field guide to diffractive optics (spie field guide vol. fg21)

$: Field Guide to Diffractive Optics (SPIE Field Guide Vol. FG21)$
Field Guide to

Yakov G. Soskind

Field Guide to

Diffractive OpticsDiffractiveOptics

SPIE PRESS | Field Guide

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DiffractiveOptics

Field Guide to

Yakov G. Soskind

SPIE Field GuidesVolume FG21

John E. Greivenkamp, Series Editor

Bellingham, Washington USA

Library of Congress Cataloging-in-Publication Data Soskind, Yakov. Field guide to diffractive optics / Yakov Soskind. p. cm. -- (The field guide series ; 21) Includes bibliographical references and index. ISBN 978-0-8194-8690-5 1. Diffraction. 2. Optics. I. Title. QC415.S67 2011 535'.42--dc23 2011018209

Published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: +1.360. 676.3290 Fax: +1.360.647.1445 Email: [email protected] Web: http://spie.org Copyright © 2011 Society of Photo-Optical Instrumentation Engineers (SPIE) All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher. The content of this book reflects the work and thought of the author. Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. For the latest updates about this title, please visit the book’s page on our website. Printed in the United States of America. First printing

Introduction to the Series

Welcome to the SPIE Field Guides—a series of publicationswritten directly for the practicing engineer or scientist.Many textbooks and professional reference books coveroptical principles and techniques in depth. The aim ofthe SPIE Field Guides is to distill this information,providing readers with a handy desk or briefcase referencethat provides basic, essential information about opticalprinciples, techniques, or phenomena, including definitions anddescriptions, key equations, illustrations, application examples,design considerations, and additional resources. A significanteffort will be made to provide a consistent notation and stylebetween volumes in the series.

Each SPIE Field Guide addresses a major field of opticalscience and technology. The concept of these Field Guides is aformat-intensive presentation based on figures and equationssupplemented by concise explanations. In most cases, thismodular approach places a single topic on a page, and providesfull coverage of that topic on that page. Highlights, insights,and rules of thumb are displayed in sidebars to the maintext. The appendices at the end of each Field Guide provideadditional information such as related material outside themain scope of the volume, key mathematical relationships,and alternative methods. While complete in their coverage, theconcise presentation may not be appropriate for those new tothe field.

The SPIE Field Guides are intended to be living documents.The modular page-based presentation format allows themto be easily updated and expanded. We are interested inyour suggestions for new Field Guide topics as well as whatmaterial should be added to an individual volume to makethese Field Guides more useful to you. Please contact us [email protected].

John E. Greivenkamp, Series EditorOptical Sciences Center

The University of Arizona

Field Guide to Diffractive Optics

The Field Guide Series

Keep information at your fingertips with all of the titles in theField Guide Series:

Field Guide to

Adaptive Optics, Tyson & Frazier

Atmospheric Optics, Andrews

Binoculars and Scopes, Yoder, Jr. & Vukobratovich

Diffractive Optics, Soskind

Geometrical Optics, Greivenkamp

Illumination, Arecchi, Messadi, & Koshel

Infrared Systems, Detectors, and FPAs, Second Edition,Daniels

Interferometric Optical Testing, Goodwin & Wyant

Laser Fiber Technology, Paschotta

Laser Pulse Generation, Paschotta

Lasers, Paschotta

Microscopy, Tkaczyk

Optical Fabrication, Williamson

Optical Lithography, Mack

Optical Thin Films, Willey

Polarization, Collett

Special Functions for Engineers, Andrews

Spectroscopy, Ball

Visual and Ophthalmic Optics, Schwiegerling



Recent advancements in microfabrication technologies as wellas the development of powerful simulation tools have led to asignificant expansion of diffractive optics and the commercialavailability of cost-effective diffractive optical components.Instrument developers can choose from a broad range ofdiffractive optical elements to complement refractive andreflective components in achieving a desired control of theoptical field.

Material required for understanding the diffractive phe-nomenon is widely dispersed throughout numerous literaturesources. This Field Guide offers scientists and engineers a com-prehensive reference in the field of diffractive optics. Collegestudents and photonics enthusiasts will broaden their knowl-edge and understanding of diffractive optics phenomena.

The primary objectives of this Field Guide are to familiarize thereader with operational principles and established terminologyin the field of diffractive optics, as well as to provide acomprehensive overview of the main types of diffractiveoptics components. An emphasis is placed on the qualitativeexplanation of the diffraction phenomenon by the use of fielddistributions and graphs, providing the basis for understandingthe fundamental relations and the important trends.

I would like to thank SPIE Press Manager Timothy Lamkinsand Series Editor John Greivenkamp for the opportunity towrite a Field Guide for one of the most fundamental physicaloptics phenomena, as well as SPIE Press Senior Editor DaraBurrows for her help.

My endless gratitude goes to my family: to my wife Eleanora,who had to bear additional duties during my work on this book,as well as to my children, Rose and Michael, who learned thematerial while helping with proofreading the manuscript.

Yakov G. SoskindAugust 2011


vii

Table of Contents

Glossary of Symbols xi

Diffraction Fundamentals 1The Diffraction Phenomenon 1Scalar Diffraction 2Paraxial Approximation 3

Fresnel Diffraction 4Fresnel Diffraction 4Apertures with Integer Number of Fresnel Zones 5Fresnel Zone Plates 6Fresnel Zone Plate Properties 7Fresnel Phase Plates 8Comparing Fresnel Plates and Ideal Lenses 9Efficiency of Fresnel Plates and Ideal Lenses 10Talbot Effect 11Fractional Talbot Distributions 12

Fraunhofer Diffraction 13Fraunhofer Diffraction 13Diffraction of Waves with Finite Sizes 14Diffraction on Ring-Shaped Apertures 15Energy Redistribution within Diffraction Rings 16Diffraction on Noncircular Apertures 17Rectangular and Diamond-Shaped Apertures 18

Apodized Apertures 19Apodized Apertures 19Apodized Apertures with Central Obscuration 20Field Obstruction by an Opaque Semiplane 21Apodization with Serrated Edges 22Serrated Apertures as Apodizers 23

Diffraction by Multiple Apertures 24Diffraction by Multiple Apertures 24Effects of Aperture Spacing 25Aperture Fill Factor 26Aperiodically Spaced Apertures 27

Resolution Limit in Optical Instruments 28Resolution Limit in Optical Instruments 28Superresolution Phenomenon 29


viii

Table of Contents

Superresolution with Two-Zone Phase Masks 30Point Spread Function Engineering 31Adjusting Diffraction-Ring Intensity 32Amplitude and Phase Filter Comparison 33Vortex Phase Masks 34Combining Amplitude and Vortex Phase Masks 35

Diffractive Components 36Diffraction Gratings 36Volume Bragg Gratings 37Polarization Dependency of Volume Bragg Gratings 38One-Dimensional Surface-Relief Gratings 39GRISM Elements 40Two-Dimensional Diffractive Structures 41Holographic Diffusers 42Multispot Beam Generators 43Design of Fan-Out Elements 44Diffractive Beam-Shaping Components 45Digital Diffractive Optics 46Three-Dimensional Diffractive Structures 47

Grating Properties 48Grating Equation 48Grating Properties 49Free Spectral Range and Resolution 50Grating Anomalies 51Polarization Dependency of Grating Anomalies 52Gratings as Angular Switches 53Gratings as Optical Filters 54Gratings as Polarizing Components 55

Blazing Condition 56Blazing Condition 56Blazed Angle Calculation 57Optimum Blazed Profile Height 58

Scalar Diffraction Theory of a Grating 60Scalar Diffraction Theory of a Grating 60Diffraction Efficiency 61Blaze Profile Approximation 62


ix

Table of Contents

Extended Scalar Diffraction Theory 63Extended Scalar Diffraction Theory 63Duty Cycle and Ghost Orders 64Extended Scalar versus Rigorous Analysis 65

Gratings with Subwavelength Structures 66Gratings with Subwavelength Structures 66Blazed Binary Gratings 67Relative Feature Size in the Resonant Domain 68Effective Medium Theory 69Scalar Diffraction Limitations and Rigorous Theory 70

Rigorous Analysis of Transmission Gratings 71Analysis of Blazed Transmission Gratings 71Polarization Dependency and Peak Efficiencies 72Peak Efficiency of Blazed Profiles 73Wavelength Dependency of Efficiency 74Efficiency Changes with Incident Angle 75Diffraction Efficiency for Small Feature Sizes 76

Polychromatic Diffraction Efficiency 77Polychromatic Diffraction Efficiency 77Monolithic Grating Doublet 78Spaced Grating Doublet 79Monolithic Grating Doublet with Two Profiles 80Diffractive and Refractive Doublets: Comparison 81

Efficiency of Spaced Grating Doublets 82Efficiency of Spaced Grating Doublets 82Sensitivity to Fabrication Errors 83Facet Width and Polarization Dependency 84Sensitivity to Axial Components Spacing 85Frequency Comb Formation 86

Diffractive Components with Axial Symmetry 87Diffractive Components with Axial Symmetry 87Diffractive Lens Surfaces 88Diffractive Kinoforms 89Binary Diffractive Lenses 90Optical Power of a Diffractive Lens Surface 91Diffractive Surfaces as Phase Elements 92Stepped Diffractive Surfaces 93Properties of Stepped Diffractive Surfaces 94


x

Table of Contents

Multi-order Diffractive Lenses 95Diffractive Lens Doublets 96

Diffractive Surfaces in Optical Systems 97Diffractive Lens Surfaces in Optical Systems 97Achromatic Hybrid Structures 98Opto-thermal Properties of Optical Components 99Athermalization with Diffractive Components 100Athermalization with SDSs 101

Appendix: Diffractive Raytrace 102

Equation Summary 105

Bibliography 111

Index 114


xi

Glossary of Symbols and Acronyms

ASMA aperiodically spaced multiple aperturesB base length of a PRISMCR incident wave obliquity factorCS diffracted wave obliquity factorCGH computer-generated hologramd diameter of central obscurationdB Bragg plane spacingdg grating period or groove spacingdi step width of ith zoneD aperture diameter or lateral sizeDA Airy disk diameterDn material dispersionD0 lens clear aperture diameterDC duty cycleDDO digital diffractive opticsDLS diffractive lens surfacee aperture obscurationE(ρ,ϕ) complex electric field in polar coordinatesE⊥ electric field normal to the grating groovesE∥ electric field parallel to the grating groovesf focal length of a lensf D0 nominal focal length of a diffractive surface

FDTD finite difference time domainFWHM full width at half maximumFPP Fresnel phase plateFZP Fresnel zone plateh grating profile depth or heighthi step height of ith zonehm profile height of a multi-order diffractive lenshopt optimum grating profile depth (height)hSDS step height of SDSHOE holographic optical element→i unit vector codirectional with x axisI(r) radial intensity distributionIR infrared→j unit vector codirectional with y axis

J0(ρ) Bessel function of the first kind of the zero orderJ1(ρ) Bessel function of the first kind of the first order


xii


→k unit vector codirectional with z axisk(x, y) wave vector of the propagating wavefrontk0 wavenumberL observation distanceLm

T Talbot distance of order mLED light-emitting diodem diffraction ordern refractive index of optical materialn1 refractive index before optical interfacen2 refractive index after optical interfacend refractive index of diffraction grating layernp refractive index of prism materialn⊥ effective index for the electric field E⊥n∥ effective index for the electric field E∥N number of binary levelsNF Fresnel zone numberNg number of grating (groove) facetsNk number of kinoform zonesOPD optical path differencePSF point spread function→q vector orthogonal to the grating plane of

symmetry at the point of intersectionQ grating “thickness” parameter→r vector normal to the grating surface at the

incoming ray intersection point→r12 vector connecting points in two lateral planesR0 substrate radius of curvature→S propagation direction vector before

diffractive surface→S′ propagation direction vector after

diffractive surfaceSDS stepped diffractive surfaceSPDT single point diamond turningt axial spacing of a grating doubletT effective thickness of volume phase gratingtb thickness of a binary lens levelTg geometrical transmission pattern of a facetTE transverse electric


xiii


TM transverse magneticU(x, y, z) complex field amplitudeUV ultravioletVBG volume Bragg gratingVLSI very large-scale integrationW(x, y) propagating wavefrontWg grating width→z vector normal to the two reference planesα coefficient of thermal expansionαB angle of the incident light after refraction into the

volume phase mediumαd deflection angleβ diffracted angle inside the volume phase mediumγ angle between the Bragg planes and the incident

lightδ minimum feature size of the diffractive

componentε grating minor (secondary) facet angleζ fill factor of radiationη normalized diffraction efficiencyηm diffraction efficiency in mth diffraction orderηM diffraction efficiency of a grating with M facetsηP diffraction efficiency of P-polarized lightηS diffraction efficiency of S-polarized lightθd diffraction angleθi angle of incidenceθm mth order diffraction angleθL

m mth order diffraction angle in Littrow mountθλlm mth order diffraction angle of the wavelength λl

θλsm mth order diffraction angle of the wavelength λs

θϕ incidence angle with respect to the grating facetλ wavelength of lightλb blazing wavelengthλL

b blazing wavelength in Littrow configuration(mount)

λs the shortest wavelength within the spectral rangeλl the longest wavelength within the spectral rangeλ0 design wavelength


xiv


ξ opto-thermal coefficient of a surfaceρ radial coordinateυ volume phase grating parameterϕ grating primary facet angleϕb grating facet blaze angleϕi facet angle of ith kinoform zoneϕp grating passive facet angleϕmax maximum value of facet blazed angleφ optical path differenceΦAH optical power of an achromatic hybridΦD optical power of a diffractive surface or lensΦH optical power of a hybrid surface or lensΦR optical power of a refractive surface or lensΦSDS optical power of SDSΦSDS

ef f effective optical power of SDS

ψ(r) radial phase profile of a diffractive surfaceψ(x, y) phase profile of a diffractive surface∆ fchr axial or longitudinal chromatic aberration∆Hchr lateral or transverse chromatic aberration∆λ spectral bandwidth∆λFSR free spectral range of a grating∆n refractive index modulationΛ grating parameter


Diffraction Fundamentals 1

The Diffraction Phenomenon

Diffraction is a fundamental wave phenomenon that explainsnumerous spatial radiation propagation effects that cannot beexplained by geomet-rical optics, includ-ing the “bending” oflight that leads tolight presence in geo-metrical shade be-hind opaque objects.It also describes thepropagation of waveswith a finite spatialextent.

