how the particles of the atmospheric aerosol affect the generation of filaments in a laser beam

How the particles of the atmospheric aerosol affect the generation of filaments in alaser beam

E. P. Kachana� and V. O. Militsin

M. V. Lomonosov Moscow State University, Moscow�Submitted February 20, 2006�Opticheski� Zhurnal 73, 38–44 �November 2006�

The generation of filaments when a laser pulse propagates in the atmospheric aerosol is numeri-cally investigated. It is shown that interference intensity maxima in high-power pulses can be-come filament-generation centers when radiation is coherently scattered at aerosol particles. Theextent to which the particles affect the generation of filaments in this case substantially dependson the particle size and the peak pulse power. A numerical analysis is carried out on the basis ofa stratified model in which nonlinear-optical interaction of laser radiation with a disperse me-dium is represented as coherent scattering at a sequence of statistically independent “aerosol”screens, between which self-focusing occurs in a nonturbid air medium. © 2006 Optical Societyof America.

INTRODUCTION

The rapidly developing femtosecond nonlinear optics ofthe atmosphere covers a wide range of problems on the in-teraction of laser pulses having a width 20–100 fs and apeak power of 1010–1012 W with components of theatmosphere.1,2 A significant place among these problems isoccupied by nonlinear optics of the atmosphere aerosol, inwhich two main approaches can be distinguished. The first ofthese consists of the development of methods of femtosec-ond probing of the atmosphere for detecting contaminants inthe environment. This approach includes work on the fluo-rescence of particles stimulated by a powerful femtosecondlaser pulse and the emission spectroscopy of an aerosol inthe laser plasma of a femtosecond pulse.3,4

The second approach of femtosecond nonlinear opticscovers the investigation of the filamentation that accompa-nies a powerful femtosecond pulse propagating in an atmo-sphere made turbid by an aerosol. The interaction of laserradiation with aerosol particles can have a substantial influ-ence on the phenomenon of filamentation of the laser pulseand the generation of a supercontinuum. In a pulse whosepeak power is several orders of magnitude greater than thecritical self-focusing power, filaments are generated becausea powerful light field in a medium with cubic nonlinearity isspatially unstable. The filament-generation centers are smallperturbations on the intensity distribution in a cross sectionof the pulse. Along with the initial perturbations in the crosssection of the pulse at the output of the laser system andphase perturbations initiated by refractive-index fluctuationsin a turbulent atmosphere, there will be intensity perturba-tions caused by coherent scattering of the laser radiation atthe particles when the radiation propagates in an aerosol.

Experiments were carried out in Ref. 5 on the interactionof a filament with separately placed aerosol particles. Largewater drops up to 95 �m in diameter were located on thepath of a propagating filament. It was found that a particlethat is comparable in size with the diameter of the filamenthas a negligible effect on the subsequent process of filamen-tation. The peripheral regions of the cross section dominate

772 J. Opt. Technol. 73 �11�, November 2006 1070-9762/2006/

in the process of forming a filament behind a drop, while thenear-axis regions play a smaller role. Our analysis is con-firmed by laboratory experiments with opaque disks locatedon the path of the pulse.6

In experiments on filamentation of a laser pulse in anatmospheric aerosol,5,7 it is shown that the energy of a radia-tion pulse in a disperse medium with a high particle concen-tration �105 cm−3� is reduced as a consequence of scattering.

A multifilament 80-fs pulse with energy to 220 mJ wasinvestigated in the scaled full-scale experiment of Ref. 8; thecross-sectional diameter was 3 cm when it propagated for50 m in an artificial aerosol having high density, with a con-centration of 6.7�104 cm−3. It was established that the mini-mum peak power at which one filament is formed under suchconditions is 28 GW, and this is about a factor of 9 greaterthan the critical self-focusing power. The number of fila-ments formed in an aerosol is less than in a nonturbid atmo-sphere, and Méjean et al. concluded that the influence of adense aerosol is similar to linear attenuation of the laserpulse energy.

The full-scale experiment of Ref. 9 on the filamentationof a 150-fs pulse with a power of 2.5�1012 W in rain wascarried out on a horizontal track 75 m long with a drop con-centration of 104 m−3. It was established that the rain at thebeginning of the track, where self-focusing is the determin-ing factor, does not impede multifilamentation of the pulse,whose power is many times as great as the critical self-focusing power. The formation of diffraction rings was re-corded when the pulse was scattered at the water particles.The assumption was expressed that diffraction perturbationscan cause filaments to be generated.

