for larger bubbles is strongly reduced when small sizes are included,and the backscattering ratio increases by a factor of two for distributions that varyas size to power of 4. Since bubbles contribute strongly to scattering at large angles,these results have significant implications for the remote observation of ocean color.

Our investigations also confirmed that phytoplankton has very low backscat-tering efficiency (Morel and Ahn 1991; Stramski and Kiefer 1991) and that it isnecessary to consider other particles, such as bubbles or mineral particles (resus-pended particles in this case), to explain the amplitude of light backscattered fromthe ocean (Brown and Gordon 1973; Morel and Ahn 1991; Ulloa et al. 1994; Zhanget al. 1998; Twardowski et al. 2001). Although in coastal waters resuspension mightbe important, it is shown that in the open ocean bubbles are a dominantbackscatterer, although contribution from atmospheric dust may play an importantrole (Twardowski et al. 2001). It is now recognized that the angular distribution oflight scattering in the ocean is at least as important as the integral quantity (scat-tering) and the absorption coefficient in determining the amplitude, the spectral andthe angular distribution of water-leaving radiance, which are the basis for currentocean color applications (Mobley et al. 2002). The facts that bubbles in the oceanproduce a significant fraction of backscattered light and their concentration andperhaps size distribution are variable in time and space are of considerable sig-nificance for an accurate definition of biological processes through remote obser-vations of ocean color (Zhang et al. 2002).

Testing and calibrating the VSF meter at Dalhousie University using clean waterand solutions with well-known properties showed that the instrument was workingaccording to expectation. The VSF meter is able to make measurements withangular resolution 0.6° in broad angle range. The instrument can be used for fieldobservations and for laboratory measurements. The sensitivity of the VSF metercovers the dynamical range up to very clean water. Absolute calibration of theinstrument can be based on measurements of solutions with well-known opticalproperties.

However, there was the need to improve the sensitivity of the instrument in allranges of angles and especially for small angles and angles close to 180°.Normalization of the raw signal for the scattering volume still has low accuracy dueto oscillations of the measured curve at angles close to 180°.


4.3.2 Intercomparison with Scattering InstrumentationDeveloped in U.S

Later, after the tests at Dalhousie University revealed VSF meter disadvantages,detailed and very thorough testing was continued within the framework of anOptical Scattering Workshop in USA. The Optical Scattering Workshop resultsfrom a collaboration between the MHI Optics Department scientific team, and U.S.scientists in the Electro Optic and Special Missions Sensors Division of the NavalAir Warfare Center Aircraft Division (NAVAIR), Patuxent River, MD and theOcean Optics Section at the Naval Research Lab (NRL), Stennis Space Center, MS.The NAVAIR and NRL scientists are specialized in oceanographic optical mea-surements, calibration, deployment, and scattering equipment at discrete angles.

During a two weeks workshop, the MHI VSF meter capability and performancerelative to other scattering instrumentation developed by U.S. commercial andacademic laboratories were evaluated. Simultaneous measurements were made withthe following instrumentation: VABAM, ECO-VSF, and ac-9 (WETLabs, Inc);LISST (Sequoia Scientific, Inc.); a–b, c–b, Hydroscat-6 (HOBI Labs, Inc.); GASM(U. Miami), and VSF meter (MHI NASU and Satlantic Inc.). Primary goal of thedata analysis was determining under what conditions the measurements with thedifferent instruments agree and how well a given instrument can be used todetermine the near forward and near backward regions of the optical volumescattering function. The long term objective include developing a series of tests todetermine the causal factors (e.g. optical geometry, source/receiver stability, range/sensitivity limitations) of disagreement between techniques, and to ensure theappropriate instrument protocols are available to researchers across the broad rangeof oceanographic and limnologic disciplines (Prentice et al. 2002).

Definitively, an evaluation of the range, precision and accuracy of the availableinstrumentation is necessary to compare and interpret ocean field data as well as toestablish the validity of deriving other parameters from the measured values.

4.3.3 Measurements of VSF of Monodisperse Latex BeadsSolution in Pure Water

The objectives of the experiment were:

• assessment of the instrument capability and performance using measurementsperformed with testing solutions with known parameters;

• examination of stability and homogeniety of testing solutions and ranges of theirusage for VSF meter calibration;

• comparison of instrument capability with other scattering instrumentationdeveloped by U.S. commercial and academic laboratories in researchenvironment.


Experiments were conducted with uniform latex micro-spheres of different size,biological particles and Maalox™ (antacid medicine) in reagent grade water (Leeet al. 2003a).

A polystyrene monodisperse microspheres with refractive index n = 1.59 andradii of 0.596, 0.93, 3.063, 5, 10, 20, 40, 80 и 160 µm were used for the experi-ments. Five series of measurements with solutions having different scatteringcoefficient in six narrow visible bands were made. Additions of micro-spheres weremade to sequentially change the scattering regime from the cleanest water to singleand multiple scattering solutions. Scattering in solutions was preliminarily com-puted and then measured with an AC-9. The measuring chamber of the instrumentwas filled with free flowing test solution through a pipe.

Incremental bead additions were made to 400–450 L of Type A reagent gradewater from a Barnstead E-Pure System. Beads were first mixed in 20 L of water andthen added to the central tank. Simultaneous measurements were obtained byremoving a 20 L volume to fill the LISST chamber, VABAM, and two AC-9, bothinterconnected through a flow through system with shielded Tygon®™ tubing.A 5L volume was used for the VSF meter while GASM, HS-6, Beta Instruments,and ECO-VSF were immersed in the tank. The solution in the tank was mixed withan SBE 3000 pump. Measurements were made in the flow through system. Duringmeasurements flow was stopped by valve termination of the tubing.

The known parameters of the solutions were used for the numerical prediction ofthe volume scattering function. The comparison of theoretical and empirical curvesis presented on Fig. 4.10.

Figure 4.10 shows satisfactory agreement of theory and measured data. Somedifferences are explained by the finite aperture of photodetector. However, essentialdifferences are observed for angles smaller than 2° and grater of 170°. The strongmaximum in angle range of 0°–2° is explained by scattering in filtered water itself.The angular scattering coefficient of filtered water was also measured during theexperiment, and it was found close to the typical values of scattering coefficient ofclear ocean water. The problem of subtraction of this maximum from measuredscattering by latex beads and plankton cells requires separate discussion. Because ofthis, the data have to be interpreted as scattering by particles plus the water itself.

A comparison of measurements simultaneously performed with the advancednephelometer (GASM) designed by Petzold (1972) was also made (Prentice et al.2002). For 0.596 lm polystyrene miscrospheres GASM and VSF meter follow thetheoretical Mie curve well between 32° and 146° (note: Mie curve was calculatedon the base of AC-9 measurements). Deviations at near forward angles are due toinstrument limitations near the incident beam. In the backward direction GASMsensitivity is lower at angles greater than 150°. Inversion of the VSF measurementsin this region is caused by instrument ‘glint’ resulting from inefficient light trap-ping. However, GASM sensitivity is limited at angles from 10° to 150°, and theVSF meter tracks the scaled Mie curve from 2° to 175°. ECO-VSF reads sub-stantially above range at 100°. Beta Boy single point values align with VSF, anddiscrepancies in HS-6 and Beta Boy measurements are evident.


For 0.930 lm polystyrene miscrospheres GASM and VSF meter track the the-oretical Mie curve between 27° and 115°. Deviations at near forward angles are thesame as for the 0.596 lm beads. Both instruments detect the leveling out of theVSF between 110° and 140°. In the backward direction, GASM sensitivity islimited at angles greater than 150°. The VSF meter tracks the scaled Mie curve upto 175°. ECO-VSF reads below range at 100°. Overall tracking of the Mie curve bythe three point measurements are improved compared to the 0.596 lm bead test.Beta Boy and HS-6 measurements appear coincident and accurate. (note: additionalattenuation corrections to avoid instrument design influence have not been applied.)

For 3.063 lm polystyrene miscrospheres GASM and VSF meter show excellentagreement with the scaled theoretical Mie curve throughout their respective mea-surement ranges. ECO-VSF reads slightly below the expected value at 150°. Valuesat 100° and 125° agree well with the Mie. HS-6 and Beta measurements at 140°

Fig. 4.10 Comparison of theory and measured data for solution of latex beads


track the magnitude of the adjusted Mie but appear slightly high. For large particlesdespite of the good agreement of the shape of curves, discrepancies at 90° and 145°angles are notable. These discrepancies are explained by sedimentation andaggregation of large beads, causing the concentration of solutions to be unknown,even though constant during a single experiment.

The comparison of instruments has shown the capability of the MHI instrumentof performing measurements in a wider angular range and with higher resolution.The VSM is able to make measurements with angular resolution 0.6° in angle rangefrom 0.5° to 178°. The sensitivity of the VSM covers the dynamical range up to avery clean water. Absolute calibration of the instrument can be done based onmeasurements of solutions with well-known optical properties. Nevertheless, itshould be noticed that the size of test particles should not exceed of 20 µm, becausesolutions obtained with larger particles do not offer sufficient stability in time for thegiven instrument (it takes 1.5 min for full angle range scanning).

4.3.4 Volume Scattering Functions of PhytoplanktonMonocultures

To study the instrument potential to determine plankton biophysical condition andto support the inverse problem of bio-optics, scattering measurements of phyto-plankton monocultures were performed. The measurements were carried out with15 different algae cultures. The cell suspension was diluted until values ofextinction were about 1–2 m−1. The main conclusion: in spite of theoretically lowbackscattering ratio for algae it can be possible to distinct not only size of particles(small angles) but inner cells structure (general and backward angles). According toMie theory the probability of backscattering by biology particles is smaller by oneorder of magnitude than for seawater. However, the measured data analysis hasshown that after eliminating the Rayleigh, backscattering curves were transformedinsignificantly. The Rayleigh scattering was calculated as:

bW Hð Þ ¼ 1:38 � 10�4 � k=500ð Þ�4:32� 1þ 0:835 � cos2 H� � � 1þ 0:3 � S=37ð Þ; ð9Þ

where k is wavelength in nm and S is water salinity in parts per thousand.In Fig. 4.11 the difference between theoretical and experimental scattering in

microalgae monocultures is illustrated. Cultures of Pycnococcus and Pyramimonasparkeae were used for the analysis. For computation purposes it was assumed thecell size distribution was given by the Gaussian distribution law:

r ¼ 0:5 � ðrmin þ rmaxÞ; r ¼ 0:25 � ðrmax � rminÞ; ð10Þ

where rmin ¼ 0 and 5 µm, and rmax ¼ 1:5 µm and 8.5 µm for Pycnococcus andPyramimonas respectively. The refractive index was assumed equal to 1.05 + 0.0i.


It is noticed that:

• agreement with the theory is observed at angles from 0° to 1.2° only;• for smaller cells of Pycnococcus, agreement with the theory till 90° angles can

be achieved by adjusting the distribution function;• significant discrepancy (up to 4 times for some angles) is detected in backward

direction, even for small cells;• all intermediate local maxima are not present (from 0° to 175°).

Absence of local maxima is explained by the fact that the light going through acell suffers an additional scattering and beam deflections, caused by interiorheterogeneities of a cell. Cell refractive index used by the Mie theory is effective,found experimentally from phase shift of a light wave going through a particlecenter. It is obvious that constancy of average do not guarantee zero dispersion, soscattering becomes greater for all angles. But major affects become apparent at largeangles. A detailed cells scattering study, including spectral dependence, for anglesmore than 90°, has shown the existence of specific repeated peculiarities for eachmicroalgae species. Thus the volume phase function of plankton cells is not onlydetermined by size distribution, but, to the greater extent, by peculiarities of cellsstructure.

Investigations of scattering in phytoplankton monocultures have confirmed thehypothesis that the main scattering variance takes place in small angle range,because phytoplankton cells are optically large particles. Additionally spectralbackscattering and measurements at a variety of angles allow to investigate internalmicroalgae structure.

Fig. 4.11 Comparison of theMie theory and measuredscattering coefficient forPycnococcus andPyramimonas parkeae


4.4 Features of the Optical Properties of Pure WaterDiscovered Through Volume Scattering Meter

Nanopure water or Milli-Q water is typically used to calibrate different marineinstruments. The water is usually filtered through micropores with size ofapproximately 0.2 µm. After that reverse osmosis method is applied. This allows toobtain highly pure water practically with no admixtures. The measurements ofvolume scattering function were carried out with the new version of the instrumentby performing light scattering measurements at angles from 0.5 to 179° with aresolution of 0.25°, at eight wavelengths k = 443, 490, 510, 532, 555, 565, 590 and620 nm.

The analysis of these VSF measurements showed that they did not agree with thetheoretically predicted values and actually question the assumption of homogeneityof water (Shybanov 2008). For example, VSF at small angles are more than100,000 times higher than the value calculated according to thermodynamic fluc-tuations theory. This can be explained by the fact that the scattering theory, which isbuilt on the assumption of an uncorrelated equilibrium position of scattering par-ticles in a fluid, cannot interpret properly the nephelometric measurements ofnanopure water. In terms of theoretical approach of mutually independent scatteringby particles, an increase of the scattering anisotropy in cleaner waters is generallyattributed to the presence of large particles of biological origin in the seawater, andto contamination of pure water during the filling of the instrument measuringchamber. An analysis of time evolution of the scattering index (Shybanov and Lee2003) of filtered water flowing through the nephelometer chamber during 6 h wasmade. Water was purified using filters with a pore diameter d = 0.2 lm. It wasshown that, although scattering decreased in whole angular range, the scatteringanisotropy increased. The observed pattern can not be explained by the assumptionthat the main contribution in the light scattering at small angles in pure water is dueto the large particles. Growing scattering anisotropy with decreasing scattering inpure water is consistent with a similar trend observed in the clear ocean waters withhigh transparency (Kopelevich 2001). According to Kopelevich, this effect can beexplained theoretically by presence of large particles with a refractive index close toone, in other words, the electromagnetic properties of the particles are only slightlydifferent from the electromagnetic properties of water. Since these hypotheticalparticles have not been identified by alternative methods (e.g. filtration), theirphysicochemical properties are also only slightly different from the properties ofwater. In other words, they are the water itself.

On the base of angular scattering measurements in nanopure water it was con-cluded that optical properties of pure waters are determined by liquid-like aggre-gates consisting of H2O molecules and impurity elements. The revealedexperimental regularity indicates that the additional inhomogeneities (so-calledliquid quasi-particles) with size exceeding the wavelength of visible light alwaysexist in the water. This hypothesis adequately explains the results of all knownexperiments regarding pure water. Results from the experiment and their analysis


show that in “pure water” at negligibly small admixture concentrations not quan-titative but qualitative changes of scattering properties of water itself are observed.This means that the linear law of angular scattering coefficient dependence onadmixture concentration has limited application (only for large angles). That is whyit is questionable to apply the “pure water” case in hydrooptics as the reference forscattering characteristics.

New additional evidences of existence of two-dimensional correlated disorder inwater on the scales 10−7–10−3 m are obtained supplementary measurements of thevolume scattering function. By mixing pure water with small amount of filteredwater at different temperature or salinity, light scattering increases by 1.8–4.5 timesover the whole angular range. It is shown that this effect is not related to eitherturbulence or formation of dissipative structures. Additionally dimensions ofnonuniformities influencing spectral angular index of light scattering were esti-mated (Shybanov et al. 2011).

To analyze the effect of the structural inhomogeneities on water scatteringproperties, an experiment on the measurement of the spectral volume scatteringfunction of pure water mixed with small amounts of the same water having differentdensity due to a different temperature or salinity was carried out at the OpticalLaboratory, Joint Research Centre of the European Commission, Ispra, Italy(Shybanov et al. 2010, 2011). The experiment was based on the idea that if the bulkdiffusion rate is lower than the impurity propagation rate in the space of structuraldefects, the refractive index of defects changes by a value of 3 � 10−3 and thetwo-dimensional pattern becomes clearer. Fresh water prepared with a MilliQsystem ensuring ion concentration of less than 10−11 mol/l was used as the initialmedium. This so-called high-ohmic water passed through a filter with a porediameter of 0.2 lm. Six test solutions were prepared. The samples were prepared inthe following manner. To 4 L of initial water with 23 °C temperature we addedabout 20 ml (0.5–0.6%) of water with a different density, namely, fresh water attemperatures of 2.5 and 64 °C, and seawater at temperatures of 3.5, 23, 35, and 60 °C passed through the same filter. The calculated relative refractive indices of therespective impurities were 1.00097, 0.99337, 1.00563, 1.00503, 1.00396, and1.00001. The analyzed sample was preliminarily mixed for 1–2 min before fillingthe measuring chamber of the instrument. The instrument was filled with water inabout 2 min. The measurements were performed after 1–2.5 min successively infour spectral ranges of 5 nm centered at 625, 490, 412, and 380 nm. The angularscanning time at each center-wavelength was about 1 min. The measurements witheach sample were performed for 10–15 min (three or four measurement series).When analyzing the measurements, a noticeable increase in the signal for smallscattering angles in water with impurities was observed with respect to the signal inthe initial water. The ratio of the detected radiance at angles in the range of 15°–40°to the radiance related the initial water for various water samples ranged from 1.8 to4. The largest effect was observed when the fresh cold water was added. To esti-mate the time evolution, measurements of the scattering coefficient were performedat 380 nm wavelength. Filtered seawater at room temperature was used as animpurity. The additional scattering intensity due to the presence of a different liquid


decreased almost exponentially with time. The relaxation times of the optical signalin the range of 15°–40° were 9.5 and 12.5 min at the concentrations of seawater of0.5 and 1.25 vol.%, respectively. A peculiar time behavior was observed whenseawater was added in an amount of 5 vol.%. The scattering coefficient had no timetrend for 15 min and, after that, began to decrease. A large amount of impuritiescannot apparently be located in a limited space of defects. A part of the impurityliquid before the propagation to the region of defects is in the bulk phase until theoccurrence of the diffusion of the impurity located in the defects.

As presently accepted in hydrooptics, the thermodynamic equilibrium approxi-mation is applicable for all scattering angles, and the volume scattering function oflight depends on the physical parameters of suspended particles. Water free of largeparticles could seemingly be used as reference water. However, many researchers,beginning with Raman, pointed to difficulties in obtaining water with the scatteringindicatrix close to Rayleigh one. Even in the presence of the stray light perturba-tions, small traces of impurity were found to affect the signal at small angles. It isworth noting that the use of ultrapure water as a reference would be technologicallycomplicated in view of its capability to absorb atoms and molecules from theenvironment (Pope and Fry 1997). At the same time, the processing of measure-ments related to pure water reveals a paradox of the singularity in the opticalproperties of water: the experimental phase function is always anisotropic exceptfor a unique water sample and, when a more pure sample is taken as the reference,the preceding reference water sample becomes “bad”. The real reason of thisparadox is the interpolation of the thermodynamic fluctuation approximation to theregion of small scattering angles.

To explain the inconsistency between theoretical predictions and experimentaldata of the volume scattering function in clear water, the hypothesis of fractalstructure of the spatial adjustment of optical inhomogeneities in water was proposedShybanov (2007a, 2008; Shybanov et al. (2010, 2011). The revision of the theory oflight scattering in a liquid is essential to explain correctly experimental data and topredict the results of more complex experiments. Contrary to common assumptionsthat molecules under equilibrium conditions satisfy the Maxwell–Boltzmann dis-tribution, it was supposed that the relation between the potential energy of themolecules in condensed matter and the density is ambiguous. Considering cor-puscular structure of matter and the shape of interaction potential in dense medium,different arrangements of the molecules can have the same density but differentenergies or almost the same energy but different local densities. The densityinhomogeneities in the liquid are due to the features of the redistribution of theenergy between the molecules, most of which undergo nonlinear vibrations withrespect to their temporal equilibrium positions. Thus, the molecules in the liquid arelocally ordered and the order is broken stepwise. Since the interaction between themolecules is nonlinear, the molecular dynamics processes are classified into slowand fast. The rearrangement of the skeleton (or its fragments) formed by theequilibrium positions and the energy exchange between the molecules of theskeleton and isolated molecules can be classified as slow processes (due to thesmallness of the ratio of the masses of the molecule and skeleton). Individual and


corporative (acoustic) vibrations of the molecules of the skeleton, as well as themotion of isolated molecules and their collisions with each other, are fast processes.The set of the trajectories of isolated molecules and boundaries between the locallyordered molecular clusters should be treated as the system of optical inhomo-geneities in the liquid. The volume scattering function is the space–time average ofthe square of the sum of the amplitudes of the partial electromagnetic waves. Forisotropy reasons, a system of spheres (whose refractive indices depend on the radiusof the sphere) scatters light equivalently to real inhomogeneities of arbitrary shapesin water. Such spherical clusters can generally overlap with each other. Analyticalsolution in approximation of fractal structure by spherical surfaces was derived byShybanov (2007a, 2008); Shybanov et al. (2010, 2011).

Figure 4.12 shows the experimental volume scattering functions at a wavelengthof 490 nm. It can be seen that the effect is pronouncedly multiplicative at least up to70°.

Setting relative refractive index m = 1.15 (characteristic of mineral particles), weobtain a concentration of about 6 � 10−3 mol/l, which is much higher than theconcentration of impurities in Milli-Q water; this fact indicates the adjustment offluctuations of the permittivity of water itself. The spectral features of the simulatedand experimental volume scattering functions are compared in Fig. 4.13. Theexponent of the spectral dependence of the volume scattering function c(h) isobtained from the approximation bs(h, k) � bs(h, k0)[k0/k]

c(h). Good agreementbetween the experimental and model curves for angles larger than 30° is apparentlynot observed in view of the weak sensitivity of the instrument and differencebetween the relaxation times of the optical signal for small and large angles.However, the presence of a local minimum of the exponent in the experimental datais confirmed by model calculations, because this local minimum is not reproducedin any existing model. In particular, according to the two-component mixture model(Kopelevich 1983) widely used in modern hydrooptics, c(h) increases monotoni-cally with h. It is worth noting that the difference between the refractive indices ofclusters or their nonspherical shape is optically similar to bulk local

Fig. 4.12 VSFs of purewater with the addition of 0.5vol.% of impurity liquid at490 nm


inhomogeneities. This means that the curve for the true exponent should liebetween two model curves obtained by Shybanov et al. (2010).

A theory based on the thermodynamic equilibrium principle cannot describe theoptical properties of water, because inhomogeneities in a liquid are spatiallyadjusted. In the proposed model, optical inhomogeneities in water are treated as atwo-dimensional pattern with contrast depending on the concentration of impurities.The performed experiments and simulations indicate that the fraction of moleculesthat are located in defects and exchange energy primarily between each other isabout 1%. This is apparently responsible for the long relaxation time of the opticalsignal in water mixed with small additions of water with a different density.According to estimates of the size distribution function of inhomogeneities, anincrease in the scattering intensity at all angles cannot be attributed to turbulence.The assumption of the adjustment of deviations of the refractive index from thevolume-averaged value makes possible to explain spectral-angular features of thevolume scattering function of water and its variation in the presence of low con-centration of impurities. The most fundamental conclusion is that most inhomo-geneities are nonlocal.

4.5 Field Measurements of Volume Scattering Function

While measurements of absorption and attenuation coefficients of natural watershave become routine, measurements of the volume scattering function haveremained elusive. But recent years renewed interest in the volume scatteringfunction, and instruments now exist to measure this missing quantity in situ rela-tively easily. One of these instruments is Multi-Spectral Volume Scattering Meter(MVSM), built at MHI. Measurements of the VSF and other optical properties areshown for a variety of areas, from clear ocean waters to turbid rivers and largephytoplankton blooms.

Fig. 4.13 Experimental andsimulated exponents of thepower-law angulardependence of VSF


4.5.1 Volume Scattering Function Measurementsin the Middle Atlantic Bight

Since 2000 an advanced instrument to measure an angular scattering coefficient ofnatural waters was employed in several Long-term Ecological Observatory(LEO-15) expeditions near the coast of New Jersey, and expedition in the northernGulf of Mexico. The Mid-Atlantic Bight, because of its wide shallow shelf and highinput of nutrients, has one of the largest productivity rates. A large fraction of thein-water biogenic material sinks to the bottom and supports benthic microbialactivity. The data analyzed were collected in the vicinity of the LEO-15 on the NewJersey shelf, which is located in the Middle Atlantic Bight (Fig. 4.14).

Optically, this region is interesting for several reasons that add complexity to theinterpretation of ocean color data: (a) large sources of organic and inorganic par-ticles, (b) distribution of particulate material is strongly influenced by advection aswell as processes specific to particles (aggregation, sinking, floating, growth),(c) terrigenous sources of dissolved and particulate organic material. It provides achallenge to inversion algorithms designed to obtain particle properties from opticalmeasurements. The results of these measurements show that coastal waters can bedivided into two distinct types: one—similar to waters near California coastaccording to Petzold findings, and the other—biologically more pure type (Haltrin1999) experimentally characterized during LEO-15 experiment in 2001. The maindifference between Petzold-type coastal waters and biologically pure waters is thetype of dependence between backscattering and beam scattering coefficients. Theratio of these coefficients, or backscattering probability, for the biologically purecoastal waters is about three times smaller than for the Petzold-type waters. Thisresult is essential for algorithms applied to process optical information gatheredfrom satellites and aircrafts and for modeling light propagation in water. (Haltrin

Fig. 4.14 Location ofstations sampled in July of2000 (circles) at the NewJersey coast as well as thetransect performed on 27 July2000 (pluses). Crosses denotethe position of the two nodesmoorings of the LEO 15observatory. Thick circlesdenote the shallow waterstations (depth *2–6 m)where the highest values ofbackscattering (>0.03) wereobserved


et al. 2002, 2003). The LEO-15 experiment added a significant number of VSFs(Fig. 4.15).

VSFs measured in the Marine Field Station polygon of Ruthert University showthe order of magnitude of the variability of scattering at high angles, greater than inthe middle angle range. Significant variability is noted for the slope of volumescattering functions at small angles. At 40–150° angles the shape of curves isbasically similar. Typical local increasing of scattering is observed at angles of160–180°. This is probably due to peculiarities of scattering for large mineralparticles. Because of the great variability of the volume scattering functions at smallangles and near 180°, the shape of curves can not be described by asingle-parametric model over the full angle range.

Figure 4.16 (left) shows data from an offshore station, where vertical changesare largely due to variations in particles of organic origin. Figure 4.16 (right) isfrom a shallow station, where variations with depth are mostly caused by resus-pension of mineral particles.

These data show a significant environmental variance in VSF for small angles(from tenths of a degree to several degrees) in the forward direction, and from 170°to 177.6° in the backward direction, i.e., in regions, where reliable measurementswere not performed earlier. These observations are of fundamental importance tothe accurate modelling of the propagation of radiation in the ocean.

The analysis of this dataset, based on processing with DataDesk software andmanual study of measurement conditions, allows us to divide all 869 phase func-tions into two categories: (1) typical oceanic waters, including a majority of Petzoldand Mankovsky phase functions and a bulk of LE0-15 phase functions obtained in2000; (2) biologically stable oceanic waters, including a majority of LE0-15 phasefunctions measured in 2001 plus some phase functions from other experiments. Thesecond category exhibits different kind of relationship between backscattering and

Fig. 4.15 Examples ofvolume scattering functionmeasured on polygon on theAtlantic shelf


scattering coefficients (Lee et al. 2003b). This type of water belongs to the class ofbiologically stable waters and can be perfectly modeled. The content of smallinorganic particles in this type of waters is smaller than in typical oceanic watersdue to processes of sedimentation and biological absorption of inorganic particlesby phytoplankton. The rough weather conditions usually destroy the biologicalequilibrium by releasing absorbed inorganic particles in open ocean and by raisingterrigenous sediments from the bottom in shallow water. A new set of threeregression relationships that couples backscattering coefficient with scatteringcoefficient of marine (and lake) waters is proposed (Haltrin et al. 2002). This set ofequations is valid in the range of scattering coefficient 0.002–10 m−1 and is basedon experimental measurements of 875 scattering phase functions in diverse areas:Atlantic, Pacific, Indian and Southern oceans, Mediterranean and Black seas, andlake Baykal. These regressions could be used to derive optical products fromremote sensing data, and to model visibility and laser propagation in natural waters.

In situ measurements of oceanic optical properties in more than 99% of cases donot include measurements of VSF or backscattering coefficient. For that reason it isvery important to have some reasonable estimates of these inherent optical prop-erties. The estimation of scattering phase function based on measurements ofscattering and attenuation was proposed by Haltrin (2000). The estimation ofbackscattering coefficient based on VSF value at 140° was proposed by Maffioneand Dana (1997). The relationship between b(140°) and bb is based primarily ontheoretical estimates (Maffione and Dana 1997). For that reason it is practicallyimportant to derive similar relationship based on our extensive experimental data(Haltrin et al. 2003). The plot of bb as a function of b(140°) for all 869 VSFs isshown in Fig. 4.17.

Square root of angle [(degree)1/2] Square root of angle [(degree)1/2]

Fig. 4.16 Vertical variations in the volume scattering function in waters off the New Jersey coast


Chang and collaborators (Chang et al. 2002) analyzed the variability of opticalproperties near LEO and related them to the underlying physics. They found thevariability in optical properties is dominated by the tides and the presence of acoastal density front with an associated geostrophic current. Here we focus on aspecific optical parameter, the backscattering ratio, its distribution and the addi-tional information it provides for the study of particulate dynamics.

Particulate scattering and backscattering are two quantities that have traditionallybeen used to quantify in situ particulate concentration. The ratio of backscatteringby particles to total scattering by particles (the particulate backscattering ratio) isweakly dependent on concentration and therefore provides us with information onthe characteristics of the particulate material, such as the index of refraction. Theindex of refraction is an indicator of the bulk particulate composition, as inorganicmaterials have higher indices of refraction relative to oceanic organic particles suchas phytoplankton and detrital material that typically have high water content. Forinstance measurements collected near the Rutgers University Long-term EcosystemObservatory in 15 m column of water in the Mid-Atlantic Bight were used toexamine the backscattering ratio (Boss et al. 2004). Using four different instru-ments, the HOBILabs Hydroscat-6, the WETLabs ac-9 and EcoVSF, and a pro-totype VSF meter, three estimates of the ratio of the particulate backscattering ratiowere obtained and found to compare well.

Cross-comparison of the total scattering coefficient derived from our scatteringmeasurement vs data measured by AC-9 (Fig. 4.18) shows a very good agreementof data. Correlation coefficient is 0.98.

This is remarkable because the application of instruments with major differencesin design and calibration. The backscattering ratio can be used to map different

Fig. 4.17 Relationshipbetween volume scatteringfunction at 140° andbackscattering coefficient


types of particles in the nearshore region, which suggests that it may act as a tracerfor water moves. We find a significant relationship between the backscattering ratioand the ratio of chlorophyll to beam attenuation. This implies that these traditionalmeasurements may be used to identify where phytoplankton or inorganic particlesdominate. In addition, it provides an independent confirmation of the link betweenthe backscattering ratio and the bulk composition of particles.

Boss and collaborators (Boss et al. 2004) have shown, through the use of fourinstruments, and three estimates of the backscattering ratio, that the latter can becomputed routinely for the waters sampled in this study within approxi-mately ±0.003. The backscattering ratio intercomparison has never been attempted,and suggests that the current technology satisfy requirements for such characteri-zations. In addition, the large dynamic range of the particulate backscattering ratio(about seven times), suggests that errors of approximately 20% in either the value ofscattering or backscattering are not likely to affect the derived bulk index ofrefraction of the particle assemblage. It was found (Boss et al. 2004) that thisinversion is sensitive to absorption for highly refractive particles when slopes ofhyperbolic particle size distribution are less than about 3.5.

Therefore, without additional information on bulk particulate absorption, theinversion model is limited in ability to retrieve the bulk index of refraction ofparticle populations dominated by highly refractive, relatively large particles, towithin ±0.02. Such particles can be present in coastal regions in waters withrecently resuspended sediments. Despite the limitation of the model in providing atight range for estimating the index of refraction of highly refractive inorganicparticles, it does appear to provide a robust differentiation between particles withdifferent bulk composition (Fig. 4.19).

The data set presented here provides evidence of usefulness of the backscatteringratio as a biogeochemical particulate tracer. Traditionally, both backscatteringmeters and beam transmissometers have been used to quantify the concentration ofparticulate material. Backscattering is more sensitive to the index of refraction thanbeam attenuation, and we therefore observed a stronger signal of the refractive

Fig. 4.18 Intercomparison ofVSM and AC-9measurements of “b”


sediment particles in the particulate backscattering. The ratio of backscattering tobeam attenuation of particles removes the effects of concentration, and providesinformation regarding particulate composition. In summary, using four differentinstruments, three estimates of the ratio of the particulate backscattering ratio wereobtained and found to compare well. The backscattering ratio is found to haveinformational content not easily seen in either attenuation, backscattering, orchlorophyll concentration (chl). Finally, by relating the backscattering ratio with theratio of (chl) to beam attenuation, an additional link between particle compositionand the backscattering ratio was established.

4.5.2 Volume Scattering Function Measurements in OtherUS Coastal Waters

Volume scattering function measurements were held during Naval ResearchLaboratory cruise on RV “Pelican” in Gulf of Mexico in May 17–26, 2002. Thewater optical properties correspond to coastal type. The VSF meter was used in 2ways: (i) flow through; (ii) discrete water samples (1–CTD surface, 2-near bottom).Figure 4.20 shows typical VSF for surface waters of the Gulf of Mexico.Significant backscattering compared to artificial samples is noticed.