Diffraction imposes fundamental limits to the resolution andto the power density at the focus of an optical system. Thediffraction phenomenon occurs over a broad range of theelectromagnetic spectrum, including ultraviolet (UV), visible,infrared (IR), and radio waves. Therefore, in this field guide theterms “light” and “radiation” will be used interchangeably.

The term “diffraction” originates from the Latin word diffrin-gere meaning “to break into pieces” and refers to wavefragmentation after interaction withobjects. The diffraction phenomenon isoften manifested by intensity ripples inthe propagating field due to the coherentsuperposition of the diffracted and non-diffracted fractions of the propagatingradiation. Diffraction has been studiedby several prominent physicists, includ-ing Isaac Newton, Augustin-Jean Fres-nel, Christiaan Huygens, Thomas Young,and Joseph von Fraunhofer.


2 Diffraction Fundamentals

Scalar Diffraction

From experimental observations it is known that longerwavelengths are diffracted at larger angles, and that tighterfocal spots are obtained from larger-aperture lenses. Thishas led to the formulation of the fundamental relation for adiffraction angle θd being proportional to the wavelength oflight λ, and inversely proportional to the lateral dimension Dof the propagating wave:

θd ∝ λ/D

Solutions to a diffraction problem consider the spatial evolutionof finite-sized waves and waves whose propagation wasdisrupted by amplitude or phase objects. Rigorous solutionsto diffraction problems satisfy Maxwell’s equations andthe appropriate boundary conditions. A simpler approach isbased on the Huygens-Fresnel principle, which defines thefoundation for scalar diffraction theory.

Scalar diffraction theory assumes that the propagating fieldcan be treated as a scalar field. The propagation of a fielddescribed by its complex amplitude U (x, y, z) in free spacefrom the object plane (z = 0) to the observation plane is governedby the Helmholtz equation:

∇2U (x, y, z)+k20U (x, y, z)= 0

in which k0 = |k0| = 2π/λ0 isthe free space wave number.According to Huygens’ principle, thepropagating field at the apertureis considered as a superpositionof several secondary point sourceswith spherical wavefronts. Fresnelstated that intensity distributionafter the aperture is the result ofinterferometric interaction betweenthe Huygens point sources.


Diffraction Fundamentals 3

Paraxial Approximation

Monochromatic field dis-tribution at the outputplane U(x2, y2) in accord-ance with the Huygens-Fresnel principle is cal-culated based on the fieldat the input plane U(x1,y1, z1) employing Kirch-hoff’s diffraction inte-gral:

U (x2, y2, z2)∝Ï

1+cos(z,r12)2iλr12

exp(ikr12)U (x1, y1, z1)dx1d y1

where z is the vector normal to the two reference planes, andr12 is a vector connecting the points (x1, y1, z1) and (x2, y2, z2) inthe two planes.

The paraxial approximation assumes small propagationangles. In the case of the paraxial approximation, cos(z, r12)∼= 1,and |r12| ∼= z12 yield the Fresnel diffraction integral:

U (x2, y2, z2) ∝ exp(ikz12)iλz12

Ïexp

{ik

2z12

[(x2 − x1)2 + (y2 − y1)2

]}×U (x1, y1, z1)dx1d y1

The sufficient condition for the above Fresnel approximationis defined as

(z2 − z1)3 Àπ/4λ[(x2 − x1)3 + (y2 − y1)3

]2max

Large propagation distances of z12 À π(x2

2 + y22)

/λ0 lead toFraunhofer approximations as a Fourier transform of theobject field, also called Fraunhofer diffraction or far-fielddistribution:

U (x2, y2, z2)∝ exp(ikz12)iλz12

exp

[ik

(x1

2 + y12)

2z12

]×Ï

exp[

ik (x2x1 + y2 y1)2z12

]U (x1, y1, z1)dx1d y1


4 Fresnel Diffraction

Fresnel Diffraction

The Fresnel diffraction phenomenon is observed at finitedistances from the objects that interact with the propagatingfield. For fields with circular symmetry, the on-axis intensitydistribution depends on the number of Fresnel zonescontained in the propagating field, as observed from the givenon-axis observation point.

The Fresnel zone number NF contained within the emittingaperture as viewed from the observation point at distance Lz isdefined as a number of half waves contained in the optical pathdifference between the outer and inner margins of the emittingstructure with respective radii rm and r0:

NF = 2λ

(√Lz

2 + rm2 −

√Lz

2 + r02)∼= rm

2 − r02

λLz

The number of Fresnel zones in an unobstructed circularemitting aperture (r0 = 0) with outer diameter D is

NF∼= D2/4λLz

The outer diameter D of a circular unobstructed emittingaperture (r0 = 0) containing NF Fresnel zones, as well as theinner diameter for a Fresnel zone with respective index (NF +1)for the observation point at a distance Lz is

D =√

NFλ (4Lz +NFλ)∼=√

4NFλLz


Fresnel Diffraction 5

Apertures with Integer Number of Fresnel Zones

The axial distance from an unobstructed circular emittingaperture with the outer diameter D containing NF Fresnelzones is found by

Lz =[D2 − (NFλ)2

]/4NFλ

Fresnel diffraction patterns produced by emitting apertureswith an odd number of zones exhibit characteristic on-axismaxima:

Fresnel diffraction patterns produced by emitting apertureswith an even number of zones exhibit characteristic on-axisminima:

In the presence of a central obscuration, the distance tothe observation point depends on the number of Fresnel zonescontained in the emitting aperture:



Fresnel Zone Plates

A Fresnel zone plate (FZP) represents an amplitude maskthat consists of alternating opaque and transparent rings. Eachring size corresponds to a Fresnel zone as defined by theobservation point. FZPs are often employed in lieu of lenses toconcentrate the propagating field into a tight on-axis spot.

Increasing the number of zones progressively reduces the width,as well as increases the peak intensity and the total powercontained in the central peak of the diffraction pattern.



Fresnel Zone Plate Properties

The increase in peak on-axis intensity for FZPs occurs for everytwo consecutive Fresnel numbers NF .

Zone plates with an equal number of transparent zones and thesame aperture size produce an on-axis distribution with equalintensity and different width.

For the number of transparent FZP zones exceeding 5 andthe total number of zones NF over 11, the width value of theintensity distribution stabilizes around the Airy distributionwidth produced by an ideal lens:



Fresnel Phase Plates

A Fresnel phase plate (FPP) represents a phase mask thatconsists of transparent ring-shaped zoneswith alternating phase shifts by half awavelength. Each ring size correspondsto a Fresnel zone as defined from theobservation point. The zone sizes of FPPsand FZPs are identical. FPPs may beemployed in lieu of lenses to concentratethe propagating field into a tight on-axisspot.

Increasing the number of zones progressively reduces the widthand increases the peak intensity and the total power containedin the central peak of the pattern.



Comparing Fresnel Plates and Ideal Lenses

When the number of zones in the Fresnel plates exceeds 10,the relative shape of the central peak in the normalized fielddistributions of the Fresnel plates and an equivalent ideal lensproducing Airy distribution are comparable with each other, asshown for the case of Fresnel number NF = 12:

The main difference between the three distributions is in thefraction of the radiation contained outside the central peak.This is shown in logarithmic scale for the above case of NF = 12over a broader radial distance range:



Efficiency of Fresnel Plates and Ideal Lenses

FPPs concentrate significantly more energy in the centralpeak than do amplitude FZPs containing the same number ofzones. The improvement in diffraction efficiency comes withincreased fabrication complexity of the phase plates. At thesame time, FPPs are less efficient than focusing lenses withequal apertures and F-numbers. In the case of FZPs and FPPs,the light outside the central peak is spread over a broad area:

The central peaks of FZP, FPP, and an ideal lens contain 7.8%,34.1%, and 83.8% of the total radiation power propagatingthrough the aperture, respectively.

The differences between Fresnel plates and an ideal lens aresummarized in the following table:



Talbot Effect

The Talbot effect, often called Talbot imaging, occurs whena periodic array of apertures is illuminated with a coherentsource. The Talbot effect is observed in the Fresnel domain andwas discovered in 1836 by Henry Talbot.

Observation planes that contain Talbot images are calledTalbot planes. An array of apertures with spacing d producesTalbot images of integer order m at Talbot distances Lm

T ,defined as Lm

T = 2md2/λ.

The figure shows the initial near-field array (left), the phase(center), and intensity distribution (right) of an image locatedat a Talbot distance of L1

T = 2d2/λ.

Fractional Talbot distributions are produced at particularfractional Talbot distances.

Talbot half distributions with intensity maxima laterallyshifted by d/2 from the original array are observed at thedistances

L0.5mT = Lm

T −d2/λ= 2(m−0.5)d2/λ

The figure below shows the initial near-field array (left), thephase (center), and the intensity distribution (right) of an imageat a half Talbot distance L0.5

T = d2/λ:



Fractional Talbot Distributions

Fractional Talbot distributions with higher spacing fre-quencies in the image planes can be produced. Two sets ofdouble-frequency distributions with spacing of d/2 are producedat quarter distances:

L0.25T = L0.5

T ±d2/2λ.

The first figure shows the phase (center graph) and intensitydistribution (right graph) of a double-frequency image with d/2spacing at a quarter Talbot distance L0.25

T = d2/2λ.

The second figure shows a quadruple-frequency distribution atthe Talbot distance L0.125

T = d2/4λ with d/4 intensity maximaspacing.

In practical applications, the finite-aperture size and spacingin the array leads to spatial degradation of the Talbot images,which is especially noticeable at the margins of the fielddistributions.


Fraunhofer Diffraction 13

Fraunhofer Diffraction

Far-field distribution can be conveniently observed in thevicinity of the focal plane of a lens.

The lens transfer function is defined as the following phasefactor:

U (x1, y1, z1)l = exp[−i

k2 f

(x1

2 + y12)]

The field in the back focal plane of a lens is found bysubstituting the above phase factor into the Fresnel diffractionintegral:

U (x2, y2, f )∝ exp(ik f )iλ f

exp[

ik

2 f

(x2

2 + y22)]×Ï

exp[− ik (x2x1 + y2 y1)

2 f

]U (x1, y1, z1)dx1d y1

The shape and size of the Fraunhofer diffraction patternsdepend on the transfer function of a lens and are used in opticalmetrology for alignment purposes, as well as to identify lensimperfections. During a star test, an image of a source locatedat infinity, such as a star, produces a point spread function(PSF) in the focal plane of a lens.

The PSF size and shape from a well-corrected lens aredominated by diffraction effects, and the lens performance iscalled diffraction limited. The PSF produced by a diffraction-limited lens is called an Airy pattern.


14 Fraunhofer Diffraction

Diffraction of Waves with Finite Sizes

Propagating wavefronts with finite lateral dimensions produceFraunhofer diffraction patterns that are observed in the farfield and depend on the pattern shape and size. The amplitudeand phase of the field in the Fraunhofer zone are defined by aFourier transform of the propagating field.

One of the most common Fraunhofer diffraction patterns isproduced by circular apertures. The field distribution is calledan Airy pattern after George Airy, who was the first toanalytically define the intensity distribution.

A plane wave with diameter D, intensity I0, and wavelength λ

produces an Airy pattern in the focal plane of a lens with focallength f , whose radial intensity is

I (r)= I0

(πD2

2λ f

)2 J1

(πDλ f r

)(πDλ f r

)2

Parameter r is the radial coordinate, and the peak value ofthe above distribution I (0) = I0

(πD2/4λ f

)2 is found on axis.The near-field aperture distribution and the corresponding Airypattern are shown below:

The central disk of the pattern, the Airy disk, has a size ofDA

∼= 2.44λ f /D and contains about 84% of the total patternenergy. A fraction of the total energy contained within a circleof radius ρ=πDr/λ f is calculated as

E (ρ)= 1− [J0 (ρ)]2 − [J1 (ρ)]2

where J0 (ρ) and J1 (ρ) are Bessel functions of the first kind, ofthe zero and the first order, respectively.



Diffraction on Ring-Shaped Apertures

Central beam obscuration reduces the width of the centrallobe of the diffracted pattern and may be employed to improvethe resolution of an optical system. For an aperture diameterD and the central beam obscuration diameter d, the apertureobscuration value can be defined as e = d/D.

The graph below shows changes in the total field power, therelative size of the central core, and the relative fraction of thetotal power diffracted outside the central core.

The reduction in power contained in the central core of thediffracted pattern significantly outpaces the reduction in thecentral core size.