Theoretical studies of the filamentation of a femtosecondpulse in an aerosol were based on a model in which a particleis replaced by an opaque, radiation-absorbing disk.10 Thefilament is re-established behind the drop as a consequenceof the dynamic restoration of the energy in it from the radia-tion at the beam’s periphery, which does not interact with theparticle. A simple model of this type was used in Refs. 5, 6,and 8 to interpret the experimental results.

772110772-06$15.00 © 2006 Optical Society of America

The model of an absorbing disk neglects scattering at thedrops. However, absorption in water at a wavelength of �=0.8 �m is negligible, and the physics of the interaction ofradiation with a water particle is such that the radiation isattenuated as a consequence of scattering, which predomi-nates over absorption. The scattering occurs forward, sincethe drop size can be greater than the wavelength, the scat-tered component is large, and it remains in the beam. Coher-ent scattering at particles can therefore have a substantialeffect on the formation of filaments in a powerful femtosec-ond laser pulse.

A stratification model of the atmosphere that describesthe coherent scattering of a laser pulse in a disperse mediumwas proposed in Ref. 11. The model reproduces the forma-tion of perturbations on the profile of a laser beam, caused bycoherent scattering at an ensemble of particles. The results ofnumerical experiments obtained in this model for low-powerpulses agree with the known regularities of radiation propa-gation in an aerosol medium.

This paper discusses the contribution of coherent scatter-ing at aerosol particles to the generation of filaments in apowerful laser pulse. The study is carried out numerically onthe basis of a stratified model of an atmospheric aerosol thatdescribes the coherent scattering of laser radiation in a me-dium with Kerr nonlinearity.

STRATIFIED MODEL OF FILAMENT GENERATION

Filaments are generated at the initial stage of filamenta-tion, where the energy is redistributed in the cross section ofthe pulse because of self-focusing, and nonlinear foci areformed. A laser plasma begins to be generated when the in-tensity exceeds 1013–1014 W/cm2,12 i.e., when a nonlinearfocus is formed and the intensity in it increases by a factor of10–100 by comparison with the initial value, which is usu-ally 1011–1012 W/cm2. The nonlinearity of the induced laserplasma is therefore insubstantial at the initial stage of fila-mentation. Moreover, the dispersion of the group velocitymay be neglected, since this factor determines the change ofthe pulse in time. As shown by an estimate, when the pulse ismore than 40 fs wide with �=0.8 �m, the dispersion lengthin air significantly exceeds the self-action length at a peakpower of P0�10Pcr.

In the approximations used here, it is possible to distin-guish the following main effects, which determine the propa-gation of laser radiation in the atmospheric aerosol: coherentscattering at the particles of the water aerosol, diffraction,and self-focusing of laser radiation because of Kerr nonlin-earity in air. In this case, we use the method of slowly vary-ing amplitudes for pulses of the width considered here.13

The generation of filaments in a pulse can thus be de-scribed by the following equation in running coordinates:

2ik � E/�z = �2E/�x2 + �2E/�y2 + �nnl��E�2�E + nE , �1�

where k is the wave number, E is the complex amplitude ofthe pulse envelope, and �nnl is the nonlinear addition to therefractive index. The third term describes the change of thefield amplitude associated with irregular refraction at waterparticles. In this case, a beam of Gaussian profile is regarded

773 J. Opt. Technol. 73 �11�, November 2006

as the initial condition at z=0, with power P0 equal to thepeak power of the femtosecond pulse:

E�x,y,z� = E0 exp�− �x2 + y2�/2a02� ,

P0 = �a02�E0�2cn0�0/2, �2�

where a0 is the beam radius at the e−1 level, and �0=8.85�10−12 F/m. The initial conditions are so chosen becausethe temporary layer of the pulse with the maximum power isfocused in a nonlinear medium at the shortest distance,thereby determining the beginning of the filament.14

In problems of the propagation of directed laser radia-tion, there is a specific coordinate along which propagationoccurs. This coordinate is the evolution coordinate; i.e., itvaries in only one positive direction, and the processes ofscattering, diffraction, and nonlinear-optical interactionevolve as this coordinate varies. The existence of the evolu-tion coordinate makes it possible to construct stratified mod-els of the propagation, in which radiation successively passesthrough the layers of a continuous medium one after theother in the forward direction. Approximate models of thepropagation can be used for an individual layer in this case,and this significantly simplifies the analysis as a whole. Astratified model, or a method by which things are split up interms of physical factors, makes it possible to use the mostefficient numerical methods and algorithms to analyze theindividual processes �in our case, scattering, diffraction, andnonlinear interaction with an air medium.