In September 2006 Naval Research Laboratory participated in large collabora-tive effort to measure physical, optical, and biological properties of waters ofMontego Bay, California (Gray et al. 2007). MVSM, LLIST, Hydroscat-6, andECO-VSF were used for measurements. During the experiment one of the largest“red tide” events occurred in the region. Measurements of the oceanic VSF weremade in a “red tide” algal bloom by several instruments. The overall agreementbetween instruments is outstanding. Interesting angular structure caused by thedominating presence of single-celled organisms was observed.

Fig. 4.19 Backscatteringratio at 532 nm versus ratio ofchlorophyll to beamattenuation (660 nm) basedon more than 3100 0.5 mbinned data points. Thecorrelation coefficient isr = −0.75


The shape of the VSF is shown to be surprisingly variable, depending on bothparticle types and sizes (Gray et al. 2008). It can be concluded that this shape canaffect the measurements of other optical instruments designed for measurements inseawater.

4.5.3 Volume Scattering Function Measurementsin European Seas

4.5.3.1 Adriatic Sea

Measurements of the volume scattering function with MVSM were performedduring different seasons in the northern Adriatic Sea on board the “Acqua Alta”Oceanographic Tower (AAOT) located off Venice. Figure 4.21 showing the posi-tion of the “Acqua Alta” Oceanographic Tower site and transects sampled duringthe campaigns of October 2004, July 2005 and April 2006.

Observed differences with the commonly used Petzold’s functions are signifi-cant, in particular for the “open ocean” and “coastal” types in the backwarddirections. The use of an empirical relationship for the derivation of bb(k) from asingle measurement b(h,k) at h = 140° for the Hydroscat-6 was validated for thiscoastal site. Finally, the use of the Kopelevich VSF model (Kopelevich 1983)together with a measurement of bp(k) at k = 555 nm allowed the reconstruction ofthe VSF to within about 35%. As a part of the processing the contribution of seawater is subtracted from b providing the VSF for particles only, bp, and conse-quently, b and bb. Such measurements, collected during three different seasons,were performed in order to complete the optical characterization of this coastal siteat which a comprehensive timeseries of bio-optical measurements has been col-lected since 1995 for optical modeling and validation of satellite ocean color data.The data constitute the first representation of the Volume Scattering Function forthe Adriatic Sea waters (Berthon et al. 2007b).

Fig. 4.20 Example of VSFmeasured in surface waters ofthe Gulf of Mexico


Figure 4.22 illustrates the individual VSF measured at the AAOT site and alongthe transects for 510 nm together with the Petzold’s bp. It can be noticed that almostall bp functions lie between the typical “coastal” and the “harbor” functions mea-sured by Petzold. A few actually measured near the Venice Lagoon are closer to the“harbor” type.

The VSF modeled by Kopelevich is also presented in Fig. 4.22 for the extremesituations (minimum and maximum values of scattering) observed during themeasurement campaign. It can be seen that they rather well describe the shape andintensity of the VSF measured in the northern Adriatic (Berthon et al. 2007a).

The estimation of bb(k) thruogh Hydroscat-6 instrument relies on a singlemeasurement of b(140, k) and application of an empirical relationship linking thetwo quantities. Using the measured VSF, linear relationships between total bb(k)and b(h, k) for h from 90° to 170° with an increment of 10° were investigated andpresented in Fig. 4.23 for k = 510 nm and h = 120 and 140°. As already observed

Fig. 4.21 “Acqua Alta” Oceanographic Tower (left) and map of the northern Adriatic Sea fieldexperiments (right)

Fig. 4.22 VSF measured inthe Adriatic at 510 nm (thinblack lines; dashed = stationsnear the Venice lagoon) andPetzold’s VSF at 514 nm(thick black lines;solid = open ocean,dashed = coastal,dotted = harbor). The filledcircles correspond to theapplication of the model byKopelevich


by Oishi (1990), relationships are robust as confirmed by the extremely high r2,although the slopes of the linear regressions are slightly different from those heprovided (note that for h = 140° the local relationship is very close to that used inthe Hydroscat-6 data processing). When considering all wavelengths, the r2 areslightly higher for h = 120° (from 0.998 to 0.999) than for h = 140° (from 0.995 to0.998).

Important differences were observed with respect to the commonly usedPetzold’s functions whereas the Fournier-Forand’s analytical formulation provideda rather good description of the measured VSF. The comparison of derived scat-tering [p(k)] and backscattering [bbp(k)] coefficients for particles with the mea-surements performed with an AC-9 and a Hydroscat-6, showed agreement within20%. The use of an empirical relationship for the derivation of bb(k) from b(h,k) ath = 140° was validated for this coastal site although h = 118° was confirmed to bethe most appropriate angle. The low value of the factor used to convert b(h,k) intobb(k) during the Hydroscat-6 processing partially contributed to the underestima-tion of bb(k) with respect to the MVSM. Finally, the use of the Kopelevich modeltogether with a measurement of bp(k) at k = 555 nm allowed to reconstruct theVSF with average RMS differences of 8–15%.

4.5.3.2 Tyrrhenian Sea

International field experiment AOPEX 2004 was conducted in Mediterranean Seain June 29–August 17, 2004 on board of RV “Le Suroit” within the framework ofCNRS programs (France). Measurements were taken at two sites exhibiting dif-ferent bio-optical conditions and where long-term observations are repeatedlymade. First site, Boussole (N 43°22′, E 7°54′), is located in mesotrophic waters 60miles south-east from Nice. Second site (N 40°20′, E 11°30′) lies in oligotrophicwaters in Tyrrhenian Sea between Italy and Sardinia. Measurements at intermediatestations were also performed.

Fig. 4.23 Value of bb versus b(h) at 510 nm for h = 120° (left panel) and h = 140° (right panel)as derived from the measured VSF. Filled circle—individual samples, solid line—linearregression, dashed line—Oishi, dotted line—Hydroscat-6 processing


Our interest to light scattering in clear natural water was encouraged by set ofcontradictions between traditional theory of scattering and experimental data.Concerning scattering function of water the inconsistency is observed at smallangles, namely, as water becomes more pure, the angular slope of scattering isincreased. The lack of experimental data of spectral properties of VSF at smallangles was partially filled during AOPEX cruise.

The measurements were carried out at 8 wavelengths and usually were per-formed at 3 depths, i.e., at the surface, chlorophyll maximum and below it(sometimes quite deep). The output results included particulate volume scatteringfunctions from 1.25° to 179° and additionally such integral parameters as bp, bbpand backscattering ratio. Figure 4.24 demonstrates typical curves for 2 investigatedregions.

The variability is observed mainly in the backscattering. So bbp(k) can beapproximated by a power law with exponents −1.31 and −1.77 for Boussole andTyrrhenian, correspondingly, while bp(k) is approximated by power law withexponents −0.51 and −0.49 (Fig. 4.25).

The spectral differences of bbp(k) and bp(k) result in a wavelength dependence ofthe backscattering ratio which was spectrally continually decreased: from 0.0011 to0.00085 for Boussole and from 0.0011 to 0.0007 for Tyrrhenian Sea.

As also can be seen from Fig. 4.24, the difference between turbid and clear waterVSFs is more substantial at general angles, and in clear natural water the angularfunction is more forward peaked. Generally, the scattering properties of water areinterpreted as the effect of varying concentration of big biological and mineralparticles. In this case in clear Tyrrhenian water the influence of phytoplanktonwould be essential based on VSFs at small angles. However the spectral shape ofbackscattering does not flatten but vice versa (see the slope 1.7 against 1.3).Alternatively if the small particles determine backscattering features in clear waters,

Fig. 4.24 Averagedparticulate VSFs in twodifferent regions of theMediterranean Sea


in order to explain strong forward scattering we have to believe that their positionsinside water body are really correlated. So our conclusion is that small particles playdominating role in the process of light scattering in clear natural water.

4.5.3.3 Southern Baltic Sea

Particle volume scattering functions measurements were made during the cruiseonboard the RV “Oceania” in May 2006 in Gulf of Gdańsk and in Southern BalticSea. Measurement sites are shown in Fig. 4.26.

We intended to display the variability of water properties on the path from openwaters of Baltic Sea to turbid waters near the Vistula Mouth. Measured functionswere compared with Petzold Average-Particle Phase Function. Spectral variationsand instability of measured scattering and backscattering coefficients are presented.

400 450 500 550 600 650

0.000

0.001

0.002

0.003

0.0

0.1

0.2

0.3

wavelength, λ, nm

back

scat

teri

ng b

y pa

rtic

les,

bbp

, m-1 scattering by particles, b

p , m-1

Tyrrhenian Sea, 5m

average bp

average bbp

bp ~ λ-0.49

bbp ~ λ-1.77

Fig. 4.25 Example of spectral integral VSF characteristics for the Tyrrhenian Sea

Fig. 4.26 The investigated area with marked points of VSF measurements


The instability has not been mentioned in the literature before. It is possible toexplain the effect of this instability by dissipation and reabsorption of adsorbedyellow substance during water flowing through the instrument. In deep layers and atthe stations near the Vistula Mouth the effect of instability is minimal. Apparentlythis effect depends on the ratio of dissolved to suspended fractions (Freda et al.2007).

The spectral variation of measured particle volume scattering functions isnoticeable. Generally, the values of VSF decrease with increasing wavelength.However inverse relation vas observed in general angular range at station P1(Fig. 4.27 left) and for angles close to backward direction at station ZN2 (Fig. 4.27right). These changes are probably caused by local variability of suspended parti-cles composition.

The Petzold Phase Function was measured at only one wavelength, 514 nm,with wide bandwidth of 75 nm. This explains why this function was compared withphase functions determined from measured VSFs for the closest wavelength—490 nm. Some results of this comparison are presented in Fig. 4.28.

The shape of volume scattering functions is generally in accordance with thePetzold’s one. All of them exhibit a strong forward peak. Moreover they also showlow values of backscattering ratio and a smoothed shape. However differencesbetween measured particle volume scattering functions and the Petzold’s one havebeen observed both at small angles and at backward direction. Values of magnitudeof measured phase functions in the range of 0.5°–10° are about two times higherthen the magnitude of Petzold’s function, and about 1.7 times in the backwarddirection. For angles comprised between 170° and 180°, the Petzold Phase Functionis almost flat whereas functions measured in Southern Baltic display a clearincrease.

As it was mentioned above, the instruments for VSF measurements are complex.So, the possibility of estimating bp and bbp from VSF measurements at a singleangle was tested (Shybanov et al. 2007b). Figure 4.29 illustrates the result of thisanalysis performed at for 555 nm.

0.0001

0.001

0.01

0.1

1

10

100

1000

0 20 40 60 80 100 120 140 160 180Scattering angle, degrees

VSF,

m-1

sr-1

443490555620

P1

0.0001

0.001

0.01

0.1

1

10

100

1000

0 20 40 60 80 100 120 140 160 180Scattering angle, degrees

VSF,

m-1

sr-1

443490555620

ZN2

Fig. 4.27 The VSFs of surface waters measured for four wavelengths at stations P1 and ZN2


Ratio of bp(555)/b(4°) calculated for whole data set provides value of 0.119 sr,and ratio bbp(555)/b(140°) gives 6.96 sr. These values are close to data of previousinvestigators (Man’kovsky 1971; Maffione and Dana 1997). The correlationscoefficients are high r2(4°) = 0.995, and r2(140°) = 0.999, thus it is possible toaccurately estimate the particulate scattering and backscattering coefficients in theBaltic Sea using single angle measurement. Particle scattering coefficient bp andparticle backscattering coefficient bbp have significant spectral variations. For opensea and for coastal stations the shape of spectral curve of bp is more flat than forintermediate stations. The averaged slope for bbp spectra is greater than for bp. Thespectral slope of particulate scattering coefficient bp(k) averaged over the whole

Fig. 4.28 Phase functions measured at 490 nm and Petzold Average-Particle Phase Function

0 10 20βp(4°), [m-1sr-1]

0.0

0.5

1.0

1.5

2.0

2.5

b p, [

m-1

]

bp = 0.1189βp(4°)R2 = 0.9947

0 0.002 0.004 0.006βp(140°), [m-1sr-1]

0.00

0.01

0.02

0.03

0.04

b bp,

[m-1]

bbp = 6.9647βp(140°)R2 = 0.9987

Fig. 4.29 Relationship between the particulate VSF at 4°, bp(4°), and particulate scatteringcoefficient bp (left), and for the particulate VSF at 140°, bp(140°), and particulate backscatteringcoefficient bbp (right)


data set can be approximated by power law with exponent −0.47 and standarddeviation of 0.29. The slope of the particulate backscattering coefficient bbp(k) hasexponent −0.91 and standard deviation of 0.34. Figure 4.30 presents the angularvariation of the averaged spectral slope of particulate VSF.

On the slope curve a maximum is observed near 8°, a rather flat section from 30°to 90° and then monotonous growth up to 180°. The highest values of variationcoefficient appear in the angle range 30–100° and for angles lower than 4°. Themonotonous growth of the spectral slope is consistent with theory and modeling(Kopelevich 1983) predicting that contribution of small particles in scatteringincreases with the angle. The maximum in the region of 8° does not have a theo-retical explanation. It should be noted that for some samples with high particulatescattering coefficient this maximum was not observed or was negligible.

It is the first time that a significant amount of data on the spectral volumescattering function (angular range from 0.5° up to 178° and 4 wavelengths) wascollected in the Baltic Sea. The waters of the investigated area have a wide range ofscattering properties, and the mineral particles are responsible for a large portion ofwater scattering. The integral parameters such as total scattering and particulatebackscattering have a spectral dependence. The slope calculated for averaged bpspectra is lower than for bbp spectra. Both parameters are spatially variable. Usingthe scattering coefficient at 4° and at 140° it is possible to accurately estimate theparticulate scattering and backscattering coefficients in different types of waters.Our results showed that the spectral properties of the scattering coefficients weresignificantly influenced by absorption of particles, especially at 443 nm. Forangular range 40–180° the spectral slope of VSF is close to theory, but a theoreticalexplanation of maximum near 8° would be required.

4.5.3.4 Black Sea

A field experiment was carried out in summer 2002 on an oceanographic platformnear the coast of Crimea, in the Black Sea. Example of results is shown onFig. 4.31. Our analysis revealed that the mineral particles are the primary

Fig. 4.30 Averaged spectralslope of VSF with standarddeviation and coefficient ofvariation (CV) versus ofscattering angle


component influencing the scattering and backscattering coefficient in the studyarea. Good correlation was obtained between backscattering coefficient bbp andnonalgal particles absorption coefficient. The ratio Chla/cp (where Chla is thechlorophyll a concentration and cp is the beam attenuation coefficient) did notcorrelate with the backscattering ratio and thus could not be used in this experimentas an alternative proxy to estimate the bulk composition of the particles.

Spectral variations of both volume scattering function and the phase function ofparticles were analyzed as a function of the scattering angle. The spectral variationof bp and bbp was less steep than for the open ocean waters. That is explained by theinfluence of the absorption on the scattering process, especially in short-waverange, as a consequence of the anomalous dispersion. The averaged backscatteringratio ~bbp varied spectrally within 4%. Nevertheless, a high spectral variability of ~bbp(around 30%) was also observed suggesting that the use of a flat spectral variation isnot accurate in coastal zones (Chami et al. 2005).

We examined the VSF at 140° because it is often used to derive the backscat-tering coefficient of particles. The data showed lower values of bp(140°) at 443 nmrelative to 555 nm, especially in the second half of the field experiment. This wasascribed to the occurrence of absorbing nonalgal particles in the study area. Theangular distribution of the spectral ratio of VSF showed complex features. The threetrends prevailing during the experiment were: (i) a monotonic decrease with h,(ii) minima near 20° and 160°, and (iii) a maximum near 120°. Besides usingabsorption, the only way to reconcile the observations with theory was to assumethat the size distribution of the particles was the sum of a Junge power law and alognormal function. In particular, the introduction of modal distribution of particles

Fig. 4.31 Example of VSFobtained at the Black Seaoceanographic platform


allowed to retrieve successfully the minima and maxima observed in the spectralratio of the VSF. Despite the fact that we were unable to validate rigorously theabsolute values of the parameters derived from theory, the order of magnitude of theretrieved complex refractive index was consistent with other ancillary optical data,such as the chlorophyll a concentration and the magnitude of the backscatteringratio. The occurrence of monodisperse particles was also consistent with the highspectral variability observed in the backscattering ratio. In a similar fashion thespectral variation of particulate phase function was highly dependent on the scat-tering angle. The angular variation of bp spectra was also sensitive to absorptioneffects and to size distribution of particles. We measured departure as high as 40%at 140°. Therefore our results suggest caution in the use of spectrally neutral phasefunctions in radiation transfer modeling. As a result of this study, future effortsshould be directed toward routine spectral measurements of the VSF over the fullrange of scattering angles for better understanding of the directional effects ofparticles and for robust prediction of the water leaving signal for remote sensingpurposes (Chami et al. 2006).

During July 2011 and September 2012 four bio-optical cruises were carried outin the Western and Central Black Sea in the frames of EUROFLEETS BIO-OPTproject and NATO “Science for Peace” project. First expedition was conducted inthe shelf waters of the Romanian coast on board of the RV “Mare Nigrum”(Constanta, July 01–05, 2011). Next field activities were continued on board theRV “Akademik” (Varna, July 08–12, 2011; July 13–22, 2011; August 31–September 15, 2012). Marine stations were chosen to represent seawaters likelycharacterized by different bio-optical regimes in the Romanian, Bulgarian,Ukrainian and Turkish waters. Measurements of the VSF on discrete water sampleswere performed with MVSM. The instrument had twelve filters withcenter-wavelengths 380, 400, 412, 435, 456, 490, 532, 560, 590, 625, 683, 780 nm.Eight (380, 412, 435, 456, 590, 625, 683, 780 nm) filters were used during 2011campaigns and ten (380, 412, 435, 456, 490, 532, 560, 590, 625, 780 nm) filterswere used at 2012.

Spectral VSF measurements in a wide spectral and angular range were per-formed at more than 230 stations. The main variation of VSF consisted in changingof absolute values of the light scattering, i.e. bpðkÞ. Coefficient bpð380Þ increasedfrom 0.2 m−1 in the Central part of the sea up to more than 4 m−1 in coastal turbidwaters. Backscattering ratio was in range 0.007–0.02 and grew with decreasingwavelength (Shybanov and Lee 2013).

The spectral dependence of VSF was rather complex. The function c hð Þdescribes the spectral slope of VSF, characterizing a selectivity of light scattering inthe water versus the angle. This function has some local minima that could beexplained by spatial inhomogeneity of the water structure (Shybanov et al. 2010).Figure 4.32 shows the angular dependence of the spectral slope function.


4.6 Volume Scattering Function Variability

To illustrate the variety of VSFs an examples of VSFs from different regions of theWorld Ocean are presented, including deep waters of Tyrrhenian Sea and AtlanticOcean and shelf zones of the Black Sea (Fig. 4.33). The variation of VSF exceedstwo orders of magnitude.

The sensitivity of the VSM covers the dynamical range up to very clean water. Itallows to conduct measurements all over the World Ocean.

Fig. 4.32 Angular dependence of VSF spectral slope for some Black Sea stations

Fig. 4.33 VSFs in differentregions of the World Ocean


Measurements made by the new polar nephelometer have shown that broadeningthe angle range creates new opportunities in studying the scattering properties ofseawater. The data displayed above show convincingly that the main variance ofscattering occurs at small angles (forward directions) and for angles close to 180°(backscattering), i.e. in such regions, where reliable measurements were not per-formed earlier.

4.7 Conclusion

An advanced instrument for VSF measurements was designed and a new shadowmethod for small angle scattering measurements was successfully developed byscientific team of Department of Marine Optics and Biophysics of MarineHydrophysical Institute, Sevastopol, Russia. A Multi-spectral Volume ScatteringMeter (MVSM) measures volume scattering function in 8 spectral bands and inangular range 0.5°–178° with angular resolution 0.25°. The instrument can be usedfor field observations and for laboratory measurements. Testing and calibration ofMVSM were made using clean water and solutions with known optical properties.Comparison of instrument capability with other scattering instrumentation devel-oped by U.S. commercial and academic laboratories has shown the capability of theMVSM instrument with a wider angle range and resolution.

The sensitivity of the VSM covers the dynamical range up to very clean water. Itallows to conduct measurements all over the World Ocean. Empirical relationshipsconnecting integral characteristics of scattering and backscattering to values of VSFat different angles were obtained for different water types. Regression coefficientsdepend on both the wavelength and water turbidity.

Measurements made by the new polar nephelometer have shown that broadeningthe angle range creates new opportunities for studying the scattering properties ofseawater. Investigation data listed above show convincingly that the main varianceof scattering is noted at small angles (forward directions) and for angles close to180° (backscattering), i.e. in regions where reliable measurements were not per-formed earlier.

Elaborated studies of seawater VSF of seawater made possible to propose asolution for the radiation transfer equation solution and to obtain an improvedoptical model for seawater reflectance.

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Chapter 5Remote Sensing of Crystal Shapesin Ice Clouds

Bastiaan van Diedenhoven

5.1 Introduction

Archaeological findings reveal a deeply rooted human obsession for atmosphericoptical phenomena such as rainbows and ice cloud halos. Our pre-historic ancestorslikely attached divine or meteorological meanings to such optical phenomena assuggested by ancient rock drawings (Ping-Yü and Needham 1959; Sassen 1991,1994). Detailed observations of rainbows and ice cloud halos were later describedby Aristotle and other scholars of antiquity. It took until the age of enlightenment,however, for the first qualitatively correct interpretations of these phenomena to bedocumented (Nussenzveig 1977; Greenler 1990; Tape 1994; Tape and Moilanen2006). For example, it was the French physicist and priest Edme Marriotte who, in1691, was probably the first to correctly attribute the commonly observed halo at anangle of 22° around the sun to randomly oriented hexagonal ice crystals, althoughhis ideas did not gain much attention until being revived 116 years later by ThomasYoung (Tape and Moilanen 2006). It was later realized that this hexagonal icemodel also offers explanations of parhelia (‘sundogs’) and other arcs, when crystalsare assumed to be oriented. One can thus argue that Mariotte’s interpretation of thehalo observations was the first successful remote sensing of ice crystal shape.

While liquid drops generally take shape of virtually perfect spheres, the observedvariety of cloud ice crystal shape is seemingly limitless (Baran 2009). Under mostatmospheric conditions and undisturbed, water ice grows as hexagonal prismsowing to its fundamental hexagonal molecular structure (generally denoted as Ih),where the growth rate of the basal and prismatic planes of the prisms are determined

B. van Diedenhoven (&)Center for Climate System Research, Columbia University, 2880 Broadway,New York, NY 10025, USAe-mail: [email protected]

B. van DiedenhovenNASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA


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by temperature and humidity (Pauling 1935; Bailey et al. 2012; Harrington et al.2013). However, because of varying atmospheric conditions throughout crystalevolution, aggregation, fracturing, riming, and other stochastic effects, observedcloud ice crystals are generally far more complex than simple hexagonal prisms(e.g., Baran 2009; Bailey and Hallett 2009; Bailey et al. 2012). Moreover, recentadvances in electron microscope technology allows imaging of the microstructureof ice surfaces, which reveals that roughness structures are generally prevalent(Pfalzgraff et al. 2010; Neshyba et al. 2013; Magee et al. 2014). Mixing of thehexagonal ice molecular structure with the cubical structure (denoted as Ic) can leadto such microscale roughness structures on the surfaces, in addition to more mac-roscale deviation of ice crystals from their fundamental hexagonal form (Kuhs et al.2012; Malkin et al. 2012, 2015; Murray et al. 2015; Hudait and Molinero 2016).Furthermore, sublimation and riming changes the surfaces of ice crystals (Ono1969; Ávila et al. 2009; Pfalzgraff et al. 2010; Magee et al. 2014).

The shape and size of cloud particles greatly affects their radiative and micro-physical properties (Yang et al. 2015; Furtado et al. 2015; Fridlind et al. 2016;Baran et al. 2016; Liou and Yang 2016). The omnipresence of clouds on Earthmeans that seemingly small changes in cloud particle properties can have profoundeffects on the radiation balance and precipitation (Stephens et al. 1990). In turn,cloud radiative properties affect the evolution of clouds owing to interactionsbetween radiation and cloud microphysics (e.g., Gu and Liou 2000; Gu et al. 2011;Russotto et al. 2016; Baran et al. 2016). While the dependences of radiative andmicrophysical properties on particle size are fairly well understood, knowledge onhow they are affected by the shape of ice crystals is still limited. The currentincomplete knowledge on the natural variation of ice crystal shape contributes toour inability to reliably represent ice clouds in global circulation models (GCMs).Biases in ice optical properties can lead to direct biases in modeled ice cloudradiative effects (Yi et al. 2013; van Diedenhoven et al. 2014a; Liou and Yang2016).

Furthermore, the relatively large uncertainties of cloud physics and the largeinfluence of clouds on the radiation balance means that weakly constrained vari-ables in GCM cloud physics parameterizations are generally adjusted to bring thesimulated global long- and shortwave radiation in agreement with measurements(e.g., Schmidt et al. 2014). Through this process, any biases in ice particle opticalproperties would thus be falsely compensated by inadvertently introducing biases incloud physical parameters. Moreover, biases in cloud particle shape and relatedmicrophysical quantities such as fall speeds can lead to biases in modeled cloudproperties such as thermodynamic phase, precipitation and cloud fraction (Furtadoet al. 2015). A new paradigm in ice microphysical modeling is emerging in whichice crystal shape characteristics are explicitly predicted (Hashino et al. 2007, 2011;Harrington et al. 2013), although a lack of observations for evaluation leave thesemodels largely unconstrained. In addition to ice crystal shape, global constrains onthe ice crystal size are also crucial in order to evaluate and improve the represen-tation of clouds in models. However, the large uncertainty on the variation of iceparticle shape, and thus optical properties, in turn lead to substantial uncertainties in

198 B. van Diedenhoven

ice particle size retrieved using satellite-based shortwave infrared measurements,such as by the Moderate Resolution Imaging Spectroradiometer (MODIS) (vanDiedenhoven et al. 2014b; Holz et al. 2016). In situ and laboratory measurements ofthe past decades have lead to an increasingly detailed picture of the macro- andmicroscale structure of ice under varying conditions and nucleation processes(Bailey and Hallett 2009). However, this picture is incomplete and likely biased,given the many technical and practical limitations of in situ and laboratory mea-surements and the uncertain and complex variation of ice nucleation and growthconditions and processes occurring in Earth’s ice-containing clouds (cf. Bailey et al.2012). Remote sensing observations of ice shape on global and regional scales arecrucial for the evaluation and refinement of parameterizations of ice crystalnucleation and evolution derived from in situ and laboratory observations andtheory.

This chapter offers a review of the current state of remote sensing of ice crystalmacro- and microscale structure. We focus on remote sensing techniques based onpassive total and polarized reflectance measurements. In addition, the potential oflidar with polarization capability to infer information about ice crystal shape isreviewed. A discussion on polarimetric radar is excluded here since currentpolarimetric radars are mostly sensitive to precipitation rather than cloud ice, andmuch of the polarimetric radar signal is related to particle orientation rather thanshape (Miao et al. 2003). First, in Sect. 5.2 we will briefly summarize the in situ andlaboratory observations that serve as foundations of the retrieval approaches. Sinceby definition remote sensing necessitates the interpretation of measured electro-magnetic radiation, we will review the optical properties of ice crystals in Sect. 5.3.Active and passive techniques for the remote sensing of ice crystals shape arereviewed in Sect. 5.4. A prospective will be given in Sect. 5.5, before concludingthe chapter in Sect. 5.6.

5.2 In Situ and Laboratory Measurements

In 1665, English natural philosopher Robert Hooke included some of the firstdetailed drawings of snow crystals in his book Micrographia (Hooke 1665). Thefirst detailed photographs were collected around the turn of the 20th century byenthusiasts such as Wilson Bentley who documented many snow crystals of variouscommon and uncommon shapes (Bentley 1927). The first known in situ aircraftobservations of ice crystal shape were performed during the second world war byGerman meteorologist Ludwig Weickmann, who performed about 100 meteoro-logical flights in an open cockpit airplane holding lacquer-coated shingles to cap-ture and replicate ice crystals (Bailey and Hallett 2009). Weickmann publishedresults and their interpretation after the end of the war (Weickmann 1945; AufmKampe et al. 1951). To date, many in situ and laboratory observations of ice crystalshape have been conducted. Here, we will give a brief overview of the general anddetailed shape characteristics measured in situ and in laboratories. Such

5 Remote Sensing of Crystal Shapes in Ice Clouds 199

observations are essential to direct and constrain remote sensing applications and tointerpret their results.

5.2.1 General Classification of Ice Habits

Crystal shapes are commonly classified according to the scheme of Magono andLee (1966) that contains 80 classes of snow and ice crystals. This scheme in turn isbased on previous work by Nakaya (1954), while a proposed expansion of theclassification was recently offered by Kikuchi et al. (2013). The identified icecrystal shapes are commonly referred to as ice ‘habits’. Nakaya (1954) and Magonoand Lee (1966) also showed that the occurrence of certain habits could be linked tothe meteorological conditions in which they grew and they provided diagrams ofoccurrence of certain ice habits as a function of temperature and humidity. Aroundthe same time, laboratory experiments in cloud chambers were performed thatsuccessfully reproduced many of the observed transitions of habits at certaintemperatures and humidities (Aufm Kampe et al. 1951; Mason 1953; Hallett andMason 1958; Mason et al. 1963). More recently, the most comprehensive habitdiagram for atmospheric ice crystals was published by Bailey and Hallett (2009),which is shown in Fig. 5.1. As is apparent from Fig. 5.1, there are two majordifferentiating characteristics of ice crystals: (1) plate-like versus columnar struc-tures and (2) single crystals versus polycrystalline particles. Plate-like crystals formwhen the basal planes grow faster than the prismatic facets, while the oppositerelation of facet growth rates lead to columnar crystals. Polycrystals can form forvarious reasons, for example, when several crystal components grow relativelyindependently from a common core, when crystals aggregate, or when conditionsvary during crystal growth. Comparing habits in different cloud types, one generalobservation is that compact and aggregated ice crystals with plate-like componentsoften occur in tropical deep convection (e.g. Noel et al. 2004; Connolly et al. 2005;Um and McFarquhar 2009; Baran 2009), while bullet rosettes are often found innon-convective (“in situ”) cirrus (e.g., Lawson et al. 2010; Fridlind et al. 2016).

The diagram produced by Bailey and Hallett (2009) (Fig. 5.1) is partly based onimages of individual ice crystals taken in clouds by the Cloud Particle Imager(CPI) aircraft probe (Lawson et al. 2001, 2006). The highly successful CPI probewas designed in 1997 by SPEC inc. and can image and count particles in the sizerange of 15–2500 lm, with a nominal resolution of 2.3—lm. As discussed byBailey and Hallett (2009), CPI images reveal many complex, polycrystalline shapesthat were not represented by laboratory studies before. Automatic habit classifica-tion applied to the images generally yield a dominance of unclassified or ‘irregular’crystals. Recently developed, more advanced classification schemes (e.g., Lindqvistet al. 2012) add more classes to previous CPI classification programs, but still yieldan abundance of unclassified crystals. The general dominance of crystals classifiedas ‘irregular’ may be partly attributed to the issues with the automatic classificationof 2D images (Stoelinga et al. 2007). Moreover, the CPI resolution is generally


Fig. 5.1 Diagram showing the general occurrence of ice crystal habits at varying temperaturesand water vapor supersaturations with respect to ice. Habit names are in the top diagram, whileexample images taken by the CPI instrument are in the lower diagram. See Bailey and Hallett(2009) for more details. Figure reproduced from Bailey and Hallett (2009). ©AmericanMeteorological Society. Used with permission.


insufficient to classify small crystals into classes other than “small irregular” or‘quasi-spherical’. Although true quasi-spherical crystals have been observed(Järvinen et al. 2016), for most cases the term ‘quasi-spherical’ is arguably amisnomer, since on closer inspection these crystals generally appear as compactpolyhedra with many small facets (Gonda and Yamazaki 1978; Bailey and Hallett2009) or irregular shapes with bulges and protuberances (Nousiainen et al. 2011;López and Ávila 2012).

In summary, in situ observations are found to follow the habit diagrams as afunction of temperature and humidity, but seemingly only for crystals that havegrown under relatively static conditions. Given the variability of atmospherichumidity, the fact that growing ice tends to fall to warmer temperatures and thepresence of up- and downdrafts in clouds, static growth conditions are an exceptionrather than the rule, especially in convectively-driven clouds. Furthermore, pro-cesses such as crystal aggregation, fracturing and riming add more complexity tonatural crystals.