For example, when the obscuration is e = 0.7, the fraction of thetotal incident energy in the diffraction rings outside the centralcore is 73.3%, and the width of the central lobe is 0.73DA ,where DA was defined earlier as the Airy disk diameter for anunobstructed aperture.



Energy Redistribution within Diffraction Rings

The Fraunhofer diffraction pattern for a circular aperture withcentral obscuration e is defined analytically as

I (q)= I0

[(1− e2) qD

2

]2 [J1 (q)

q− e

J1 (eq)q

]2

With an increase in obscuration e, the peak intensity in thefirst diffraction ring increases, and the maxima shift toward thecenter:

Intensity in the higher-order diffraction rings depends on theobscuration e and is not monotonic. For obscurations e < 0.3,the peak intensity of the second ring is reduced, while thepeak intensity of the third ring is increased with an increasein obscuration:

For high obscuration values (e > 0.7), the peak intensity in thediffraction rings is reduced with the ring order.



Diffraction on Noncircular Apertures

For near-field elliptical distributions, the respective Fraun-hofer pattern is also elliptical, with the longer ellipse axis in thediffraction pattern orthogonal to the longer aperture axis in thenear field:

Deviation of the aperture shape from circular symmetrymanifests as a distortion of the Fraunhofer pattern and maybe effectively used during the inspection process.

The offset of the obscuration also causes Fraunhofer patterndistortions.

Decentering of the obscuration is manifested by contrastreduction between the central core and the first diffractionring.



Rectangular and Diamond-Shaped Apertures

The lateral size of the far-field diffraction pattern is inverselyproportional to the size of the near-field distribution. Adiffraction pattern produced in the focal plane of a lens bya rectangular aperture with lateral dimensions a and b isdefined as

U (x2, y2, f )= (ab)2 sinc2(ikax2

2 f

)sinc2

(ikay2

2 f

)

Radiation in the Fraunhofer pattern is spread in directionsnormal to the edges of the near-field apertures, as shown in thefollowing graphs:


Apodized Apertures 19

Apodized Apertures

Diffraction on a finite-sized aperture causes a fraction of thepropagating energy to spread outside the central core of theFraunhofer diffraction pattern into the outside rings or nodes.In many applications, this may also lead to adverse effectssuch as difficulties in detecting weak signals, channel crosstalk,changes in photodetector responsivity, etc.

Apodization is an important technique employed to reducethe amount of light diffracted into the diffraction rings ornodes. Soft-edge apertures with the amplitude transmissionfunction gradually changing in the vicinity of the edge from100% transmissive to completely opaque are commonly used asapodizing structures. The following figures show a reduction inthe number of diffraction rings with an increase in width of thetransition edge zone of the soft-edge aperture.

Soft-edge apertures with different transition widths:

Corresponding far-field patterns:


20 Apodized Apertures

Apodized Apertures with Central Obscuration

Apodized apertures with central obscuration significantlyreduce both the amount of light outside the central core and thecentral core diameter.

Apodized apertures Far-field patterns

An increase in central obscuration leads to a reduction in thecentral core size and an increase in the relative amount ofenergy in the diffraction rings. For a given obscuration valuee, the energy outside the central core of the soft-edge apodizedapertures is significantly lower than for the apodized shapes.



Field Obstruction by an Opaque Semiplane

Diffraction of a field obstructed by an opaque semiplane witha straight edge occurs in numerous devices, including beamprofilers, variable attenuators, and optical shutters. Replacingthe sharp hard edge with a soft apodized edge significantlyreduces diffractive energy spread orthogonal to the edge.

Apodizers with soft edges can significantly reduce the energyspread outside the central core into diffraction rings and nodes.


22 Apodized Apertures

Apodization with Serrated Edges

Serrated edges can be effectively used for suppressingdiffracted light in the vicinity of the central core of far-fielddiffraction patterns obstructed by opaque objects. Serratededges can be employed in high-power laser applications,whereas absorbing soft-edge apodizers can be easily damagedunder similar conditions.

The figures below show the influence of edge serration onFraunhofer diffraction patterns for apertures with 50% obstruc-tion. A diffraction pattern for an aperture with a straight edge(left column) is compared to the respective diffractive patternsproduced by two different serrated edges.

Near-field distributions

Straight edge

Far-field distributions

Saw-tooth serration

Sinusodial serration

The saw-tooth-shaped serrations produce the least amount oflight diffracted in the direction normal to the semi-apertureedge.



Serrated Apertures as Apodizers

Serrated apertures can also be employed to suppress thediffraction of light outside the central core of the far-fielddiffraction pattern. This technique is especially importantin high-power laser applications, when absorbing grayscaleapodizers with soft edges can no longer be employed.Serrations may take various shapes, and radial serrations aremost common.

The figures below present the field distributions for circularapertures with progressively increasing radial serration depths.

Near-field distributions

Far-field distributions

Two effects are noticeable. An increase in the serration depthleads to a significant reduction of light diffracted outside thecentral core. At the same time, the increase in the serrationdepth causes an increase in the diameter of the central core inthe Fraunhofer diffraction pattern.


24 Diffraction by Multiple Apertures

Diffraction by Multiple Apertures

Far-field patterns produced by diffraction on multiple aper-tures depend on the size, shape, and quantity of the apertures.

Fraunhofer diffraction by multiple rectangular slits


Diffraction by Multiple Apertures 25

Effects of Aperture Spacing

Effects of aperture spacing between multiple apertures areillustrated by observing diffraction on five identical rectangularapertures with width w and height h. The aperture centers arespaced a distance d from each other.

An increase in the aperture spacing d is associated with arespective increase in the number of maxima in the diffractionpattern. The spacing between the maxima and the lateralmaxima widths is inversely proportional to the aperturespacing.


26 Diffraction by Multiple Apertures

Aperture Fill Factor

The near-field aperture duty cycle, also known as a fill factorζ, is defined as the ratio of the aperture width w to the aperturespacing value d (ζ= w/d). The effects of the duty cycle changeson the shape of a diffraction pattern are shown for threerectangular apertures:

An increase in the duty cycle leads to a reduction in the numberof high-peak-intensity nodes, as well as an increase in theenergy concentration in the central diffraction spot.

In the limiting case of ζ= 1, the pattern becomes identical to anentirely filled single rectangular aperture with a width of 3wand a height h.


Diffraction by Multiple Apertures 27

Aperiodically Spaced Apertures

Aperiodically spaced apertures (ASAs) represent anensemble of apertures with a variable duty cycle. ASAs areemployed to suppress secondary radiation maxima in the farfield.

An exemplary ASA structure consisting of 10 apertures withconstant aperture width and gradually varying spacing as wellas the associated far-field distribution is shown below.

Respective field distributions from a periodic structure consist-ing of 10 equally spaced apertures:


28 Resolution Limit in Optical Instruments

Resolution Limit in Optical Instruments

An extended object can be considered a superposition ofindividual points with varying intensities across the object. Fora diffraction-limited lens, each object point produces a PSFin the image plane. Intensity distribution in the image planeproduced by incoherent illumination represents a convolutionof the object intensity and the PSF.

The Rayleigh resolution criterion is based on distinguishingtwo closely spaced point objects of equal intensity when the PSFmaximum of one object coincides with the first PSF minimumof the second object. The minimum separation between the tworesolvable incoherent sources at the focal plane of a diffraction-limited optical system based on the Rayleigh criterion can bedescribed as

dRmin = 1.22λ f /D

The Sparrow resolution criterion is based on an intensityminimum appearing in the joint intensity function between thetwo objects. It can be applied to visual observations based onthe high sensitivity of the human eye to intensity differences,and in the case of two objects with equal intensity, is defined as

dSmin = 0.95λ f /D

In the case of coherent illumination, the combined fielddistribution is defined based on amplitude addition of the PSFs.The resolution of two objects now depends on the phase delaybetween the objects. For the two in-phase objects (zero phasedelay) with equal intensity, the Rayleigh criterion no longerresolves the objects. The Sparrow criterion requires 1.5 timeslarger separation and is defined as

dSmin = 1.46λ f /D

For two objects with a phase delay of λ/2, the coherently addedpattern has the minimum intensity value between the objects,regardless of the separation d. Phase masks are commonlyemployed to improve resolution in applications using coherentillumination, such as projection lithography.


Resolution Limit in Optical Instruments 29

Superresolution Phenomenon

Resolution enhancements in the far field are commonlyachieved using pupil filters or pupil masks. The pupil filtersare located at the lens aperture stop, or at the entrance or exitpupil locations of the lens.

Lenses employing apertures with central obscuration reducethe width of the PSF central peak below the width of the centralpeak in the Airy pattern (also known as the Airy disk). PSFswith central peak width lower than the widths of the Airy discare referred to as superresolved PSFs.

Optical system employing a phase pupil filter

Amplitude masks, phase masks, and their combinations areemployed to produce superresolved PSFs. The idea of usingpupil phase masks with alternating phase delays between theneighboring ring zones to reduce the PSF width was proposedby Toraldo di Francia in 1952.

Reduction in the PSF central peak width is associated with areduction in the Strehl ratio, which is defined as the ratioof the PSF peak value to the PSF peak value of a diffraction-limited distribution. Reduction in the PSF central peak widthis also associated with diffraction of a significant fraction of thepropagating energy to the outside of the central peak and anincrease in the peak intensity of the diffraction rings.



Superresolution with Two-Zone Phase Masks

The simplest phase mask consists of two transparent circularzones with a relative phase delay.

PSF shapes produced by a two-zone pupil phase mask with arelative phase delay of π producing an optical path difference(OPD) of λ/2 are shown in the figure below for differentnormalized inner-zone radii R. Airy distribution correspondsto the inner-zone radius R = 0. The figure shows the tradeoffbetween the central peak width and the relative energycontained in the diffraction rings.

Reduction in the PSF central peak width is associated withreduction in the Strehl ratio, defined as the ratio of the PSFpeak intensity to the peak intensity of the Airy disc.



Point Spread Function Engineering

The intensity in the first diffraction ring reaches the intensityof the central peak when the zone radius R = 0.569. When therelative zone size reaches R ∼= 0.7, the central peak vanishes,and the PSF becomes ring shaped. For relative zone sizes over0.84, the PSF no longer has a depression in the center of thedistribution, and instead produces a flattened central peak.

Instrumentation fields that benefit from superresolved PSFsinclude confocal scanning microscopy and optical data storage.In the case of scanning confocal microscopy, a desirable increasein the axial resolution is achieved by producing PSFs withreduced axial extent as compared to diffraction-limited PSFs.

Resolution-enhancement techniques constitute a subset of moregeneral point spread function engineering techniques thatrepresent an active topic in contemporary optical research. Avariety of specially designed PSFs are used in both imaging andnonimaging instruments.

Vortex phase masks have been employed in solar corona-graphs, in high-resolution fluorescent depletion microscopy, andin optical tweezers for particle trapping and manipulation.

Another use of phase structures was found in extended depthof field imaging systems as well as for producing a special classof asymmetric Airy beams.



Adjusting Diffraction-Ring Intensity

It is often desirable to reduce the peak intensity in the PSFdiffraction rings. In the case of two-zone phase masks,the reduction in core width is inevitably associated with anintensity increase of the first diffraction ring.

Increasing the number of the phase mask zones providesadditional degrees of freedom in PSF design. A three-zone phasemask can significantly reduce the peak intensity in the firstdiffraction ring of a superresolved PSF.

PSF engineering can significantly reduce the peak intensitiesin the diffraction rings. For a three-zone phase mask with thesecond radius R2= 0.886 and Strehl ratio of 0.5, a 3× reductionin the secondary peak intensity values is achieved. The peakintensity in the diffraction rings is below 0.6% of the centralpeak intensity value.



Amplitude and Phase Filter Comparison

The changes in PSF shape over the inner zone radius occurmore rapidly in the case of phase pupil filters. The followingfigure presents the power contained in the PSF central core asa function of the inner zone radius:

When differences in the rate of change in the PSF centralcore power are accounted for, the PSF widths produced usingboth pupil amplitude filters and phase filters result incomparable performance:



Vortex Phase Masks

Vortex phase masks are extensively employed with coherentradiation to produce doughnut-shaped beams in opticaltweezers and in fluorescence depletion microscopy.Vortex masks are also employed as pupil phase filters to alterthe PSF of an optical system.

For an optical system with a uniformly illuminated vortex phasemask located at the pupil, the field distribution can be writtenas

E (ρ,φ)={

E0eimφ, when ρ≤ Rmax

0, when ρ> Rmax

where ρ is the radial coordinate, φ is the azimuthal coordinatein the transverse plane, m is the topological charge, Rmax is themaximum pupil radius, and E0 is the pupil amplitude.

The pupil phase profiles and the respective doughnut-shapedfar-field intensity distributions for topological charges m = 1and m = 2 are shown below:



Combining Amplitude and Vortex Phase Masks

The doughnut-shaped field size gradually increases with anincrease in the topological charge of the vortex phase mask:

The combination of a vortex phase mask with topological chargem = 2 and an elliptical amplitude mask at the pupil of anoptical system produces elongated superresolved PSFs. Thefigure below shows the transition from a doughnut-shaped PSFwith circular amplitude mask (ellipticity ε= 1) to an elongatedsuperresolved PSF (ε= 10):

The top, central, and bottom rows correspond respectively to theamplitude masks, combinations of the two masks, and the PSFsfor the ellipticities ε= 1, ε= 2, and ε= 10.