The main idea of the stratified model for the problemunder consideration is that the aerosol medium is representedas a series of layers of width �z, whose particles are concen-trated in flat aerosol screens �Fig. 1�. Free diffraction andnonlinear interaction of the radiation with the medium occursbetween the screens. Using the concept of screens, the strati-fication model can be represented as a chain in which a pairof flat screens, the nonlinear and aerosol screens, are locatedsuccessively one behind the other at a spacing of �z alongthe direction in which the pulse propagates. In our model, thenonlinear screen is placed close to the aerosol screen, and theproblem of diffraction in a linear medium without particles is

FIG. 1. Stratification model. All the particles of the water aerosol are con-centrated in the aerosol screens—layers of finite width dz, where coherentscattering occurs. The nonlinear-optical interaction is simulated by the non-linear screens. L is the transverse size of the test sample, and �z is thedistance between adjacent aerosol screens.

773E. P. Kachan and V. O. Militsin

solved on the interval �z between the aerosol and nonlinearscreens.

The aerosol screen in the aerodisperse medium has afinite thickness dz and consists of two parallel planes. Thefirst plane includes the ensemble of water particles at whichthe field is scattered; the radiation scattered by the dropsinterferes with the unperturbed field at the next plane. Theparticles are located randomly at the nodes of a network withuniform distribution. The problem of the diffraction of scat-tered radiation is solved inside the aerosol screen for eachdrop by means of the Kirchhoff integral:

E�r,z� = − i/�� SD

�exp�− i��,�� − 1�

�exp�− ikl�,,r,z��E0/l�,,r,z�dd , �3�

where E0 is the incident field, � and k are the wavelength andwave number, and l� , ,r ,z� is the distance from point� ,�, which lies in the diametrical plane of particle z=0, tothe observation point �r ,z�. Parameter �� ,�, equal to thephase lag that the ray undergoes when it passes through aparticle at the point with coordinates � ,�, is determined inaccordance with the anomalous diffraction method.15 The in-tegral is taken along the plane of cross section SD of thespherical particle.

The following parameters of the calculational network inthe plane of the cross section were used in the numericalexperiments: the step �x /a0=�y /a0=0.004, the size of theregion L /a0=8, and the number of nodes 2048�2048. Thedistance between the aerosol screens is �z=15 cm. Scatter-ing causes angular deviation of wave vector k and conse-quently causes field components to appear that diverge fromthe axis in a plane perpendicular to the propagation direction.In a numerical experiment, these components are reflectedfrom the boundaries of the calculational region. To suppressthe effects of the reflection of radiation from the boundariesof the numerical lattice, an absorption coefficient thatsmoothes the field at the boundary to zero is introduced intothe near-boundary region of the network. Such absorptioneliminates the scattered field from the region of consider-ation and is thereby responsible for reducing the energy ofthe field during scattering.

AN INDIVIDUAL PARTICLE

To analyze how an aerosol influences the generation offilaments, let us consider the model problem of the stationaryself-focusing of a beam in a Kerr medium when it is scat-tered at an individual drop located on its axis. The radius ofthe drop in this case was varied from 2 to 15 �m. The radiusof the beam was a0=2.5 mm, and its power exceeded thecritical self-focusing power in air Pcr=6 GW by a factor of100.14 The initial peak intensity in a beam of this power isI0=3.6�1012 W/cm2.

As an example, Fig. 2a shows the intensity profile in thebeam at a distance of 2 mm behind a drop with radius r=2and 15 �m. It can be seen that the maximum intensities inthe distribution are significantly greater behind the large dropthan when scattering occurs at the small drop. Such a size


dependence of the intensity distribution behind a drop willintroduce significant differences in the process of generatinga filament. This confirms the analysis of the variation withdistance of the maximum intensity of a beam scattered by adrop �Fig. 2b�. It can be seen that the peak intensity in thebeam tends to increase as it approaches the nonlinear focus.Therefore, the nonlinear focus length zf can be determinedwith satisfactory accuracy from the distance to the plane inwhich the intensity increases by a factor of 10–100 by com-parison with the initial value I0.1� It follows from these re-sults that, when scattering occurs at the large particle, withradius 15 �m, the self-focusing distance zf of the beam ap-preciably decreases by comparison with the case of a nontur-bid medium, whereas zf increases when scattering occurs atthe small particle.