5.2.2 Aspect Ratios of Ice Crystals

Traditionally, ice crystal classifications and observations have focused on thelarge-scale appearance of the crystals, that is, on the habits. As will be discussed inSect. 5.3, smaller scale details of the crystals are also of importance for their opticaland microphysical properties, if not more important than habit. One such property isthe relation between the thickness and length of particles or components of poly-crystals. For a hexagonal prism, the fraction between crystal basal plane width Wand prism length L is commonly referred to as aspect ratio a, which can be definedas

a ¼ L=W : ð5:1Þ

In this definition the aspect ratio of a column is greater than unity and that of aplate smaller than unity. However, this choice is arbitrary and confusingly somestudies define aspect ratio as the inverse of Eq. 5.1. Alternatively, van Diedenhovenet al. (2016a) proposed a definition that limits the aspect ratio to below unity forboth plates and columns, viz.

a� 1 ¼ minfL;WgmaxfL;Wg : ð5:2Þ

When this definition is used, it needs to be specified separately whether crystalsare column- or plate-like. Also note that the term ‘aspect ratio’ is sometimes usedfor the ratio between two perpendicular dimensions of complex ice crystals (e.g.,Korolev and Isaac 2003) instead of the ratio between dimensions of components ofthose complex crystals, as the definition used here.


Several early studies on dimensional relationships of ice crystals were reportedon by, e.g., Mason (1953), Ono (1969) and Auer and Veal (1970) (See Um et al.2015 for a completer list). Auer and Veal (1970) deduced power-law relationshipsbetween the orthogonal dimensions of relatively simple particles such as plates,columns and dendrites that are still much used today as references for opticalproperty calculations (e.g., Yang et al. 2015). Dimensions of more complex par-ticles, such as bullet rosettes and aggregates of plates, were estimated by, e.g.,Heymsfield and Andrew (1972), Mitchell and Arnott (1994), Xie et al. (2011), Umand McFarquhar (2007, 2009), Um et al. (2015) and Fridlind et al. (2016).However, most of these estimates are based on very limited data and unconstrainedassumptions and different techniques can lead to conflicting results (Um et al.2015). For plates, reported aspect ratios range from 0.01 to 0.5, while aspect ratiosof columns generally range from 1.5 to 5 (Auer and Veal 1970; Um et al. 2015). Ingeneral, aspect ratios increasingly deviate from unity as the maximum dimensionincreases. For bullet rosettes and their aggregates, reported aspect ratios of the armsalso range from 1.5 to 5.0 and are found to decrease towards unity with decreasingnumber of bullets attached to the rosettes (Um et al. 2015; Fridlind et al. 2016). Umet al. (2015) found that the aspect ratios of plates, columns and bullets in rosettesare not or only weakly sensitive to temperature, although a substantial temperaturedependence may be expected from theory (Chen and Lamb 1994; Harrington et al.2013). Note that the crystals for which the dimensions can be determined generallycomprise only a very limited subset of crystals in the observed cloud. Furthermore,the values listed above are mostly derived by the functional relationships betweencrystal (component) length and width that are determined from the data by theindividual studies. These relationships, however, do not reflect information aboutthe spread of data from which they are derived, which can be considerable (e.g.,Um et al. 2015; Fridlind et al. 2016). For other complex crystal habits, such asaggregates of plates or columns, determining the dimensional relationships of theircomponents from imagery is more problematic causing their aspect ratios to bemore uncertain (e.g., Um and McFarquhar 2009).

5.2.3 Microscale Structure of Ice Crystals

A perfect, ‘pristine’, ice crystal exclusively consisting of hexagonal molecularstructure (Ih) will grow smooth basal and prismatic surfaces. However, it has beenlong known that imperfections in the Ih mesh of growing water ice are prevalent,even under stable growth conditions (Pauling 1935; Mason et al. 1963; Cross 1969;Mizuno and Yukiko 1978). Electron microscopic observations of ice crystals grownin the laboratory under different conditions reveal varying imperfections orroughness structures on the ice surfaces (Pfalzgraff et al. 2010; Sazaki et al. 2010;Neshyba et al. 2013; Magee et al. 2014). As discussed in Sect. 5.3.2, such surfacedistortions strongly affect the scattering properties of ice crystals and need to beconsidered in techniques for the remote sensing of ice crystal shapes. Ice crystal


surface roughness generally decreases asymmetry parameters and depresses halosin phase functions at 22° and 46° (Yang et al. 2008a; Neshyba et al. 2013; vanDiedenhoven 2014; van Diedenhoven et al. 2014a; Yang et al. 2015).

An example of roughness structures observed on laboratory-grown ice crystalsunder various conditions is shown in Fig. 5.2 (Magee et al. 2014). Magee et al.(2014) found that such roughness structures are prevalent and not confined to anarrow range of macroscopic morphology, substrate, temperature, humidity, orgrowth rates under most conditions. Although they did not detect a systematicdependence on the degree of supersaturation or the rate of growth, clear surfacemorphology differences between growing and sublimating crystals were observed.Specifically, sublimating crystals develop scalloped depressions and sharp ridges,while more linear structures were observed for growing crystals (see Fig. 5.2, alsocf. Pfalzgraff et al. 2010). Indirect observations of roughness are provided by theSmall Ice Detector-3 (SID-3) probe mounted in cloud chambers and on aircraft.This probe measures the light scattered in near-forward direction by single icecrystals. As discussed also in Sect. 5.3.2, roughness structures will randomize thescattering angles at which light is scattered forward resulting in ‘speckled’ imagesmeasurement by the SID-3 probe, while pristine particles lead to more organized,symmetric measurements patterns. A subjective, arguably arbitrary roughnessparameter is deduced from these measurements to quantify the degree of roughness.Using the SID-3 probe mounted on aircraft sampling cirrus and convective clouds,Ulanowski et al. (2014) and Schmitt et al. (2016), respectively, found that rough-ness is ubiquitous. Somewhat in contrast with the results of Neshyba et al. (2013)and Magee et al. (2014), cloud chamber experiments with the SID-3 measurementsconducted by Schnaiter et al. (2016) indicate a clear correlation between thesmall-scale crystal complexity and the volume mixing ratio of available condens-able water vapor determining the growth rate. Furthermore, Ulanowski et al. (2014)reported on similar roughness in the growth and sublimation zones of cirrusobserved by the SID-3 probe, which also seems to contradict conclusions from

Fig. 5.2 High-magnification images of hexagonal ice crystals acquired by environmentalscanning electron microscopy revealing roughness structures on the crystal surfaces. The leftimage shows the crystal under vapor growth conditions. The center and right images show thecrystal at the initial and a progressed stage of sublimation, respectively. For details see Magee et al.(2014). Images courtesy of Dr. Nathan Magee at The College of New Jersey


electron microscope observations (Magee et al. 2014). Clearly, more research needsto be conducted to investigate the presence and level of roughness in the varyingtypes of ice clouds and under varying conditions.

Although roughness structures are found to be prevalent on natural ice crystals,this appears to be in conflict with the frequent sighting of halos around the sun at22° that are attributable to hexagonal crystals with smooth surfaces (Sassen et al.2003; Tape and Moilanen 2006; Verschure 1998). For example, Sassen et al. (2003)reported that 37.3% of the daytime ice cloud observations in their *10 year recordover Salt Lake City, Utah, showed indications of the 22° halo, with bright andprolonged halos occurring in 6% of the record. Van Diedenhoven (2014) reconciledthe abundance of rough ice surfaces detected with the prevelance of halos byconsidering cirrus clouds to contain mixtures of crystals with varying roughnesslevels. Van Diedenhoven 2014) concluded that the contribution by pristine crystalsto the total scattering cross section needs to be greater than only about 10% in thecase of compact particles or columns, and greater than about 40% for plates for the22° halo feature to be present in scattering phase functions. These results indicatethat frequent sightings of 22° halos are not inconsistent with the observed domi-nance of rough ice crystals. From SID-3 measurements in a halo producing cloud,Schmitt et al. (2016) also concluded that it does not require high concentrations ofhalo producing particles to produce an observable halo. The occurrence of 22° haloand other optical phenomena attributed to pristine crystals in some clouds but thelack there-of in other clouds illustrates that ice crystal microscale morphology ishighly variable. Note that the contrast and brightness of halos also depends on thecloud optical thickness (Kokhanovsky 2008).

Roughness structures can be effectively observed by present-day instruments,but the quantification of roughness remains elusive. One metric to quantifyroughness from electron microscope imagery was proposed by Neshyba et al.(2013) and relationships between this roughness metric and roughness parameter-izations used for calculations of optical properties of rough ice crystals, as discussedin Sect. 5.3.2, were also provided. Unfortunately, Magee et al. (2014) found thatthis metric proposed by Neshyba et al. (2013) is dependent on image magnificationand resolution and is thus rather subjective. As surface roughness has profoundeffects on the scattering properties of ice crystals, and an increasing number ofstudies of the formation roughness structures under varying conditions are con-ducted, it is advisable that a more universal definition of roughness ought to bedeveloped.

Apart from roughness structures on crystal surfaces, other small-scale ice crystalimperfections occur. Specifically, hollow endings in ice columns or bullets arefrequently reported (Heymsfield et al. 2002; Schmitt et al. 2007; Bailey and Hallett2009; Smith et al. 2015). Bailey et al. (2004) found that hollowness appears atrelatively high supersaturations of 50% and the depth of hollowness increases withincreasing supersaturation. The hollowness generally appears to be conical(Heymsfield et al. 2002), but step-wise hollowness is also reported (Smith et al.2015). Other observed crystals imperfections include air bubbles andsoot-inclusions (e.g., Hong and Minnis 2015; Panetta et al. 2016).


5.3 Ice Crystal Optical Properties

For the interpretation of ice cloud remote sensing observations, appropriate opticalproperties of ice crystals are essential. Because ice crystals have non-spherical,complex structures and are generally much larger than solar wavelengths, exactmethods for the calculation of optical properties, such as Lorenz-Mie theory (Mie1908) or the T-matrix approach (Mishchenko 1991), are not suitable. Commonly,ice optical properties are approximated using geometric optics (GO) calculations,sometimes in combination with other methods. Already in the 17th century, GOprinciples were applied by Edme Mariotte and others to explain ice cloud halos.Modern applications are based on Monte Carlo ray tracing techniques that combinecalculated paths of a large, finite number of light rays through the particle of interestassuming random orientations of the incoming ray. Early implementations of raytracing were presented by Jacobowitz (1971) and Wendling et al. (1979), but moreversatile computer codes that can be applied to complex particles and take intoaccount light polarization are provided by Takano and Liou (1989) and Macke(1993) and are still much used today with some modifications. Although conven-tional GO has several shortcomings, mostly these issues are limited to its appli-cation to small and pristine particles with smooth surfaces (Bi et al. 2014; Yanget al. 2015). A comprehensive review of optical property calculations using con-ventional and modified geometric optics and other methods is given by Yang et al.(2015). Here, we aim to summarize the optical properties that are needed for thesimulation of remote sensing data and to give a brief review of the dependency ofoptical properties on particle shape characteristics.

5.3.1 Definitions of Optical Properties

In order to solve the radiative transfer problem in the case of an ice cloud, the cloudlayer optical thickness, single scattering albedo and phase matrix are needed (van deHulst 1957). The dimensionless optical thickness s of a uniform ice cloud layer ofthickness DZ is given by

s ¼ DZ Ntot re ð5:3Þ

where Ntot is the total number concentration of ice crystals and re is the mean icecrystal extinction cross section. The extinction cross section for individual crystalscan be written as a combination of absorption ra and scattering rs cross sections, i.e.

re ¼ ra þ rs: ð5:4Þ

The fraction of light that is scattered rather than absorbed is given by the singlescattering albedo x defined as


x ¼ rsre

: ð5:5Þ

The extinction cross section of an ice crystal is largely determined by the pro-jected area Ap of the ice crystal, i.e..

re ¼ Qe Ap; ð5:6Þ

where Qe is the extinction efficiency. The projected area is defined as the crystal’saverage projection when an infinite number of random orientations are applied(Macke 1993). For convex particles, Ap is equal to the total surface area divided by4 (Vouk 1948), while for particles with concave parts the projected area can becalculated using a Monte Carlo approach (Macke 1993). In the conventional GOapproximation, Qe is 2 by definition (Berg et al. 2011), but actual values depend onparticle size and shape (Yang et al. 2015). For particles with sizes comparable to theconsidered wavelength, ray interference leads to oscillations of Qe around 2 thatdecrease in strength with increasing particle size. For particles much smaller thanthe wavelength, Qe strongly decreases as it enters the Rayleigh scattering regime.Note however that generally the extinction coefficient is not very relevant for mostremote sensing applications as the optical thickness is generally retrieved directly.However, the extinction coefficient is important for radiative transfer calculationsbased on model results (e.g., van Diedenhoven et al. 2012b; Bi et al. 2014; Baranet al. 2016).

The intensity and polarization state of light can be described by the Stokes vectorS ¼ ðI;Q;U;VÞ, where I is the total intensity, Q and U describe the linear polar-ization state and V describes the circular polarization state (van de Hulst 1957). Foreach scattering event on randomly oriented ice crystals, the alteration of propaga-tion direction and polarization state of incoming light is described by the symmetricphase matrix P, which can be defined through (van de Hulst 1957).

IsQs

Us

Vs

0BB@

1CCA ¼ rs

4pr2

P11 P12 0 0P12 P22 0 00 P33 �P34

0 0 P34 P44

0BB@

1CCA

IinQin

Uin

Vin

0BB@

1CCA; ð5:7Þ

where subscript “in” denotes the incoming Stokes vector, while “s” denotes theStokes vector after scattering. Furthermore, r is the distance between the particleand the location at which Ss is evaluated. Note that all elements are a function ofscattering angle H. The first element of the phase matrix P11 is referred to as thescattering phase function and is the only element to be considered when polar-ization is ignored. The other elements describe the alteration of the light’s polar-ization state at each scattering event. For example, the P12 element determines thelinear polarization of light after one scattering event for incoming unpolarized light.The degree of linear polarization (DoLP) is given by P12=P11. Circular polarization


can usually be ignored (Stamnes et al. 2016), but is relevant for lidars emittingcircular polarized light (e.g., van Diedenhoven et al. 2009).

In the geometrics optics regime, the total scattering phase function includescontributions from refraction into and out of the particle and internal reflections andcontributions from diffraction (Macke et al. 1996b), viz.

P11ðHÞ ¼ 12x

½ð2x� 1ÞPRTðHÞþPdifðHÞ�; ð5:8Þ

where Pdif and PRT are the contributions by diffraction and internal refraction andreflections, respectively. For radiative transfer applications that aim to approximatethe upward and downward fluxes of light, e.g., so-called two-stream applications(Coakley and Chylek 1975), the full phase function is not needed and only its firstmoment, the asymmetry parameter g, is considered. The asymmetry parameter isdefined as

g ¼Zp

0

PtotðHÞcosðHÞsinðHÞdH: ð5:9Þ

The reflection of an ice cloud layer at a given wavelength and scatteringgeometry is primarily determined by the optical thickness, single scattering albedoand asymmetry parameter of the ice crystals. The detailed shape of the phasefunction is of second order importance.

The optical properties of single ice particles can be calculated using GO or othermethods (Yang et al. 2015). Real ice clouds obviously consist of numerous icecrystals that have certain distributions of sizes and shapes. Recipes for integratingoptical properties over such distributions are given by Baum et al. (2005).

5.3.2 Dependence of Optical Properties on Crystal Shape

The optical properties of ice depend on wavelength and ice size, shape and ori-entation. Here, we limit the discussion to randomly orientated crystals. Althoughhorizontally oriented crystals are often present in clouds, it is estimated that theirnumbers are small and their influence on the integrated cloud properties is generallyminimal (Bréon and Dubrulle 2004; Zhou et al. 2012). Furthermore, essentially allradiative transfer and remote sensing applications are practically limited to ran-domly oriented particles.

The extinction efficiency mainly depends on the particle size parameter. It iscommonly assumed that errors due to the GO approximations (i.e., Qe ¼ 2) aresufficiently small for most applications for particles with a size parameter greaterthan 100 (Baran 2009; Bi et al. 2014). Following Bryant and Latimer (1969), herethe size parameter v at wavelength k is defined as


v ¼ ðmr � 1Þ 2pk

VAp

; ð5:10Þ

where V is the volume of the particle and mr is the real part of the refractive indexof ice. This definition of size parameter depends on the volume-to-area ratio of thecrystals. Confusingly, various other definitions of the size parameter that dependon, e.g., maximum diameters or the radius of an equivalent-area sphere, are used inthe available literature (e.g., Wyser and Yang 1998; Zakharova and Mishchenko2000; Um et al. 2015). Bryant and Latimer (1969) showed that Qe is a function of vwith a minimal dependency on shape and refractive index when the size parameteris defined as in Eq. 5.10. Occasionally ice extinction efficiencies are shown to beseemingly substantially dependent on shape, but at least part of theshape-dependency of the ice extinction efficiency reported in the literature is causedby using a definition of size parameter other than Eq. 5.10 (e.g., in Zakharova andMishchenko 2000; Yang et al. 2013; Liu et al. 2014; Um and McFarquhar 2015).For example, extinction efficiencies presented as a function of maximum dimen-sions for different shapes may show functional differences, but the differences willbe small if extinction efficiencies are instead presented as a function of the sizeparameter defined as by Eq. 5.10 (cf. Wyser and Yang 1998).

The influence of particle shape on single scattering albedo is also limited (Wyserand Yang 1998; Key et al. 2002; van Diedenhoven et al. 2014a) since it mainlydepends on the absorption size parameter as defined by van Diedenhoven et al.(2014a), i.e.,

vabs ¼miVkAp

; ð5:11Þ

where mi is the imaginary part of the refractive index. For a distribution of icecrystals, the integrated single scattering albedo at a given wavelength is mainlydetermined by the particle size distribution effective radius defined as

Reff ¼ 34Vtot

Atot; ð5:12Þ

where Vtot and Atot are the total volume and projected area of all particles in thecloud volume (Baum et al. 2005). This dominant influence of size, rather thanshape, on the single scattering albedo make measurements that depend on the singlescattering albedo, i.e. at shortwave infrared wavelengths, very suitable in retrievingparticle effective radius (Nakajima and King 1990).

In turn, the ice crystal scattering phase matrix is largely determined by particleshape, rather than size, especially at solar wavelengths where ice absorption isminimal. In fact, in the GO approximation and at non-absorbing wavelengths, theray tracing part of the phase matrix is scale invariant by definition. Scattering phasematrices for crystals with size parameters smaller than about 100 do feature sub-stantial size dependencies that are not accurately modeled by classical GO but these


can be approximated by corrections to GO and/or by other techniques (Yang et al.2015). However, most cross-sectional area is generally contributed by naturalcrystals larger than this limit when evaluated at solar wavelengths (Baran 2009). Ascan be inferred from Eq. 5.8, the phase function is also affected by the singlescattering albedo for non-absorbing wavelengths. Confusingly, substantialsize-dependencies of scattering phase matrices for large particles at non-absorbingwavelengths are frequently presented in publications (e.g., Key et al. 2002; Baumet al. 2011, 2014). However, these variations of the phase matrices with size mostlystem from the assumed crystal geometries (e.g., the aspect ratios of ice crystalcomponents) that depend on size. Although geometries might be expected tochange with size for natural ice crystals, it is important to separate the dependenceof optical properties on size from their dependence on crystal shape when con-sidering the remote sensing of crystal shape. The strong dependence on shape of icephase matrix elements at solar wavelengths and the minimal dependence on size areexploited by the methods to remotely sense information about ice shape usingsignatures in the scattered light, as discussed in Sect. 5.4.

Thus, out of the fundamental optical properties, only scattering phase matricesare substantially dependent on shape. For the purpose of remote sensing of icecrystal shapes, it is important to determine to which shape characteristics the phasematrix is particularly sensitive. Early applications of ray tracing calculations alreadydetermined that the phase matrix of single hexagonal prisms show a strongdependence on their aspect ratio (e.g., Macke et al. 1996b). For example, thedependence of the asymmetry parameter at a wavelength of 865 nm on the aspectratio hexagonal prisms is shown in Fig. 5.3. This figure also depicts the dependence

Fig. 5.3 Asymmetry parameter of hexagonal prisms at a wavelength of 865 nm as a function ofaspect ratio (x-axis) and roughness parameter (y-axis). Figure reproduced from van Diedenhovenet al. (2012a)


of the asymmetry parameter on crystal surface roughness, as discussed later. Asseen in Fig. 5.3, the asymmetry parameter is lowest for crystals with an aspect rationear unity. Also the increase of asymmetry paramneter with aspect ratio is rathersymmetric for plates and columns, implying that asymmetry parameter is largelydetermined by the aspect ratio a\1 as defined by Eq. 5.2 (van Diedenhoven et al.2016a). Asymmetry parameters increase with aspect ratio owing to the increase ofparallel surface areas leading to greater probability of light passing through theparticle with low orders of refraction þ reflection and a minimal change ofdirection (Yang and Fu 2009). For the same reason, the increase of asymmetryparameter with decreasing aspect ratio a\1 is slightly weaker for columns than forplates owing to the larger parallel surfaces of plates (cf. Macke et al. 1996b; Yangand Fu 2009). Examples of phase functions and the degree of linear polarization forhexagonal ice prism with varying aspect ratios and for various complex ice habitsare shown in Figs. 5.4 and 5.5, respectively. The applied size distributions aredescribed by van Diedenhoven et al. (2012a). From right panels in Figs. 5.4 and5.5, one is tempted to conclude that ice optical properties strongly depend on thepolycrystalline structure or habit. However, on closer inspection, the variation ofphase function between different habits is mostly caused by the variation of theaspect ratio of their components. For example, aggregates of columns have

(a)

(b)

Fig. 5.4 Phase functions of hexagonal columns with varying aspect ratios (AR, top left) androughness parameters (lower left) and phase functions of complex crystals (Yang et al. 2015) withsmooth (upper right) and severely roughened (r ¼ 0:5, lower right) surfaces


components with an average aspect ratio of 1.5 (Fu 2007) and their phase matrixelements show resemblances with compact hexagonal columns (see Figs. 5.4 and5.5), while bullet rosettes have thinner components and their phase matrices aresimilar to those of thin columns. Moreover, as was recognized by many authors(e.g., Iaquinta et al. 1995; Um and McFarquhar 2007, 2009; Fu 2007; Baran 2009;van Diedenhoven et al. 2012a, 2014a), the phase function of an ice crystal con-sisting of multiple, identical hexagonal components largely resembles the phasefunction of the individual single components. Comparisons of phase functions ofcomplex crystals and their components are shown previously for bullet rosettes andtheir aggregates (Iaquinta et al. 1995; Um and McFarquhar 2007) and aggregates ofplates (Baran 2009; Um and McFarquhar 2009) and columns (Baran 2009). Forvarious crystal habits, Fu (2007) showed that the asymmetry parameters of thecomplex crystals are approximated to within about 0.01 by the asymmetryparameters of their individual components. Um and McFarquhar (2007) showedthat the asymmetry parameter of individual bullets at visible wavelengths are about0.01 larger than rosettes with 6 arms, and about 0.02 larger than an aggregate ofbullet rosettes consisting of 18 components. For aggregates of plates, Um andMcFarquhar (2009) showed that the asymmetry parameter at visible wavelengthsdecreases with roughly 0.01 for every 5 plates attached to an aggregate, althoughthe decrease was found to be non-monotonic. Furthermore, they found that dif-ferences between asymmetry parameters of aggregates of plates and those of thecomponent plates increase as the distances between the centres of mass of the plates

(a)

(b)

Fig. 5.5 Same as Fig. 5.4 but for the degree of linear polarization


within the aggregates decrease. Essentially, increasing the compactness of aggre-gate of plates leads to stacking of the plates and finally to morphing into a thickersingle plate and hence to a lower asymmetry parameter. To put the sensitivity of theasymmetry parameter to aggregation into perspective, a small change in the aspectratio of a single plate, from 0.6 to 0.5, increases the asymmetry parameter at awavelength of 550 nm by 0.01, i.e., from 0.79 to 0.80. (Here the parameterizationof van Diedenhoven et al. (2014a) is used to calculate the asymmetry parameter.)Thus, although the larger-scale structure of such complex particles influences theirsingle-scattering properties, the influence of natural variations in aspect ratios ofcrystal components may be expected to be greater.

As discussed in Sect. 5.2.3, aside from general shape of ice crystals and theaspect ratios of their components, microscale structures on the crystal surfaces areof importance for optical properties. In order to compute the optical properties of agiven ice crystal, an idealized definition of its facets is needed. However, thecomputational effort sharply increases as the number of facets required to define aparticle with microscale surface roughness increases (Liu et al. 2013). Moreover,these complex or rough particles are determined by stochastic processes and allparticles in a cloud volume essentially have unique complexity or roughnessstructures. To calculate the optical properties of such collections of particles,basically the optical properties of many random members have to be determinedand then appropriately averaged. Since this is impractical, stochastic approaches toapproximate the effects of particle roughening have been developed. A commonlyused method is based on the work of Peltoniemi et al. (1989), Macke et al. (1996b)and Yang and Liou (1998) and takes crystal roughness of ice crystals into accountin a statistical manner by perturbing the normal of the crystal facet surface from itsnominal orientation by an angle that, for each interaction with a ray, is variedrandomly with a given distribution. This was found to be an effective and accuratetreatment of the effects of surface roughness on ice crystal scattering properties (Liuet al. 2013). Practically, codes that use this approach (Macke et al. 1996b; Yang andLiou 1998; Shcherbakov et al. 2006) vary in the assumed distribution of facet tiltangles (Neshyba et al. 2013; Geogdzhayev and van Diedenhoven 2016). Forinstance, the code by Macke et al. (1996b) varies the tilt angles randomly withuniform distribution between 0° and d � 90�, where d is referred to as the roughnessparameter. Yang and Liou (1998) and Shcherbakov et al. (2006) vary tilt anglesaccording to Gaussian and Weibull distributions, respectively, where the mean tiltangle is determined by roughness parameter r. However, Geogdzhayev and vanDiedenhoven (2016) and Neshyba et al. (2013) showed that roughness parameterswith these different definitions, for a given value, lead to rather similar phasefunctions and asymmetry parameters (i.e. d � r). However, these roughnessparameterizations currently need to be considered as merely effective methods tosimulate the effects of roughness structures on the scattering matrices, since they aredifficult to relate to physical characteristics of the structures observed on real icecrystals, although attempts have been made (Neshyba et al. 2013). Furthermore, Liuet al. (2014) have shown that the simulated effects of microscale roughness featuresand larger scale geometric irregularities of ice crystals are similar. Indeed, irregular


crystals such as Koch fractals (Macke et al. 1996b) and Voronoi particles (Letuet al. 2016) have phase matrices similar to rough hexagonal prisms. Increasing thenumber of impurities within ice crystals also has a similar effect (Macke et al.1996a; Hong and Minnis 2015; Panetta et al. 2016). Thus, the roughness param-eterization can be interpreted as representing the randomization of reflection andrefraction on and in the particles caused by any of such deviations from pristine,solid, smooth crystals. Since the roughness parameter reflects more than justroughness structures, some authors prefer the term “distortion parameter” instead.Using the definition of roughness of Macke et al. (1996b), Fig. 5.3 shows thedependence of the asymmetry parameter on roughness and aspect ratio, while thebottom left panel of Figs. 5.4 and 5.5 show the variation of phase functions anddegree of linear polarization, respectively, on roughness for columns. Also, thebottom right panels of Figs. 5.4 and 5.5 show the phase functions and the degree oflinear polarization, respectively, of severely roughened complex particles (r ¼ 0:5).Angular features in the phase functions and degree of linear polarization functionsare smoothed out as roughness increases. Most notable, the halos are diminished asroughness increases and asymmetry parameters systematically decrease withroughness (cf. Yang et al. 2008a; van Diedenhoven 2014; van Diedenhoven et al.2014a; Geogdzhayev and van Diedenhoven 2016).

Another property of ice crystals that can have substantial impact of scatteringproperties is the presence of hollow crystal endings, or cavities (Schmitt et al. 2007;Yang et al. 2008b; Smith et al. 2015). Generally, asymmetry parameters increase ascavity depth increases, albeit this increase becomes smaller for crystals with aspectratios increasingly deviating from unity (Schmitt et al. 2007). As can be seen inFig. 5.4 (right panels), the hollow structures increase the scattering phase functionvalues at scattering angles smaller than 20°, but decrease values at side- andbackscattering angles of 60 to 180° (cf. Schmitt et al. 2007; Yang et al. 2008b). Oneway to interpret the observed influence of hollowness on columns is that, since thewalls of the hollow parts of the crystals are thin, these essentially resemble columnswith high aspect ratios, leading to an increased asymmetry parameter (vanDiedenhoven et al. 2012a). This interpretation is also supported by the resemblanceof phase functions and degree of linear polarization of hollow particles to those ofcolumns with large aspect ratios seen in Figs. 5.4 and 5.5.

In summary, the most important features of ice particle shape influencing theiroptical properties are the aspect ratios of the crystals or the components of complexcrystals as well as the microscale surface roughness, distortion or hollowness. Thelarger scale polycrystaline structure of the ice crystal is of lesser importance. Thus,the different characteristics of ice particle shape can be grouped in order ofimportance for the scattering properties as follows:

1. Aspect ratios of simple crystals or components of complex crystals;2. Microscale structure, surface roughness, distortion, impurities or cavities;3. Polycrystalline structure or habit.


These crystal shape characteristics are relevant targets for the remote sensingtechniques described in the next section.

As discussed in Sect. 5.2, myriads of shapes are found in real cloud volumes.Moreover, the aspect ratios and roughness levels may be expected to vary perparticle (Um et al. 2015; Fridlind et al. 2016; Schnaiter et al. 2016). However, mostpublications about optical properties of ice crystals and remote sensing of ice crystalshapes consider calculations for individual crystals with a particular shape ratherthan mixtures of crystals with large variations in shapes, aspect ratios and roughnesslevels. Various habit mixtures have been presented (e.g., Baum et al. 2005, 2011;Baran and C.-Labonnote 2007) but these usually contain just a handful of iceshapes. Furthermore, remote sensing applications generally aim to find a singleshape or simple mixture that fits the measurements best. The question arises how tointerpret remote sensing results that conclude one crystal shape or a limited mixtureto be consistent with the measurements. In other words, which part of the ensembleis represented by the single inferred crystal shape? In order to estimate the asym-metry parameter of an ensemble of hexagonal ice crystals with a distribution ofaspect ratios, Fu (2007) defined an ensemble-average aspect ratio as the averageaspect ratio weighted by the orientation-averaged projected area of the crystals inthe distribution. Van Diedenhoven et al. (2016a) showed that the definition ofaspect ratio limiting the values below unity for both plates and columns (i.e.,Eq. 5.2) needs to be used in order for an ensemble-average aspect ratio to ade-quately represent the optical properties of the ensemble. Furthermore, they showedthat even for ensembles containing both columns and plates, the ensemble asym-metry parameters are generally represented consistently by a single crystal with anaspect ratio equal to the ensemble average, especially if geometrical averaging isused. Tests on mixtures of plates and columns yielded root-mean-squared differ-ences between ensemble-average asymmetry parameters and those calculated fromensemble-average aspect ratios were 0.006. Furthermore, van Diedenhoven et al.(2016a) showed that effective asymmetry parameters based on arithmetic averagesof roughness parameters are also largely consistent with ensemble-average asym-metry parameters. As discussed by Fu (2007) and van Diedenhoven et al. (2016a),these conclusions are likely qualitatively applicable to ensembles of more complexstructures such as aggregates of columns, aggregates of plates and bullet rosetteswith a range of aspect ratios of their components, as well as to internal mixtures ofplate-like and column-like components, such as asymmetric bullet rosettes,aggregates of varying plates and/or columns or plate-capped columns. Thus, theconclusions by Fu (2007) and van Diedenhoven et al. (2016a) suggest that shapesinferred from remote sensing applications represent an area-weighted ensembleaverage of the ice crystals in the observed cloud volume. Further discussion on thisis included in Sect. 5.2.4.

In summary, it is practically impossible to consider all geometries of ice crystalsformed in natural ice clouds for remote sensing applications. Any ice model usedfor remote sensing purposes will be a highly idealized simplification of natural icecrystals. Often, crystal shapes that are found to match observations are presentedmerely as “radiative equivalent effective shapes” (e.g., McFarlane and Marchand


2008; Cole et al. 2013), and no direct relation between the inferred shape and theice crystal shapes in the observed clouds is claimed. However, using idealizedmodels, it is still possible to infer useful information about shape characteristics ofnatural ice crystals in clouds. Here, a famous quote from the late George Box isfitting, namely that “essentially, all models are wrong, but some are useful”(Box and Draper 1987). In order to infer information about the variation of icecrystal shape in ice clouds, a set of ice models is needed that systematically variesthe physical and scattering properties and is able to fit the full range of availableremote sensing observations. George Box provides further inspiration by statingthat “since all models are wrong, the scientist cannot obtain a ‘correct’ one byexcessive elaboration. On the contrary, following William of Occam [she/]heshould seek an economical description of natural phenomena” (Box 1976).Following such wisdom, one can conclude that for remote sensing it may beadvised to focus on the variation of the ice shape characteristics that are mostlydetermining the radiative properties as listed above, i.e., component aspect ratiosand microscale structure, and less on crystal habits. Note that shape characteristicssuch as aspect ratios, roughness levels and cavity depths are quantitative parametersthat suit systematic variation in contrast to ‘habit’. Focusing on such shapeparameters provides an “economical description” and reduces the remote sensingproblem substantially to quantifying these parameters without considering thevirtually infinite number of possible large-scale shapes. Based on the equivalence ofscattering properties of complex crystals and their components, as discussed above,such inferred parameters, e.g., aspect ratios and roughness parameters, represent theaveraged properties of the components of the crystals in the observed ice clouds.Given the variation of ice crystals observed and the myriad of conditions in whichice crystals form and evolve, we can assume that values of aspect ratios androughness levels occur continuously over rather large ranges, which need to bespanned by the models considered for remote sensing applications to infer icecrystal shapes.