36 Diffractive Components

Diffraction Gratings

Diffraction gratings are periodic diffractive structures thatmodify the amplitude or phase of a propagating field. Lineargratings represent the simplest periodic diffractive structures.

Amplitude gratings are based on the amplitude modulationof the incident wavefront and are employed in spectral regionswhere nonabsorbing optical materials are not available. Theamplitude modulation is associated with transmission lossesintroduced by the grating.

Phase gratings are based on the phase modulation of theincident wavefront by introducing a periodic phase delay tothe individual portions of the propagating wavefront. Phasegratings are designed to work in transmission, reflection, orin a bidirectional manner.

Surface-relief phase gratings are based on wavefront-divisioninterference principles and introduce periodic phase delays tothe fractions of the incident wavefront due to periodic changesof the substrate thickness.

Reflective surface-relief phase grating withsinusoidal profile

Transmissive surface- relief phase grating with triangular profile

Reflective surface-relief phase grating with

triangular profile


Diffractive Components 37

Volume Bragg Gratings

Volume Bragg gratings(VBGs) are based on ampli-tude division interferenceand are designed to perform inreflection or in transmission.VBGs introduce a periodicphase delay to the propagatingwavefronts associated with theperiodic modulation ∆n of therefractive index of the grating material. VBG diffraction effici-encies of S-polarized (TE-polarized) and P-polarized (TM-polarized) light are calculated as

ηS = [sin(υ)]2

ηP = {sin[υcos(2γ)]}2

where the angle 2γ is between the incident light and diffractedlight inside the volume phase medium, and the parameter υ isdefined as υ= π∆nT

λp

CRCS.

T is the effective grating thickness, CR is the incident waveobliquity factor, and CS is the diffracted wave obliquity factor:

CR = cos(2γ)

CS = cos(2γ)− λ

ndBtan

(β−αB

2

)where dB is the Bragg plane spacing, αB is the incidence angleafter refraction into the phase medium, and β is the angle afterrefraction into the volume phase medium.



Polarization Dependency of Volume Bragg Gratings

VBGs can be designed to operate either as polarization-inde-pendent or polarizing components, depending on the relationbetween the peak diffraction efficiency of the S-polarizationand P-polarization states.

The polarization-independent operating condition is

cos(2γ)=−2p−12s−1

The S-polarizing operating condition is

cos(2γ)=− 2p2s−1

The P-polarizing operating condition is

cos(2γ)=−2p−12s

The refractive index modulation ∆n corresponding to theS-polarization peak is calculated as

∆n = λ (s−0.5)T

√cos(2γ)

[cos(2γ)− λ

ndBtan

(β−αB

2

)]The relation between the angles αB and β is found from thegrating equation

sin(αB)+sin(β)= λ

ndB

The peak efficiency maxima for the S- and P-polarization statesare satisfied when either of the following two possible conditionsis valid:

β= cos−1(

2p−12s−1

)−αB

β= 180−cos−1(

2p−12s−1

)−αB



One-Dimensional Surface-Relief Gratings

The evolution of diffraction grating technology has led to avariety of grating profiles and structures. The choice of aspecific profile or structure is governed by fabrication costs andperformance requirements.

Surface-relief gratings withtriangular grooves are form-ed on top of aluminum- orgold-coated substrates by a di-amond tool of a ruling engine.The grooves are defined by thegroove spacing dg and the facet angle ϕ. Alternatively, tri-angular grooves are fabricated by using a grayscale mask pho-toresist exposure and a subsequent transfer etching into thegrating substrate.

Lamellar gratings are com-posed of rectangular ridges ofwidth w and height h spaced ata distance dg from each other.They are well suited for fab-rication using well-establishedlithographic techniques. Lamel-lar grating structures oftenhave ridges that are compara-ble with the operational wave-length λ and are therefore designed using rigorous diffractiontechniques.

Sinusoidal grating profilesare usually etched into thesubstrate after being exposedto a pattern produced by two-beam interference.



GRISM Elements

The term GRISM refers to an inte-grated optical component that combines adiffraction grating and a prism in a singleelement. Direct-view GRISM spectrome-ters with constant dispersion combine thegrating and the prism dispersions whileproviding a cancellation of the dispersionslopes.

The GRISM can be designed to satisfythe zero-deflection condition and toavoid deflection of the propagating ra-diation. The zero-deflection conditionis found for the grating blazing con-dition when θi =ϕ. The deflection an-gle αd at the exit of a planar grating with the substrate refrac-tive index nd is

αd = sin−1[nd sin(ϕ)]−ϕThe optimum step height hopt of the grating facet for the blazedwavelength λb in the mth diffraction order is

hopt = mλb

(nd −1)

A zero-deflection GRISM is pro-duced when the grating interface isapplied to a surface of a prism witha properly defined vertex angle ϕp.The vertex angle of a prismϕp for azero-deflection GRISM is calculatedas

tan(ϕp

)= nsin(ϕ)−sin(ϕ)√(np

)2 − [nd sin(ϕ)]2 −cos(ϕ)



Two-Dimensional Diffractive Structures

Two-dimensional diffractive structures are composed oftwo-dimensional arrays of microstructures. The shape, size, andspacing may differ in the two lateral directions. Fabricationof two-dimensional gratings is performed using lithographictechniques.

Two-dimensional diffraction gratings areoften made with feature sizes smallerthan the operating wavelengths and arecalled photonic crystals or artificialdielectrics. The features may be in theform of cavities etched into the substrate orpillars elevated above the substrate level.

Rigorous diffraction techniques, such asfinite difference time domain (FDTD),

are commonly employed to simulate the performance ofphotonic crystals. The two most common applications oftwo-dimensional gratings include antireflection surface-reliefmicrostructures and microstructured surfaces for increasedlight-emitting diode (LED) light extraction and increasedphotovoltaic cell efficiency.

The figure shows an example of a two-dimensional diffractivestructure that has antireflection properties when the featuresize is less than the wave-length. An effective refrac-tive index increases gradu-ally from the index of air tothe index of the substrate.



Holographic Diffusers

Holographic diffusers (or diffractive homogenizers) areemployed to control light distribution in illumination and lasersystems. Diffusers represent far-field shaping componentsdesigned to transform coherent and incoherent radiation usingmultiple diffraction orders. Traditional opal glass or grounddiffusers are limited to Lambertian scatter. Holographicdiffusers allow substantial flexibility in controlling spatialillumination patterns, producing nonrotationally symmetricspatial shapes as well as desired angular distributions.

Diffusion angles can be selected within a broad range of0.5–80 deg. Circular, oval, square, and line-shaped patterns arecommon. More than 90% of the incident light can be directedinto the specified illumination pattern.

Holographic diffusers are cost-effectively replicated in highvolume using injection or compression molding or can beembossed onto surfaces of other optical components. A mastermold for a holographic diffuser is produced as a surface-relief structure by holographic recording with subsequentlithographic fabrication.

Illumination patterns produced by 10 LEDs

Illumination patterns produced by 10 LEDs with a holographic diffuser



Multispot Beam Generators

Multispot beam generators, also known as array beamgenerators or fan-out elements, are diffractive elementsthat split an incoming laser beam into a finite number ofbeams with a specific intensity distribution. They belong to theclass of diffractive far-field shaping components, or Fouriergratings. The field distribution in the angular space θx afterthe fan-out element is given by

U (θx)=N∑

n=1An exp(iφn)δ (θx −θn)

where An is the amplitude, φn is the phase, and θn is theangular coordinate of a one-dimensional spot array.

Far-field beam shapers are composed of multiple diffractivephase cells significantly smaller than the size of the laserbeam employed during multispot beam generation. The fan-outelements are relatively insensitive to the size and position ofthe laser beam at the diffractive element.

Due to the small size of the phase cells, the far-field conditionis satisfied starting from distances close to the multispotgenerator. The formed multispot pattern will scale in the farfield based on the divergence angles of the generated beams.

Multispot beam generators are used as image replicators, asbeam splitters or beam combiners, and as spatial multiplexorsin multichannel optical interconnects.



Design of Fan-Out Elements

The design of fan-out elements is based on multiple approaches,including Dammann gratings and multilevel binary andkinoform structures.

Dammann gratings represent binary phase grating struc-tures. Each grating period is divided into multiple segmentswith phase shifts of 0 and π. The fan-out elements basedon Dammann gratings are simple and require only a singlemask during the fabrication process. However, the diffractionefficiency of fan-out elements based on Dammann gratings isaround 80%. The figures below show the phase profile and fielddistribution of a three-beam fan-out element with transitionpoints at 0.47 and 0.70.

Kinoform structures with continuous profiles produce fan-out elements with the highest diffraction efficiencies. Fan-out elements designed to achieve uniformity of the resultingbeam intensities at <1% have efficiency over 92%. Diffractionefficiencies of 98–99% are achieved when the intensityuniformity requirement can be relaxed. The profile shape ofa kinoform fan-out element designed to split the propagatingbeam into nine beams is shown.



Diffractive Beam-Shaping Components

Diffractive beam-shaping components are employed tomodify the field distribution of coherent radiation. Diffractivenear-field shaping components introduce spatially varyingphase delay across the propagating laser beam and remap thebeam onto a plane of regard using a single diffractive order.Field distributions produced with the near-field diffractivebeam-shaping technique are sensitive to variations in phaseand intensity of the input beam. A phase corrector placed in theplane of the remapped distribution combined with a Fouriertransform lens is employed to extend the axial range of theshaped field along the propagation direction.

Far-field shaping components can be made as pixilatedphase structures. They are referred to as digital diffractiveoptical elements and can produce intricate radiation patterns.Local intensity variations due to interference effects, knownas speckle, are present in the field distributions generatedusing diffractive beam shapers. The average speckle size d atthe observation plane located a distance L from the diffuser isestimated to be

d = Lλ

D

where λ is the wavelength of the propagating radiation, and Dis the input beam diameter. Speckle reduction in the generatedpattern is achieved by employing input radiation with reducedcoherence and by translating the input beam across the diffusersurface, as well as by performing spatial filtering of the pattern.

The figures below show diffractive beam shaping of a Gaussianlaser beam into a square-shaped beam with uniform intensitydistribution.



Digital Diffractive Optics

Digital diffractive optical (DDO) elements representperiodic two-dimensional discrete phase structures that aredesigned using iterative procedures. Digital diffractive opticalelements are also called computer-generated hologramsand represent versatile structures, allowing for the creationof arbitrarily shaped field distributions. The phase profile ofa DDO is found by solving an inverse problem wherein theinput beam and the desired field distribution are used as inputparameters.

Compared to holographic optical elements, which representdiffractive structures with continuous changes in phase profileand are recorded through interference of at least two beams,DDOs provide significantly higher flexibility in definingcomplex diffraction patterns.

DDOs are made by fabrication of the designed diffractionpattern onto the element substrate. The patterns are oftentransferred into the substrate during lithographic fabricationusing a mask set. DDOs are widely used for pattern generationand beam shaping and can be cost-effectively replicated in highvolumes.

The figure below shows a phase-distribution fragment of a DDO(left) employed to produce a square-shaped Fraunhofer fielddistribution (right).



Three-Dimensional Diffractive Structures

Three-dimensional diffractive structures represent themost complex microstruc-tures that exhibit volu-metric anisotropy.

Photonic crystals andvolume Bragg gratings(VBGs) are produced bymultiple exposures to thesubstrate and are twoexamples of three-dimen-sional diffractive struc-tures. In both cases,diffraction is caused by theBragg phenomenon.

Three-dimensional diffractive structures present significantchallenges with respect to analysis and fabrication. Simulationtechniques developed for photonic crystals, such as band dia-grams, Bloch states, and Brillouin zones, are incompatiblewith ray-tracing techniques commonly used in optical design.Isofrequency diagrams, or wave vector diagrams, can beemployed only to define the number of propagating diffractionorders. Photonic crystal designs are concerned with confinementof the propagating field. The operating wavelength and the rel-ative feature size of the crystal, known as the lattice constant,are selected below the photonic bandgap, ensuring an evanes-cent nature of all diffraction orders in reflection and in trans-mission.

The volumetric anisotropy of photonic crystals leads to several“unusual” propagation phenomena, such as the superprismeffect, supercollimation, and negative refraction.


48 Grating Properties

Grating Equation

The grating equation defines the propagation directions ofradiation after interactionwith the grating structure.For a plane wave incidentonto the grating structureat an angle θi, the diffrac-tion angle θm is found fromthe following general equa-tion:

n2 sinθm = n1 sinθi +mλ/dg

where n1 and n2 define the refractive indices of the materialbefore and after the grating interface. When the diffractionorder is zero (m = 0), the grating equation reduces to the well-known Snell’s law. For a reflection grating located in air (n1 =−n2

∼= 1),

sinθm +sinθi = mλ/dg

When the grating structure is applied to one of the surfaces of aplane-parallel plate made of refractive material, the diffractionangle at the plate exit θm is

sinθm = sinθi +mλ/dg

Littrow mounting, also known asautocollimation, is a specific conditionin which the incidence and diffractionangles of a reflection grating are equal(θi = θL

m). The grating equation for auto-collimation is

sinθLm = mλ/2dg

The grating classical mount corresponds to the incident planebeing normal to the grating grooves. The conical mount occurswhen the incident plane is at an angle to the grating grooves sothat diffraction orders deviate from the incident plane, forminga cone.


Grating Properties 49

Grating Properties

The grazing incidence mountcorresponds to steep angles of in-cidence required to achieve higherresolution. Grazing mounts areoften used in autocollimation.