It can be concluded that the large scattering angle atsmall particles causes a powerful beam to be attenuated, andthe filaments are generated later, whereas the coherent scat-tering at large particles results in the appearance of intensitymaxima, which accelerates the generation of filaments.

FIG. 2. Scattering at a drop with radius r=15 �1� and 2 �m �2�, located onthe axis of a beam with radius a0=2.5 mm, and in the absence of drops �3�.The intensity profile I in the near-axis region of the beam at a distance of2 mm behind the drop �a�. Maximum intensity Imax vs distance z with self-focusing of a beam having power P0=100Pcr, scattered at the drop �b�. I0 isthe peak intensity in the initial beam.


A MONODISPERSE MEDIUM

To investigate how the particle size affects the genera-tion of filaments, let us consider the propagation of a pow-erful laser pulse in monodisperse aerosols that differ in par-ticle size. The particle concentration n in the media wasidentical and equalled 100 cm−3. Each aerosol screen con-tained 4000 particles on the average. A collimated Gaussianbeam with radius a0=2.5 mm was considered whose powerP0 was a factor of 100 greater than the critical self-focusingpower Pcr. The wavelength is �=0.8 �m.

In a disperse medium, perturbations are induced that ap-pear when there is scattering on a large number of randomlylocated particles, and the generation of filaments is substan-tially stochastic. Figure 3 shows the variations with distanceof the intensity distribution in the central cross section of thepulse, obtained in individual implementations of a dispersemedium. This figure also shows for comparison the intensityvariation when there is self-focusing of the beam in the ab-sence of particles. It can be seen that, when a pulse propa-gates in a medium with large particles, with a radius of r=15 �m, intensity perturbations �z=1 m� appear in its crosssection �Fig. 3a�. This increases the induced maxima in thenear-axis region of the pulse as a consequence of the modu-lation instability of a powerful light field in a medium withcubic nonlinearity.16 Several intensity peaks randomly lo-cated in the cross section are formed at a distance of z1.76 m. Some of them become filament-generation cen-ters. As a result, filaments are generated earlier than in amedium without particles �Fig. 3c�.

Small particles �2 �m� have virtually no effect on thefilamentation process �Fig. 3b�. In a finely dispersed aerosol,the magnitude and transverse size of the intensity perturba-tion in the cross section of a pulse is significantly smaller

FIG. 3. Intensity distribution I�x ,y� in the central cross section of a pulse at2 �m �b� and in the absence of particles �c�.


than in a medium with large particles. One filament2� isformed on the axis of the pulse, which is generated at thesame distance as in a medium without particles �Fig. 3c�.

The way that the size of the scattering particles affectsthe generation of the filaments depends on the peak pulsepower. Thus, in a pulse whose power P0 exceeds the criticalpower Pcr by a factor of 30, the generation of filaments in amonodisperse aerosol with particles up to 15 �m is virtuallyno different then the case of filamentation in a nonturbidmedium. This is confirmed by Fig. 4, which shows the de-pendence on distance z of the maximum intensity in the cen-tral layer of pulses with peak power P0=100Pcr and 30Pcr

when there is self-focusing in monodisperse media with par-ticle radii of r=2 and 15 �m.

us distances z for a monodisperse aerosol with particles of radius 15 �a� and

FIG. 4. Maximum intensity Imax vs distance z for a monodisperse aerosol. ris the radius of the drops, P0=100Pcr, 30Pcr. Radius of the drops 15 �1, 4�,2 �m �2, 5�, and in the absence of particles �3, 6�.

vario


The resulting dependence is physically explained by thefact that, in a pulse with peak power P0=100Pcr, when thereis scattering at particles with radius r=15 �m, inhomogene-ities that contain the critical self-focusing power appear, andthis causes spatial instability in the beam,16 initiates nonlin-ear focusing, and consequently generates a filament. Whenthe peak power of the pulse is reduced to P0=30Pcr or as theradius r of the scattering particle is reduced, the power arriv-ing at the perturbations that occur falls below Pcr, and thescattering at the aerosol particles has no effect on the gen-eration of filaments.