5.4 Remote Sensing of Ice Crystal Shapes

Essentially two different observational approaches have particular potential forinferring information about particle shape, namely active lidar observations andpassive multi-directional measurements of total and/or polarized reflectances. Here,these approaches are discussed separately in Sects. 5.4.1 and 5.4.2.

5.4.1 Lidar Measurements

Lidars (i.e., laser radars) probe clouds by emitting laser beam pulses and measuringthe power of the returned signals (Weitkamp 2005). By measuring the delay


between the emitted and detected beams, the distance between the lidar and thecloud volume on which the beam was scattered back can be determined, resulting inheight-resolved information. When the emitted beam is polarized and the polar-ization state of the detected signal is measured, the depolarization of the beamcaused by backscattering on particles can be measured. The altitude-dependentlinear depolarization ratio dl of the returned signal is usually defined as (Schotlandet al. 1971)

dlðzÞ ¼ b?ðzÞbkðzÞ

; ð5:13Þ

where b is the backscattering cross sections in the planes of polarization perpen-dicular (?) and parallel (k) to the laser’s reference plane. In Eq. 5.13 it is assumedthat atmospheric extinction is independent of the polarization state of the propa-gating radiation. Other definitions for lidar depolarization are discussed byGimmestad (2008).

Since lidar signals are mainly determined by singly scattering light, the depo-larization of a lidar signal returned from a cloud layer can be simulated in a straightforward manner from the assumed scattering phase matrix, namely by

dlðzÞ ¼ P11ð180�Þ � P22ð180�ÞP11ð180�ÞþP22ð180�Þ : ð5:14Þ

Lorenz-Mie theory applied to spherical particles shows that no depolarization iscaused by single back-scattering on purely spherical cloud droplets, althoughmultiple scattering on liquid cloud drops can lead to some depolarization of thelidar signal (Sassen and Petrilla 1986; Hu et al. 2001). However, singleback-scattering of the laser light on non-spherical particles such as spheroids orhexagonal prisms leads to substantial depolarization of the signal that is largelydependent on particle shape.

The first ground-based lidar depolarization measurements of clouds werereported by Schotland et al. (1971) and many polarized ground-based lidars aredeployed world-wide today. The satellite-based Cloud-Aerosol Lidar withOrthogonal Polarization (CALIOP) on the Cloud-Aerosol Lidar and InfraredPathfinder Satellite Observation (CALIPSO) platform (Winker et al. 2007), laun-ched in 2006, is particularly relevant for ice cloud studies as it is providing globalstatistics and observes cirrus otherwise obscured by lower lying liquid clouds whenviewed from the surface. Similar statistics are expected from the Atmospheric Lidar(ATLID) on the Earthcare platform to be launched in 2018 (Lefebvre et al. 2016).Furthermore, the cloud physics lidar (CPL) has been providing a wealth of datasince the year 2000 while mounted on the high altitude NASA ER-2 aircraft duringmany field campaigns targeting clouds (Yorks et al. 2011). General conclusionsabout the depolarization measured by such lidars is that the depolarization ratio inice clouds typically ranges from 0.2 to 0.6, and generally increases with increasingcloud height or decreasing temperature (Sassen and Benson 2001; Reichardt et al.


2002; Noel et al. 2004; Sassen and Zhu 2009; Yorks et al. 2011; Baum et al. 2011;Sassen et al. 2012). Furthermore, CALIOP statistics presented by Sassen et al.(2012) show a globally, vertically and yearly averaged ice cloud depolarizationratio of 0.37 and generally lower values at higher latitudes. In addition, Martinset al. (2011) found no significant variations in depolarization ratios for ice clouds inrelation to updraft strength or horizontal windspeed. Note that the depolarizationratios quoted here reflect the values measured at an off-nadir angle, since hori-zontally orientated ice crystals strongly decrease depolarization measured at directnadir owing to specular reflection on the crystal surfaces (Del Guasta et al. 2006;Zhou et al. 2012; Sassen et al. 2012).

Although the variation of depolarization ratios in ice clouds at different altitudes,temperatures and latitudes are fairly well documented, a definite quantitativeinterpretation of the values in terms of ice crystals shape variations is lacking. Oneinterpretation is offered by Noel et al. (2002) who showed that the simulateddepolarization ratio of single hexagonal prisms depends on their aspect ratio. Asalso shown by Yang and Fu (2009), the simulated depolarization ratio of hexagonalcrystals with smooth surfaces is about 0.2 for thin plates, oscillating between 0.3and 0.4 for thicker plates and about 0.4–0.6 for compact particles and columns.Based on this relation between depolarization ratio and aspect ratio, Noel et al.(2004) concluded that the depolarization ratios observed by the CPL during theCRYSTAL-FACE campaign conducted near Florida in 2002 are consistent with adominance of compact or ‘irregular’ crystals and an increasing contribution byplate-like crystals with increasing cloud top temperature. Measurements on icecrystals grown in the laboratory presented by Amsler et al. (2009) andAbdelmonem et al. (2011) seem to support this quantitative interpretation ofdepolarization ratios, although modeled depolarization ratios were found to begenerally larger than the measured ones. However, as discussed in Sect. 5.2, mostcrystals observed in situ have more complex, aggregated shapes than simplehexagonal prisms. Although large scale complexity generally has a weaker effect onthe scattering phase matrix than aspect ratio, as discussed in Sect. 5.3.2, it is unclearwhether or how this complexity affects the depolarization ratios specifically(Reichardt et al. 2008). A fundamental problem for simulating lidar backscatteringand depolarization on ice crystals is that the phase matrix computed based ongeometric optics exhibits a singularity at the exact backscattering direction forrandomly oriented crystals (Borovoi et al. 2005, 2014). Usually values for anglesclose to 180 degrees are used instead or they are extrapolated towards 180 degrees(e.g., Smith et al. 2016). Furthermore, the dependency of depolarization on aspectratio presented previously by Noel et al. (2004) and others are based on solidcrystals with smooth surfaces. Although calculations presented by Smith et al.(2016) suggest that the effects of crystal hollowness on lidar depolarization isgenerally small, reported estimates of the effects of surface roughness on lidardepolarization are inconclusive. Mostly, roughness or distortion of ice crystals isshown to lead to a decrease of depolarization (Del Guasta 2001; Baum et al. 2010,2011), although the opposite trend has been shown as well (Smith et al. 2016;Konoshonkin et al. 2016). Furthermore, rounded crystal shapes caused by crystal


sublimation are sometimes associated with a decrease of depolarization (e.g.,Martins et al. 2011), although increasing depolarization from sublimation is sug-gested by Schnaiter et al. (2012). Also, low depolarization in ice clouds aresometimes interpreted as caused by compact or ‘quasi-spherical’ particles (e.g.,Choi et al. 2010). However, these associations may not be supported by observa-tions and calculations of optical properties. For example, Mishchenko and Sassen(1998) showed that depolarization ratios of spheroids and deformed spheroids withaspect ratios close to unity are comparable with, or larger than, the values ofhexagonal plates, even if such shapes can be considered quasi-spherical. Sassen(1977) reported depolarization ratios above 0.5 for frozen rainwater drops with aregular spherical or spheroidal appearance. In contrast, recent work presented byJärvinen et al. (2016) and Schmitt et al. (2016) shows that sublimation of icecrystals can lead to smooth frozen droplets that have optical properties similar tospheres including near-zero depolarization ratios. Other factors complicating theinterpretation of lidar depolarization values are the presence of oriented ice crystals(Del Guasta et al. 2006; Zhou et al. 2012) and the co-existence of ice crystals andliquid drops in mixed-phase clouds (Bourdages et al. 2009; van Diedenhoven et al.2011).

In conclusion, systematic variations of lidar depolarization in clouds areobserved, but their interpretation is currently inconclusive. As put by Sassen et al.(2012), “these [variations in depolarization ratios] must reflect the different iceparticle shapes that depend on the cirrus cloud formation mechanism (e.g.,convective-anvil, orographic, synoptic, etc.), or more specifically on the basiccloud-particle forming aerosol available for crystal formation and the microphysicaleffects of typical cloud updraft velocities.” However, currently, interpretations oflidar depolarizations in ice clouds are more qualitative than quantitative. Toquantitatively relate the measured depolarization ratios to microphysical propertiesof the ice crystals, more research is needed, including (1) further development ofmethods for calculating exact backscattering properties of complex ice crystals andcrystals with randomly roughened surfaces; (2) investigations on the effects ofcrystal complexity on lidar depolarization; (3) further statistical studies on measuredlidar depolarization ratios with colocated in situ ice crystal shape observations.

5.4.2 Multi-angular Measurements

As shown in Figs. 5.4 and 5.5, the shapes of the scattering phase function anddegree of polarization of ice crystals for visible wavelengths depend on the crystalshape. These angular variations in scattering phase function and degree of polar-ization lead to angular features in the total and polarized reflectances of ice cloudsthat can be observed by remote sensing instruments that make observations atmultiple viewing angles per instrument footprint. Here, we first sumarize thetechniques used to infer ice crystal shape from multi-angular total reflectances(Sect. 5.4.2.1) and multi-angular polarized reflectances (Sect. 5.4.2.2). Then some


discussion about data selection and availability is offered in Sect. 5.4.2.3 beforesummarizing results in Sect. 5.4.2.4.

5.4.2.1 Multi-angular Total Reflectances

The measured reflectance of a cloud layer can be defined as

Rðl0; l;D/Þ ¼pIðl0; l;D/Þ

l0F�; ð5:15Þ

where l0 and l are the cosine of the solar zenith angle and viewing angle,respectively, F� is the solar irradiance, and I is the radiance measured by aninstrument. Since this definition refers to the reflectance of light irregardless thepolarization state, it is commonly referred to as “total reflectance”.

Ice crystal shape can be inferred from total reflectances because angular featuresin the single scattering phase function of ice crystals, as seen in Fig. 5.4 forexample, are preserved in the directional measurements. The directional reflectanceof singly scattered light is determined by the phase function (van de Hulst 1957).Because of the large extinction in cloud tops, however, singly scattered lightemerging from the cloud is only a small fraction of the total reflection, adding aweak signal of directionality to the measured total reflectances. As discussed inSect. 5.3, the magnitude of the total reflectance of an ice cloud is mostly determinedby its optical thickness and the asymmetry parameter of the scattering phasefunction (Coakley and Chylek 1975). One might expect that multiply scattered lightemerging from a cloud is fully isotropic. However, as explained by Zhang et al.(2009), at any scattering event, the highly peaked phase functions of ice crystalslead to a large portion of the rays scattered forward without a substantial change indirection. Thus, a likely path for light rays is one where one or more forwardscattering events are followed by a single sideward scattering event, which is thensubsequently followed by one or more forward scattering events again before lightemerges from the cloud and is measured by an instrument. Although such pathsdescribe multiply scattered light, the angular features in the scattering matrix of icecrystals are also preserved in the directional reflectance measurements for thesepaths, similarly as for the singly scattered light. This effect enhances the signal tonoise ratio and the information content of multi-directional total reflectance mea-surements for retrieving ice crystal shape.

For a plane-parallel homogeneous cloud layer, its visible reflectance at a par-ticular viewing geometry is fully determined by the scattering phase function andthe optical thickness. Thus, when multi-directional measurements are available for acloud layer, an optical thickness can be retrieved from each angular measurementassuming a particular phase function. If the assumed phase function is correct thenthe optical thickness retrieved at each angle would be exactly the same, i.e., thecorrect value. However, an incorrect phase function assumption leads to an angulardependence of the retrieved optical thickness. This angular variation of retrieved


optical thickness, or spherical albedos derived from them (e.g., Doutriaux-Boucheret al. 2000), is the basis of methods to retrieve ice shapes from multi-angular totalreflectances (e.g., Doutriaux-Boucher et al. 2000; McFarlane and Marchand 2008).Generally, the angular variation of retrieved optical thicknesses or spherical albedosis determined for a selection of assumed crystal shapes and crystals leading to thesmallest angular variations are presented as the most consistent with real icecrystals.

Examples of instruments that make measurements of total reflectances at mul-tiple viewing angles per footprint are NASA’s Multi-angle ImagingSpectroRadiometer (MISR, Diner et al. 2002) and ESA’s Along-Track ScanningRadiometer 2 (ATSR-2) and the Advanced ATSR (AATSR) (Sayer et al. 2011).The ATSR-2 and AATSR instruments have nadir pointing cameras as well as oneforward pointing camera observing each footprint. MISR has 9 cameras at nadir,forward and aft pointing angles. Also multi-angle polarimeters as discussed inSect. 5.4.2.2 measure total reflectances in addition to the polarized reflectances.Overlapping footprints of geostationary satellite imagers have also been used tocreate dual-view observations (Chepfer et al. 2002). Furthermore, global statisticsof single-view imager data can be used to infer the angular variation of cloudreflectances (Zhang et al. 2009; Wang et al. 2014).

5.4.2.2 Multi-angular Polarized Reflectances

Similar to the definition of total reflectances (Eq. 5.15), polarized reflectances Rp

are generally defined as

Rpðl0; l;D/Þ ¼p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2 þU2

pl0F�

: ð5:16Þ

The dependency of Q and U on l0, l and Du is omitted in Eq. 5.16 for clarity.Note that both Q and U are signed according to the orientation of the measuredelectromagnetic wave relative to a reference plane. This information about polar-ization direction is lost by the definition of Eq. 5.16, but can be analyzed separately(e.g., Sun et al. 2015) or a sign can be added to the polarized reflectancesaccordingly, as described by C.-Labonnote et al. (2001). Normalizations differentfrom the one used in Eq. 5.16 are sometimes applied too (e.g., C.-Labonnote et al.2001).

Ice crystals shape characteristics can be inferred from the polarized reflectancesby matching the measured angular variation of polarized reflectances with valuessimulated using a radiative transfer model that includes multiple scattering andpolarization. Since multiple scattering depolarizes light, polarized reflectances aredominated by singly scattered light and have pronounced angular dependencies thatare determined by the polarization properties of the ice crystals, as shown inFig. 5.5. Furthermore, for cloud optical thickness values above about 2–5


(depending on the asymmetry parameter of the phase functions), polarized reflec-tances do not depend on cloud optical thickness anymore (van Diedenhoven et al.2012a), which is convenient when retrieving information about ice crystals shape invarious thick cloud types. However, measuring the polarization state of reflectedlight is challenging, as per footprint and per viewing angle it requires a minimal setof three simultaneous observations at three well-determined polarization angles(Tyo et al. 2006).

Relevant multi-angular polarized reflectances from space were measured by thePOLarization and Directionality of the Earth’s Reflectances (POLDER) instruments(Deschamps et al. 1994; Fougnie et al. 2007). The POLDER imager acquiredoverlapping images at different orbital locations that yield observations for eachgiven footprint at a maximum of 16 viewing angles (Fougnie et al. 2007). For thePOLDER wavelength bands at 490, 670, and 865 nm, three acquisitions are per-formed through a polarizer oriented at −60°, 0°, and 60° relative to a given ref-erence from which the Stokes parameters ½I;Q;U� are retrieved. In addition to anairborne version of the POLDER instrument (Chepfer et al. 1998), threesatellite-based versions were deployed. The first two (POLDER-1 and POLDER-2)were mounted on the Advanced Earth Observing Satellite (ADEOS- 1 andADEOS-2, respectively), but both only lasted for about 6 months owing to platformfailures. The third, mounted on the CNES/Myriade Polarization & Anisotropy ofReflectance for Atmospheric Sciences coupled with Observations from a Lidar(PARASOL) microsatellite, was highly successful and lasted for nearly 9 years,from 2004 to 2013. POLDER-PARASOL was part of NASA’s afternoon train(A-train) constellation until 2010, allowing its data to be combined with that ofCloudsat, CALIPSO, MODIS and other A-train instruments. No multi-directionalpolarimeter other than POLDER has flown in space to date. Unfortunately, thelaunch of the Glory satellite carrying the Aerosol Polarimetry Sensor (APS,Mishchenko et al. 2007) failed in March 2011. Several airborne multi-anglepolarimeters are regularly deployed during field campaigns, often on high altitudeaircraft capable of flying sufficiently above ice clouds. The Research ScanningPolarimeter (RSP, Cairns et al. 2003) is an airborne version of the APS and pro-vides simultaneous measurements of total and polarized reflectances in 9 spectralbands from the visible to the shortwave infrared. The RSP scans along track,providing measurements of each pixel at 152 different viewing angles at 0.8 degreesintervals. Other deployed airborne polarimeters include the Airborne Multi- angleSpectro Polarimetric Imager (AirMSPI, Diner et al. 2013) and AirborneSpectropolarimeter for Planetary Exploration (SPEX, Rietjens et al. 2015).

5.4.2.3 Data Selection

For the remote sensing of ice crystal shape, employing relevant data selectioncriteria is crucial. Obviously, one important selection criteria is that ice clouds areselected without interference of liquid clouds. For studies on POLDER data, this isgenerally efficiently achieved by testing for rainbow features in polarized


reflectances. However, as noted by, e.g., C.-Labonnote et al. (2001) and Cole et al.(2014) this generally filters out clouds with low optical thicknesses (below opticalthickness 1–5). This inadvertent selection of thick clouds is often not noted bystudies using POLDER data, although evidently statistics of POLDER polarizedreflectances generally show saturated reflectances, which only occur at aboutoptical depths of about 4–5 (Chepfer et al. 2001; van Diedenhoven et al. 2013).Since cirrus typically has an optical thickness below 4 (Rossow and Schiffer 1999),this selection has the important implication that most cirrus are not included in thestudies on ice crystals shapes using POLDER data. Alternatively, colocatedMODIS thermodynamical phase determination can be used to select ice-toppedclouds. However, this cloud phase retrieval is shown to be less reliable formixed-phase and multi-layered clouds (e.g., Riedi et al. 2010). Interference ofliquid drops to the multi-angular total or polarized reflectances is expected to lead tospurious angular dependence of the measurements and thus to biases the retrievalsof ice shape towards more pristine particles.

Since the simulations of ice optical properties and radiative transfer calculationsgenerally assume random orientation of the ice crystals, Chepfer et al. (2001) andSun et al. (2006) concluded that the POLDER data needs to be screened for thepresence of oriented ice crystals. Specular reflection on oriented crystals is highlypolarizing and leads to sharp angular peaks in the polarized reflectances at thespecular reflection angles (Chepfer et al. 1998; Breon and Dubrulle 2004). Thispeak can be used to screen the data for particle orientation, but it requires mea-surements at and around the scattering angle associated with specular reflection fora given solar angle. As noted by Sun et al. (2006), this requirement excludes mostdata, especially for instruments with relatively low angular resolution such asPOLDER. For example, Sun et al. (2006) noted that the stringent data selectioncriteria used in their study on POLDER data lead to only 0.37% of the 5-monthPOLDER dataset passing these criteria. However, analyzes of global POLDER datasuggests that that the relative number of oriented crystals is generally low and thatthe typical effective contribution to particle area from oriented plates in clouds issmaller than 1% (Bréon and Dubrulle 2004). Thus, the total and polarized reflec-tances from ice clouds outside of the specular reflection geometry are dominated byrandomly oriented crystals, which implies that the screening for random orientationis not needed. It may be needed to remove data affected by sunglint on oceansurfaces (van Diedenhoven et al. 2013), although this will only be an issue foroptically thin clouds that are generally already excluded from POLDERmeasurements.

As the crystal shape information is inferred from angular variations in total andpolarized reflectances, the information content depends on the scattering angles thatare sampled by the instrument. In turn, the sampled scattering angles are dependenton the solar zenith angle and thus on latitude when analyzing data from polarorbiting satellites. It is important to note that for the statistics on polarized reflec-tances derived from global POLDER measurements, data from higher latitudescontribute more towards the small scattering angles, while tropical clouds con-tribute more to the larger scattering angles (Buriez et al. 2001). This also implies


that any latitudinal variations in retrieved ice model or roughness level may stemfrom the latitudinal variation of available scattering angles and the sensitivity of theretrieval approaches to scattering angle range. Results from van Diedenhoven et al.(2012a, 2013) show that an angular range including samples between scatteringangles of at least 120° and 150° is needed for ice crystal shape retrievals. However,the retrieval performance is shown to be rather insensitive to number of anglessampled within that range or to any reasonable random error or bias on the mea-surements (van Diedenhoven et al. 2012a).

5.4.2.4 Overview of Results

An overview of the most notable studies on inferring ice crystal shape along withthe targeted locations and the data used is given in Table 5.1. Some studies use totalreflectances R, some use polarized reflectances Rp and some use both. Most studiesanalyze data averaged over time and space, while some infer crystal shapes fromdata at the instrument’s pixel resolution. The table also lists a description of thedominating ice shape found by the particular studies. From this overview it is clearthat most data are fitted best by crystals with roughness, impurities or other dis-tortions. Some studies, especially early ones, did not include particle roughness orother distortions and their results should be treated with caution.

Many of the studies listed in Table 5.1 aim to find ice crystal models that bestrepresent globally averaged data. Some studies compare total reflectances withmodeled reflectances in order to identify crystal shapes that lead to most realisticangular features on average (e.g., Doutriaux-Boucher et al. 2000). Studies on globalpolarized reflectances usually present observation density plots of the POLDERmeasurements as a function of scattering angles to which simulated polarizedreflectances are compared (e.g., Knap et al. 2005; Baran and C.-Labonnote 2006;Cole et al. 2013). These density plots represent the statistics of polarized reflec-tances observed at any specific scattering angle. The range of measured polarizedreflectances can be assumed to stem from the natural variation in ice crystalsshapes. It is generally concluded that the ice shapes that lead to simulated polarizedreflectances that are closest to most observations at all of the scattering angles is agood model to represent natural ice clouds. However, it is important to note thatalmost all models included in such studies generally lead to simulated polarizedreflectances that fall within the observation envelopes at most scattering angles.This implies that all such models could represent a subset of the measurements. Forexample, Cole et al. (2013) evaluate simulated polarized reflectances for habitmixtures with smooth or rough surfaces, as well as several habits with severelyrough surfaces (droxtal, solid bullet rosette, hollow bullet rosette, hollow column,solid column, plate, compact aggregate of columns, small spatial aggregate ofplates, and large spatial aggregate of plates) and all simulations fall within theobservation envelopes. The results of Baran and C.-Labonnote (2006) show thatpolarized reflectances simulated assuming smooth bullet rosettes and smooth


Table 5.1 Overview of studies on remote sensing of ice crystal shapes

Reference(s) Region or target Data used Dominatingshapes

Notes

Baran et al. (1998,1999)

Tropical anvil,cirrus

ATSR-2 dual view Koch fractals No roughnessor inclusions

Chepfer et al.(1998)

EUCREX’94cam-paign,mid-lat. cirrus

Airborne POLDERR and Rp

Pristine thinplates,a = 0.05–0.1

No roughnessor inclusions

Doutriaux-Boucheret al. (2000), C.-Labonnote et al.(2000, 2001)

Global ocean &land

POLDER-1 R and Rp Columns withinclusions,a = 5


Global ocean andland

Pixel levelPOLDER-1 Rp

Koch fractalsand columnsat lowlatitudes;Plates at highlatitudes



Continental USA GOES West and Eastdual view

Compacts,columns, andbullet rosettes


Knap et al. (2005) Global ocean POLDER-2 R and Rp Columns withinclusions orroughness, a= 2.5

Knap et al. (2005) Tropical anvilout-flow

ATSR-2 dual view Moderatelyroughenedcolumns

Sun et al. (2006) Global ocean Pixel levelPOLDER-1 Rp

Plates andhollowcolumns

Few roughparticlesconsidered

Baran and C.-Labonnote (2006)

Global ocean POLDER-2 R and Rp Roughcrystals, d= 0.4

McFarlane andMarchand (2008)

Southern GreatPlains

Pixel level MISR+MODIS

Roughaggregate andbullet rosettes

van Diedenhovenet al. (2012b)

TWP-ICEcampaign

POLDER- PARASOLRp

Rough plates:a = 0.7 atT < −38 °C;a = 0.15 atT > −38 °C

Only 2 plateaspect ratiosconsidered

van Diedenhovenet al. (2013)

CRYSTAL-FACEcampaign Florida

Pixel level RSP Rp Roughcompact andplate-likecrystals

Cole et al. (2013) Global ocean POLDER-PARASOLR and Rp

Rough habitmixture

(continued)


chain-like aggregates of columns, as well as rough and distorted crystals, fall withinthe global statistics of measured polarized reflectances. In addition, the range ofobserved angular variations of total reflectances is also large in comparison to thevariation in modeled angular features for different ice models (Doutriaux-Boucheret al. 2000; Baran and C.-Labonnote 2006; Baran 2009; Cole et al. 2013), againsuggesting that many considered models may fit well to a subset of the measure-ments. Many such studies aim to select and test optical models for global retrievalsof ice cloud optical thickness and effective radius and selecting a model that fit bestto most of the globally averaged data is very much justified in this case. However,as pointed out by McFarlane and Marchand (2008) “using a featureless phasefunction will likely result in the correct scattering properties on average, howeverindividual cases may have large errors.” To avoid such biases, simultaneousretrievals of ice crystal shape and size can be employed on a pixel level (McFarlaneand Marchand, 2008; van Diedenhoven et al. 2014b). For the purpose of collectinginformation about how ice crystal shape varies with, e.g., temperature, cloud typeand atmospheric state, it is important to study the variation in ice shape and crystal

Table 5.1 (continued)

Reference(s) Region or target Data used Dominatingshapes

Notes

van Diedenhovenet al. (2014b)

TWP-ICEcampaign

Pixel levelPOLDER-PARASOLRp

Rough plateswith d = 0.4–0.7

Propertiesvary withheight andconvectivestrength

Cole et al. (2014) Global ocean Pixel level POLDER-PARASOL Rp

Roughaggregate ofcolumns

Roughnessvaries withlatitude

Wang et al. (2014) Global ocean andland

MODIS over opticallythin cirrus

Mixture ofrough andsmoothcrystals

Differencesbe-tweenocean andland

Baum et al. (2014) Global ocean POLDER-PARASOLRp

Rough habitmixture

Baran et al. (2015) Off coast ofScotland

Pixel levelPOLDER-PARASOLR

Rough crystalmixtures

Datainterpreted asroughnessvarying withhumidity

Letu et al. (2016) Global ocean POLDER-PARASOLR

“Voronoi”habit

Hioki et al. (2016) Global ocean Pixel levelPOLDER-PARASOLRp

Roughaggregate ofcolumns(only habitconsidered)

Unphysicallylargeroughnessparametersfor 74% ofdata


roughness leading to the ranges of observed angular total and polarizedrelfectances.

Studies that focus on the variation of ice crystals shapes are presented by, e.g.,Chepfer et al. (2001), Sun et al. (2006), Cole et al. (2014) and Baran et al. (2015).Analyzing global POLDER-1 data, Chepfer et al. (2001) concluded that“polycrystals” [i.e., Koch fractals (Macke et al. 1996b)] and hexagonal columnsseem to dominant at low latitudes, whereas the hexagonal plates seems to occurmore frequently at high latitudes. However, other than the Koch fractal, no otherdistorted or roughened crystals are included in that study. More recently, Cole et al.(2014) included 9 different shapes (Yang et al. 2015) and a mixture (Baum et al.2011) with a large range of simulated roughness levels in their global study. Theyfound that the aggregate of columns dominates at all latitudes and plates are theleast representative of the POLDER-PARASOL polarized reflectances globally.Interestingly, particles with smooth surfaces (no or low roughness levels) werefound to be more prevalent at high latitudes, while severely rough crystals (r ¼ 0:5)were observed most frequently in the Tropics. This suggests that, in general, icephase functions and polarization functions have some more angular features in thehigh latitudes than at low latitudes, which may be consistent with the findings ofChepfer et al. (2001) that smooth hexagonal plates fit the POLDER-1 data morefrequently at high latitudes than elsewhere. Few roughness parameter valuesr[ 0:5 were found by Cole et al. (2014). Somewhat in contrast with these resultsare the results of Hioki et al. (2016). Applying an algorithm based on empiricalorthogonal function analysis of the modeled and measured POLDER-PARASOLpolarized reflectances and assuming aggregates of columns with varying roughness,Hioki et al. (2016) found roughness values varying with latitude, but they obtainedr[ 1 for most data, which is unrealistically large. They conclude that the unex-pected results indicate that the roughness retrieval is sensitive to an assumed par-ticle shape, although the same aggregates of columns model with r� 0:5 is foundto be matching the data well by Cole et al. (2014). The results by Sun et al. (2006),analyzing global POLDER-1 data, also appear to be in conflict with the studies byChepfer et al. (2001) and Cole et al. (2014) and arguably every other study listed inTable 5.1, since rough particles were found to fit virtually none of the measure-ments and smooth plates and hollow columns were inferred from the data mostfrequently. Smooth plates and hollow columns lead to strong angular variation ofpolarized reflectances that are not often seen in the POLDER data and it may hencebe surprising that Sun et al. (2006) found these habits to fit most of the includeddata. The stringent data selection applied in this study as discussed in Sect. 5.4.2.3may have biased the data. On a regional spatial scale, Baran et al. (2015) investi-gated the local variation of crystals roughness inferred from multi-directional totalreflectances in relation to relative humidity. Using a rather limited dataset off thecoast of Great Britain, they concluded that the occurrence of pristine crystal mix-tures are associated with relatively humid conditions. However, these interestingconclusions are based on only 12 POLDER pixels with inferred pristine particlesand need to be confirmed using a larger dataset. In addition, biases from interfer-ence of lower liquid clouds could not be convincingly excluded.


From the list of dominating shapes in Table 5.1 and the discussion of remotesensing studies above, it is clear that the inferred shapes are highly dependent onwhich shapes are included in the investigation. For example, Cole et al. (2014) findaggregates of columns to be dominating globally but do not consider Koch fractals,while the opposite is true for Chepfer et al. (2001). Also, Cole et al. (2013) found arough general mixture to fit most global data and severely roughened aggregates ofcolumns to be a poor fit, while Cole et al. (2014) applied a wider range of roughnessto all particles and came to the exact opposite conclusion, i.e., a dominance ofaggregates of columns and hardly any data to be more consistent with the generalmixture of habits with any roughness applied. Furthermore, Cole et al. (2013, 2014)found the inferred shapes and roughness values to be dependent on assumed crystalsize. However, as also discussed in Sect. 5.3, this size dependency largely stemsfrom the fact that most of the assumed ice crystal model geometries (i.e., aspectratios of the crystals or their components) depend on size. Thus, assuming a dif-ferent size essentially changes the set of models considered in this case. As dis-cussed in Sect. 5.2, the geometry of the components of such complex crystals, andparticularly their size dependency, is very poorly constrained. A general aspect ofthe studies discussed above is that they focus on the inference of either ice crystalshabit, crystal roughness or a combination of the two. However, as discussed inSect. 5.3, the aspect ratio of the crystals or their components greatly affects thescattering properties and needs to be taken account in a systematic way wheninferring ice crystals shapes from remote sensing data. Without the systematicinclusion of aspect ratios and roughness values in the retrieval products the resultsfrom such remote sensing studies are difficult to interpret as demonstrated above.

A retrieval algorithm to infer the aspect ratios of crystals or their components inaddition to particle roughness level from polarized reflectances is presented by vanDiedenhoven et al. (2012a) and applied to measurements of RSP (van Diedenhovenet al. 2013) and POLDER-PARASOL (van Diedenhoven et al. 2014b). In essence,this approach uses hexagonal plates and columns as proxies for the components ofmore complex particles. Other than most other studies in Table 5.1, the methodincludes a large, nearly continuous range of aspect ratios and roughness values.A look up table of simulated polarized reflectances is used based on a database ofoptical properties for hexagonal plates and columns, which is calculated using thegeometrics optics code developed by Macke et al. (1996b). The aspect ratio ofcolumns is varied between 1 and 50 with 26 geometrically increasing steps. Theaspect ratios of plates are the inverse of those for columns, for a total of 51 aspectratios. The roughness parameter, as defined by Macke et al. (1996b), is variedbetween d ¼ 0 and d ¼ 0:7 in steps of 0.05. The aspect ratio and roughnessparameter values that produce the simulated polarized reflectances that lead to thebest fit to the measurements are considered the retrieved values. Furthermore, theasymmetry parameter is derived from the retrieved aspect ratio and roughnessparameter per Fig. 5.3.