For a constant angle of incidence, the grating angulardispersion is inversely proportional to the cosine of thediffracted angle θm:

∂θm

∂λ= m

dgn2 cosθm

The angular dispersion of a blazed reflective grating in Littrowmounting is reduced by half. The dispersion is a function ofthe blaze angle ϕb and is no longer dependent on the order ofdiffraction m:

∂θLm

∂λ= m

2dg cosθLm

= tan(ϕb)

λLb

Linear dispersion at the focal plane of a focusing objectivewith a focal length f is the product of the focal length and theangular dispersion of the grating.

An alternative form of grating angular dispersion is obtainedusing the incidence and diffraction angles:

∂θm

∂λ= n2 sinθm −n1 sinθi

λn2 cosθm

When accounting for the blazing condition, the angulardispersion becomes

∂θm

∂λ= 1[

dgn1 cos(θi)m − λb

tan(ϕb)

]From the grating equation it follows that longerwavelengths are diffracted at larger angles.For the two wavelengths λs < λl , the relationbetween the respective diffraction angles isθλsm < θλl

m .



Free Spectral Range and Resolution

The free spectral range∆λFSR of a grating in a givendiffraction order m definesthe largest bandwidth thatdoes not overlap with thesame bandwidth in adjacentorders:

∆λFSR = λs

m= λl

m+1

For gratings operating in higher orders, the free spectral range∆λFSR is significantly reduced.

Grating resolution is proportional to the product of thediffraction order m and the total number of illuminated gratinggrooves Ng:

λ

dλ= |m|Ng

The resolution of a uniformly illuminated grating is propor-tional to the grating width Wg:

λ

dλ= |m| Wg

dg= Wg

λ|sinθm +sinθi|

For a given grating groove spacing dg, the resolution can beincreased either by increasing the order m, or by enlarging thegrating width Wg. The upper limit of the grating resolution isdefined as the number of half-wavelengths contained within thegrating width: (

λ

dλ

)max

= 2W g

λ

For a given diffraction order, the grating resolution is constantacross the working spectral range. A coarse grating with a fewgrooves designed to work in a high diffraction order may havethe same resolution as a fine grating with a large number ofgrooves working in a low diffraction order.



Grating Anomalies

Diffraction orders m satisfying the condition |sinθm| < 1are called propagating orders. When varying the angle ofincidence, some of the diffraction orders, called passing-offorders or cut-off orders, may propagate along the gratingsubstrate. This is called the threshold condition:∣∣∣∣ 1

n2

(n1 sinθi + mλ

dg

)∣∣∣∣= 1

Small changes in the angle of incidence lead to the anomaliesassociated with the appearance or disappearance of the cut-offorder and cause abrupt changes in the diffraction efficiency ofthe propagating orders. The disappearing orders that satisfy thecondition |sinθm| > 1 are called evanescent orders.

Grating anomalies, also known as Wood anomalies, manifestas rapid variations in efficiency that occur within either narrowspectral or angular intervals. The wavelengths for the cut-offorder are

λ= dm

[sgn(m)−sin(θi)]

The figure below shows the strong anomalous behavior of asinusoidal reflective grating at a 30-deg angle of incidencefor TM (S-polarized) light diffraction efficiency. The anomalyis caused by the emergence of the m = −3 order and thedisappearance of the order m = 1.



Polarization Dependency of Grating Anomalies

Rayleigh anomalies represent resonance phenomena due tothe simultaneous occurrence of positive and negative diffractiveorders propagating in opposite directions along the gratingsurface.

Grating anomalies have strong polarization dependencyor polarization anisotropy. For reflective gratings, theanomalies are most prominent in transverse-magnetic or TM-polarized light. TM-polarized light has the electric field vec-tor oriented parallel to the incident plane, also known as thetangential plane, and is therefore called P-polarized light.Anomalies for transverse-electric, or TE-polarized light, occur-ring when the electric vector is oriented in the sagittal plane(S-polarized) perpendicular to the incident plane, are observedin gratings with small groove spacing and deep grooves.

The first figure shows strong TM-polarized anomalies in thefirst diffraction order for a gold-coated grating at an incidenceangle of 20 deg.

Grating anomalies can be observed in any diffraction order. Thesecond figure shows 0th-order diffraction anomalies for a gold-coated grating at a 20-deg angle of incidence with two efficiency“notches” around 1.15 µm and 1.24 µm.



Gratings as Angular Switches

Binary phase gratings can be em-ployed to spatially redirect incoming ra-diation. The figure shows two binarygratings with a duty cycle of 0.5 later-ally shifted by half of the grating pe-riod with respect to each other. The grat-ing combination is equivalent to a plane-parallel plate, and the incoming radia-tion propagates unaltered through thegrating pair.

When the two gratings are relatively off-set in the lateral direction by a quarterof the grating period, the grating combi-nation becomes equivalent to a gratingwith triangular grooves. The incomingradiation is split into two beams at theexit of the grating pair.

Resonance phenomena can be effectively employed to performangular switching of radiation. The following figure presentsangular switching between the zero and first diffraction ordersof a reflective grating as a function of the incidence angle. Byrotating the grating, the output of TM-polarized light can berapidly switched between the two diffraction states.

A change in the grat-ing angular orienta-tion by 1 deg causesa change in the firstdiffraction order froman evanescent stateto 75% output effi-ciency.



Gratings as Optical Filters

The resonance phenomenon of grating anomalies can beeffectively employed to alter grating properties. One practicaloutcome of using grating anomalies that is not obvious fromgrating behavior is the design of efficient band-pass filters.

The following figure shows the efficiency graph for TM-pol-arized light reflected into the first diffraction order with afull width at half maximum (FWHM) bandwidth of 0.54 µmcentered at 1.6 µm.Reflection of TE-pol-arized light into thefirst diffraction orderis also shown for com-parison.

Resonance anomaliesin subwavelengthtransmission gratingsallow for the design ofsharp notch filters working in the 0th diffraction order. Thefollowing graph shows a grating structure producing a narrow-band etalon effect in reflection to block the transmission of alaser beam at 1.06 µm, while transmitting neighboring wave-lengths. The theoretical pass-band FWHM of the notch reso-nance is 0.14 nm.



Gratings as Polarizing Components

The design of reflective diffractive polarizers employsanomalous grating behavior. The figures below present apolarization grating performance designed at ∼1.4 µm.

The grating effectively reflects TM polarization into thediffraction order m = 1 and reflects TE polarization into thediffraction order m = 0.

The polarization extinction ratio for TE-polarized radiationexceeds 200, and for TM-polarized radiation is greater than1000.


56 Blazing Condition

Blazing Condition

Blazed gratings are composedof individual grooves that areshaped to concentrate the inci-dent radiation into the diffractionorder of interest.

The blazing condition occurswhen the propagation directionθm of the diffracted radiationcoincides with the propagation direction through the gratingmicrostructure defined by the laws of reflection or refraction.The blazing condition for a grating comprises grooves with ablazed facet angle ϕb is defined by a system of two equations:{

n1 sin(θi +ϕ)= n2 sin(θm +ϕb)n2 sin(θm)= n1 sin(θi)+mλ/dg

The angle of diffraction into the mth diffraction order θm can befound by

n2 cos(θm)= n1 cos(θi)−mλ/[dg tan(ϕb)]

The blazing wavelength λb is calculated as

λb = dg

m

(n2 sin

{sin−1

[n2

n1sin(θi +ϕb)

]−ϕb

}−n1 sin(θi)

)The angle ϕp is the passive facet angle. For reflectivegratings located in air, the blazing wavelength λb is found by

λb = 2dg

msin(ϕb)sin(θi +ϕb)

In Littrow mounting, the incident wave propagates at normalincidence to the grating facet, so that ϕb =−θi, and the blazingcondition becomes

λLb = 2dg

msin(ϕb)


Blazing Condition 57

Blazed Angle Calculation

The blazed facet angle at a given blazing wavelength λb, intoa diffraction order m, and with a groove spacing dg, is a functionof the angle of incidence θi:

tan(ϕb)=(mλb/dg

)n1 cos(θi)−

√(n2)2 − [

n1 sin(θi)+(mλb/dg

)]2

The following graph shows the blazed angle ϕb as a functionof the incidence angle θi for different diffraction orderspropagating from a medium with a lower refractive index into amedium with a higher refractive index.

When propagating from a medium with a higher refractiveindex into a medium with a lower refractive index, the absoluteblazed angle values ϕb as a function of the incident angle θiexhibit higher values:


58 Blazing Condition

Optimum Blazed Profile Height

The optical path differenceφ between the neighboring facetsof a blazed transmission grating produced as an interfacebetween two materials with indices of refraction n1 and n2 isdefined as

φ= dg [n2 sin(θm)−n1 sin(θi)]= mλb

The optimum profile height hopt of a blazed grating surfaceoperating in the mth diffraction order is derived as

hopt =∣∣∣∣mλb

/{√(n2)2 − [


)]2 −n1 cos(θi)}∣∣∣∣

The first graph shows the optimum profile height as a func-tion of the incidence angle for m = 1, λ = 0.5 µm, and threedifferent grating peri-ods when propagatingfrom a less dense op-tical medium into amedium with a higherdensity when n1 < n2.

The optimum profileheight as a function ofthe incidence angle form = 1, dg = 10 µm,and three different blazing wavelengths when n1 < n2 is shownin this graph:


Blazing Condition 59

Optimum Blazed Profile Height (cont.)

The first graph showsthe optimum profileheight hopt for m = 1,λ = 0.5 µm, and threedifferent grating peri-ods when propagatingfrom a higher to a lowerdensity optical medium,such that n1 > n2.

The optimum profileheight as a functionof the incidence anglefor m = 1, dg = 10µm, and three blazingwavelengths is shownin the second graph.

The optimum profileheight at a normalangle of incidence fora blazed grating in themth diffraction order is simplified to


/[√(n2)2 − (

mλb/dg)2 −n1

]∣∣∣∣The optimum angle of the passive facet angle ϕp must beparallel to the direction of the diffracted field θm:

ϕp = cos−1 [n1 sin(θi) /n2 +mλb/

(n2dg

)]This requirement cannot al-ways be satisfied due to lim-itations associated with thegrating profile fabrication pro-cesses.


60 Scalar Diffraction Theory of a Grating

Scalar Diffraction Theory of a Grating

Scalar diffraction theory enables an accurate definitionof the propagation directions of diffracted light based onthe grating equation. It also permits diffraction efficiencycalculations when the grating period d significantly exceedsthe operating wavelength λ. The gratings described by scalardiffraction theory have shallow facet angles. Therefore, thecondition d À h and the scalar diffraction theory are oftenreferred to as the thin-element approximation. Diffractiveoptics that satisfy the condition d À h belong to the scalardomain.

Scalar diffraction theory allows for the transition from a partialdifferential wave equation to an integral equation form.Diffraction efficiency in the mth transmitted order for gratingsin the scalar domain can be defined as

ηm =∣∣∣∣ 1dg

∫ dg

0t (x) e

i 2πλ

{[n1 cos(θi)−n2 cos(θm)] f (x)− mλ

dgx}dx

∣∣∣∣2where f (x) is the grating profile function, t (x) is the localtransmission Fresnel coefficient, and x is the coordinate alongthe grating surface normal to the facets.

The relative grating facet size in the scalar domain is typicallyd/λ ≥ 15. Polarization and shadowing effects on the gratingfacets are ignored in the thin-element approximation.

The grating period in general is a function of the lateralcoordinates d(x, y). The wavefront quality of a grating in thescalar domain is determined by the accuracy with which thespacing d(x, y) is reproduced and is usually very good. Inmany cases the optical quality is limited by the quality of thesubstrate rather than by the errors associated with the groovespacing d(x, y). Even for gratings with large apertures that havestitching errors, the wavefront quality is usually good, and theerrors manifest in the presence of weak “ghost” orders.


Scalar Diffraction Theory of a Grating 61

Diffraction Efficiency

Diffraction efficiency in a given propagating order representsa fraction of the incident power contained within the order. Therelative diffraction efficiency of a diffraction grating in thescalar domain is defined with respect to a perfectly reflectingor transmitting substrate. The diffraction efficiency ηm of anideal blazed grating with blazing wavelength λb correspondingto the diffraction efficiency peak value depends on the operatingwavelength λ and the diffraction order m:

ηm ={

sin[mπ

(λb

λ−1

)]/[mπ

(λb

λ−1

)]}2

The figure shows changes in diffraction efficiency as a functionof the operating wavelength for diffraction gratings optimizedat 500 nm and three different diffraction orders.

The normalized diffraction efficiency of a grating consistingof a finite number M of identical facets is a product of aninterference term H from M beams and an intensity term Idof a single facet with width d:

ηM = HIS = sin

(M kdp

2

)M sin

(kdp

2

)2 sin

(kdp

2

)kdp

2

2

where p = n2 sinθm −n1 sinθi.