ATMOSPHERIC AEROSOL

Atmospheric clouds and fogs are polydisperse aerosolsin which the concentration of various particles is determinedby their size-distribution function ��r�.18 When filamentationof a pulse occurs in a polydisperse aerosol, there is compe-tition of the attenuation of its power due to scattering atsmall particles and the increase of the intensity perturbationsinitiated by large particles. To analyze this, let us considerthe filamentation of a pulse in an aerosol model that corre-sponds to the microphysical parameters of type-Cl atmo-spheric clouds.18 Function ��r� is assumed to be in the formof a generalized gamma distribution. The particle concentra-tion is n=100 cm−3, the most probable radius is rm=4 �m,and the attenuation factor in clouds is �=16.7 km−1. In mod-elling a polydisperse medium, the number of particles of acertain radius r, r+dr at the aerosol screen was given inaccordance with distribution function ��r�. The laser pulsehad a peak power P0=100Pcr. As earlier, the field in thecentral layer of the pulse was considered.

Figure 5 shows the intensity profile over the two “hot-test” points in the plane of the cross section at a distance of2 m in a polydisperse aerosol. It can be seen that, despite thechaotic arrangement of the scatterers—drops of varioussize—in the aerosol medium, intensity maxima that exceedthe initial peak intensity by a considerable factor are formedat a comparatively small distance because of Kerr self-focusing in air. It can be expected that, under conditions of

FIG. 5. Intensity profile I�l� of a laser beam propagating in a polydisperseaerosol, in a cross section that passes through the two hottest points, at adistance of 2 m.


strong nonlinearity, in a pulse with a high peak power, fila-ments randomly arranged in space are formed at these inten-sity maxima.

Statistical studies using the Monte Carlo method werecarried out to analyze how the particle concentration affectsthe generation of filaments in a polydisperse aerosol. Theensemble of random implementations of the intensity distri-

bution I�x ,y� in the plane of cross section z was determinedby solving Eq. �1� for a model with statistically independentsequences of aerosol screens. The mean numbers of fila-ments, Nf�, and distances to the start of the filament, zf�,were obtained from an ensemble made up of ten implemen-tations for various concentrations of particles in a polydis-perse aerosol �see Table I�. It can be seen that, as the particleconcentration increases, there is an insignificant increase ofthe number of filaments Nf� and a decrease of the distancezf� to the place at which they form.

CONCLUSIONS

The influence of the particles of an atmospheric aerosolon the generation of filaments in a high-intensity laser pulsedepends on their size and the power of the pulse.

Coherent scattering at large particles results in the ap-pearance of intensity maxima, which accelerate the genera-tion of filaments in the pulse, whereas the large angular scat-tering at small particles causes power attenuation, and thefilaments are generated later. The influence of large particleson the generation of filaments is substantial in high-intensitypulses, in which the power arriving at the diffraction maximawith coherent scattering exceeds the critical self-focusingpower.

The scattering in the polydisperse aerosol of atmosphericclouds stimulates the appearance of a multitude of randomfilaments in a powerful laser pulse. As the particle concen-tration increases, the mean number of filaments increases andthe distance to the site at which they form decreases.

The authors are very grateful to Professor V. P. Kandidovfor supervising the work and for inestimable help in the dis-cussion and preparation of this paper.

The authors thank the Russian Foundation for Basic Re-search �Grant No. 03-02-16939� for financial support.

a�E-mail: [email protected]�In practice, a computational experiment shows that determining the coor-

dinate of the nonlinear focus from the plane in which the intensity reaches1000 results in a difference of no more than 0.5%.

2�Note that the formation of one filament in a pulse of peak power P0�Pcr,for example P0=100Pcr, is an idealization. Under actual conditions, thereare always perturbations on the beam profile of the laser system or inho-

TABLE I. Mean number of filaments, Nf�, and mean distance to the sitewhere the filaments start, zf�, in a pulse with peak power P0=100Pcr whenit propagates in a polydisperse cloud with various particle concentrations n.


mogeneities of the medium, which initiate generation of chaotically lo-cated filaments.17

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how the particles of the atmospheric aerosol affect the generation of filaments in a laser beam

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