The retrieval technique was evaluated by van Diedenhoven et al. (2012a) usingsimulated measurements based on optical properties of smooth, moderatelyroughened and severely roughened solid plates, solid and hollow columns, solid


and hollow bullet rosettes, droxtals, aggregates of columns and aggregates of plates,as well as several mixtures of these habits (Baum et al. 2005, 2011; Yang et al.2015). The evaluation showed that particles with plate-like, column-like, smoothand rough components are generally correctly identified. For all particles‚ theretrieved roughness parameters increase with increasing roughness of the particlesassumed in the simulated measurements, as expected. As seen in Fig. 5.6, the icecrystal asymmetry parameters are generally retrieved to within 5%, or about 0.04 inabsolute terms, largely independent of calibration errors, range and samplingdensity of scattering angles and random noise in the measurements. Since theasymmetry parameter is largely determined by the aspect ratio and roughness of thecrystal components, this good agreement between retrieved and true asymmetryparameters suggests that aspect ratio and roughness parameters of the componentsof these complex particles are retrieved well by the method, although they were notexplicitly evaluated by van Diedenhoven et al. (2012a). A detailed study on sim-ulated clouds consisting of bullet rosettes with arms of varying aspect ratios androughness values showed that the algorithm retrieves aspect ratios with a bias of20% on average and the roughness parameter within 0.05 when cloud opticalthickness is above 5 (van Diedenhoven et al. 2012a). Errors on retrieved aspectratios increase with decreasing optical thickness. Interestingly, the method finds thatthe polarized reflectances of clouds consisting of hollow columns and hollow bullet

Fig. 5.6 Asymmetry parameters retrieved with the method of van Diedenhoven et al. (2012a)from simulated data based on optical properties of several complex crystals plotted against the trueasymmetry parameters of the complex crystals. Black, green and red colors are results forsimulated data assuming smooth (r ¼ 0), moderately rough (r ¼ 0:03), and severely rough(r ¼ 0:5) particles respectively. For each combination of roughness and habit, 10 different sizedistributions are applied. See van Diedenhoven et al. (2012a) for further details. The solid lineshows the 1–1 line. Dotted lines indicate the targeted 5% accuracy limits. Figure reproduced fromvan Diedenhoven et al. (2012a)


rosettes most closely resemble those consisting of columns with high aspect ratio(a[ 15). This result makes sense since the walls of the hollow parts of the crystalsare thin, resembling columns with high aspect ratios (see Sect. 5.3 and Figs. 5.4and 5.5). In addition, van Diedenhoven et al. (2012a) showed that asymmetryparameters of mixtures of smooth and rough complex particles as defined by (Baumet al. 2011) are also mostly retrieved within 5%. For mixtures of hexagonal columnsand plates with varying aspect ratios and roughness values, van Diedenhoven et al.(2016a) showed that the average absolute errors between retrieved and theensemble-average aspect ratio a� 1 are generally below 0.1. Furthermore,ensemble-average roughness parameters are generally retrieved within 0.1.Generally, the approach tends to be somewhat biased toward retrieving column-likecrystals, although for about 75% of the test cases the dominating geometry wascorrectly determined. Furthermore, only considering mixtures that are dominatedfor over 3/4 by either plates or columns yielded the correct dominating geometry inabout 90% of the cases.

Van Diedenhoven et al. (2013) further evaluated this approach applied tomeasurements of the RSP instrument collected during the CRYSTAL-FACEcampaign based in Florida in 2002. Four case studies were analyzed: two cases ofthick convective clouds and two cases of thinner (detached) anvil cloud layers. Inall cases the measurements indicate roughened ice crystals, consistent with previousfindings. Retrieved aspect ratios in three cases were found to be close to unity,indicating that compact particles dominate the radiation, qualitatively consistentwith CPI images where available. Retrievals for one contrasting anvil case indicateice crystals consisting of plate-like components with aspect ratios around 0.3,consistent with the increased number of aggregates of plates seen in the CPI imagesobtained in this cloud layer.

An example of crystal properties varying per cloud type and conditions isillustrated in Fig. 5.7, which is derived from (previously unpublished) data of RSPand the Cloud Physics Lidar (CPL), both mounted on NASA’s high-altitude ER-2aircraft during the Studies of Emissions and Atmospheric Composition, Clouds andClimate Coupling by Regional Surveys (SEAC4RS) campaign based out ofHouston, Texas in 2013 (Toon et al. 2015). This figure also shows retrievals of iceeffective radius (Eq. 5.12) using the shortwave infrared measurements on RSP(Nakajima and King 1990; van Diedenhoven et al. 2014b, 2016b) as well as anindex quantifying the strength of observed specular reflection in the RSP data (cf.Bréon and Dubrulle 2004). For completeness, the cloud top height derived by RSP(Alexandrov et al. 2012; Sinclair et al. 2017) and CPL are also given, in addition tothe CPL lidar depolarization and penetration depth. Here, CPL penetration depthindicates the physical depth at which CPL signals saturate, which is related to themean extinction at cloud top (cf. van Diedenhoven et al. 2016b). Three differentcloud conditions were observed on September 2nd, 2013. The top row of Fig. 5.7shows retrievals for a frontal cloud system that was sampled, indicating a ratherlarge variability in effective radius with a peak near 35–40 lm, cloud top heights ataround 11 km and almost exclusively rough compact crystals with aspect ratiosnear unity, roughness parameters around 0.5 and asymmetry parameters near 0.75.


The second row shows retrievals for the following hours, when the cloud systemwas dissipating. Interestingly, the tops of these clouds seem to sublimate first,leaving the lower ice cloud layers visible to the RSP. The retrievals indicateplate-like particles with lower aspect ratios and a broader range of roughness valuescompared to the previous case. Observations of specular reflection indicates hori-zontally oriented ice plates were present in some regions. CPL depolarization ratiosfor these cloud layers are also lower compared to the earlier measurements, whichmight be caused by a change in ice habit, but may also be due to horizontallyoriented crystals, especially in the case of depolarization values below 0.25. RSPeffective radius values for these clouds range between 15 and 45 lm and consid-erable differences are seen between sizes retrieved with 2.26 and 1.59 lm bands,which implies substantial vertical variations of ice sizes (van Diedenhoven et al.2016b). Finally, the third part of the day was devoted to sampling convection overland. The retrievals, shown in the third row, generally yield much smaller effectiveradii around 20 lm with almost no difference between retrievals using differentspectral bands, indicating little vertical variation. This is consistent with relativelyshallow lidar penetration depths for this case indicating compact and opaque cloudtops. The aspect ratios and asymmetry parameters show more variation than for therest of the day, although lidar depolarization at cloud top has a narrower

Fig. 5.7 Histograms of ice cloud properties retrieved from measurements of RSP and CPLobtained on 2 September, 2013, during the SEAC4RS campaign. From left to right respectively,the panels show the effective radii, asymmetry parameters, aspect ratios, roughness (or distortion)parameters d, specular reflection index, cloud top heights, lidar penetration depths anddepolarization ratios. Effective radii retrieved with the RSP channels at 1.59 and 2.26 lm areshown in yellow and blue, respectively. CPL retrieved quantities are shown in green. The top andmiddle panels show retrievals obtained during flight legs over a frontal cloud system in developingand dissipating stages, respectively. The bottom panel shows data for convective clouds over land.See text for more details


distribution. Furthermore, roughness parameters are somewhat larger for theseconvective clouds compared to the frontal system shown in the top row. This casestudy demonstrates the complex variability that can be observed in ice shapes andother properties of ice clouds. Furthermore, it shows the benefits of combiningdifferent retrievals techniques and instruments crucial to obtain a more completeview of ice cloud properties.

Van Diedenhoven et al. (2014b) presented retrievals of ice crystals shape andasymmetry parameters from POLDER measurements collected off the north coastof Australia in relation with the Tropical Warm Pool—International CloudExperiment (TWP-ICE) campaign in 2006 (May et al. 2008). The data are dividedinto periods of 4–9 days with alternating weak and strong convection, indicated byobserved rain rates. Furthermore, the data is presented as a function of cloud toppressure and temperature, as ice crystal properties are generally observed to varywith temperature (e.g., Lawson et al. 2010; Noel et al. 2004, see also Sect. 5.2).Only clouds with an optical thickness larger than 5 are included. The mean resultsshown in Fig. 5.8 indicate that mostly plate-like particle components with meanaspect ratios (a� 1) around 0.6 and low asymmetry parameters characterize stronglyconvective periods, while weakly convective periods generally show particles withlarger asymmetry parameters, lower component aspect ratios, somewhat lowerroughness parameters and more column-like crystal components. The abundance ofcompact plate-like crystals in the tops of convective clouds is consistent withprevious observations of the dominance of compact and aggregated ice crystalswith plate-like components observed in tropical deep convection (e.g., Noel et al.2004; Connolly et al. 2005; Um and McFarquhar 2009; Baran 2009, see alsoSect. 5.2). There appears to be a trend towards lower aspect ratios and morecolumn-like particles at warmer temperatures. The more column-like ice crystalswith component aspect ratios further deviating from unity as indicated by theobservations during the weakly convective periods may be consistent with astronger contribution of particles grown in situ, which are more likely to form ascrystals with column-like components, such as bullet rosettes, at the observedtemperatures (Bailey and Hallett 2009; Baran 2009; Gallagher et al. 2012; Fridlindet al. 2016, see also Fig. 5.1). Comparing the results for the three strongly con-vective periods shows that microphysical parameters observed during the first twoare very similar, while the third period shows somewhat greater roughness, fewercolumn-like crystals and lower asymmetry parameters. As shown by vanDiedenhoven et al. (2014b), this later period also had substantially larger effectiveradii, especially at the warmer temperatures. The meteorological quantity thatpossibly distinguishes the third strongly convective period from the other periodswith strong convection is the middle-to-upper tropospheric zonal wind shear, whichis much weaker for the third period and may have affected crystal evolution. Forstrongly convective periods, the roughness parameter values are about 0.55 onaverage and decrease significantly with increasing cloud top temperature, whileasymmetry parameters increase. These results suggest systematic variations ofcrystals shape characteristics in relation to cloud top heights and atmosphericconditions. Such relationships need to be substantiated with more data. As


demonstrated by van Diedenhoven et al. (2014b), such observed variations havesignificant impacts on the radiative properties of convective clouds, which need tobe better understood to improve their representation in climate predictions.

In order to compare convective clouds observed in different atmosphericregimes, Fig. 5.9 shows histograms of ice crystal asymmetry parameters, aspectratios and roughness parameters retrieved over convective clouds using(1) POLDER-PARASOL data collected in relation with the TWP-ICE campaign(May et al. 2008; van Diedenhoven et al. 2014b); (2) POLDER-PARASOL datacollected in relation to the TC4 campaign (Toon et al. 2010) in 2007 near CostaRica (previously unpublished); and (3) RSP data collected during the SEAC4RScampaign (Toon et al. 2015) based out of Houston, Texas in 2002 (cf. vanDiedenhoven et al. 2016b). Only ice clouds with optical thicknesses larger than 5are included in all data sets. Statistics of asymmetry parameters, aspect ratios and

Fig. 5.8 Mean ice crystal-component aspect ratio (a� 1, top left), roughness parameter (top right)and asymmetry parameter (bottom right) retrieved from POLDER-PARASOL data off the northcoast of Australia between 16 January and 20 February, 2006. The percentage of retrievedcolumn-like aspect ratios (i.e., a[ 1) is shown in the bottom left panel. Data within 25–hPa– widecloud top pressure bins are averaged to produce profiles for five different periods (indicated bycolors) with alternating strong and weak convective strengths. See text and van Diedenhoven et al.(2014b) for further details


roughness parameters derived from these different datasets are very consistent. Thedata indicate crystals with mostly plate-like components (a\1), which is consistentwith in situ measurements in convective clouds (e.g., Um and McFarquhar 2009).Also, aspect ratios close to unity are mostly found, indicating the dominance ofcompact particles. Roughness values are generally large with maxima greater at 0.5,which is largely consistent with the roughness statistics found in tropical regions byCole et al. (2014). Asymmetry parameters peak at around 0.76, but the distributionshows a substantial tail toward larger values.

In summary, results from retrieval approaches that focus on retrieving specificice crystal habits are generally inconclusive and highly dependent on the shapesincluded in investigations. One general conclusion that can be derived from thevarious studies is that particle roughness is prevalent. However, roughness levelsare found to vary with location, cloud top temperature, and atmospheric conditions.Systematic retrievals of crystal component aspect ratios show convective cloudregimes generally have crystals with aspect ratios close to unity and mostlyplate-like particles, but the particle properties depend on multiple factors, e.g.,cloud type, cloud height, convective strength and possibly other dynamical quan-tities, such as wind shear and humidity. More global and local studies are needed tountangle such relationships between ice crystals shape and cloud type, cloud heightand atmospheric conditions.

5.5 Prospective

Although many studies on remote sensing of ice crystals shapes using lidar andmulti-directional reflectance data have been performed over the past few decades,the discussions in Sects. 5.4.1 and 5.4.2 show that it remains difficult to extractsystematic conclusions from these studies. Specifically, robust quantitative inter-pretations of lidar depolarization measurements are still not available. As discussed

Fig. 5.9 Histograms of the retrieved asymmetry parameters (left), aspect ratios (middle) androughness parameters (right) for convective clouds observed in relation to the TWP-ICE (red), TC4

(green) and SEAC4RS (yellow) campaigns


in Sect. 5.4.1, more research on optical properties calculations will likely improvethe prospective of gaining quantitative information about ice crystals shapes fromlidar measurements. Furthermore, measurements of a multistatic lidar, as proposedby Mishchenko et al. (2016), measuring the backscattered signal at additionalangles and thus probing the depolarization properties at two or more angles‚ couldyield increased information content for the retrieval of ice crystal shape from lidars(cf. Smith et al. 2016). Combining colocated lidars at multiple wavelengths is notexpected to increase potential for ice shape retrievals since the ice scatteringproperties are largely wavelength independent. Also high spectral resolutioncapability, such as included in ATLID and the airborne HSRL (Burton et al. 2015),is not expected to increase information content on ice crystal shapes, although itwill provide valuable measurements of ice cloud extinction.

Currently, the POLDER instruments have been the only multi-directionalpolarimeters deployed in space. As polarimetry has great potential for cloudretrievals, as well as for the inference of aerosol properties, polarimeters are con-sidered for many future satellite missions. For instance, the Multi-viewing,Multi-Channel Multi-Polarization Imaging instrument (3MI) is a follow-up versionof POLDER and is planned to be included on the European MetOp series (Marbachet al. 2013). NASA’s upcoming Plankton, Aerosol, Cloud, ocean Ecosystem(PACE) mission will likely include a multi-directional polarimeter. Furthermore,the Hyper-Angular Rainbow Polarimeter (HARP) instrument is a cubesat missionthat is slated be be launched soon. Also the Multi-Angle Imager for Aerosols(MAIA) instrument is selected by NASA for further development and spacedeployment. It is expected that all these multi-directional polarimeters havepotential for the retrieval of ice crystal shapes that is similar to or better thanPOLDER’s. Interestingly, the 3MI instruments will include a channel at around1370 nm that is located on a strong water vapor absorption band. This limits theband’s sensitivity to the surface and to liquid clouds in the lower atmosphere andincreases the sensitivity to thin cirrus (cf. Gao et al. 1993). The RSP band at1880 nm has similar capabilities and was used by Ottaviani et al. (2012) to infer theice shape of a thin cirrus layer over the Deepwater Horizon oil spill site in 2010.Furthermore, the SPEX airborne polarimeter (Rietjens et al. 2015) yieldsmulti-directional polarization measurements in the oxygen A-band that similarlyshields the lower atmosphere and surface. Further statistical evaluation of suchmeasurements will provide valuable statistics of thinner cirrus clouds, which arelargely excluded from the current remote sensing results.

Ice crystal shape retrieval approaches have been largely limited to finding bestfits to the measurements within look up tables of simulated measurements. It isadvisable that more systematic inversion techniques are employed since these allowfaster data processing, non-discrete solutions, and, more importantly, better errorestimations (Rodgers 2000). For example, Hioki et al. (2016) recently developed analgorithm based on empirical orthogonal function analysis of polarized reflectancesto infer ice crystal roughness levels from such measurements. Approaches thatretrieve quantifiable parameters such as ice crystal component aspect ratios androughness parameters are especially suitable for the implementation in an algorithm


employing such an inversion technique. The results of van Diedenhoven et al.(2016b) suggest that such an approach could infer the relative contribution ofplate-like and columnar components to the ice crystal distributions, in addition tothe mean aspect ratio (a� 1) and roughness parameter.

To date, few systematic investigations on the variation of crystal shape, aspectratios and crystals roughness and their relation with cloud type, height and atmo-spheric conditions have been performed (e.g., Baran et al. 2016; Cole et al. 2014;van Diedenhoven et al. 2014b). More global and local studies are needed tountangle such relationships between ice crystals shape and cloud type, cloud heightand atmospheric conditions. Such relationships will provide observational con-straints for improved parameterizations of ice cloud optical properties (e.g., Baranet al. 2016) and for microphysical packages for cloud simulations, especially thoseexplicitly prediction ice crystal shapes (e.g., Hashino et al. 2007, 2011; Harringtonet al. 2013).

5.6 Conclusions

Improved constraints on the natural variation of ice crystal shapes is important sincethe shape of ice greatly affects their radiative and microphysical properties.Theoretical and laboratory studies show that ice crystal shapes largely depend ontemperature and humidity of the environment in which they grew. However, in situmeasurements in real ice clouds generally show complex mixtures of shapes andlarge contributions of irregular, complex, aggregated crystals. Also, the aspectratios of components of these crystals are found to vary substantially. In addition,high magnification imaging of ice crystals show roughness structures of variousforms and levels on the ice surfaces of growing and sublimating ice crystals. All ofthese macro- and microscale ice shape characteristics substantially affect theradiative properties of ice clouds and better constraints on how these shape char-acteristics vary with cloud type, temperature, humidity, locations, availability ofaerosols, etcetera, are crucial to improve the representation of ice clouds in climateprojections. This chapter reviews the current state of remote sensing of ice crystalmacro- and microscale structure.

The radiative properties of clouds are determined by the number of ice crystals,their extinction cross sections, single scattering albedos and scattering phasematrices. Of these optical properties, the scattering phase matrix is especially rel-evant for the remote sensing of ice shapes, since it is substantially dependent to icecrystal shape but relatively independent to the size of ice crystals at non-absorbingwavelengths. The angular features in the scattering phase function and the linearpolarization phase function that depend on ice crystals shape are used by remotesensing studies. Also the depolarization of backscattered polarized lidar signals isused to obtain information on ice crystal shape. Reviewing the dependencies ofscattering phase matrices on ice shape reveals that the phase matrices are mostlydetermined by the aspect ratios of components of complex crystals as well as by the


microscale structures such as crystal roughness, while the macroscale shape (i.e.,habit) is of lesser importance. Furthermore, while ice particle macroscale shape hasa seemingly endless variability and is not a quantifiable parameter, crystal com-ponent aspect ratio and roughness level are quantifiable and can be systematicallyrelated to variations in the phase matrix. For example, phase function asymmetryparameters increase as aspect ratios deviate from unity and decrease as roughnesslevels increase. It is therefore advised that remote sensing studies focus onretrieving information about crystal component aspect ratios and microscalestructure, rather than on inferring the occurrence of specific ice habits.

A review of the literature on lidar depolarization measured in ice clouds showsthat depolarization is generally found to increase with increasing cloud height andalso varies with latitude. This variation is generally linked to the variation of icecrystal shape. However, the interpretation of the depolarization remains largelyqualitative and inconclusive. For simple, smooth hexagonal prisms, lidar depolar-ization is shown to vary with crystal aspect ratio, but studies on the effects of crystalmacroscale complexity, microscale roughness and hollowness are lacking, contra-dictory or inconclusive. More research on the relation between lidar depolarizationand ice crystal shape is advised in order to move toward more quantitative infer-ences of ice crystal shape properties from lidar measurements.

Numerous studies evaluated the angular variation of total and/or polarizedreflectances of ice clouds in order to infer information about ice crystal shape fromthem. A general conclusion is that ice crystal surface roughness or crystal distortionis prevalent. However, the conclusions about the dominating ice shapes are oftencontradictory. Furthermore, the data suggests that ice crystals shape and roughnessis highly variable. Perusing the various studies, it is clear that the inferred shapesare strongly dependent on which shapes are included in the investigation and thatsuch selections of shapes can be interpreted as rather arbitrary. Moreover, thespecific geometries to define these shapes, most importantly the aspect ratios oftheir components, are very unconstrained. Since the inferred crystal shapes often donot agree with what is expected from in situ measurements and theory, the retrievedshapes are often interpreted as merely “radiative equivalent effective shapes” withno direct relation to ice crystal shapes in the observed clouds. However, focusingremote sensing applications on retrieving crystal component aspect ratio and par-ticle roughness, rather than ice habit, yields useful physical information about theseshape characteristics. Retrieval approaches that focus on inferring aspect ratios and/or roughness reveal that ice shapes depend on cloud height, latitude, cloud type,convective strength and possibly on humidity and dynamical quantities as windshear. Statistics of ice crystal component aspect ratios, roughness parameters andasymmetry parameters of convective clouds in several different regimes are verycomparable, suggesting generalized conclusions could be derived from such mea-surements. The data for these convective clouds indicate crystals with mostlycompact plate-like components with high roughness levels are prevalent.Asymmetry parameters peak at around 0.76, but the distribution is substantiallywide with a tail toward large values.


As is generally the case with all remote sensing products, the results in thecurrent literature represents a subset of all ice clouds. Especially studies usingmulti-directional measurements are often biased to relatively optically thick iceclouds and thus excludes most cirrus. Improved data selection (e.g., Wang et al.2014) and analysis of measurements obtained at wavelengths with substantialatmospheric gas absorption (e.g., Ottaviani et al. 2012) could yield more infor-mation about thin cirrus. Furthermore, the multi-directional total and polarizedreflectances as well as lidar depolarization only yield information of the top 1–3optical depths of clouds, while ice crystal shape is likely to vary vertically in clouds.This limitation also applies to retrievals of other cloud particle properties such asphase and size (Platnick 2000; van Diedenhoven et al. 2016b). Hence, it isimportant to augment such retrievals with, e.g., in situ studies of the verticalvariation of cloud particle properties. For convective clouds, however, statistics ofcloud top trends of ice crystal properties can generally be used as surrogates fortrends with height within convective cloud tops (referred to as time-space inter-changeability, Lensky and Rosenfeld 2006; van Diedenhoven et al. 2016b).Furthermore, cloud top properties are very relevant as the top of clouds can beconsidered as radiatively the most relevant part.

Lidar depolarization and multi-angular measurements, in addition to in situobservations, consistently reveal that ice crystal shape varies considerable withcloud top height, cloud type, location and atmospheric state. Ignoring these vari-ations leads to biases in retrievals of ice effective radius and optical thickness thatare sensitive to shape as well as to biases in simulations of ice cloud properties andtheir radiative effects. Research has begun to untangle such relationships betweenice crystals shape and cloud type, cloud height and atmospheric conditions, butmore studies are needed to reach systematic conclusions. Given the high potentialof polarized lidar and multi-angle polarimeters for cloud and aerosol remotesensing, such instruments are planned or considered for many future satellitemissions providing many future opportunities to further study the global variationof ice crystal shapes.

Acknowledgements Bastiaan van Diedenhoven is supported by NASA under project numbersNNX14AJ28G and NNX15AD44G. I would like to thank Dr. Nathan Magee at The College ofNew Jersey for providing the electron microscope images of hexagonal ice crystals images. I amgrateful to Dr. Ping Yang for providing the optical properties of complex ice crystals.

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Chapter 6Light Scattering in Combustion:New Developments

Alan R. Jones

Abbreviations

ADT Anomalous diffraction theoryCCD Charge coupled deviceCRD Cavity ring downDDA Direct dipole approximationDf Fractal dimensionFFT Fast Fourier transformFII Fourier interferometric imagingGA Genetic algorithmGPD Global phase DopplerG-RDG Generalised RDGILIDS Interferometric laser imaging for droplet sizingLDV Laser Doppler velocimetryLIDAR Light detection and rangingLIF Laser induced fluorescenceLII Laser induced incandescencePAS Photoacoustic spectroscopyPDA Phase Doppler anemometryPIV Particle imaging velocimetryRg Radius of gyrationRDG Rayleigh-Debye_GansRDG-FA Rayleigh-Debye-Gans-fractal-aggregateSDV Shadow Doppler velocimetrySLIPI Structured light illumination planar imagingSMPS Scanning mobility particle sizingTEM Transmission electron microscopyTIR-LII Time resolved LIITRFPA Time-resolved fluorescence polarization anisotropy

A. R. Jones (&)Department of Chemical Engineering, Imperial College, London, UKe-mail: [email protected]


251

6.1 Introduction

This review is an update to that published in Jones (2006). In the intervening yearsa very large literature has been published on monitoring particulates in combustion.By far the bulk of this has been studies on soot because it is a primary product ofcombustion and has consequences for heat transfer, pollution and atmosphericaerosol. Apart from developments in the more traditional optical techniques forcharacterising soot there has been considerable interest in laser induced incandes-cence (LII), a method based on sizing via the mass obtained through cooling rates.Apart from particle size other properties to be reported include refractive index,aggregation (and fractal analysis), heat transfer from fires and soot mixed in withwater drops.

Coal remains a major contributor to combustion and it is evidently important tounderstand the burning process and the formation of pollutants, including ash. Inaddition there are liquid fuel sprays for which the rate of burning must be estab-lished in terms of heat transfer, trajectories and velocities. Methods for carrying outthese studies will be discussed. These include velocimetry in two phase flow bycombining traditional methods with fluorescence, out of focus interferometry(ILIDS).

There is also the question of non-spherical particles and methods to explore this.Particles in nature are not generally regular, such as spheres, cylinders, cubes,polyhedra and so on. These regular shapes are usually restricted to small individualparticle or crystal structures. More often particles may be irregular, in the sense thattheir shape cannot be predicted from one particle to another. However it must besaid that some authors refer to non-spherical particles as irregular, which may notbe the case. True irregularity should be treated statistically, or simply by taking anaverage over a large number of randomly oriented particles. For example a statis-tical approach was attempted by Jones (1987), for Fraunhofer diffraction, and by AlChalabi and Jones (1995). Alternately, numerical methods such as the direct dipoleapproximation (DDA) (Draine 1988) or T-matrix method (Mishchenko and Travis1994) may be used. These require knowledge of the actual particle shape, but anumber of different shapes may be generated and the scattered intensities added.This is computationally extensive. A review has been provided by Mishchenko(2009). For the case of small agglomerates that may be treated in theRayleigh-Debye-Gans (RDG) approximation the fractal geometry approach may beused (Sorensen 2001).

As a general comment, some sizing techniques rely on intensity measured as aparticle crosses a laser beam. Often difficulty arises due to the Gaussian nature ofthe beam so that the scattered intensity is dependent on the position of the particle.To avoid this, a so-called “top hat profile” may be employed in which the laserbeam is expanded and only a small central region accepted. This method has thedisadvantage of not making full use of the laser illumination. An alternative is abeam homogeniser, such as that described by Pfadler et al. (2006).

252 A. R. Jones

A review of scattering by arbitrary shaped beams by non-spherical particles hasbeen given by Gouesbet and Lock (2015).

Flames and combustion products are very hostile environments. The tempera-tures involved are normally well in excess of 1000 °C. This, combined with thehigh flow rates and bombardment by particles, is damaging to any probes that areinserted into the gas stream. This is one of the reasons why optical techniques havebeen developed into powerful tools for combustion diagnostics. Apart from thewindows necessary in some cases, all the optical components are external to thecombustion system. In addition, electromagnetic radiation at moderate intensitiesdoes not significantly interfere with the object under study, unlike, for example, theinsertion of a probe.

The interaction of radiation with solid particles or liquid drops is covered underthe generic term “scattering”. The nature of the interaction depends upon the par-ticle size, shape, structure, concentration and refractive index. In principle, there-fore, scattering can be used to measure all of these variables. The nature of thescattering process can be a simple rebound not involving a change in frequency,other than Doppler shift. This is elastic scattering. Alternatively, there may befrequency shifts due to absorption and re-emission or due to non-linear effects. Thisis inelastic scattering. This chapter will be restricted to elastic scattering.

Interaction of light with gases can be problematic. Most generally, it is tem-perature gradients that cause difficulty because they cause deflections of the lightthat may lead to uncertainty in the position and size of the measurement space andspreading of the incident beam. There may also be a loss of intensity due toscattering out of the beam before it reaches the measurement point. A further dif-ficulty in industrial combustors is that access may be limited and often only oneport is available. Fogging of windows is also an important problem that will affecttransmission of the light in and out of the combustor, leading not only to a loss ofintensity but also false results. For this reason, methods that do not rely on absoluteintensity are to be preferred. Another problem is the presence of thermal radiation.This occurs at all wavelengths and can create difficulty in separating out the scat-tered signal.

When preparing this review a literature search was conducted from 2005onwards to 2015. It became apparent that there was a very large bulk of publica-tions, and it would be necessary to be selective to some degree to keep the articlewithin reasonable bounds. With this in mind it will concentrate on significantdevelopments of existing techniques and new measurement methods. New resultson the nature and development of combustion will not be covered. Evidently theselected references are a very personal choice and apologies are proffered to anyauthors who feel they should be included.

6 Light Scattering in Combustion: New Developments 253

6.2 Soot

6.2.1 Laser Induced Incandescence (LII)

During the ten year period covered here this technique has been by far the mostpopular subject in the literature. The basic concept is simple. The absorption of apulse of laser radiation causes heating of the particles and thermal emission. Theproperties are deduced from the temporal profile of the emission, and, in particular,after the pulse the rate of cooling is observed. This rate is inversely proportional tothe mass, or volume, of the particles and thus the size may be obtained. The conceptoriginated with Melton (1984). Advantages of the technique include its apparentsimplicity and excellent sensitivity, estimated to be better than one part per trillion(2 lm/m3). The first measurements based on the temporal profile appear to be dueto Will et al. (1995). Other useful references prior to 2005 are given in the earlierreview.

Analysis of the measurements is dependent upon a suitable theory to explain theheat transfer characteristics between the particles and the laser beam and hot gases.A number of models have been proposed dependent upon the prevailing conditions.

It is critical to accurately model the heat transfer between the particle and theambient gas during the cooling phase. At atmospheric pressure this is almostentirely due to free molecular heat conduction. Here gas molecules travel ballisti-cally between the equilibrium gas and the particle surface without undergoingintermediate intermolecular collisions. Then the overall conduction rate depends onthe energy transferred when a gas molecule collides with the particle surface. Thisquantity is specified via the thermal accommodation coefficient. There has beenmuch uncertainty in the literature about the value of this parameter.

Other aspects that have been highlighted include the presence of polydispersityand radiative shielding. Also, a problem can arise if the soot temperature is too highwhen significant evaporation and sublimation can occur. To avoid this, the laserflux should be kept below 0.2 J cm−2.

References prior to 2005 that cover the above in more detail are given in theearlier review (Jones 2006).

Critical reviews were published by Schulz et al. (2006) and Michelsen et al.(2007). The former was based on a workshop that brought into question the statusof LII up to that date. They indicated that the understanding of LII at that time waslimited enough to lead to a wide variability in model predictions and experimentalresults, even under well-defined conditions. The predictions of nine models werecompared, each using the same input data. Similarly, Michelsen et al. compared tenmodels.

Very significant differences were found for predicted cooling rates, with evidentconsequences for measured particle volumes. The studies highlighted the uncer-tainties in the understanding of the mechanisms influencing LII signals, and con-clude that considerable work was needed to narrow these uncertainties. Theaccuracy of the data obtained was significantly influenced by many experimental

254 A. R. Jones

parameters, e.g. excitation and detection wavelengths, laser fluence and spatialbeam profile, temporal detection issues, and calibration methods. Many of thedifferences between the models could be attributed to the values of a few importantparameters, such as the refractive-index function and the thermal and massaccommodation coefficients. Shock tube experiments were proposed that mayclarify the value of the thermal accommodation coefficient. Generally agreementbetween the models was better at low fluences, possibly because of the effects ofevaporation and sublimation at the higher fluences. It was suggested that furtherexperiments were needed to identify species present when soot evaporates. Bladhet al. (2009) concluded that although there is some influence of the spatial laserenergy distribution on the evaluated particle sizes both in modelling and experi-ments, this effect is substantially smaller than the influence of the uncertainties ingas temperature and the thermal accommodation coefficient.

Michelsen et al. (2008) considered aspects of sublimation, especially thepotential effects of oxidation. Their model suggests that recovered particle sizesmay be underpredicted by as much as 9% if expansion work and oxidation areneglected. A thermal accommodation coefficient of around 0.3–0.25 was also foundfor carbon oxidizing to carbon monoxide.

Memarian and Daun (2014) and Memarian et al. (2015) discussed reasons fordiscrepancies between models and experiment for high fluences where sublimationwas present. This has variously been ascribed to back flow of sublimed species andthe formation of shock waves. They confirmed the presence of back flow, but shockwaves were not predicted.

In 2009 Michelsen examined the influence of temperature on the thermalaccommodation coefficient. A polynomial in terms of soot and gas temperatureswas developed for NO gas. This suggested values of approximately 0.8 at very lowtemperatures, reducing to about 0.4 at 1000 K and 0.18–0.19 at 1900 K. The lattertemperature is thought to be particularly appropriate in LII analysis and thesecoefficients were significantly smaller than is typically used. Michelsen’s resultsfurther suggested that re-condensation of sublimed species on the originatingnanoparticle was not significant. Allowing for condensation of sublimed speciesoriginating from neighbouring soot particles enhanced the role of re-condensationof sublimed species in slowing down the soot particle temperature decay. However,it was still not sufficient to be considered as a plausible cause for the discrepancybetween modelled soot temperature and those measured by two-colour pyrometryand high-fluence LII.

In combustion a complex mixture of gases will surround the particles. Daunet al. (2008) explored the thermal accommodation coefficient for a variety of gases.They extracted soot from an ethylene flame. The sizes of the sampled soot particleswere measured by electron microscopy. Other samples were entrained into amoving stream of a variety of gases. LII measurements were carried out, and thecoefficient was inferred from the observed temperature decay rate, given that thesize was known. The experimental results showed that the thermal accommodationcoefficient increases monotonically with molecular mass for monatomic gases, andbecomes smaller as the structural complexity of the gas molecule increases for


gases having similar molecular masses. Interestingly their results indicated a verylow value in hydrocarbons of between zero and 0.1.