62 Scalar Diffraction Theory of a Grating

Blaze Profile Approximation

The fabrication of high-fidelity triangular profile blazed grat-ings is costly and time consuming. Reduction in manufacturingcosts is achieved by employ-ing grating structures withapproximated blazed pro-file shapes. Lithographictechniques are often em-ployed to reduce gratingfabrication costs. Binary gratings approximate a continuousgrating profile with a staircase shape and are fabricated usinga succession of lithographic etching steps. The diffraction effi-ciency η of a multilevel binary grating that employs N binarysteps is calculated as

η=sin

[mπ

(λbλ −1

)]mπ

(λbλ −1

)

2 sin(πλbλN

)(πλbλN

)2

The peak diffraction efficiency depends on the number of binarysteps approximating the blazed profile:

Diffraction efficiency as a function of the operating wavelengthfor diffraction gratings optimized at 500 nm with differingnumbers of binary steps is shown below:


Extended Scalar Diffraction Theory 63

Extended Scalar Diffraction Theory

Extended scalar diffraction theory improves the accuracyof efficiency calculations by accounting for the shadowingeffects, the fill factor of the propagating field after diffraction(also called the duty cycle ζ of the diffracted field), and theFresnel reflections at the substrate interfaces. The figureshows the appearance of gaps in the propagating wavefront andthe reduction in fill factor after diffraction.

The field shadowing by the grating profile reduces the duty cycleζ and diffraction efficiency. Changes in diffraction efficiency dueto shadowing are accounted for by introducing the duty cycle ζinto the diffraction efficiency calculations:

ηFF ≈ ζηscalar

The duty cycle of a transmission grating is found by

ζ= dm

dg= 1−

[n1 sin(θi)+mλ/dg

]tan(ϕ)√

(n2)2 − (n1 sinθi +mλ/dg

)2

Accounting for the blaze angle ϕ, the shadowing becomes

ζ = 1− 1√(n2)2 − (

n1 sinθi +mλ/dg)2

×[n1 sin(θi)+mλ/dg

](mλ/dg

)(∣∣∣∣n1 cos(θi)−√

(n2)2 − [n1 sin(θi)+mλ/dg

]2∣∣∣∣)


64 Extended Scalar Diffraction Theory

Duty Cycle and Ghost Orders

When propagating from the substrate into air at normalincidence θi = 0, the duty cycle due to shadowing is

ζ= 1−(mλ/dg

)2√(n2)2 − (

mλ/dg)2

(∣∣∣∣n1 −√

(n2)2 − (mλ/dg

)2∣∣∣∣)

The influence of the shadowing on the diffraction efficiency ofthe propagating field can be explained by observing the farfield of the diffracted radiation. Shadowing effects lead to theappearance of several secondary ghost orders in the vicinityof the primary diffraction order, as shown below for the dutycycle ζ ranging from 0.25 to 0.95.


Extended Scalar Diffraction Theory 65

Extended Scalar versus Rigorous Analysis

Extended scalar diffraction theory provides an expresscalculation technique for the amount of light directed into dif-ferent diffraction orders without the need for computationallyextensive rigorous algorithms. The following graph comparesthe diffraction efficiencies of a blazed transmission grating inthe first diffraction order as a function of the relative featuresize d/λ for both the extended scalar and rigorous diffractionanalysis techniques.

For a relative feature size of d/λ > 15, the difference inthe diffraction efficiency calculated using both techniques ispractically unnoticeable. For a feature size of 2.5 < d/λ < 15,the diffraction efficiency calculated using the extended scalartheory provides optimistic efficiency results, while diffractionefficiency predictions for the feature size d/λ < 2.5 becomeoverly pessimistic.

Diffraction efficiency of gratings with a feature size d/λ< 15 areapproximated using the following equation:

ηrig ≈ ηscalar [1−Cm (n,θ)λB/d]

where ηscalar is the maximum efficiency in scalar domain,and the coefficient Cm (n,θ) is a function of both the index ofrefraction and the angle of incidence.


66 Gratings with Subwavelength Structures

Gratings with Subwavelength Structures

Advancements in high-resolution lithography and microfabri-cation have enabled diffractive structures with subwavelengthfeature sizes. As the period-to-wavelength ratio decreases, thediffraction efficiency predictions based on scalar diffraction be-come inaccurate. When the feature size is of the order of thewavelength or less, the scalar diffraction theory is no longervalid.

The design of subwavelength diffraction structures oftenemploys rigorous electromagnetic propagation techniques thatare based on the solution to Maxwell’s equations. Rigorousdiffraction analysis techniques can accurately account for thediffraction efficiency and polarization states of the propagatingradiation.

In the case of both transmission and reflection gratingsdesigned to blaze in TM polarization, the efficiency in theTE polarization state is always less than 100%. The blazingcondition for both polarizations may occur when one of thepolarization states propagates in transmission while the otherstate propagates in reflection.

These grating structures areemployed as polarizing com-ponents in visible and IRphotonics applications.

The size of the subwave-length grating period dg isoften selected to be smallerthan the structural cutoffdc, which is defined as the period below which all nonzero re-flected and transmitted diffraction orders become evanescent.The necessary condition for the structural cutoff is

dλ< 1√

max(n1,n2)+n1 sin(θi)


Gratings with Subwavelength Structures 67

Blazed Binary Gratings

Due to practical limitations associated with mask misalign-ments and etching depth nonuniformity, an increased numberof binary levels does not necessarily lead to improved gratingperformance. For example, a 32-level binary grating may havelower efficiency and higher scattering as compared to a 16-levelgrating.

Blazed binary grating pro-files define surface-relief struc-tures approximating the blazeprofiles and requiring a singlelithographic step for their fab-rication. Blazed binary gratingshave features smaller than thestructural cutoff, defined as a feature size below which thegrating behaves as a homogeneous layer.

Blazed binary gratings represent artificial dielectrics and aredesigned using the effective medium theory. They operatein the resonant domain, where the grating period is equal toonly a few wavelengths. Each grating period contains a series ofridges with continuously varying widths that change the localfill factor. When properly fabricated, the efficiency of the blazedbinary gratings may exceed the efficiency of the blazed gratingsoperating in the scalar domain.

An alternate design for a blazedbinary grating consisting of anumber of individual “pillars”with progressively decreasinglateral dimensions is shown inthe figure.



Relative Feature Size in the Resonant Domain

The grating performance in the resonant domain, when thegrating period d is comparable to the operating wavelength λ,rapidly changes with the relative feature size d/λ. In thecase of a gold-coated reflection grating with a groove spacing of1.42 µm and TE-polarized light propagating at a 20-deg angleof incidence, several diffraction orders coexist when the relativefeature size d/λ> 1.5:

For the relative feature sizes 0.75 < d/λ < 1.5, only the zeroand negative first diffraction orders coexist, and the totalreflected energy is redistributed between the two orders. Forsubwavelength feature sizes d/λ< 0.75, the negative first ordervanishes, and the grating performs as a mirror.


Gratings with Subwavelength Structures 69

Effective Medium Theory

As shown earlier, for a gold-coated grating with subwavelengthfeatures, the structural cut-off condition for TE-polarized lightis satisfied when dg/λ< 0.75 and the grating surface behaves asa mirror. Similarly, for a transmission grating operating belowthe structural cutoff, all propagating diffraction orders exceptthe zero order become evanescent. These types of gratings arecalled zero-order gratings. Zero-order gratings are employedas antireflection coatings, wave plates, and artificial distributedindex media.

Based on effective medium theory, the effective index ofrefraction n⊥ for a subwavelength binary surface structureand the electric field E⊥ orthogonal to the grating grooves isdefined as

n⊥ = n1n1√(n1)2 ζg + (n2)2

(1−ζg

)where n1 and n2 are the refractive indices of the gratingsubstrate material and the medium surrounding the grooves,respectively; w is the groove width, and ζg = w/dg is the gratingduty cycle. The effective index of refraction n∥ for the electricfield E∥ parallel to the grating grooves is defined as

n∥ =√

(n1)2 ζg + (n2)2(1−ζg

)The difference between the two effectiveindices of refraction 4n = n⊥ − n∥depends on the profile of the diffractivestructure, which behaves as an artificialdielectric material that exhibits formbirefringence.

The figure shows a grating profile withthe effective index gradually changingover the grating period dg.



Scalar Diffraction Limitations and Rigorous Theory

Fresnel reflections at the air–grating interface cause the for-mation of forward- and backward-propagating diffraction or-ders, limiting the theoretically achievable diffraction efficiencyin the blazed transmission order. In practice, uncoated trans-mission gratings transfer no more than 90% of the propagatingradiation in the first diffraction order and about 80% in the sec-ond order. With a reduction in groove spacing, scalar theory be-comes progressively less accurate, and rigorous simulation tech-niques are applied to optimize the grating structure.

Limitations of scalar diffraction theory become apparentwhen accurate efficiency calculations are required for gratingswith smaller periods, or when changes in the polarizationstates of propagating light need to be accounted for.Larger discrepancies in efficiency predictions are observed forpropagating light with the electric vector orthogonal to thegrating grooves.

Rigorous diffraction analysis techniques produce accuratediffraction efficiency calculation results and are based onsolving Maxwell’s equations. Major rigorous techniques includecoupled wave analysis based on space-harmonic expansion,modal analysis based on modal expansion, finite differencemethods, as well as integral methods. With continuousadvancements in computer speed, the finite difference timedomain (FDTD) technique is establishing its place as apowerful practical technique for rigorous diffraction analysis.

The figure shows a high-aspect-ratio grating struc-ture requiring rigorousdiffraction analysis tech-niques for performanceevaluation.


Rigorous Analysis of Transmission Gratings 71

Analysis of Blazed Transmission Gratings

Rigorous analysis of transmission gratings provides valuableinsight into the physics of diffraction phenomena. Thediffraction efficiency of blazed transmission gratingsgradually decreases with a reduction in the grating period dg.When the grating profile height h is set to be constant, the peakdiffraction efficiency shifts toward the shorter wavelengths, asshown in the following figure for TM-polarized radiation in thefirst diffraction order m = 1:

Even for grating feature sizes as small as 0.6 µm, the peakdiffraction efficiency remains above 30%.

For the diffraction order m = 2, the respective efficiency graphsare shifted toward the shorter wavelength:


72 Rigorous Analysis of Transmission Gratings

Polarization Dependency and Peak Efficiencies

For gratings with periods of dg ≥ 1.0 µm at normal incidence,the relative difference in the peak diffraction efficiencies forTM- and TE-polarization states does not exceed 2.4%. Ingeneral, at normal incidence a transmission grating diffractionefficiency has low polarization dependency, as shown in thefollowing graph for the grating period dg = 3.0 µm in the firstdiffraction order:

When the grating profile height is adjusted in accordance withthe optimum height value hopt defined based on extended scalartheory, the diffraction efficiency curves no longer shift towardthe shorter wavelengths with a reduction in the grating perioddg. This is shown in the following figure for TM-polarizedradiation:



Peak Efficiency of Blazed Profiles

Changes in the peak diffraction efficiency of a blazedtransmission grating as a function of the feature size are shownin the next graph.

In the so-called scalar domain, when the relative featuresize d/λ > 10, the peak diffraction efficiency variation is lessthan 2%.

Outside of the scalar domain, the diffraction efficiency of ablazed transmission grating rapidly degrades with a reductionin the relative feature size.

The peak diffraction efficiency and position of the peak wave-length as a function of the grating facet spacing change asfollows:



Wavelength Dependency of Efficiency

The diffraction efficiency of blazed transmission gratings in ascalar domain (d/λ > 10) exhibits a characteristic red spectralshift as a function of the incidence angle. The figure showsthe diffraction efficiency of a blazed transmission grating withfeature size d = 50 µm and TM-polarized light at several anglesof incidence as a function of the operating wavelength λ. Thepeak diffraction efficiency for TM-polarized radiation remainspractically unchanged over a wide range of incidence angles(+/−60 deg).

The peak diffraction efficiency for TE-polarized radiationgradually diminishes with an increase in the angle of incidence,as shown in this figure:



Efficiency Changes with Incident Angle

Diffraction efficiency as a function of the incident angle of TM-polarized light for blazed transmission gratings in a scalardomain (groove spacing d = 50) at several discrete wavelengthsis shown in the following figure:

An increase in the operating wavelength above 400 nm leads tosignificant diffraction efficiency enhancements over the rangeof the incident angles, forming the batwing-shaped efficiencycurves.

Diffraction efficiency curves may also exhibit polarizationanisotropy at higher angles of incidence, as shown in the figurefor a wavelength of 700 nm:



Diffraction Efficiency for Small Feature Sizes

In the case of blazed gratings with small feature sizes (d/λ< 15),the facet angle is relatively steep, and the peak diffractionefficiency depends on both the angle of incidence and thewavelength of the incident light. The figure presents TM-polarized light diffraction efficiency for a blazed transmissiongrating with feature size d = 3 µm as a function of the operatingwavelength λ:

Diffraction efficiency curves exhibit asymmetry with respectto the angle of incidence, as shown for TM-polarized lightdiffracted by a blazed transmission grating with feature sized = 3 µm:

An increase in the operating wavelength leads to a peakefficiency angular shift.


Polychromatic Diffraction Efficiency 77

Polychromatic Diffraction Efficiency

The diffraction efficiency of a single grating surface (diffrac-tive singlet) in a given diffraction order can be maximized onlyat a single operating wavelength, where the in-phase conditionfor the individual diffracted fragments of the propagating wave-front is satisfied.

An appropriately designed diffractive doublet based onthe combination of two grating surfaces can produce highpolychromatic diffraction efficiency over an extendedoperating spectral range, known as broadband blazing,shown for the wavelength range 350 nm< λ< 650 nm:

The polychromatic diffraction efficiency of the grating doubletshown in the graph is 98.7% over the spectral range from 350 to650 nm, as compared to the polychromatic diffraction efficiencyof 88.8% for a single diffraction structure with efficiencymaximum at 480 nm.

The employment of diffractive singlets in imaging systemsoperating over an extended spectral range offers significantbenefits in correcting the system’s aberrations. These benefitsare often outweighed by the image contrast degradationassociated with reduced polychromatic diffraction efficiency ofa diffractive singlet. Diffractive doublets can provide similaraberration correction benefits with higher image contrast.


78 Polychromatic Diffraction Efficiency

Monolithic Grating Doublet

In the case of a diffractive singlet, the blazing condition issatisfied only for a single wavelength λb, and the polychromaticdiffraction efficiency is reduced.The broadband blazing conditioncan be achieved by employing a com-bination of two diffractive structuresthat form a grating doublet, eithermonolithic or spaced. A monolithicgrating doublet is shown in thefigure.

The optical path difference (OPD) introduced by a mono-lithic grating doublet with spacing dg and step height h madeof optical materials with refractive indices n1 (λ) and n2 (λ) isfound based on the blazing condition

OPD= mλb = h [n2 (λ)cos(θm)−n1 (λ)cos(θi)]

The broadband blazing condition requires the OPD to beproportional to the individual wavelengths over the extendedspectral range. Provided that the incident angle does notchange, the broadband blazing condition for the monolithicgrating doublet is defined as

ddλ

[n2 (λ)]cos(θm)− ddλ

[n1 (λ)]cos(θi)= m

Grating doublets can be made with various profiles, includingtriangular, lamellar, sinusoidal, etc.

While monolithic grating doublets are relatively insensitiveto fabrication errors, the broadband blazing condition for amonolithic grating doublet is difficult to satisfy due to thelimited choice of materials satisfying the blazing requirements.



Spaced Grating Doublet

Material constraints aregreatly relaxed for spacedgrating doublets. The OPDintroduced by grating stepheights h1 and h2 made ofoptical materials with refrac-tive indices n1(λ) and n2(λ)and a spacer with index n3(λ)is defined as

OPD = mλbl = h1 [n3 (λ)cos(θm1)−n1 (λ)cos(θi1)]+h2 [n2 (λ)cos(θm2)−n3 (λ)cos(θi2)]

Broadband blazing for the spaced grating doublet is defined as

h1

{d

dλ[n3 (λ)]cos(θm1)− d

dλ[n1 (λ)]cos(θi1)

}+

h2

{d



}= m

For an air-spaced grating doublet with d [n3 (λ)] /dλ∼= 0, thebroadband blazing condition is reduced to

ddλ

[n2 (λ)]h2 cos(θm2)− ddλ

[n1 (λ)]h1 cos(θi1)= m

The broadband blazing condition is satisfied for at least twooperating wavelengths. A grating doublet significantly extendsthe broadband diffraction efficiency as compared to a gratingsinglet:


80 Polychromatic Diffraction Efficiency

Monolithic Grating Doublet with Two Profiles

In order to eliminate the material constraints associated withmonolithic grating doublets while maintaining the lowdoublet sensitivity to variations in the profile height, a seconddiffraction profile is formed at the doublet exterior. The secondexterior profile can be formed, for example, by compressionmolding or etching.

A dual-profile monolithic grating doublet has a significantlylower sensitivity to fabrication errors as compared to an air-spaced grating doublet. The depth errors ∆h1 in the gratingsubstrate structure with refractive index n1 (λ) lead to OPDerrors proportional to the refractive index difference betweenthe two doublet materials:

∆OPD=∆h1 [n1 (λ)−n2 (λ)]

These OPD errors are several times smaller than similarerrors of an air-spaced doublet, which are proportional tothe difference between the refractive index of the substratematerial and that of air:

∆OPD=∆h1 [n1 (λ)−1]

The exterior profile of the monolithic grating doublet is usuallyshallower than the substrate profile(s) employed in air-spacedgrating doublets.

The dual-profile monolithic grating doublet provides anadditional benefit of reduced Fresnel losses due to a reducednumber of optical interfaces.



Diffractive and Refractive Doublets: Comparison

There is a similarity between diffractive and refractivedoublets. Both types of doublets are intended to reduce thespectral dependency of one of the key performance parameters.In the case of a diffractive doublet, this parameter is thepolychromatic diffraction efficiency, while in the case ofa refractive doublet, it is the size of the polychromatic pointspread function.

The net refractive optical power of a refractive doublet isthe sum of the positive and the negative optical powers of thetwo refractive elements composing the refractive doublet. Thenet optical power of a refractive doublet is less than the opticalpower magnitude of the two lens components.

Diffractive optical power is proportional to the spectraldispersion of the propagating field. For the diffractive doublet,the net diffractive power is the difference of the powers ofthe two diffractive components. The net diffractive power inthe diffractive doublet with high polychromatic diffractionefficiency is lower than the powers of the individual diffractivecomponents.

The axial thickness of a refractive doublet and the sensitivity ofits components to surface shape variations significantly exceedthose of a refractive singlet of equal optical power.

The diffractive profile heights and the sensitivity to profileheight variations of the doublet grating components consider-ably exceed those of a diffractive singlet of equal diffractivepower.

Surface profile errors in the refractive components causedistortions that lead to an increase in the refractive doubletPSF. Surface profile height errors in diffractive doublets leadto reduced polychromatic diffraction efficiency in the workingorder.


82 Efficiency of Spaced Grating Doublets

Efficiency of Spaced Grating Doublets

The diffraction efficiency of a spaced grating doubletis a function of several parameters, including polarization,wavelength, angle of incidence, material properties, stepheights, and the grating spacing.

In most cases, a rigorous diffraction analysis techniqueis required to design and optimize the performance ofgrating doublet structures. The following analysis shows theperformance of an air-spaced grating doublet designed tooperate in the visible spectrum and made of two dissimilarmaterials with refractive indices n1 = 1.4623 and n2 = 1.6588.Diffraction efficiency exhibits a strong asymmetric angulardependency, as shown in the following graphs for TE-polarizedlight and positive and negative angles of incidence.


Efficiency of Spaced Grating Doublets 83

Sensitivity to Fabrication Errors

The fabrication of spaced grating doublets with broadbanddiffraction efficiency requires a relatively high fabricationaccuracy of individual grating components constituting thedoublet. The following graphs show the grating doubletdiffraction efficiency sensitivity to changes in step height ofone of the grating components for TE- and TM-polarized light,respectively.



Facet Width and Polarization Dependency

The broadband diffraction efficiency of air-spaced gratingdoublets depends on the facet width of the gratings composingthe doublet. For grating doublets with a broadband diffractionefficiency optimized over the visible range, significant efficiencydegradation is observed for the facet widths of less than 50 µm.

The broadband diffraction efficiency of air-spaced gratingdoublets has a significant polarization dependency. TE-polarized light has higher diffraction efficiency than TM-polarized light over the operating spectral range. This can beexplained qualitatively by the differences in Fresnel reflectionlosses of the two polarization states.


Efficiency of Spaced Grating Doublets 85

Sensitivity to Axial Component Spacing

Polychromatic diffraction efficiency of an air-spaceddiffraction doublet depends on the axial spacing between thedoublet components. Changes in axial spacing t between thediffraction doublet components cause lateral shifts of thepropagating field fragments diffracted by the first grating withrespect to the facets of the second grating.

An increase in the axial spacing t between the two gratingcomponents leads to an oscillatory spectral response ofthe doublet. The polychromatic efficiency modulationincreases with an increase in the axial spacing t, as shown inthe figure for TE-polarized light.



Frequency Comb Formation

The lateral shift of the field diffracted by the first grating inthe doublet leads to wavefront division by the facets of thesecond grating. Interference of the wavefront fragments afterthe second grating causes formation of a frequency comb inthe spectral response of the doublet, as shown in the figure fortwo different values of the axial spacing.

Nodal locations in the transmitted frequency comb do notchange with adjustments in the axial separation between thedoublet components. The oscillatory spectral response of thegrating doublet is observed in different diffraction orders. Thefigure shows alternating extrema locations for the spectralefficiency of TE-polarized light diffracted by the doublet withan air gap of 250 µm into the orders 0, 1, and 2.


Diffractive Components with Axial Symmetry 87

Diffractive Components with Axial Symmetry

Diffractive components with axial symmetry constitute one ofthe largest groups of diffractive structures and include am-plitude masks, phase plates, diffractive lens surfaces, steppeddiffractive surfaces (SDSs), hybrid diffractive-refractive struc-tures, as well as lens diffractive doublets. Axially symmetricdiffractive components are found in a variety of applications,including imaging objectives operating in the UV, visible, andIR, as well as intra-ocular lenses for vision correction and inillumination and laser systems.

Amplitude masks with axial symmetry, such as Fresnel zoneplates, are based on amplitude modulation of the incidentradiation. Amplitude masks are employed in the spectralregions where nonabsorbing optical materials are not readilyavailable, such as in the extreme UV. Employment of amplitudemasks is associated with net transmission losses introduced bythe masks.

Annular phase plates, such as Fresnel phase plates, arebased on the phase modulation of the incident wavefrontby introducing phase delay to the different portions of thepropagating wavefront. Phase plates are designed to workin transmission or reflection, or to produce bidirectionalpropagation. With an appropriate choice of coatings and phaseplate material, minimal transmission losses are introduced intothe system.

Diffractive lens surfaces include kinoform, binary, and multi-order diffractive lenses and represent phase structures thatcontribute to the net optical power of their optical systems.

Stepped diffractive surfaces (SDSs) represent diffractivephase structures with axially symmetric zones and zero opticalpower at the design wavelength. An SDS cross section is shapedlike a staircase. Each SDS zone produces a phase delay that isan integer multiple of 2π.


88 Diffractive Components with Axial Symmetry

Diffractive Lens Surfaces

Diffractive lens surfaces (DLSs) are composed of annulargrooves and contribute to the net optical power in an opti-cal system. The group of DLSs includes diffractive kino-forms and binary surface structures, as well as refractive-diffractive and reflective-diffractive hybrid structures fabri-cated into a curved substrate.

Diffractive kinoforms are grating phase structures consistingof circular phase zones with radially varying blazed facetangles. Diffractive kinoforms are based on the phase modula-tion of the incident wavefront by introducing a radially variablephase delay to the different portions of the propagatingwavefront.

Longer wavelengths focus closerto the diffractive surface.

Locations of the focal points forthree wavelengths

λr > λg > λb

satisfy the inequality

fr < fg < fb

Binary surface structures are made as multistep approxima-tions to the kinoform zone profiles. Each binary level producesa phase delay that is a fraction of a 2π kinoform zone delay(usually ranging from π/32 to π).

The hybrid refractive-diffractive and reflective-diffractive sur-face structures combine the properties of diffractive kinoformswith the refractive or reflective properties of the substrate.

Diffractive lens surfaces are designed for use in imagingand nonimaging applications, including illumination and laseroptics.



Diffractive Kinoforms

Diffractive kinoforms are surface-relief struc-tures fabricated on a planar substrate that com-prise several concentric zones with continuousprofiles. The term “kinoform” was introducedin 1969 by researchers from IBM. The opticalpower of a kinoform is proportional to the num-ber of zones Nk and the design wavelength λ0 andis inversely proportional to the square of the clearaperture diameter D0:

ΦD (λ)= 8Nkλ0/ (D0)2

A diffractive kinoform resembles the shape of a conventionalFresnel lens, and its cross section looks like a series ofsawtooth-shaped ridges with variable radial spacing di andfacet angle ϕi. In contrast with Fresnel lenses, diffractivekinoforms precisely control the OPD introduced by the zonesand are composed of a significantly larger number of zones.

The OPD φ between two neighboring kinoform zones is aproduct of the diffraction order m and the design wavelengthλ0:

φ= mλ0

For a given design wavelength λ0 and refractive index n0 ofthe substrate, the optimum zone height h of the diffractivekinoform operating in the first diffraction order m = 1 is afunction of the incident angle θi:

h (θi)= λ0√(n0)2 − (sinθi)2 −cosθi

The kinoform zone thickness profile t (r,θi) is found by

t (r,θi)= h (θi) (φ (r) mod 2π)

For on-axis propagation (θi = 0), the zone height is

h0 = λ0/ (n0 −1)



Binary Diffractive Lenses

Binary diffractive lenses are surface-relief structures thatare fabricated based on very large-scale integration (VLSI)lithographic techniques. The name “binary” reflects the codingscheme employed to create the sequence of photolithographicmasks for fabrication. Binary lens coding provides a step-wiseapproximation to the radial phase profile ψ (r) of a diffractivekinoform.

The figure compares a three-zone diffractive kinoform andthe respective three-zone binary lens with four binary levelsapproximating the diffractive kinoform.

To reduce the number N of processing cycles and the respectivenumber of masks required during binary lens fabrication, abinary coding scheme is used. The number of binary levels isdefined as 2N , and the thickness tb of each binary lens level forthe design wavelength λ0, zone height h0, and refractive indexn0 is found by

tb = h0

2N= λ0

(n0 −1)2N

The diffraction efficiency of binary lenses is less than theefficiency of the respective kinoforms. Efficiency reduction iscaused by profile approximations, additional losses associatedwith interfacial roughness, and fabrication imperfections of thebinary steps. Increasing the number of binary steps to greaterthan 32 may not necessarily produce a binary lens with higherefficiency, due to increased interfacial roughness, etch depthvariations, and mask misalignments.