The influence of gas conduction on modeling has been reviewed by Kuhlmannet al. (2006). They looked at discrepancies between various models and found theyare probably due to uncertainties in the thermal accommodation coefficient. Basedon data from electron microscopy an effective value was found to be 0.25 forisolated particles, and for fractal aggregates 0.45. Maffi et al. (2011) explored thethermal accommodation coefficient in rich premixed flames. They found a figure of0.22 for young soot and 0.4 for mature soot. As above, this may have been due toaggregation.

When considering aggregates it is often assumed that there is point contactbetween the primary particles. However, electron microscopy indicates that there isactually a finite area of contact, or bridging. The potential impact of this wasexplored by Johnsson et al. (2013) by considering overlapping spheres. Theyconcluded that the primary particle diameters could be overestimated by up to 9% ifbridging was not taken into account. Skorupski and Mroczka (2014) also examinedthis problem, but limited to two and five monodisperse soot particles and assuminga cylindrical connector. The scattering was modelled using DDA. In general theinfluence on relative absorption cross section was small, but could be as high as3.75%.

Yon et al. (2015a) further investigated the influence of overlapping. They alsofound that overlapping and necking significantly affected the absorption and scat-tering properties of soot aggregates. This was especially so in the near UV spectrumdue to enhanced multiple scattering effects within an aggregate. They concludedthat by accounting for the effects of multiple scattering, the simple RDG theory forfractal aggregates could reproduce reasonably accurate radiative properties ofrealistic soot aggregates.

Calibration procedures have been considered by a number of authors. Forexample Pastor et al. (2006) considered two different calibration methods in orderto obtain the most suitable procedure for sooting conditions in a laminar diffusionflame.

Lehre et al. (2005) also reviewed the theory and pointed to the difficulty ofknowing precisely the extent to which the laser energy was absorbed. The hotparticle radiates thermally and they sidestepped the problem by measuring theparticle radiation at two wavelengths. The temperature was then obtained using twocolour pyrometry via Planck’s Law (De Iuliis et al. 2005).

It is known that the refractive index of soot varies with wavelength and that thiscan influence the recorded temperature by the two colour method. The influencethat this may have when applying LII has been discussed by Goulay et al. (2010).Depending upon the choice of the functional form of the emissivity, the maximumtemperature reached by the soot during the laser pulse at 1064 nm was calculated tospan a range of 3475–3816 K with a fluence of 0.1 J cm−2 and 4115–4571 K witha fluence of 0.4 J cm−2. It was acknowledged that the results may vary dependingupon the particular flame conditions. The spread in temperature can lead to large

256 A. R. Jones

uncertainties regarding the physico-chemical processes occurring at the surface ofthe soot during the laser heating.

Further information on refractive index was provided by Therssen et al. (2007).They measured E(m,1064 nm)/E(m,532 nm) where

E m; kð Þ ¼ �Imm kð Þ2�1

m kð Þ2 þ 2

!; m kð Þis complex refractive index

They employed LII and observed radiation at the two wavelengths, taking careto ensure that the fluence was sufficiently low that it would not disturb the result.The method was demonstrated on an acetylene/air flame and validated againstextinction measurements performed by cavity ring-down spectroscopy.

Eremin et al. (2011) discovered that the refractive index of carbonaceous par-ticles can vary significantly as they grow. They applied two-colour LII to carbonparticles growing in acetylene behind a shock wave. Small carbon particles of about1–14 nm in diameter had a low value of E(m,k) similar to 0.05–0.07, which tendsto increase up to a value of 0.2–0.25 during particle growth up to 20 nm. Also ameans of following the cooling as a function of time is desired. This is achieved byusing a pulsed laser (in their case of 13 ns duration) and a single shot streak camera,which displays the received radiation as a function of time. (The method is nowreferred to as two colour time resolved LII—or two colour TIR-LII.)

A means of increasing the sensitivity of the two colour method, and with it theaccuracy, has been proposed by Flugel et al. (2013). A major drawback of tradi-tional LII is the use of narrow band interference filters which greatly reduce theavailable signal. When carbonaceous particles are studied this may not be a problembecause they are strong radiators due to their high emissivity. Other materials mayexhibit significantly lower emissivity. This can partly be addressed using bandpassfilters with wider transmission windows. Earlier, Lehre et al. (2005) had attemptedthis approach.

Flugel et al. (2013) suggest increasing the detected signal by a careful selectionof a combination of shortpass and longpass filters. These effectively form edgefilters which can cut off the incident laser radiation while permitting much more ofthe emitted radiation to pass. The signal intensity is improved by a factor ofapproximately 20. However, the authors point to the need to take into account thespectrum of other emissions and, in particular, avoid the transmission offluorescence.

There are two riders. Particle size distribution is obtained assuming a log-normalvariation and fitting the parameters to the time signal using a regression analysis.Also there is a need to fully understand the evaporation process and, thereby, thecontribution of latent heat. The authors point to the need to develop the ability tomeasure the signals at lower soot temperatures, preferably below 2000 K.

An important factor in TIR-LII is the time response of the detection system. Thishas been investigated by Shaddix and Williams (2009), Charwath et al. (2006) andMichelsen (2006), in the latter case using 65 ps laser pulses and employing a streak


camera. A LII model (Michelsen 2003) gives good agreement with the nanoseconddata at fluences less than 0.2 J cm−1. However, the picosecond temporal profilesincrease significantly faster and earlier in the laser pulse than predicted by themodel. Michelsen hypothesises on the physical reasons for this discrepancy.

The work of Lehre et al. (2005) measured at a point. A two-dimensional methodcan be achieved by expanding the laser into a sheet using a combination ofspherical and cylindrical lenses (Xin and Gore 2005), so-called planar LII. Theauthors used this technique to study the spatial distributions of soot in vertical andhorizontal planes in turbulent fires. A number of factors can influence the mea-surements, not least the Gaussian laser profile and attenuation by soot within theflame. The laser sheet had an effective height of 3.5 cm. because of the energyvariation across this only 2.0 cm around the laser axis was actually used in datacollection. To further correct for the beam profile a blackbody source was used toprovide uniform illumination to calibrate all the pixels of the camera. Ultimately,the results were collected following a calibration against the known soot volumefractions of a laminar co-flow ethylene/air flame (Gore and Faeth 1986.)

Laser attenuation effects had been considered by Shaddix and Smyth (1996).More recently radiation attenuation has been explored by Migliorini et al. (2006) forthe case of two-colour LII and an axisymmetric flame. Here the flame is dividedinto a number of concentric circles and an iterative procedure applied. Initially theabsorption in the outer ring is calculated. Moving inwards there are two rings andthat of the outer ring is subtracted to find the value for the next ring. This procedureis then continued up to the centre. The authors concluded that owing to the com-peting effect of different absorption at the two wavelengths, the variation in the sootvolume fraction profile in an axisymmetric flame is small. Further numerical testsdemonstrated that soot concentration and flame size, although influencing thecorrection of the LII signals, have little effect on measured volume fractions.

The effect of soot absorption was also investigated by Liu et al. (2008a, b). Thediscrete ordinates method of solution of the radiative transfer equation was used.The effects of absorption and scattering on LII intensities were found to be sig-nificant especially at the shorter wavelengths and when the soot volume fractionwas higher. This can lead to a lower estimated soot particle temperature from thetwo colour LII method. Also the corresponding soot volume fraction derived fromthe absolute LII intensity technique is overestimated. In the later paper (Liu et al.2009) the authors used radiative transfer theory to explore the problem. The con-tribution of scattering to signal trapping was found to be negligible in atmosphericlaminar diffusion flames. They also confirmed that the errors are smaller in a 2D LIIsetup where soot particles are excited by a laser sheet. The simple Beer-Lambertexponential attenuation relationship holds in LII applications to axisymmetricflames as long as the effective extinction coefficient is adequately defined. Liu andSmallwood (2010a) extended the study to examine the effects of particle aggre-gation. According to their model aggregation affects the calculated soot temperaturein laser-induced incandescence mainly in the low laser fluence regime. At high laserfluences, the effect diminishes due to the enhanced importance of soot sublimationcooling.

258 A. R. Jones

The influence of aggregation on two-colour LII was further considered by Yonet al. (2015b). They corrected for soot the influence of aggregate emission due tomultiple scattering on LII signal at two excitation wavelengths. For wavelengthsshorter than 532 nm E(m,k) increases significantly with decreasing wavelength. Forwavelengths longer than 532 nm, the wavelength dependence of E(m,k) becomesvery small and can be neglected. Their analysis showed that the soot absorptionfunction varied with the height above the burner exit and could be correlated withthe degree of soot maturation. Bambha and Michelsen (2015), however, mademeasurements on time-resolved laser-induced incandescence (LII) and laser scatterfrom combustion-generated mature soot. Their results demonstrated a stronginfluence of aggregate size and morphology on LII and scattering signals.Conductive cooling competed with absorptive heating on these time scales and theeffects were reduced with increasing aggregate size and fractal dimension (Df).Their results also revealed significant perturbations to the measured scatteringsignal from LII interference and suggested rapid expansion of the aggregates duringsublimation.

Sun et al. (2015) examined LII for turbulent flames. Here they found that, inaddition to radiation trapping by absorption in the soot, beam steering was a severeproblem. The influence of local focusing and de-focusing of the laser could result inan underestimate of the averaged LII signal by 30%, even when operating withinthe so-called plateau regime of laser fluence. Even at low turbulence levels(2000 < Re < 3000), beam steering effects can be significant.

Attenuation was further explored by Chen et al. (2007) for dense media. In manystudies it is assumed that the soot particles are so small that scattering can beignored relative to scattering. These authors allowed for multiple scattering basedon Lorenz–Mie theory, coupled with absorption and emission, by the use of aMonte-Carlo technique. They predicted that for low soot particle concentrations, thedetected spectral flux increased with the soot concentration. However for moderateto high concentrations the detected flux decreased with the concentration, thedecrease being a function of the particle size parameter. The deduced temperaturesbased of two-colour pyrometry tended to lower values when increasing the con-centration of soot particles.

Ochoterena (2009) also examined dense media and pointed to the presence ofscattering especially for larger and aggregated particles and the influence of mor-phology. In addition, the form, soot concentration gradients and optical thickness ofthe flame caused uneven laser fluence across the measuring volume that affected thegeneration of the LII signal. Because of these effects an appropriate calibrationprocedure was essential. Among possible methods the author found that the use ofan in situ calibration procedure using a laser sheet that propagated through thecomplete measuring volume was the most adequate among the calibration routinesstudied.

In a similar vein Bladh et al. (2008) considered the influence of particle sizebased on a heat and mass transfer model. They emphasised that for LII applications,it is desired that there is a linearity between LII signal and soot volume fraction,without any particle size influence on this relationship. To minimise such deviation


they recommended the use short prompt gates and longer detection wavelengths.They also noted that for atmospheric flame conditions, the particle size influencesthe relationship between prompt LII signal and volume fraction to a lesser degree inthe low fluence regime, but has a clear influence in the high fluence regime. Inaddition their model predicted that the laser beam profile has little effect in the highfluence regime. At lower fluences the spatial profile would have a somewhat higherimpact. They also considered that aggregation has little impact at atmosphericpressure, and also at increasing pressures in the high fluence region. However, theimpact would be higher in the low fluence region. A high degree of aggregation waspredicted to decrease the uncertainties introduced due to the size dependence as itdecreases the heat conduction rate.

There is an open question about the extent to which the soot itself changes due toexposure to the strong laser radiation. This has been investigated by de Iuliis et al.(2011) by exposing the soot to repeated laser pulses and observing the changes inthe LII signal over time. Transmission electron microscopy (TEM) analysis wasalso presented. The results indicated that even at low laser fluences a permanentsoot transformation was induced causing an increase in the absorption functionE(m,k). This was considered to be due to graphitization of soot particles. At highfluences the vaporisation process and a profound restructuring of soot particlesaffected the morphology of the aggregates. Similar conclusions were drawn byThomson et al. (2011).

Michelsen et al. (2009) used LII and transmittance measurements at 532 and1064 nm to examine the wavelength and temperature dependence of the absorptionand scattering cross sections of mature soot in an ethylene flame. The LII mea-surements indicated that the emissivity of soot in a flame deviates from the expected1/k dependence. They found single-scattering albedos of 0.058–0.077 at 1064 nmand 0.22–0.29 at 532 nm and values of F(m,k)/E(m,k) of 2.2–2.9 at 532 nm and2.4–3.3 at 1064 nm. These values confirmed that scattering must be taken intoaccount when interpreting extinction data at these wavelengths. Their results alsoindicated increases in the absorption cross section and decreases in the scatteringcross section with increasing fluence at low fluences. Here

F m; kð Þ ¼ m kð Þ2�1

m kð Þ2 þ 2

��2

having established the basic techniques of LII, further studies have concentrated onspecific situations that influence its use. For example, Liu et al. (2005a, b) andMcCrain and Roberts (2005) have considered elevated pressures. Due to thereduced mean free path of molecules with increasing pressure the rate of conductiveheat loss from the soot particles increases significantly. Thus the lifetime of thelaser-induced incandescence (LII) signal is significantly reduced as the pressureincreases.

Hofmann et al. (2008) conducted experiments on a high pressure combustionsystem. They developed a model for the situation and compared time resolved LII

260 A. R. Jones

(TIR-LII) results based on this with samples extracted and viewed by transmissionelectron microscopy (TEM). The extracted sizes were fitted to a log-normal dis-tribution. The results showed good agreement of the mean diameter of theparticle-size distribution obtained by TEM analysis and LII for all pressures.Throughout, a thermal accommodation coefficient of 0.25 was found to be moresuitable. Similarly, Kim et al. (2008) and Charwath et al. (2011) found that TIR-LIIcan be successfully applied at elevated pressure.

At the other extreme, in a vacuum there can be no conduction or convection andheat transfer can only be by radiation. This situation was explored by Rohling(1988) and Beyer and Greenhalgh (2006). If conduction is eliminated then incan-descent lifetimes increase and this leads to a dramatic increase in total signal level.Also for radiation-only cooling the shape of the cooling curve for the particlesshould be dominated by their optical properties and so provide the potential tore-evaluate the emissivity or complex refractive index. The latter authors found thatresults for E(m,k) of soot were consistent with a value between 0.4 and 0.6. Alsothe incandescence lifetime in a vacuum is dramatically extended to more than50 µs. They further observed that large clusters of particles tended to fragment overa time period up to 10 µs. Further considerations at very low pressures wereconsidered by Liu and Snelling (2007) and Headrick et al. (2011).

The possibility of soot evaporation has been mentioned above. It results in anenergy loss and a reduction in soot size. Van der Wal et al. (1999) showed that laserheating by a 1064 nm excitation wavelength causes mass loss through evaporationat fluences above 0.5 J cm−2 coupled with a decrease in LII signal on a time scaleof 100 ns. For 532 nm laser excitation, Dasch (1984) reported a fluence of0.23 J cm−2 as the threshold for soot evaporation effects. Michelsen et al. (2003)reported a similar threshold value of 0.2 J cm−2 for time-resolved LII experiments.To avoid the many uncertainties in soot characterisation by LII due to soot evap-oration, as enumerated above, a low-fluence LII model has been proposed, andnumerical and experimental studies have been performed (Snelling et al. 2004; Liuet al. 2005a, b).

These effects have been further explored by Yoder et al. (2005) with the primaryobjective of quantifying the effects of soot particle evaporation. LII signals wererecorded for laser fluences of 0.61 and 0.47 J cm−2. They showed that the timescale of soot evaporation is not only coincident with the laser pulse for a Gaussianprofile, but is confined to it with the majority occurring within the first half of thelaser waveform. This implies that the use of prompt LII detection schemes is notwell founded. Further use of delayed LII detection records a soot volume fractionsomewhat less than the original for incident laser fluences above the evaporationthreshold. This is commonly taken to be about 0.2 J cm−2, though these authorsfound that a value of about 0.1 J cm−2 was necessary to eliminate evaporationcompletely.

Other potential changes in soot structure due to the effects of laser illuminationare of some concern. These include graphitisation and changes in morphologyincluding aggregation. These were explored by Van der Wal et al. (1999), who


suggested fluences below the evaporation threshold should be employed to avoidthese difficulties.

A further reason to keep the fluence below 0.2 J cm−2 was highlighted byGoulay et al. (2009). They point to the potential for photochemical effects. Thesecan perturb the LII signal at wavelengths between 380 and 680 nm, suggesting thatthis detection region should be avoided for LII measurements made using a 532-nmlaser beam at fluences of 0.2 J cm−2 and above. The detection wavelength regionsto be avoided are predicted to be much more extensive than previously believed.

It was seen that analysis of the results from LII is heavily dependent upon asuitable model for the heat transfer processes that occur. For this reason calibrationagainst some known standard is usually required. For example, Stirn et al. (2009)compared of particle size measurements with LII, mass spectroscopy, and scanningmobility particle sizing (SMPS) in a laminar premixed ethylene/air flame. Theyfound good agreement between the three methods in the appropriate ranges. SMPSand LII are suitable in the mid- and upper range of the particle sizes around 2–3 nm.

Snelling et al. (2005) proposed a calibration independent technique based onmeasuring the absolute spectral intensity of the LII signal and the temperature of thesoot. This is sometimes referred to as the auto-compensating method. It relies on adetailed understanding of heat and mass transfer in time and space, commonlydescribed by the mass and energy balance of an isolated soot particle. To measurethe absolute LII intensity, the detection system must be calibrated by using aradiation source of known radiance. The temperature of the soot particles can bemeasured using the ratio of the LII signals at two different wavelengths. On theassumption of small particles within the Rayleigh limit the emitted intensity of aparticle is shown to be a function of its temperature and volume, via its absorptivityand the Planck function.

Realistically, there is always a soot particle size distribution. This results indifferent decay rates in the LII signal and the measurement has to be inverted toobtain the size distribution. This has been explored by Liu et al. (2006) assuming alog-normal size distribution. They found that in the non-sublimation regime, theinitial decay rate of polydispersed soot particles was inversely proportional to theSauter mean diameter, rather than the arithmetic mean diameter. At later times therewas divergence from this rule. They described a soot particle sizing technique basedon these two observations.

An approach to bidisperse systems has been proposed by Cenker et al. (2015).By noting that the decay of the LII signal is slower for larger particles, the signal isdominated by these at large times. Fitting the time variation to an exponentialfunction at these times enabled the extrapolation to shorter times. In that region theextrapolation could be subtracted from the measured signal to reveal that for thesmaller particles. The authors claimed to be able to discriminate between distri-butions with equivalent particle sizes of 15 and 57 nm.

Further information may be obtained by combining LII with other methods. Forexample, Snelling et al. (2011) combined LII with elastic light scattering todetermine both primary particle size and radius of gyration (Rg) of aggregates. Thelatter was obtained by measurement of elastic scattering at 34°, which is suitable for

262 A. R. Jones

Rg less than about 200 nm. They used 532 nm wavelength from a frequencydoubled Nd-YAG laser. The scattered and emitted light were collected at 34° andthe scattered light and LII signals were separated using dichroic mirrors andinterference filters. The results were found to be in better agreement with aggre-gation theory than TEM measurements, which has also been found in multi-anglelight scattering experiments. This was possibly due to insufficient microscope data.In a similar vein Crosland et al. (2013) combined LII with two angle elastic lightscattering to include soot volume fraction.

A further example has been provided by Hayashi et al. (2013). Here the idea wasto use LII to measure small soot particles and Mie scattering for burning coalparticles. They claimed that this was necessary because the soot formation processin coal burning flames has not been precisely defined. Unfortunately, the laserheated the coal particles as well as the soot and there was a problem distinguishingbetween the two LII signals. The authors found that the LII signal increased withtime and then began to decrease as usual for small particles. However in thepresence of coal it then increased again. Discrimination was obtained by noting thatthe Mie scattering signal decreased with distance from the centre line of the flameand also decreased with height above the burner. In contrast the signal from the sootreached a peak some distance from the centre and increased with height. Theposition of the soot peak was compared with fluorescence from OH radicles. Thisindicated the position of maximum soot formation relative to the flame front.

The effects of other geometries on LII are important in some circumstances. Forexample, Bambha et al. (2013) discussed the influence of volatile coatings on soot.Their measurements on soot coated with oleic acid results demonstrated a stronginfluence of coatings on the magnitude and temporal evolution of the LII signal.Higher laser fluences were required to reach signal levels comparable to those ofuncoated particles, and this effect increased with greater coating thickness up to75% by mass where saturation occurred. These effects were predominantlyattributable to the additional energy needed to vaporise the coating. Their resultssuggested negligible enhancement in absorption cross-section.

A prospect of extending LII to atmospheric remote sensing by combination withLIDAR was examined by Kaldvee et al. (2014). Light detection and ranging(LIDAR) is a measurement technique for single-ended range-resolved detection oflaser-induced light-emitting events. It utilises the fact that it takes a finite time forlaser light to propagate to the target and back. If the laser pulse is shorter than thistime then direct backward measurement can be made without interference from thelaser beam. By applying this concept to LII the authors made measurements onsooting flames with a range of up to 16 cm, dependent upon the time response ofthe detector. Further discussion of this technique for atmospheric measurements isgiven by Miffre (2015).

In summary it still seems that there is lingering uncertainty concerning numericalmodels and the value of the thermal accommodation coefficient, and calibrationmay be required. Against that there is the calibration independent method intro-duced by Snelling et al. (2005). Though this may be limited in scope it has beensuccessfully used by a number of authors. In general, good agreement has been


reported when LII is compared with other measurement methods (e.g. Thomsonet al. 2006; Crosland et al. 2013; Liu and Smallwood 2013).

A more detailed review of LII and its applications has been provided byMichelsen et al. (2015).

6.2.2 Cavity Ring Down (CRD)

An interesting technique that may prove very sensitive at low particle volumefractions is cavity ring down (CRD) (O’Keefe and Deacon 1988). We will begin byincluding some of the material from the previous report and then move on to morerecent developments.

A related technique, in so far as it employs a cavity, is photoacoustic spec-troscopy (PAS). In this a microphone is inserted into the cavity and a continuouslaser beam is modulated at an acoustic frequency. Particles within the cavity absorbenergy from the beam and heat the internal gas, which expands and contracts withthe modulation thus generating a sound wave. One advantage is that the frequencyof modulation can be adjusted to match the acoustic resonance of the cavity thusenhancing the signal. An example of this to study the distribution of soot in a flameis provided in the paper by Humphries et al. (2015).

In CRD a laser pulse is launched into a cavity formed by two mirrors thatcontains a cloud of absorbing particles. The distance between the mirrors is large incomparison to the pulse length, so that the pulse may be considered to travel backand forth many times leaking a little intensity every time it hits a mirror. On eachpass there is some loss of intensity due to scattering and absorption, the conse-quence of which is that the pulse decays in time in a manner determined by theextinction coefficient of the particles. The CRD technique measures a characteristicexponential decay of the signal. The soot volume fraction, fv, is obtained from thedecay rate with the flame on, given by

KextL ¼ ‘

cs� 1þR

� �

where Kext ¼ kextfv=k. Here kext is the specific extinction coefficient and ‘ is thespacing between the cavity mirrors of reflectivity R. L is the path length in the flameand s is the time constant of the exponential decay.

A discussion of some aspects of CRD has been given by van der Wal and Ticich(1999), who were interested in its use for the calibration of LII. Potentially CRDcan measure down to one part in 109. Also, in CRD the laser power densities aremuch less than those observed to cause soot evaporation: Typically 0.25 J/cm2 at532 nm and 5 J/cm2 at 1.06 lm, as discovered by LII measurements. Anotheradvantage of CRD is that it yields integration over path length directly. It sufferssimilar problems to LII in the presence of scattering by large aggregates andfluorescence.

264 A. R. Jones

One of the difficulties with CRD is lack of spatial resolution. A way ofimproving on this has been discussed by Tian et al. (2015). They have developed ahigh spatial resolution laser cavity extinction technique to measure the soot volumefraction from low-soot-producing flames. The cavity is realised by placing twopartially reflective concave mirrors on either side of a flame under investigation.This configuration makes the beam convergent inside the cavity, allowing a spatialresolution within 200 lm. It also increases the absorption by an order of magnitude.The measurements of soot distribution across the flame show good agreement withresults using LII in a range from around 20 ppb to 15 ppm.

It is worth noting here that Kext includes scattering as well as absorption. Thusthe technique is available for aerosols of many sorts, and not only soot. Forexample, Bulatov et al. (2006) applied CRD for the detection and characterizationof airborne particulates under ambient conditions. Their method providedtime-resolved absolute aerosol concentration, with spatial resolution along a line.They reported absorption spectroscopy of monodispersed aerosols in the size range100–200 nm. Interestingly, they measured particle concentration using theknowledge that noise in a system is inversely proportional to the square root of thenumber of particles. Their results indicated the potential of applying CRD forselective analysis of aerosols. They also discovered that a plot of the ratio ofextinction coefficient against the square root of number produced approximatelystraight lines, the gradients of which were strongly correlated with real refractiveindex. They suggest this may lead to a method for estimating real refraction indexfrom the ring down data.

Measuring the value of the extinction efficiency (Qext) at a range of particle sizeparameters allows the complex refractive index of the aerosol to be determinedthrough comparison with Mie theory. Sources of error in this measurement by CRDhave been examined by Miles et al. (2011). They considered both experimentalsources of error and those which compromise the theoretical models against whichmeasurements are compared. Their results showed that for absolute measurementsmade using single-cavity instruments, factors such as uncertainty in the length ofthe cavity and the counting efficiency can introduce an error of similar to 2.5% intothe real part of the refractive index retrieved from experiment. They noted that dueto the dependence of particle extinction efficiency on diameter, the effect of a givenerror on measurements for different particle sizes was not constant.

The complex refractive index of an aerosol can be retrieved by finding thetheoretical Mie theory curve that best fits the measured extinction efficiency for asmany aerosol diameters as possible. According to Bluvshtein et al. (2012) theretrieval of complex refractive index can be simplified using extinction measure-ments at only two carefully selected size parameters. They demonstrated this for anumber of aerosol types, though not for soot.

With relevance to atmospheric aerosol, coated soot particles were studied usingPAS by Bueno et al. (2011). The contribution of soot to the overall energy balancein the atmosphere was the question that motivated their study. The coating of sootwith sulfuric acid and subsequent hygroscopic growth due to interactions in theatmosphere lead to enhancement of the absorption cross section of the soot


particles, and may double it (Zhang et al. 2008). Accurate measurements of sootabsorption cross section are difficult to obtain because of variability in morphology,composition, mixing state, and conditioning of soot, including the effects ofatmospheric aging. There are also difficulties associated with the large number ofuncontrolled variables, such as temperature, pressure and humidity, as well as thesource of the soot. The advantage of PAS is that it measures the properties directlywithout the complications of some sampling methods.

Bueno et al. (2011) used dibutyl phthalate as a non-absorbing coating material,which is an optical surrogate for atmospheric H2SO4 and measured the change inthe absorption cross section as a function of particle size and coating thickness.They observed that their soot particles were aggregated and confirmed that theirscattering could be treated with RDG theory. The size range according to mobilityanalyser ranged from 100 to 200 nm, and coating thicknesses were up to 100 nm.They found that for the larger particle sizes and thickest coating that the absorptioncould be increased by a factor of approximately 1.8. They also claim excellentsensitivity to subtle changes, including variation in coating thickness as little as2 nm.

A combination of CRD and PAS was used by Radney et al. (2014) to explore thedependence of soot optical properties on morphology. The measurements showedthat the mass-specific absorption cross sections were proportional to particle massand independent of morphology, whereas mass-specific extinction cross sectionswere morphology dependent. The results were also compared to theoretical cal-culations of light absorption and scattering from simulated particle agglomerates.The observed absorption was reasonably well modelled, but the model was unableto satisfactorily reproduce the measured extinction, underestimating thesingle-scattering albedo for both particle morphologies.

6.2.3 Extinction and Scattering

Perhaps the most traditional method of examining soot is via extinction and elasticscattering. There are essentially two approaches. In the first extinction is measuredas a function of wavelength, the so-called spectral extinction. It is usually assumedthat individual soot particles are sufficiently small that they fall within the region ofRayleigh scattering. In that case the extinction coefficient is approximately equal toabsorption coefficient and proportional to particle volume, number density andinversely proportional to wavelength. Thus, in principle, a simple measurement ofextinction at a fixed wavelength should determine the volume fraction of the par-ticles. However, the optical properties of the soot are wavelength dependent. If thisvariation could be established then measurement of extinction at a range ofwavelengths can yield both volume fraction and size. The results of variousattempts at refractive index determination have been reviewed, for example, byJones (1993).

266 A. R. Jones

The second approach makes use of the fact that for Rayleigh particles whileabsorption coefficient is proportional to volume, scattering coefficient is propor-tional to volume squared.

Thus measurements of extinction and scattering can yield both size and volumefraction. The drawback of this is that for Rayleigh scatterers the angular distributionon intensity is independent of size. It follows that absolute scattered intensity mustbe measured with inherent difficulties.

For particles slightly larger than Rayleigh the scattering polar diagram displaysstronger forward than back scatter. Measurement of the ratio of intensities at twosymmetric angles is known as dissymmetry, and yields useful information in thisregion.

Of course, real soot particles have a size distribution. The methods above willnormally yield a mean size, but fitting the spectral extinction curve may giveinformation on the factors describing an assumed distribution function.

Much of the above is discussed in the books by Kerker (1969) and Van de Hulst(1981).

The other complicating factor is the presence of agglomeration. This will be thesubject of the next section.

An important goal is the investigation of soot formation at high pressure.A recent example of the scattering/absorption method was provided by de Iuliiset al. (2008), who used the technique to examine soot formation in shock tubes. Inthis way, the induction delay time, the soot yield and the rate of soot formation canbe evaluated for different fuels at controlled conditions. They determined d63(essentially the ratio of average volume to average area) using the refractive indexas published by Chang and Charalampopoulos (1990). They measured directtransmission and scattering at 90°. The calibration of the scattering was carried outusing a gas of known scattering cross-section. The results obtained for toluene andethylene showed that soot diameter was in the range of 25–35 nm in good agree-ment with literature measurements.

Roy and Sharma (2005) explored the direct inversion of spectral extinction datato retrieve particle size distribution. They restricted themselves to monomodaldistributions of spherical particles. The aim was to develop a comprehensiveunderstanding of how the essential features of a particle size distribution are cap-tured in the various parts of the extinction spectrum. Their approach was based onan earlier study (Roy and Sharma 1997), in which they showed that the mean valuetheorem can be used very effectively to obtain the key parameters such as moments,maximum and minimum particle size, and particle concentration of the distributionfunction. This enabled them to obtain simple and nearly accurate relations for thefirst four moments of the distribution function, greatly increasing the computationalefficiency.

Spectral extinction was also used by Sun et al. (2007). Monomodal and bimodalparticle size distributions were retrieved in the independent model and dependentmodel algorithms, respectively. The constrained least-squares inversion methoddeveloped by Phillips and Twomey was applied in the independent model, and the


genetic algorithm (GA) optimization method was employed in the dependentmodel.

Shape is important as it influences the optical properties, coagulation processes,and removal of particles from the atmosphere. Hence measurement of this propertymust be tackled. With this in mind, the use of extinction to study irregular particlesin an urban environment has been explored by Kocifaj and Horvath (2005). Theyfound that non-spherical particles scatter radiation in a more complex manner thanequal volume identical spheres. Submicron particles scatter more light in the for-ward direction in comparison. However, scattering in the other directions is reducedyielding a smaller Qext. A reduction by a factor �2 was found to be typical for sizeparameters less than 5. This was found to be almost independent of the physicalnature of the particles. In the measurement of size, distribution the modal size wasvery little different for spherical and non-spherical particles, but the shapes of thedistribution functions differed.

Characterisation of non-spherical particles of specific type was also tackled byZhao et al. (2014), although their method was limited to particles where anomalousdiffraction theory (ADT) was applicable. They found that for each type ofnon-spherical particle there exists an ADT transform pair between the size distri-bution and the complex absorption spectrum, which provides the physical basis forsolving the inverse problem.

Further techniques for direct inversion of spectral extinction include the antcolony and particle swarm methods (He et al. 2014; Qi et al. 2015; Mao and Li2015). The basis of the extinction calculations was the method employed by Zhaoet al. (2014). The inversion analysis is too complex to be covered here, butappropriate references are given in the papers mentioned above.

A simple non-spherical shape is the spheroid and the study of this by spectralextinction was considered by Wang and Sun (2012). They point out that manynon-spherical particles can be reasonably well modelled as collections of randomlyoriented spheroids, and they pursued approximate methods to calculate theextinction to minimise computation time. Their chosen method was a combinationof the Mie method, for particles smaller than 1 µm, and the generalised eikonalapproximation averaged over orientation. They determined the valid ranges of size,eccentricity and refractive index, and suggested experimental techniques forretrieval of particle size.

For soot the extinction coefficient is approximately equal to the absorptioncoefficient and so is relevant to the emissivity and radiative transfer. Mackowskiet al. (2006) observed that the radiative effects of agglomerated soot were verysignificant at very long wavelengths in the infrared. They performed laser extinctionmeasurements on an acetylene/air flame at wavelengths of 633 nm, 3.39 µm and11.2 µm. Their experiments suggested the ratio of the extinction coefficient at thelong wavelength to that in the visible to be about 0.15, which is about three timesgreater than expected on the basis of Mie theory for Rayleigh-sized sphericalparticles.

Emissivity is also important for determining the radiation from flames. This isrelevant to radiative transfer, pyrometry and emission. In the latter case,

268 A. R. Jones

tomography can be used to determine profiles of soot and/or temperature in flames.Because a high quality background radiation source is required for transmissionmeasurements, emission-based tomography systems are preferred for practicalapplications. An example is provided by Ayranci et al. (2007) who employedtomographic reconstruction of flame emission spectra to the characterization of soottemperature and volume fraction fields within an optically thin axisymmetric flame.It is worth noting here that these limitations on the nature of the flame greatlysimplify the tomographic process.

Huang et al. (2009) also applied tomography to an asymmetric diffusion flame.To obtain sufficient spectral line-of-sight emission projections, the flame should bescanned at different positions along the lateral axis and in different directions. Thewhole procedure is time-consuming and unfit for non-steady flame measurements.To overcome some of the disadvantages these authors proposed a high-resolutionstereoscopic image system, and developed a matrix deconvolution method toprocess the data. They made measurements of the soot temperature and volumefraction distributions of a turbulent asymmetric ethylene/air flame. Numericalassessments showed that for soot volume fraction measurement, the systemsignal-to-noise ratio should be larger than 62.5 dB.

6.2.4 Aggregrates

When small particles formed in flames, such as soot, are sampled and viewed underan electron microscope they usually appear as branched chain agglomerates. Theaggregate is made up of a number of primary particles. For soot, the sizes of theprimary particles are typically of the order 30–60 nm and the aggregates are up to500 nm. The properties of the aggregates that we would like to measure include thesizes of the primary particles and the aggregates, both as functions of size andposition. We would also like to follow the formation of the primary particles andthe aggregation process, ultimately leading to smoke formation.

Indeed, soot particles are rarely seen as individuals. Aggregation is very com-mon, as observed by Yang and Koylu (2005) using the scattering/extinction methodon a turbulent flame. Aggregation formation was found in the very early stages lowdown in the flame. Murphy and Shaddix (2005) also pointed out that aggregates canbe significant scatterers and this must not be ignored when calculating the emis-sivity of sooting flames.

Originally there were attempts to treat the aggregates as equivalent spheres orspheroids. The effect of fractal aggregation of smoke particles on light scatteringwas considered by Qiao et al. (2007). They found that, compared with the sphericalparticle with the same volume and other optical parameters, the fractal-likeaggregate has smaller forward scattering and larger backward scattering. Liu et al.(2010) and Zhang et al. (2010) also looked into this question by comparing rigorouscalculations for aggregates against various equivalent sphere assumptions. In all


cases the equivalent spheres resulted in significant errors. Evidently the aggregatesmust be treated as such.

The theoretical treatment of aggregates of small particles can follow two fun-damental approaches. The first of these is full interactive scattering, but it is verycomplicated and very computer intensive. Because the monomers are so small, theycan be treated as Rayleigh scatterers as a simplification and this leads to the directdipole approximation. This is less complicated, but still involves the solution of 3Nsimultaneous equations, where N is the number of monomers in the aggregate. Theadvantage of the rigorous theories is that they will predict polarisation properties ofthe aggregates. The main disadvantage is that the position of every particle must beknown.

A common approach is to treat agglomerates that are large random assemblies asfractal geometries. In this the number (N) of particles in the agglomerate is

N ¼ KRg

ap

� �Df

where Rg is the radius of gyration of the agglomeration, ap is the radius of theprimary particles and Df is the fractal dimension.

An example of rigorous scattering calculations for aggregates was given by Liuet al. (2008a, b). They employed the numerically exact superposition T-matrixmethod to perform extensive computations of scattering and absorption propertiesof soot aggregates with varying states of compactness and size. They demonstratedthat the absorption cross section tends to be reasonably constant when Df < 2 butincreases rapidly when Df > 2. The scattering cross section and single-scatteringalbedo increase monotonically as fractals evolve from chain-like to more denselypacked morphologies, which is a strong manifestation of the increasing importanceof scattering interaction among spherules. Generally the results for soot fractalsdiffer profoundly from those calculated for the respective volume-equivalent sootspheres, as well as for studies that do not allow for electromagnetic interactionsbetween the monomers.

The effect of fractal parameters on absorption by soot was also looked at byPrassana et al. (2014) using the exact T-matrix method and modelling the formationof the aggregates. They found that the absorption cross section of soot is muchhigher than the Rayleigh approximation prediction due to the high refractive index.They noted that aggregates having similar particle distance correlation functionshad similar absorption cross-sections. On this basis they were able to develop anempirical model which successfully predicted the absorption within ±5%. Theypredicted that, compared to the Rayleigh approximation, the absorption enhance-ment can be as high as 200% at low temperatures and 120% at high temperatures.

A simpler approach arises from the observation that the aggregates are usuallytenuous by nature. This leads to the prospect that the incident wave may propagateundisturbed through the structure, and that the Rayleigh-Debye-Gans(RDG) approximation may be applied. Where this is suitable the primary

270 A. R. Jones

particles may be considered to scatter independently. The resulting analysis is thenmuch more straightforward.

The positions of the primary particles cannot be predicted in any one aggregate,and all aggregates are different from each other. Overall the positions may beconsidered to be random. This suggests a statistical method, which leads to theconcept of a correlation function. This is quite easily applied in the RDG modelusing the fractal approach; the so-called Rayleigh-Debye-Gans -fractal-aggregate(RDG-FA) model.

Excellent reviews of this method have been given by Sorensen (2001) andBushell et al. (2002). The RDG approximation works reasonably well when Df\2.For Df [ 2 the agglomerate is too dense and the aggregate is better described byrigorous theory. As Df approaches 3, Mie theory may be used.

Sorensen et al. (2003) and Kim et al. (2004, 2006) pointed to the possibleformation of super-aggregates with fractal dimension as high as 2.6. Whereasearlier work was performed on laminar flames, Kearney and Pierce (2012) used LIIand electron microscopy to examine large scale turbulent pool fires. They foundvery large superaggregates up to 100 µm in size. Further, on their large scale suchaggregates occurred at much lower concentrations than in laminar flames.

Perhaps the exact opposite of superaggregration is soot burnout, a subjectstudied by Sirignano et al. (2015). They comment that correct evaluation of oxi-dation is needed to predict the final emission of particles from diffusion flames, andthat fragmentation has been proposed as a controlling step in determining soot burnout as well as the size of particles emitted. The oxidation and fragmentation of sootparticles was studied in counterflow diffusion flames with in situ optical diagnos-tics, LII and elastic light scattering. Two counterflow diffusion flames were chosen,a soot formation (SF) and a soot formation/oxidation (SFO) flame. A modellingapproach was also used to predict particle formation and burnout. Their resultsillustrated the role of fragmentation in controlling the burn-out and the size dis-tribution of particles in flames. The SF flame, where no soot oxidation occurs,produced large particles. The mean diameter suggested that coagulation waseffective and that large soot aggregates were formed. By contrast in the SFO flamethe volume fraction decreased as the oxidation zone was approached, suggestingthat soot oxidation was effective. Also, the mean diameter remained very smallsuggesting that together with the surface oxidation a fragmentation process wasactivated and coagulation was less effective.

The RDG-FA model was extended by Yon et al. (2008, 2014) to include theinteraction of large monomers. Generally, important conditions have to be obeyedfor the correct use of this theory. Aggregates and primary particles have to be verysmall in comparison to the wavelength, and internal scattering must be negligible.Also exiting studies have been usually for a polydisperse population. This studyaimed to evaluate possible corrections that could be applied to the classicalRDG-FA formula in order to take into account internal multiple scattering. Theydeveloped a new generalised form factor allowing the evaluation of the opticalproperties of an aggregate with a primary particle diameter of up to 90 nm. In thelater paper they developed an extended form of RDG-FA that took into account


multiple scattering using a scaling approach and comparison with rigorous calcu-lations. Their calculations showed that fractal dimensions can be misinterpreted bylight scattering experiments, especially at short wavelengths. Multiple scatteringeffects should be taken into account for the interpretation of absorption measure-ments that are involved in LII or extinction measurements.

The range of applicability of the RDG approximation is well understood for thecalculation of scattering and extinction coefficients. However, less work has beenperformed on the limits for angular scattering and the components of the scatteringmatrix. Yan and Lin (2009) have examined the usable range by comparison with theexact T-matrix method. They produced error contour charts of angle versus sizeparameter of the primary particles for elements of the scattering matrix. All fractalparameters were fixed and two complex refractive indices used. For size parametersup to 0.4 errors of less than 1% regions were very narrow, but up to 12% theregions were wide. Generally errors were slightly greater for the higher refractiveindex. Normalisation against intensity in the forward direction or of one componentof the matrix against another tended to result in wide regions of less than 1% error.

Somewhat similar conclusions were reached by Zhao and Ma (2009) whocompared the RDG approximations with no cut-of and with a Gaussian cut-off, bothagainst T-matrix calculations. The Gaussian model showed significant advantage.

The RDG approximation by its nature does not allow for the electrical inter-action between primary particles within the agglomerate. Karlsson et al. (2013)have described an analytical theory which allows for this interaction which theydesignate generalised RDG (G-RDG). In this the RDG method was generalised toparticles of arbitrary shape and inner structure. The polarisation was determined bynumerically solving a quasi-static problem with the entire object in an externalconstant electric field. The near-field interaction between the primary particles wasthus included. They applied the method to absorption and scattering from anensemble of randomly oriented aggregates, and obtained closed form expressionsfor the averages these led to expressions for fast. Evaluation of these averages fromensembles of aggregates with overlapping primary particles. They compared theirresults against both the T-matrix and DDA methods at three wavelengths and for upto 20 primary particles per aggregate. The G-RDG method showed a considerableimprovement over RDG.

An experimental technique for measurement of fractal aggregate properties hasbeen proposed by Holve (2011) and Holve et al. (2011). Earlier studies had shownthat scattering at two appropriate angles can be used to determine the meanagglomerate particle size and mass concentration. In these papers they show that adimensionless invariant function of the two-angle scattering ratio can be defined forcomputation of the mass concentration and mean agglomerate size. In addition, thethree soot optical properties can be combined into one overall soot property con-stant (Sp) which is almost invariant for a variety of fuel and combustor conditions.Measurements were made to validate the theory on a range of gas turbine and dieselengines. The results were consistent with the assumption that primary particle sootproperties are almost invariant for a wide range of engine operating conditions.

272 A. R. Jones

Caumont-Prim et al. (2013) also considered the use of angular scattering for thedetermination of the size distribution of submicronic fractal aggregate particles.They state that their method is independent of refractive index, the size of primaryspheres and the fractal prefactor, and that the measurement of scattered intensity atthree angles is sufficient. They show that in the RDG-FA approximation the ratio ofscattered intensities at two angles is independent of the properties of the primaryparticles. They can then define an equivalent radius of gyration Rg

* which corre-sponds at a given angle to the gyration radius of a mono-disperse population havingthe same scattering properties. They then devise a set of charts from which theradius of gyration and standard deviation of a log-normal distribution can beestablished. The drawback of their method is that the fractal dimension must beknown and determined independently.

The formation of primary soot was pursued by Bruno et al. (2008) usingfluorescence by incipient particles. In consideration of pollution and human health,they point to interest in very small particles in the range 1–5 nm. These particles aredifficult to detect with many techniques. In particular, they absorb in the ultra-violetwhich makes them insensitive to visible radiation. Also, their high diffusivityreduces the ability of filters to capture them. In this paper they use a techniqueknown as time-resolved fluorescence polarization anisotropy (TRFPA), which issensitive to the nano-organic carbon (NOC) generated by flames. This method isthought to be a good candidate for measuring in situ the average diameter of NOCparticles directly. The authors managed to determine particle diameters of 3 nm.

Soot precursors were also explored by Kobayashi et al. (2008) using a small poolfire. They used laser induced fluorescence (LIF) to identify molecular species in thevapour phase and LII to examine small soot particles. Their results suggested thatpolycyclic aromatic hydrocarbons (PAH) with smaller molecular mass, such asbenzene and toluene, remained in both the PAHs-soot transition and sootingregions, and they concluded that molecules heavier than pyrene are the leadingcandidates for soot precursor formation.

It must be noted that the primary particles themselves are aggregates of smallercomponents, or crystallites. To investigate this very short wavelengths are neededand Sztucki et al. (2007) used small angle scattering of X-rays to study sootformation in an acetylene flame. The primary particles were found to have acompact morphology, with a terminal radius of gyration (Rg) of about 27 nm. Theirgrowth dynamics were consistent with the nucleation and growth process. Theagglomeration revealed a diffusion limited growth mechanism resulting in Df � 2and Rg similar to 250 nm. A similar study was carried out by Mitchell et al. (2009).They also observed the existence of smaller sub-primary particles and largeraggregated particles that usually are classified as primary particles. Kammler et al.(2005) also used the method to study the growth of metal oxide particles in flames.

A feature of rigorous calculations on aggregrates is that the positions of theprimary particles must be known. To this end simulations of the aggregation pro-cess are undertaken (e.g. Kostoglou et al. 2006; Cui et al. 2011; Heinson et al.2012).


Numerically determined aggregates and rigorous calculations were used by Liuand Smallwood (2010b) to examine the radiative properties of fractal clusters.Attention was paid to the effect of different realisations of a fractal aggregate withidentical prefactor, primary particle diameter, and the number of primary particleson its orientation-averaged radiative properties. Their results demonstrated that theorientation-averaged absorption and total scattering cross sections and the asym-metry parameter exhibit relatively small variation from one realisation to the otherwith a maximum relative deviation of less than about 7%, especially for theabsorption cross section with a maximum relative deviation of about 1%. Theyconclude that it is reasonable to conduct numerical calculations using just oneaggregate realization for these variables. Much greater variation is noted for situ-ations dominated by multiple scattering, such as the vertical-horizontal differentialscattering cross-section. In that case a single aggregate realization cannot be used.

Liu and Smallwood (2011) extended their studies to explore the influence ofaggregation on measured temperature by the two colour LII method. They usedRDG theory and a generalised multi-sphere Mie method both for variousmonodisperse sizes and for lognormal distributions. Under conditions of typicallaminar diffusion flames at atmospheric pressure, where the primary particlediameters are generally less than about 30 nm and the aggregate size distribution isrelatively narrow from one to several hundreds, use of the RDG theory leads tosmall error in the soot temperature. The error appears to be negligible at temper-atures below 2000 K and grows with increasing temperature. The error also growsquickly with increasing the primary particle diameter, which indicates that the effectmay be important at high pressures due to much larger primary particles andpotentially wider aggregate distribution associated with enhanced soot loading.

Yin and Liu (2010) studied the effects of morphological structure, water coating,dust mixing and primary particle size distribution on the radiative properties of sootfractal aggregates in atmosphere using the T-matrix method. The radiative prop-erties of compact aggregates were found to notably deviate from that of branchedones, and the effect of morphology changes on the radiative properties in wet aircannot be neglected. In wet air, the scattering, absorption and extinctioncross-sections and the asymmetry parameter of soot aggregates coated with waternotably increased with water shell thickness. The volume fraction of dust has asignificant effect when the size parameters are above the Rayleigh limit, as did thesize distribution of the primary soot particles. The size distribution has a significanteffect on forward scattering of the phase function, while the effect can be neglectedas the size parameter approaches the Rayleigh limit. Similar conclusions werereached by Liu et al. (2012), including the observation that forward scatteringincreased while backscatter decreased.

A cautionary note has been sounded by Dastanpour and Rogak (2014). Theynote that in modelling of aggregates it is often assumed that the aggregate size is notcorrelated with that of the primary particles. However, they have found that largeraggregates can be associated with larger primary particles. The observed variationsin primary particle size can be explained if soot aggregates form and grow by

274 A. R. Jones

coagulation in small zones of the combustion chamber prior to dilution andtransport (with minimal coagulation) to the sampling system.

Patterson and Kraft (2007) considered a range of models for understanding thefractal nature of soot agglomerates as they grow in laminar diffusion flames. Theyintroduced and tested models for the aggregate structure of soot particles with fullybivariate simulations; by which they meant that mass and surface area were used asindependent variables. The predictions of their model, which reflected thenon-spherical nature of soot particles, were found to be quantitatively different fromthose of a single variable model assuming spheres. Their investigations indicatedthat several models gave very similar bivariate distributions. The importance ofincluding particle shape was emphasised, and more work was needed on theinfluence of surface curvature.

A further study of the influence of non-spherical primary particles was made byWu et al. (2016) who treated them as spheroids. Their optical properties werecalculated using the exact DDA method. The results indicated that the opticalproperties of soot aggregates composed of highly non-spherical spheroidal mono-mers were considerably different from those composed of spherical monomers,leading to larger cross sections of extinction, absorption and scattering. In extremecases with axial ratios equal to 3 (prolate) and 1/3 (oblate) for the spheroidalmonomers, the relative deviations compared to spheres reached up to 15% for theabsorption cross section, 10% for the single scattering albedo and −25% for theasymmetry parameter. There is a clear need to take primary particle morphologyinto account in scattering models and calculations.

6.3 Coal and Ash

While standard light scattering can be used to study coal particles the measurementsare complicated by the fact that such particles are not only non-spherical but alsoirregular. Thus the shape cannot be predicted from one particle to the next. Whilestandard techniques may be acceptable for large numbers of randomly orientedparticles, it is advisable to study the influence of shape on the combustion process.

One method developed to examine the shapes of individual particles is shadowDoppler velocimetry (SDV), devised by Hardalupas et al. (1994) and Morikita et al.(1995). In this, the particle passes through the fringes formed by two laser beams asin regular LDV and an offset detector measures the velocity from the frequency ofthe signal in the usual way. In addition, however, an extra lens images the particleonto a plane where a linear array detector is situated. As the shadow image of theparticle crosses this detector the array gives the length of cross-section. The shapeof the cross-section is then determined after the whole particle has traversed thearray. Other developments up to 2004 were discussed by Jones (2006).

SDV was one of the techniques employed by Hwang et al. (2005, 2006) toprovide a cross-section of the particles from which equivalent circular sizes couldbe obtained. Additionally LIF, to study OH distribution, and two-colour pyrometry


were used. Measurements were made on a laboratory scale pulverised coal burner.They observed that the mean size of the coal particles increased with height abovethe burner due to preferential combustion of the smaller particles and swelling dueto formation of volatiles. Their results indicated that the size-classified diameter andvelocity of the pulverized coal particles in the flame could be measured well bySDV. They also emphasised the importance of such techniques for the validation ofcombustion models with two-phase flow.

To obtain a complete picture of coal combustion there are a number of variablesto be measured. Toth et al. (2015) discuss a technique for determining the in situtemperature, size, velocity, and number density of a population of burning coalparticles, to yield insight into the chemical and aerodynamic behaviour of a pul-verized coal flame. They developed a method for the simultaneous,three-dimensional (3D) measurement of particle velocity, number density, size, andtemperature using a combination of stereo imaging, 3D reconstruction, multicolourpyrometry, and digital image processing techniques. The particle size was deter-mined from the absolute emitted intensity: knowledge of the temperature yields thesurface area. Detailed analysis of the uncertainties in their measurements suggestedthat particle temperature could be measured up to ±5% accuracy, while particle sizecan be measured by ±45% accuracy on average.

In general holography has the advantage of excellent optical depth and, being animaging method, is unaffected by non-spherical particles. It can also reduce prob-lems of multiple scattering in dense sprays because only the coherent part of thescattered light is reconstructed. Lee et al. (2009) have discussed the design of aninexpensive digital holographic set up for drop size and velocity measurement.They opted for an in-line system to minimise the spatial requirement, but to avoiddisturbance to the reference wave this was diverted around the spray and thenrecombined with the object wave via a beam splitter. Two pulsed lasers were usedso that two images could be recorded with a known time gap and velocity mea-sured. The holograms were recorded on a CCD camera and reconstructed digitally.

In a flow system coal particles may aggregate, a possibility that is enhanced byshear flow. Several methods to measure the structures of coal aggregates generatedunder controlled shear conditions have been compared by Liao et al. (2006). Veryfine particles of mean volume diameter 12 lm were used. The methods of analysisincluded small angle light scattering, but also image analysis, light obscuration andsettling behaviour. In the light scattering analysis allowance was made for thedubious applicability of RDG theory to coal. They commented that small anglelight scattering has the advantage that thousands of aggregates could be analysed ina fast and non-intrusive manner. The aggregates were characterised by their fractaldimension, Df. They found values of Df ranging from 1.84 to 2.19 for aggregateswith more open structures, and around 2.27–2.66 for compact ones. The wideranges of values were attributed to the different nature of the measurement tech-niques. Light scattering and obscuration measured the aggregates collectively togive average values of Df; consequently ignoring structural variation betweenaggregates, and leaving possible small contaminations undetected.

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An inevitable consequence of coal combustion is the formation of ash. While ithas its uses, e.g. in the manufacture of concrete, it can block radiative transfer andlead to slagging in furnaces. It can also carry unburned carbon away and lead todamage of turbine blades. Thus, there is a requirement to understand the formationand nature of ash and to monitor its production and the efficiency of gas cleaningtechniques. The requirements of a gas turbine necessitate a maximum particle sizeof 3 µm and a mass concentration of less than 3 mg m−3 at STP.

Practically, there is a need to measure particle sizes at both high temperature andpressure, where many combustion systems operate. Umhauer et al. (2008) havereported a novel optical particle counter for this purpose aimed at operating attemperatures and pressures up to 1400 C and 16 bar. The instrument made directmeasurements on a stream of particles extracted from the main flow at a temper-ature limited for practical reasons to 600 C. The authors discussed the mechanicaland optical considerations in detail.

They pointed to various errors. Among these is an error associated with particlecounters that use absolute scattered intensity to estimate size. Here, particlespassing near the edge of the measurement volume may not be fully illuminated andappear artificially small. To avoid this the time of flight through the test space is alsomeasured. In their instrument the authors used a two-step test space and the particlemust pass through both to be acknowledged. Measuring intensity also introducedthe difficulty of suitable calibration.

For radiation calculations it is necessary to know the refractive index of the ash.An attempt was made by Ruan et al. (2007) by embedding ash particles in a slab ofpotassium bromide and applying the radiative transfer equation. Transmission bythe slab was measured over a range of wavelengths and inverted via theKramers-Kronig relation to yield values for the complex refractive index over aspectral range of 1–25 lm. The particle sizes were 1–2 lm.

In the near infrared range of (1.0–8.0 µm), the absorption index was quite small(of the order 10−2). However, in the range of (8.0–10.0 µm), an absorption peakexisted with a value of up to about 0.5. The real part is of the order 1.5 at shortwavelengths but rose to about 1.8 at long wavelengths.

While the authors examined two types of ash, with broad agreement betweenthem, it must be remembered that a wide variety of fly ashes occur. Also the processof sampling ash may result in physical and chemical changes. Finally they com-mented that in the analysis they used the precise Mie theory, but the particles wereclearly irregular in shape and the applicability of Mie theory was questionable.

Radiation is affected by the amount of residual carbon in the ash, as are thephysical and environmental implications. Also, too much carbon in the ash points toinefficient combustion. For this reason a number of techniques have been exploredto measure this residual carbon. A light scattering method was explored by Cardand Jones (1991) and Ouazzane et al. (2002). In this it was found that the polari-sation ratio in the scattered light decreased linearly with the mass fraction of carbonin the ash. A similar approach was used by Iannone et al. (2011).

A review of unburned carbon in ash has been provided by Bartonova (2015).


6.4 Drops and Sprays

Liquid fuels are commonly atomised and burned as small drops in a spray. Thisincreases the surface area per unit mass and the efficiency with which the dropsevaporate and burn. Light scattering is an essential tool for analysing the spray,including drop size, velocity and distribution. Where the drops are sufficiently largeimaging presents an obviously simple method to obtain size directly. This basictechnique can be varied to also determine velocity and concentration.

The simplest imaging method is to take a photograph of the spray. However,when a spray is photographed many particle images may be out of focus due to thefinite depth of field of the optical system. This limits the amount of analysisavailable. To improve the situation Ju et al. (2012) developed a multi-thresholdalgorithm to recover the particle size from out of focus images. This was found tobe of particular value for smaller particles of diameters less than 50 µm. Klinnerand Willert (2012) combined shadowgraphy with tomography to determine thestructure of a spray.

Seifi et al. (2013) used digital on-line holography to study drops in sprays. Fornon-evaporating drops they found that they could measure size down to 1 µm.However, for evaporating drops the vapour caused interference with the image.This could be reduced with an optical mask, but size could only be determineddown to 5 µm.

The excellent depth of field in a hologram was used by Wu et al. (2015) todetermine 3D velocities of irregular particles. However, being non-spherical and/orirregular the particles have more than one velocity component and this may havesignificant effects on the multiphase flow. Being an imaging technique the holo-gram reconstruction can provide information on shape and a double pulse methodenables measurement of 2D rotation. The authors made measurements on ellip-soidal drops in a spray.

Koh et al. (2006) have devised a technique for studying dense sprays by com-bining an optical patternator with Mie scattering. In this a laser beam scans acrossthe spray and a photodiode records the extinction at each position. The scatteredlight is displayed as an image on a CCD camera orthogonal to the incident direc-tion. A mathematical procedure is then used to relate the attenuation in the image tothe measured extinction across the spray.

One difficulty with high pressure sprays is the high velocity of the drops. Inorder to obtain good images it is necessary to freeze the motion and very shortexposure times are required. To achieve this Purwar et al. (2015) used a customdesigned fibre laser generating 20 ps pulses at a repetition rate of 8.2 MHz at awavelength of 1.04 lm. The results showed excellent spatial resolution and con-trast. Kashdan et al. (2007) discussed the use of phase Doppler image analysiscombined with pulsed laser illumination to study fast drops.

Doppler image analysis has also been used by Kashdan et al. (2007) to inves-tigate drops containing inclusions. Here the scattered light from a phase Dopplersystem was collected at an off axis angle by a CCD camera. The resulting images

278 A. R. Jones

showed the expected interference fringes for pure drops, but in the presence ofinclusions there was an additional speckle pattern. This pattern could completelyobscure the fringes at high concentration. Using Fourier analysis the scattering fromthe pure drop could be filtered out. The resulting speckle pattern was analysed threeways. The first was based on the wavelet transform and yielded a measure for theconcentration of the inclusions. The second and third methods were the turbidityand Fourier transform analysis, which were combined to give an estimate of theinclusion parameter. However, the authors commented that their measurementswere not unique. They discussed various reasons for this including the small size ofthe sample they were able to study and flocculation of the inclusions.

Interference fringes can be used to determine the size of spherical particles.According to geometrical optics there are two primary rays, one reflected directlyoff the surface of the particle and one transmitted once. When viewed these two raysappear as bright spots on the surface; the so-called “glare spots”. A method ofparticle sizing based on imaging of the glare spots from individual particles hasbeen described by Hess and L’Esperance (2009). While a laser is employed they usean optical system to produce mixed polarisation. Their technique then takesadvantage of the differences in intensity between parallel and crossed polarisationof the spots as viewed by two CCD detectors, which obviates the problem ofoverlapping images. After correcting the images for distortions, the two sets of glarepoints were correlated and yielded separations that were within a fraction of a pixel.Large particles could be measured from the separation between glare points as longas the particles were spherical. They thus claimed to measure sizes from a fewmicrons to very large. However, non-spherical drops and spray features yieldedcomplex light intensity patterns on the image plane that required advanced imageprocessing.

When an out of focus image is formed the glare spots can be thought of as pointsources producing an interference pattern. It can be shown that the fringe spacing isinversely proportional to the particle size (Roth et al. 1991). Maeda et al. (2002) andKawaguchi et al. (2002) referred to this method as “interferometric laser imagingfor droplet sizing”, or ILIDS. They noted that conventionally circular images wereproduced, which had difficulties at high concentration due to overlapping. Theyproposed compressing the images using cylindrical lenses. They then had the formof linear images that were horizontally defocused and vertically focused keeping theinformation of the location and the size of droplets. Similarly Shen et al. (2012)proposed an optical set up employing a combination of a spherical and cylindricallens producing fringes which were rotated to an extent depending on horizontalposition. The position of a particle could be determined from the fringe rotation andthe size from the fringe spacing as usual.

The problem with overlapping images in ILIDS has been investigated by Evanset al. (2015). The emphasis was not on a precise identification of droplets, but onobtaining a good estimate of the droplet size distribution function. They devised analgorithm based on Fourier analysis and wavelet transformation of the receivedfringes. They obtained a frequency distribution of the fringes from which an esti-mation of the size distribution could be found. They then measured the distribution


of droplet sizes produced by spinning disk aerosol generators. The mean primarydroplet diameter agreed with predictions from the literature, but they found a broaddistribution of satellite droplet sizes.

Dehaeck and van Beeck (2007) performed a stringent error analysis on theILIDS technique, taking account a number of variables including calibration pro-cedures and the influence of particle shape. On the basis of this they were able torecommend optimum optical design. They concluded that for perfect spherescombined uncertainties below 1.5% were realistic. However any deviation fromsphericity would increase the error significantly.

A modification of ILIDS has been developed by Damaschke et al. (2005).Whereas ILIDS uses glare spots of different order the modification used glare spotsof the same order. This was achieved by using two laser beams as in Dopplervelocimetry and phase Doppler anemometry (PDA). Then two glare spots bothfrom surface reflections produced the interference pattern. The authors referred tothis technique as global phase Doppler (GPD). One advantage was that in thissystem the two glare spots were always of the same intensity and measurementcould be made at any angle. It could also be used for opaque particles. The authorsfurther commented that the size of the defocused image of each particle alsodepended on the position of the particle perpendicular to the laser sheet. Hence,with appropriate calibration, the third component of velocity was also accessible inprinciple.

ILIDS is inherently based upon geometrical optics and, therefore, mainly usefulfor larger particles. However, there is a requirement to measure gas velocity in aspray and to this end the system is seeded with very small particles that can followthe gas flow. Hardalupas et al. (2010) combined ILIDS, which is an out of focustechnique, with in focus particle imaging velocimetry (PIV) and were able todemonstrate that they could discriminate between droplets and ‘seeding’ particles.Then two-phase measurements in polydisperse sprays could be achieved. Using atwo colour YAG laser and different dyes in each of the liquid and gas phase tracers,Toth et al. (2009) demonstrated that the two emission wavelengths could be used toimage the two phases separately.

When the drops are much larger than the laser beamwidth a drop scatters twopulses as it passes; a so-called “dual burst”. The reason for this is that some of theradiation is reflected by the surface and some is refracted through. This was orig-inally noted by Onofri et al. (1996a, b), who showed that, for a known particle size,the refractive index could be measured from the delay between the two pulses.Further, if the drop is absorbing the extent of absorption can be obtained from theirrelative heights. Thus the full complex refractive index could be obtained.Damaschke et al. (2002) proposed applying the same method in the backscatterdirection. Because of the different path lengths there is a time delay between the twopulses that is proportional to the particle size. Generally, the separation of thefractional signals in time will be determined by the particle size, the relativerefractive index, the particle shape and the particle velocity. Even for spheres it isnecessary to know the velocity to extract the size. This can be achieved by using

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two laser beams in a LDA arrangement so that the velocity can be measured fromthe signal modulation frequency.

Schafer and Tropea (2014) referred to this as the time shift technique. Theypointed to a number of advantages, including the ability to use thermal sourcesrather than lasers and measurement at near backscatter. They also suggested a wayof determining velocity without the need for LDA. These advances enabled thedevelopment of simpler and more cost effective instrumentation. However, theparticle diameter must be large in comparison to the beamwidth. For this reasonlasers are to be preferred. Otherwise large focal lengths may be required. In theirexperiments two detectors were employed symmetrically arranged in the backwarddirection. There were a number of peaks due to multiple internal reflections, eachindependently yielding particle size. This redundancy of size enabled the deter-mination of refractive index and velocity, the latter being based on the widths of thesignal peaks.

Application to irregular rough particles was discussed by Brunel et al. (2015).By assuming that irregular rough particles could be modelled as a collection of alarge number of coherent emitting glare points located over the global form of theparticle, they could demonstrate theoretically that the 2D-autocorrelation of theshape of the particle is directly given by the 2D-Fourier transform of thespeckle-like out-of-focus image. Experiments confirmed this result well. Using amatrix transfer based-formulation, they further determined the exact scaling factorsbetween both functions, whatever nature of the imaging system.

A means of velocity measurement of both liquid drops and the surrounding gasphase has been suggested by Kosiwczuk et al. (2005). Here the liquid and gasphases were labelled with two different fluorescent dyes. These were imaged atdifferent wavelengths with separate cameras and the velocities are obtained bycorrelations of the flows.

To fully characterise a drop the refractive index should be obtained as well as thesize. This would be important where the composition is not known in advance orwhere it changes over time; e.g. due to chemical reaction or distillation. A proposalfor measuring both size and refractive index using glare spot analysis has been putforward and studied theoretically by Hespel et al. (2008). In this detection was usedin the image plane of the drop where the images of the glare spots were separated inspace and time. When two incident beams were deployed each spot produced aDoppler burst. When two detectors were used the phase differences and theintensity ratios between two signals the distance between the reflected and refractedspots could be obtained. These measured values provided information about theparticle diameter and its refractive index, as well as its two velocity components. Inparticular, the refractive index could be evaluated from the mean value of the phasedifference ratios of the two signals.

An obstacle to imaging at high concentration is shadowing and multiple scat-tering. One attempt to overcome this is so-called ballistic imaging in which onlyphotons that are directly transmitted from the target are collected. Other multiplyscattered photons that take longer paths and, therefore, greater times are removed bytemporal gating and small receiving angle.


Paciaroni et al. (2006) and Linne et al. (2009) used ballistic imaging in anatomising spray to obtain high spatial resolution, single-shot images of the liquidcore. Time series of these images revealed a flow field undergoing turbulent pri-mary breakup. Their technique provided good spatial resolution of 40–50 µm witha single laser pulse so that no averaging was required. The image highlightedpotential signatures of spatially periodic behaviour, shedding of droplets from anintact core, and small voids that seemed to appear and then coalesce with distance.The evolution of the liquid core was characterized by the growth of these structures.They claimed that classical breakup, with organized stripping from the sides andmass ejection from the end was not supported by their images. Rather, theyobserved violent ejection of mass from the sides and loss of an organized jetstructure.

Duran et al. (2015a, b) have demonstrated the use of a 15 ps laser to performballistic imaging on a diesel spray at high pressure and temperature. The resultingimages of the near-orifice region revealed dramatic shedding of the liquid near thenozzle.

To deal with moderate multiple scattering Berrocal et al. (2005) proposed atechnique for retrieving only singly scattered photons. In this two small volumeswere defined by the optical system, one being the volume of the laser beam and theother being the volume collected by the detector. They demonstrated that whenthese volumes were the same and intersected exactly the result was dominated bysingle scattering. The technique was validated by a Monte Carlo calculation.

Polarisation methods have also been used to estimate the level of multiplescattering from spheres. A true sphere scattering individually will not producecross-polarisation, in contrast to multiple scattering. Labs and Parker (2005), forexample, used the ratio of two polarisation states to estimate the amount of multiplescattering from a spray. However, caution must be applied since quite smalldeviations from sphericity can produce significant cross-polarisation.

Another route to partially overcome the influence of multiple scattering is toemploy a laser sheet. This restricts the scattering to be out of a very narrow strip andprovides good resolution in one dimension. The laser sheet is formed by the use ofcylindrical lenses, one of the earliest descriptions being by Long et al. (1979).Evidently the image is limited by the quality of the optical arrangement, so themethod will be most suitable to particles above some minimum size.

In very dense sprays multiple scattering is still a problem in laser sheet illumi-nation. A technique for reducing the influence of this was described by Berrocalet al. (2008, 2010). In this the laser sheet was modulated along the vertical direc-tion. The main idea was that photons which have experienced several scatteringevents within the sample would lose modulation information on the way to theimaging system, while singly scattered photons would not. The scattered lightcontained two components, a steady term, which involves both singly and multiplyscattered photons, and an oscillating term due to only single scattering. Theydemonstrated that by measuring the scattered light with three chosen phase shiftsthe steady term could be numerically deducted. They found that, for the case of

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averaged images, an unwanted contribution of 44% of the detected light intensitycould be removed. This enabled an increase from 55 to 80% in image contrast.

Similarly Kristensson et al. (2008) used structured illumination and a high speedlaser to record three images of the scattered light with time delays of 55 µs. In thisway they could effectively freeze the image while removing the blurring effect ofmultiple scattering.

In a later paper (Kristensson et al. 2011) discrepancies in the performance of thestructured illumination technique were discussed. They showed that photonsundergoing scattering without changing their incident trajectory could not be dis-criminated. Since large particles scatter very strongly forward the suppression ofmultiply scattered light was reduced. At the same time the authors claim that suchbehaviour allows denser media to be probed.

Light scattering methods in dense media have been surveyed by Coghe andCossali (2012). They proposed a single parameter, the optical depth, to quantify theconcept of dense spray and to indicate when multiple scattering predominates.Experimental results then become questionable. This survey was intended as acritical analysis of optical techniques capable of providing quantitative and reliabledata in dense sprays, and to point out the conditions necessary to safely obtain suchmeasurements. They explained that two experimental approaches may be pursued.The first was to attempt to directly penetrate the inner spray structure by opticaltechniques capable of yielding reliable information in such hard environments. Thesecond strategy was based on the study of the ‘‘external effects’’ produced by theinteraction between the spray and the gaseous environment. These are closelyrelated to the gas spray interaction mechanisms but only carry indirect informationabout the spray structure. However, while the former were few in number and stillunder development the indirect methods are numerous and quite reliable. Theoptical techniques critically discussed by the authors included the structured illu-mination laser sheet (SLIPI) and ballistic imaging, mentioned above. They alsoincluded the use of X-rays and molecular tagging, and the usefulness of velocitymeasurement to study the gas flow. They concluded that these techniques showpromise but still needed further development.

One difficulty with the sheet is the loss of laser intensity en route to (obscuration)and from (signal trapping) the test space. Overcoming this problem has exercisedKalt et al. (2007a, b). They performed laser attenuation experiments using sus-pensions of spherical particles in water at various concentrations. Their aim was toformulate a calibration for the effects of diffuse scattering and laser sheet extinctionboth for uniform and strongly divergent sheets. They developed a model to describethe attenuation of the laser and compared the results to the experiments. Theydemonstrated that the scattered signal may be considered proportional to the localparticle concentration multiplied by the incident laser power, which varies as afunction of the attenuation. They proposed a calibration constant Ck and suggestedmeans of estimating it. Their calibration then enabled concentration to bedetermined.

A new technique is planar fluorescence imaging. This combination of laser sheetimaging with laser induced fluoresecence (LIF) was originally suggested by Yeh


et al. (1993). The fundamental principle behind this is that while scattered intensityis proportional to the area of the particle the fluorescence intensity depends upon thevolume. For a size distribution the average squared and cubed diameters are foundand the ratio yields the Sauter mean diameter directly. The method is heavilydependent upon calibration techniques.

A detailed examination of fluorescence within drops has been given by Domannand Hardalupas (2001), Domann et al. (2002) and Frackowiak and Tropea (2010).The nature of the internal structure was verified by experimental observations. Oneconclusion was that fluorescent signal varies as size raised to the power 2.96. Inaddition, Pastor et al. (2009) performed calibration experiments under controlledtest situations to correct for the effect of scattering on the LIF and Mie signals.

Charalampous and Hardalupas (2011) investigated the accuracy of the LIF/Mieratio method in some detail. They developed an analytical model which showed thatthe technique is susceptible to sizing errors that depend on the mean droplet sizeand the spread of the droplet size distribution independently. A new data processingmethod was proposed that could improve the sizing uncertainty of the technique forsprays by more than 5% by accounting for the size spread of the measured droplets,while improvements of 25% were possible when accounting for the mean dropletsize. They also examined the sizing accuracy of the technique in terms of therefractive index of liquid, the scattering angle, and the dye concentration in theliquid. They found that the proposed approach led to sizing uncertainty of less than14% when combined with light collection at forward scattering angles close to 60°and the lowest fluorescent dye concentration in the liquid for all refractive indices.

Mishra et al. (2014) highlighted the importance of multiple scattering on the LIF/scattering ratio method. To reduce this they combined LIF with the structuredillumination system. In passing they noted the significance of multiple scatteringeven in sprays that were not deemed optically dense.

Another method for examining the internal structure of a cloud of scatterers istomography. In this the cloud is illuminated at various angles of incidence andscattering detected over a range of angles. The relationship between the detectedpattern and the structure of the object leads to a complicated set of mathematicalequations. For this reason significant computing power is required.

In early work (e.g. Sivanathu and Gore 1993; Menguç and Dutta 1994) theproblem was simplified by assuming axial symmetry. Tomography has beenapplied to aerosols by Ramachandran et al. (1994) and to sprays by Oberlé andAshgriz (1995). A Doppler tomography method has been devised by Wang et al.(1995) to map fluid flow velocity.

With existing imaging techniques it is difficult to obtain information on theinfluence of interactions between particles. To improve this situation a newapproach was suggested by Briard et al. (2013) and Saengkaew et al. (2014) basedupon Fourier interferometric imaging (FII). Their objectives were to introduce asimple model permitting fast and accurate numerical simulations. For a number ofdrops within the scattering test space the scattered amplitudes were added. Whenthis summation was squared it resulted in two terms: the added intensities fromindividual particles and a sum of terms describing the relationship between pairs of

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particles. For a large number of randomly positioned particles the latter term wasnormally assumed to sum to zero, and only intensities needed to be added.However, it was the latter term that held information about the relative positions ofthe particles. To extract the information from this term a two-dimensional fastFourier transform (2D FFT) was calculated. To obtain the required interferencepattern, they used the simplified glare spot method for large particles. The FFT wasthen found to consist of a number of bright spots. They tested their method byconsidering a single line of monodisperse drops, for which the FFT showed anumber of pairs of bright spots corresponding to interference between various pairsof particles. A mask could be used to select any one pair and inverted to produce theinterference pattern for that pair. The theory agreed with experiment thus validatingthe model. Measurement at the rainbow angle is a step towards measurement ofrefractive index.

Perhaps the simplest non-imaging technique for determining particle size is theforward diffraction method. Instruments using this method provide size distribu-tions based on the assumption of spheres. While it is generally thought that forwardscattering is insensitive to shape it must be recognised that drops in sprays may benon-spherical, especially close to the spray nozzle. The influence of this has beenexplored by Dumouchel et al. (2010). They compared experimental results of theequivalent size distribution from diffraction against the surface based length scaledistribution from image analysis. Their results said that diffraction yielded thediameter distribution of a set of spherical droplets that had the same scale distri-bution of the actual spray. Thus, diffraction performed a multi-scale analysis of thespray droplets. In consequence, the diffraction diameter distribution must containinformation on the shape of the drops.

Yu et al. (2013) have explored measurement in the primary rainbow region toexamine spheroidal drops. First they explained the relation between rainbow angleand the Brewster angle for spheres and how for a specific refractive index therainbow was completely polarised. An indication of the refractive index could thenbe obtained from the degree of polarisation. Experimentally, they observed that atthe primary rainbow the scattering pattern showed a series of orders which appearas straight line fringes. As the axial ratio increased these fringes became increas-ingly curved until they ultimately took on various catastrophic shapes. Theycompared their results against the Airy approximation and found that the predictedrefractive index agreed well with a known value. They were further exploring therelationship between the fringe curvature and the particle shape.

Vetrano et al. (2005a) asked how a refractive index gradient in a drop can affectrainbow thermometry. By exploiting a generalization of the Airy theory (Vetranoet al. 2005b), a data inversion algorithm for a single droplet, presenting a parabolicrefractive index gradient, was proposed. This algorithm was used to compute thediameter of the drop, and the refractive index at the core and the surface of asimulated burning droplet.

Saengkaew et al. (2007) also explored the influence of temperature gradient. Forhomogeneous particles they improved the accuracy of the rainbow measurementsby employing Complex Angular Momentum theory to calculate the scattering.


In this case they claimed accuracy of 0.01 µm for size and 0.0001 for refractiveindex, but they acknowledged the need to assess accuracy in the presence of atemperature gradient. To this end they employed the Mie theory for multi-layeredspheres. They then searched for an equivalent refractive index to match the rainbowfor each profile that they considered and for a variety of supposed surface values.

Standard rainbow thermometry connects the scattering angle of the main rain-bow maximum, generated by a single droplet to the droplet’s refractive index andthus to its temperature. Van Beek et al. (1999) proposed a method they called globalrainbow thermometry, which measured the average rainbow position that wascreated by multiple droplets, from which a mean temperature could be derived. Inthis method parallel light was collected from a number of particles in an extendedtest space so that the rainbow patterns from the particles interfered with each other.The collected scattered light was focused onto a CCD camera, the resulting patternbeing equivalent to a Fourier transform in which distance corresponds to angle. Itwas claimed that this technique was insensitive to particle non-sphericity. Using theglobal rainbow technique in conjunction with the Airy rainbow theory, Vetranoet al. (2006) demonstrated that the size distribution and mean temperature of a waterspray could be established.

The insensitivity of the global rainbow method to particle shape was testedexperimentally by Saengkaew et al. (2009) and numerically by Wang et al. (2011).For spherical droplets, the temperature was measured with an accuracy of about 2 °C. The associated size distributions were in agreement with size distributionsmeasured by PDA and the shadow imaging technique. For non-spherical spheroidaldroplets with randomly distributed orientations, the temperature was measured witha good accuracy of about 5 °C, but the associated size distribution (in number) wasstrongly dominated by ‘‘ghost’’ small particles. The numerical calculations werebased on the T-matrix formulation and simulations showed that the rainbow patternwas very sensitive to the particle non-sphericity. For randomly oriented spheroidsthe results were largely in line with those of Saengkaew et al. (2009). They pro-posed that the presence of the spurious, or “ghost” particles could be viewed as anindicator of the refractive index quality. When no spurious particles were extracted,the refractive index measurements could be qualified as nearly exact. However, ifspurious particles were extracted the measured temperature is overestimated.Another interesting phenomenon in the simulation was that when the ellipticity ofthe particle exceeded some limit, the extracted refractive indices became so smallthat unreasonable temperatures resulted which could also be regarded as an indi-cator for experiments. Once an unreasonable temperature was found it indicatedthat the shape of the particle departed too far away from spherical.

The potential of femtosecond pulses for rainbow measurements has been dis-cussed by Bakic et al. (2008, 2009). Because of the relatively wide bandwidth ofthe pulse (and only partial coherence) much of the detailed structure in scatteringpatterns was suppressed. Similarly the detailed structure at the rainbow was alsosuppressed. This structure was particularly problematic for particles below 20 µmdiameter, and so the use of femtosecond pulses improved rainbow measurementsfor small particles. The primary rainbow was shown to be detectable for pulse

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lengths as short as 10 fs and free of interferences with other scattering orders fordroplet diameters down to 5 µm.

The question of measuring drop temperatures in a spray has been discussed byLabergue et al. (2012). They examined two-colour laser induced fluorescencewhere the emission was collected at two spectral bands. For monodisperse dropsthis could yield the temperature directly, but it failed for polydisperse sprays. Theyproposed combining the fluorescence with PDA such that the temperature could bedetermined drop by drop. There was a complication, however, in that the ratio ofthe two fluorescent signals was itself affected by the particle diameter, due to achange in molecular structure that was a function of the radius of curvature. Thiswas particularly significant for small drops. There was also an effect of depth offield which, if too large, could cause out of focus fluorescence due to multiplescattering. The authors found that using a third spectral band in combination with along distance microscope could correct for the size effect and reduce the depth offield.

Bruchhausen et al. (2006) proposed temperature measurement in sprays usinglaser induced fluorescence (LIF) at three emission wavelength bands. By takingratios the dependence on dye concentration, the dimensions of the probe volume,the laser intensity, and the optical layout could be eliminated. It could also correctfor the wavelength-dependent scattering of the fluorescence. They indicated thatfurther analysis on the problem of re-absorption of fluorescent photons neededfurther examination.

A review of optical techniques for the measurement of temperature and com-position in sprays has been provided by Lemoine and Castanet (2013). A wideranging review of spray measurement technology has been given by Fansler andParrish (2015).

6.5 Conclusions

From the shear bulk of the literature it is evident that combustion remains a veryactive area of research. The major part of the references discusses diagnostics ofsoot, and the major part of that concerns LII. The reason for this dominance isprobably the importance of soot to radiative heat transfer from flames and fires,concern about its polluting effects and its significance to the optical properties of theatmosphere. Even if combustion of hydrocarbons was phased out completely, therewould always be fires.

The wide interest in LII is due to its potential accuracy and its novelty.Considerable questions arise, however, concerning its operation and analysis. Inparticular there are worries about the influence of laser heating on vaporisation andchange in structure. Also there is still uncertainty about the important thermalaccommodation coefficient, the value of which depends upon the composition ofthe surrounding atmosphere, pressure, liquid coatings and so on. While LII can bevery accurate, it still relies upon calibration.


Other techniques are more straightforward in principle, whether applied to soot,sprays or coal and ash. Work pursues improvements in equipment design andaccuracy, as well as tackling concerns around dense systems, multiple scatteringand particle shape and irregularity. While each of these present problems the basictheoretical and experimental techniques exist for their treatment and only appli-cation needs to be considered.

In all cases the question of experimental accuracy and error arises. Attention isdrawn to high uncertainties associated with flux density or concentration mea-surements, due to poor quantification of the detection area/volume of opticaltechniques. In addition there remain problems around comparisons between variousinstruments. More studies are needed in these areas.

Of course, given concern about global warming combustion of fossil fuels maybe considerably reduced. However, the same techniques reviewed here can beapplied to any solid or liquid particles suspended in a gaseous medium, whetherstagnant or flowing.

There are a number of reviews in the literature additional to those mentioned inthe text. Some of these are listed at the end of the Reference Section.

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Some Additional Reviews

Albrecht H-E, Borys M, Damaschke N, Tropea C (2013) Laser doppler and phase dopplermeasurement techniques. Springer, Heidelberg

Baumgardner D et al (2012) Airborne instruments to measure atmospheric aerosol particles, cloudsand radiation: a cook’s tour of mature and emerging technology. Atmos Res 102:10–29

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Gouesbet G, Grehan G (2015) Laser-based optical measurement techniques of discrete particles: areview. Inter J Multiphase Flow 72:288–297

292 A. R. Jones

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Nathan GJ et al (2012) Recent advances in the measurement of strongly radiating, turbulentreacting flows. Prog Energy Comb Sci 38:41–61

Sorensen CM (2010) Light scattering as a probe of nanoparticle aerosols. Part Sci Tech 28:442–457

Tropea C (2011) Optical particle characterization in flows. Ann Rev Fluid Mech 43:399–426[Davis SH, Moin P (eds)]


Index

AAbsorption, 1, 7, 8, 16, 17, 25, 46, 59, 60, 63,

66, 68, 73, 76, 78, 80, 86, 113, 116, 151,152, 159, 164, 174, 177, 179, 186,188–190, 206, 209, 215, 238, 253, 254,256, 258–260, 263–268, 270, 272, 274,275, 277, 280

Absorption coefficient, 59, 63, 73, 152, 164,189, 266, 267

Aerosol, 1, 2, 55–57, 59, 64–67, 72–74, 76, 77,90, 120, 217, 219, 222, 235, 236, 238,252, 265, 280, 284

Aggregates, 63, 69, 75, 78, 79, 89–92, 203,211–213, 215, 226–230, 256, 259, 260,262, 264, 269–276

Airborne particulates, 265Albedo, 32, 40, 41, 65, 72, 85, 206, 208–210,

221, 236, 260, 266, 270, 275Angular scattering, 159, 166, 170, 171, 175,

272, 273Anomalous diffraction, 60, 135, 139Artificial spherical beads/bubbles, 44, 77, 109,

159, 162–164, 205Ash, 61, 74, 252, 275, 277, 288Aspect ratio, 41, 138–140, 202, 203, 210–216,

218, 219, 225, 228–237Astigmatism, 123, 138, 143Asymmetry parameter, 204, 208, 210–215,

220, 222, 228–234, 237, 274, 275Atmospheric remote sensing, 263Auto-correlation, 281

BBackscattering, 41, 67–69, 72, 73, 77, 79, 86,

92, 109, 152, 153, 155, 164, 168, 169,175–180, 183–190, 192, 214, 217–219

Backscattering coefficient, 65–67, 69, 77, 152,177, 178, 185, 187–189

Backscattering ratio, 153, 164, 168, 178–180,184, 186, 189, 190

Ballistic imaging, 281–283Beam steering, 259Bi-disperse systems, 262

CCalibration, 154, 159, 160, 164, 165, 168, 178,

192, 229, 255, 256, 258, 259, 262–264,267, 277, 280, 283, 284, 287

Calibration independent method, 263CALIPSO, 77, 217, 222Carbon in ash, 277Cavity Ring Down (CRD), 65, 264–266Chiral media, 1, 9, 14, 43Chlorophyll, 159, 179, 180, 184, 189, 190Cholesteric phase, 45, 46Cirrus, 3, 8, 37, 38, 200, 204, 205, 217, 219,

223, 225, 226, 235, 238Clouds, 3, 4, 8, 37, 38, 40, 55, 62, 65, 73–75,

77, 119, 120, 198–200, 202, 204, 205,208, 215–219, 222, 227, 229–238

Clouds of bubbles, 119Coagulation, 268, 271, 275Coal, 252, 263, 275, 276, 277, 288Coarse structures, 124, 127, 138, 139Coastal seawaters, 190Coated particles, 263Coating of particles, 263, 266, 274Complex-Angular Momentum theory, 112Complex refractive index, 56, 57, 59, 60, 64,

65, 72, 73, 75, 79, 84, 85, 86–89, 91–93,190, 257, 261, 265, 277, 280

Composition, 31, 34–36, 64, 66, 77, 110, 119,120, 178, 179, 180, 186, 189, 230, 266,281, 287

Convection, 200, 231, 232, 261CPL, 217, 218, 230, 231

© Springer International Publishing AG 2018A. Kokhanovsky (ed.), Springer Series in Light Scattering, Springer Seriesin Light Scattering, https://doi.org/10.1007/978-3-319-70808-9

295

Critical-angle refractometry and sizingtechnique, 119

Cross section, 16–18, 60, 116, 140, 154,205–207, 210, 217, 236, 256, 260, 263,265–267, 270, 274, 275

DDepolarization, 4, 43, 70, 71, 74, 76, 77, 79,

80, 86–89, 91–93, 217–219, 230, 231,234–238

Digital imaging, 276Direct Dipole Approximation (DDA), 61, 62,

90, 93, 120, 252, 256, 272, 275Direct inversion, 267, 268Discrete ordinates, 258Dissolved colored organic matter, 159Dissymetry, 4, 45, 46, 122, 140, 215Distortion, 11, 203, 214, 218, 224, 231, 237,

279Drops, 110, 111, 197, 217, 219, 223, 252, 253,

278–281, 284, 285, 287Dual burst, 280

EEdge diffraction, 257, 277Effective radius, 209, 226, 230, 231, 238Eikonal approximation, 268Elastic light scattering, 262, 271Emissivity, 256, 257, 260, 261, 268Energy balance, 5, 37, 131, 135, 142, 262, 265Eotvos number, 110Extinction, 2, 3, 7, 14–18, 22, 26, 27, 30, 35,

38, 39, 41, 60, 63–67, 69, 116, 168,206–209, 217, 220, 230, 235, 236, 257,258, 260, 264–269, 272, 274, 275, 278,283

Extinction coefficient, 64–67, 207, 258,264–266, 268, 272

Extinction efficiency, 207–209, 265

FFar-field, 5, 63, 83, 116, 119, 131, 132, 139,

142Fast Fourier Transform (FFT), 285Femtosecond pulse, 286Fibre laser, 278Fine structure, 133, 135, 143Fluorescence, 74, 159, 252, 257, 263, 264, 273,

283, 284, 287Forward diffraction, 123, 124, 135, 138, 139,

143, 285Fourier analysis, 37, 279

Fourier Interferometric Imaging (FII), 284Fractal aggregates, 89, 256, 274Fractal dimension, 259, 270, 271–273Fractal geometry, 252Fragmentation, 271Fraunhofer diffraction, 8, 9, 252Fraunhofer’s approximation, 132, 135Fresnel coefficients, 121, 124, 126, 137, 138,

143Fringes, 124, 131, 132, 143, 275, 279, 285

GGas conduction, 256Gaussian beam, 118, 138Gaussian cut-off, 272Generalized Lorenz-Mie Theory, 117Genetic algorithm, 268Geometrical optics, 7, 13, 60, 112, 121, 132,

137, 279, 280Geometrical Optics Approximation, 112, 121,

137Geometric optics, 61, 206, 218Glare points, 279, 281Global Phase Doppler (GPD), 280Goos-Hänchen shift, 132, 134

HHabit, 200–203, 211, 212, 215, 216, 224–229,

231, 234, 237Halo, 37–39, 197, 205Hexagonal, 37, 40, 41, 197, 198, 202–205,

210–212, 214, 215, 217–219, 227, 228,230, 237, 238

High spatial resolution, 265, 282Hollow, 205, 214, 218, 224, 225, 227–230, 237Hollowness, 205, 214, 218, 237Holography, 276, 278Hygroscopic growth, 265

IIce crystals, 1, 3, 8, 37, 38, 40, 41, 197–200,

202–208, 210, 213–216, 218–224, 227,228, 230, 232, 234–238

ILIDS, 252, 279, 280Image analysis, 276, 278, 285Inclusions, 117, 205, 225, 278, 279Interactive scattering, 270

LLaser, 44, 58, 65–68, 74, 80–82, 92, 110, 152,

155, 177, 216, 217, 254–265, 268, 273,275, 276, 278–283, 287

296 Index

Laser beam profile, 260Laser Doppler Velocimetry (LDV), 275Laser flux (fluence), 255, 258–261, 263Laser induced fluorescence (LIF), 273, 287Laser induced incandescence (LII), 254,

258–260Laser sheet, 258, 259, 280, 282, 283LED, 158, 159Lidar, 56, 57, 58, 59, 66–69, 74, 77, 80, 199,

216–219, 222, 230, 231, 235–238, 263LIF/Mie ratio, 273, 275, 283, 284, 287Light, 3, 4, 37, 38, 42–46, 55–93, 110–113,

117, 124, 137, 142, 151–159, 162, 164,169–173, 175, 184, 185, 190, 204,206–208, 210, 211, 217, 220, 222, 253,262, 263, 266, 268, 269, 271, 272,275–279, 282–284, 286

Light scattering, 46, 55–66, 68–93, 110–112,152, 164, 170–172, 184, 185, 190, 262,263, 269, 271, 272, 275–278, 283

Liquid crystals, 4, 11, 45, 46Lorenz-Mie theory, 59–61, 63, 71, 72, 79, 112,

117, 206, 217, 259

MMass spectroscopy, 262Maxwell equations, 2, 4, 5, 6, 13, 45Mean free path, 260Mie, 41, 59, 60, 85, 112, 113, 162, 163,

166–169, 263, 265, 268, 271, 274, 277,278, 284, 286

Mie theory, 112, 162, 163, 168, 169, 265, 268,271, 277, 286

MODIS, 199, 222, 223, 225, 226Morphology (Shape), 57, 61, 68, 77, 79, 90,

93, 204, 205, 259, 260, 261, 266,273–275

Morton number, 110Multi-angular, 2, 219–223, 238Multi-directional measurements, 216, 220, 238Multiphase flows, 112Multiple scattering, 5–8, 15, 57–59, 62, 63, 65,

70–72, 74, 76, 79, 85, 88, 89, 119, 166,217, 221, 256, 259, 271, 272, 274, 276,282–284, 288

Multispectral Volume Scattering Meter(MVSM), 157, 158, 174, 180, 181, 183,190, 192

NNear-critical-angle, 111, 117, 120, 122, 124,

127, 132, 133, 135, 139, 140Near field, 113, 117, 131, 139, 272

Nematic phase, 45, 46Nephelometer, 72, 80, 153, 154, 166, 170, 192Non-spherical particles, 1, 8, 37, 41, 77, 217,

252, 253, 268, 276

OOceanographic platform, 188, 189Optical depth, 65, 223, 238, 276, 283Optically active media, 2–4, 7, 9, 12, 14–17,

26, 33, 43, 44Optical particle counter, 277Optical patternator, 278Optical thickness, 22, 23, 65, 85, 205–208,

220–223, 226, 229, 232, 233, 238, 259Orientation, 1, 9, 15, 37, 39, 41, 74, 83, 90, 91,

93, 199, 206, 207, 213, 215, 221, 223,268, 274, 286

PParticle Imaging Velocimetry (PIV), 280Particle size distribution, 56, 57, 65, 85, 179,

209, 257, 261, 262, 267, 274Particle swarm (ant colony), 268Particulate media, 8, 55–59, 62–72, 74–78, 80,

81, 83–89, 91, 92Periscope prism, 155, 157Phase Doppler Anemometry (PDA), 280Phase matrix, 2, 3, 7, 14, 16–18, 26, 30, 33–39,

71, 83, 206, 207, 209, 210, 212, 217,218, 236, 237

Photoacoustic spectroscopy (PAS), 264–266Photochemical effects, 262Photography, 278Photomultiplier, 156–158Physical Optics Approximation, 123, 139Phytoplankton, 151, 159, 164, 168, 169, 174,

177–179, 184Planck function, 262Plasma, 4, 11, 42Polarimeter, 43, 73, 82, 221, 222, 235, 238Polarisation ratio, 282Polarization, 1, 3, 4, 8, 9, 11, 12, 13, 15, 16, 20,

23–25, 33, 35, 37, 41–43, 45, 46, 55, 56,64, 69–80, 82, 88, 90, 116, 124, 127,129, 131, 135, 137, 139, 199, 206, 207,211, 212, 214, 217, 219, 220, 221, 222,227, 235, 236

polarized reflectance, 77, 199POLDER, 73, 222–228, 232, 233, 235Polycrystals, 200, 227Pressure, 232, 233, 254, 260, 261, 266, 267,

274, 277, 2778, 282, 287Primary particles, 256, 269–275

Index 297

Pulse delay (time shift), 281Pulsed laser, 66, 81, 257, 278

RRadiation attenuation, 35, 258Radiation transfer equation, 192Radiative shielding (trapping), 254Radii of curvature, 138Radius of gyration, 262, 270, 273Rainbow, 131, 132, 222, 235, 285, 286Rayleigh-Debye-Gans approximation (RDG),

252, 256, 266, 270, 271–274, 276Rayleigh limit, 262, 274Re-condensation, 255Reflectance, 2, 39, 44, 64, 73, 76–78, 151, 192,

199, 220, 222, 234, 238Refractive index, 6–8, 11, 15, 37, 42, 43, 46,

56, 57, 59, 60, 63, 64, 67, 72, 73, 75, 79,81, 84–89, 91, 92, 93, 111,113, 115,117, 119, 132, 133, 135, 136, 139, 158,166, 168, 169, 170, 171, 173, 174, 190,209, 252, 253, 255, 256, 257, 261, 265,266, 267, 268, 270, 273, 277, 280, 281,284, 285, 286

Reynolds number, 110Rough particles, 213, 227, 281RSP, 222, 225, 228, 230, 231, 233, 235RTT, 7, 8, 62, 72, 76, 83

SScanning mobility sizer, 262Scattering, 4, 7, 14, 16–18, 34, 37, 62, 63, 68,

73, 80, 85, 88, 112, 117, 119, 120, 131,152–154, 160, 162–164, 166, 168–170,171, 178, 184, 189, 192, 203, 205, 206,214, 217, 219, 228, 253, 256, 260, 264,270, 274, 284

Scattering anisotropy, 170Scattering at small angles, 170Scattering by particles, 61, 76, 166, 178, 185Scattering coefficient, 118, 152, 154, 166, 169,

171, 172, 175, 177, 187, 188, 267Scattering cross-section, 267, 274Scattering indicatrix, 17, 172Semi-analytical algorithms for reflectance, 183Shadow Doppler Velocimetry (SDV), 275Shapes (bubbles), 77, 109, 110, 117, 137Shock waves, 255, 257Signal modulation, 281Single scattering albedo, 206, 208, 209, 236,

260, 266, 275Singly scattered photons, 282

Size distribution, 65, 68, 77, 79, 164, 169, 174,189, 190, 229, 262, 267, 273, 274, 279,284, 286

Size parameter, 60, 63, 86, 113, 118, 120, 208,209, 265, 272, 274

Smectic phase, 45Soot, 55, 56, 65, 90, 252, 254–256, 260–262,

265, 266, 268–270, 274, 275, 287, 288Soot concentration, 258, 259Soot transformation, 260Specific absorption, 266Spectral extinction, 65, 266–268Spectrophotometer, 82Specular reflection, 112, 218, 223, 231Spheroid, 3, 11, 117, 121, 138–140, 142, 217,

219, 268, 269, 275, 286Sprays, 252, 276, 278, 280, 282–288Stationary phase method, 131Stokes parameters, 14, 15, 33, 222Stokes vector, 2, 3, 6, 7, 13–16, 19, 20, 32, 33,

70–72, 80, 207Structured illumination, 283Sublimation, 198, 204, 219, 254, 255Superaggregates, 90, 91, 271Surface roughness, 204, 205, 211, 213, 214,

218Suspended particles, 172

TTemperature, 83, 110, 171, 198, 200, 202–204,

218, 232, 236, 255, 256, 259, 260, 262,266, 276, 277, 282, 286, 287

Temperature gradient, 285, 286Thermal accommodation coefficient, 254–256,

261, 263Time shift technique, 281T-matrix method, 121, 252, 270, 272, 274Tomography, 269, 278, 284Total reflectance, 220, 221, 224, 226Tunneling phase, 124, 126, 127, 130, 132, 137Turbulence, 171, 174, 259Two colour method (pyrometry), 255, 256,

259, 275

UUnicellular algae monocultures, 168Upwelling radiance, 152

VVacuum, 261Van de Hulst geometrical approximation, 71

298 Index

Vectorial Complex Ray Model (VCRM),137–139

Vector radiative transfer, 72Virtual wavefront, 131, 133, 140, 142Volatiles coating, 263Volume Scattering Function (VSF), 151, 152,

155–163, 166, 168, 170–178, 180, 181,186, 191, 192

WWavelet transform, 279

XX-rays, 274, 283

Index 299

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