Optical Power of a Diffractive Lens Surface

The optical power ΦD of a diffractive lens surface at anyoperating wavelength λ is related to the nominal optical powerΦD

0 at the design wavelength λ0 as

ΦD (λ)=ΦD0

λλ0

The focal plane locations forthe wavelengths

λr > λg > λb

satisfy the inequality

fr < fg < fb

The difference ∆ fchr = fb − fr is called longitudinal or axialchromatic aberration. The change in image size at thedifferent wavelengths ∆Hchr = Hb −Hr is called transverse orlateral chromatic aberration.

Axial chromatic aberration of a diffractive surface ∆ f Dchr is

proportional to the spectral bandwidth ∆λ = λ− λ0 of thepropagating field and can be expressed as

∆ f Dchr =∆λ/

(ΦD

0 λ0

)For comparison, the optical power ΦR of a refractive surface canbe defined as

ΦR (λ)= n (λ)−1R0

=ΦR0

[1+ Dn (λ)∆λ

n0 (λ)−1

]where ΦR

0 = [n0 (λ)−1]/R0 is the nominal refractive surfacepower corresponding to the substrate radius R0 with refractiveindex n0 at the design wavelength λ0. The lens materialdispersion Dn (λ) is calculated as

Dn (λ)= n (λ)−n0 (λ)λ−λ0



Diffractive Surfaces as Phase Elements

For raytracing purposes, it is convenient to represent diffractivesurfaces as phase elements with respective phase profilesψ (x, y) that depend on the surface lateral coordinates x and y.

A phase profile ψ (x, y) of a diffractive lens surface forming animage I(xi, yi, zi) of a point O(xo, yo, zo) in the object space at thedesign wavelength λ0 is defined as

ψ (x, y) = 2πλ0

{[√(x− xo)2 + (y− yo)2 + (zo)2 − zo

]−

[√(x− xi)2 + (y− yi)2 + (zi)2 − zi

]}The axial distances zo and zi satisfy the sign convention andare positive to the right of the surface. The optical power ΦD (λ)of the surface is then given by the Gaussian lens formula

ΦD = 1/zi −1/zo

In many practical applications, including aberration correction,beam shaping, and athermalization, the diffractive surface doesnot form images of an object. For surfaces without rotationalsymmetry, the phase is

ψ (x, y)= mK∑

k=0

L∑l=0

Akl (x)k (y)l

For axially symmetric diffractive surfaces, the phase is afunction of the radial coordinate r and order m:

ψ (r)= mN∑

i=0A i (r)2i



Stepped Diffractive Surfaces

Stepped diffractive surfaces (SDSs) are powerless surface-relief diffractive phase structures fabricated as a set ofconcentric circular zones with planar interfaces, also referredto as “steps.” Components employing SDSs are sometimes calledstaircase lenses.

The steps are bounded by cylindrical sections. Each zone ischaracterized by a width di and is axially offset from theneighboring zone by a distance hi.

The nominal optical power of an SDS at the design wave-length λ0 is zero. Therefore, SDSs can be added to optical sys-tems without altering the first-order properties.

The shadowing effects produced by stepped diffractivesurfaces at the zone boundaries are significantly smaller ascompared to diffractive lens surfaces with the same aperturesand dispersion values, leading to increased diffraction efficiency.

While complex diffractive lens profiles require radially vary-ing facet angles with sharp corners that are difficult to repro-duce, SDS staircase profiles are easier to fabricate using directsingle-point diamond turning (SPDT) or replication tech-niques.



Properties of Stepped Diffractive Surfaces

The optical power of an SDS depends on the operatingwavelength λ and the refractive indices n1 (λ) and n2 (λ) of thematerials before and after the interface, respectively:

ΦSDS (λ)=∣∣∣∣ n2 (λ)−n1 (λ)

R0

∣∣∣∣( λλ0−1

)SDS optical power at the design wavelength λ0 is zero.

SDSs have several advantages over blazed diffractive kinoformsand their binary approximations:

1. An SDS can be added to any optical system withoutchanging the paraxial optical power of the system. It canbe employed in an optical system to correct for chromaticand/or monochromatic aberrations or to provide passiveathermalization of the system.

2. The simple staircase zone geometry of an SDS allows foraccurate microstructure fabrication by using the SPDTtechnique, leading to reduced aberrations caused byfabrication errors as compared to a diffractive lens surfacewith an equal aperture and power.

3. Compared to a binary diffractive surface, an SDS does notinvolve an approximation of the microstructure’s shapeand therefore produces higher diffraction efficiencies thanbinary diffractive lenses.

The optimum step height hSDS of an SDS is a function of theincident angle θi, the working diffraction order m, and thedesign wavelength λ0, and is found by

hSDS =

∣∣∣∣∣∣∣mλ0√

(n2)2 − [sin(θi)]2 −n1 cos(θi)

∣∣∣∣∣∣∣In the case of a normal-incidence SDS, the step height becomes

hSDS =∣∣∣∣ mλ0

n2 −n1

∣∣∣∣



Multi-order Diffractive Lenses

Multi-order diffractive lenses are designed to produce OPDsbetween the zones that are a multiple m of the operatingwavelengths. Multi-order diffractive lenses are similar toechelle diffractive gratings with high-profile depths and aredesigned to work in high diffraction orders. For a multi-orderdiffractive lens designed to operate at an infinite conjugate inthe mth diffraction order, the profile height can be defined as

hm = mλ0

n (λ0)−1

Multi-order diffractive lenses are often designed to blaze inmore than a single operating wavelength. For a multi-order lensdesigned to operate at wavelengths λB and λR in respectiveorders m and k, the blaze condition becomes

mλB

n (λB)−1= kλR

n (λR)−1

The above blaze condition may be difficult to satisfy whileachieving reasonably shallow depths of the diffractive zones. Inthat case, the profile depth is designed to blaze at the shorterwavelength λB. The graph presents changes in the diffractionefficiency for a multi-order diffraction lens designed to operatein the order m = 3 for the visible wavelength λB = 0.532 µm,as well as in the order k = 1 at a longer infrared wavelengthλR = 1.55 µm.



Diffractive Lens Doublets

High diffraction efficiency for a single diffractive surfaceis achieved only over a narrow spectral bandwidth ∆λ. Inapplications with broad radiation bandwidth, a significantfraction of radiation is directed into the spurious diffractionorders and causes a degradation of the system performance byreducing the system’s resolution and image contrast, as well asby increasing the crosstalk in the detection channels.

Increased polychromatic diffraction efficiency is achievedby designing diffractive lens doublets consisting of twodiffractive surfaces with opposite powers. The lens doublets aremade as monolithic or air spaced:

The following figure shows theoretical diffraction efficienciesfor three air-spaced diffractive lens doublets made of differentmaterial pairs and designed to operate over an extendedspectral range in the visible spectrum. All three doublets haveefficiency maxima at 400 and 550 nm. The material choiceplays an insignificant role in the achieved diffraction efficiency.Instead, other factors such as manufacturability, material cost,and required etch depth of the phase zones play dominant roles.


Diffractive Surfaces in Optical Systems 97

Diffractive Lens Surfaces in Optical Systems

The small surface-relief thickness of diffractive lens surfacesmakes them well suited for integration into optical systems.A surface-relief diffractive structure can be combined with arefractive or reflective surface to attain additional degrees offreedom in optical design.

Hybrid components combine diffractive surfaces with refrac-tive or reflective counterparts. Hybrid structures integratediffractive structures into refractive or reflective surfaces.

The total optical power of the hybrid structure ΦH containingdiffractive as well as reflective or refractive surfaces is foundas the sum of the diffractive power ΦD and the respectivesubstrate power ΦR :

ΦH (λ)=ΦR (λ)+ΦD (λ)=ΦR0

[1+ Dn (λ)∆λ

n0 (λ)−1

]+ΦD

0

(1+ ∆λ

λ0

)Material dispersion Dn (λ) depends on the operating spectralrange λ−λ0 as well as on the material’s refractive index changewith wavelength, as shown below:


98 Diffractive Surfaces in Optical Systems

Achromatic Hybrid Structures

An achromatic condition for a hybrid doublet structure∆ΦH =ΦH (λ)−ΦH (λ0) = 0 is satisfied when the relative powerof the refractive surface equals

ΦR0

ΦD0

= ∆λ [n0 (λ)−1]λ0 [n0 (λ)−n (λ)]

=− [n0 (λ)−1]λ0Dn (λ)

The relative power of the refractive surface required to satisfythe achromatic condition ∆ΦH = 0 is shown in the graph fordifferent substrate materials:

The relative diffractive power in a hybrid achromat is reducedwith an increase in the operating wavelength.

The total power of the achromatic hybrid structure ΦAH ,found as the sum of the refractive and diffractive powers, ishigher than the power of the diffractive ΦD

0 or refractive ΦR0

surfaces of the doublet:

ΦAH =ΦD0

{1− [n0 (λ)−1]

λ0Dn (λ)

}=ΦR

0

{1− λ0Dn (λ)

[n0 (λ)−1]

}In the case of an achromatic refractive doublet the tworefractive lenses have opposite powers, and the net opticalpower of a refractive achromat is reduced. Therefore,hybrid achromats yield lower monochromatic aberrations, axialthickness, and weight than respective refractive achromats.



Opto-thermal Properties of Optical Components

The thermal behavior of an optical surface can be describedin terms of an opto-thermal coefficient (OTC) ξ. An OTCdefines the thermally induced relative rate of change in opticalpower over the temperature range ∆T:

ξ=∆Φ/ (Φ∆T)

Thermally induced changes in the kinoform optical power arecaused by changes in the zone spacing. Respective changesin the substrate refractive index affect only the diffractionefficiency of the lens.

The OTC of a kinoform ξK depends only on the substratecoefficient of thermal expansion (CTE) α:

ξK =−2α

Changes in the optical power of a refractive interface are dueto changes in the surface radius and in the substrate refractiveindex. The OTC of a refractive interface ξR is calculated as

ξR (λ)= 1[n2 (λ)−n1 (λ)]

[dn2 (λ)

dT− dn1 (λ)

dT

]−α

Changes in the optical power of an SDS are due to changes inthe zone spacing and in the refractive indices n1 (λ) and n2 (λ) ofthe media before and after the interface, respectively. The opto-thermal coefficient of an SDS depends on both the CTE of thesubstrate material and the thermally induced changes in therefractive indices dn/dT:

ξSDS (λ)= 1[n2 (λ)−n1 (λ)]

[dn2 (λ)

dT− dn1 (λ)

dT

]+α

The OTC sign of an SDS is positive when the SDS is convex andnegative when the SDS is concave.

The opto-thermal coefficients of an SDS, a refractive surface,and a diffractive kinoform are related as follows:

ξSDS (λ)=± [ξR (λ)−ξK ]


100 Diffractive Surfaces in Optical Systems

Athermalization with Diffractive Components

Thermally induced relative changes in power Φ of an opticalsurface are proportional to the temperature changes ∆T andthe surface opto-thermal coefficient ξ:

∆Φ

Φ= ξ∆T

Diffractive components provide additional degrees of freedomin designing athermal lens solutions. During athermal lensdesign it is also necessary to account for the optical componentmounting scheme and the relative component shifts based onthe CTE of the housing material.

For an optical system containing N optical interfaces, theathermal condition occurs when the net change in opticalpower of the lens system, defined based on the contribution ∆Φi

from all of the optical interfaces, is zero:

N∑i=1

(∆Φi)= 0

The athermal condition for a hybrid refractive–diffractivekinoform singlet lens and for thermally invariant housing canbe written as

ξKΦK =−ξRΦ

R

The ratio of the optical powers of the kinoform ΦK and therefractive ΦR surfaces in the athermal singlet is

ΦK

ΦR=

{d

dT [n2 (λ)]− ddT [n1 (λ)]

}2α [n2 (λ)−n1 (λ)]

− 12

An athermal achromat can be constructed based on a singlehybrid lens element and an appropriate selection of mountingmaterial.



Athermalization with SDSs

Opto-thermal coefficients for refractive, diffractive, and SDSsurfaces depend on the substrate material.

The opto-thermal coefficients of diffractive components havenegative values, while the OTCs of refractive and SDScomponents can be either positive or negative.

The nominal optical power of an SDS is equal to zero. Thermallyinduced changes in SDS optical power are calculated based oneffective optical power ΦSDS

eff , defined by the SDS substrateradius R0 and the refractive indices of the materials:

ΦSDSeff (λ)=−

[n2 (λ)−n1 (λ)

R0

]The figure shows an example of anathermal hybrid singlet composed of thefront aspheric surface shaped to correctfor spherical aberrations, and the backSDS surface designed to reduce thesinglet dependency on changes in theambient temperature.


102 Appendix: Diffractive Raytrace

Appendix: Diffractive Raytrace

The diffraction of a single ray after encountering a gratingsurface can be described in vector form as

n2

(S′×r

)= n1

(S×r

)+Λq (A1)

where the unit vectors S, S′, r, and q are composed of theirrespective components in Cartesian coordinates:

S′ = L′Si+M′

S j+N ′Sk

S= LSi+MS j+NSkr= Lri+Mr j+Nrkq= Lqi+Mq j+Nqk

(A2)

Unit vectors S and S′ define the ray propagation directionbefore and after encountering the grating surface; vector rdefines the local normal to the grating surface; q is a vectorparallel to the grating grooves at the ray intersection point.

The term Λq in Eq. (A1) is responsible for the diffractionphenomenon. For a purely refractive case, the term vanishes,and Eq. (A1) reduces to Snell’s law. The grating parameter Λin Eq. (A1) is a function of the working diffraction order m, thelocal groove spacing dg, and the design wavelength λ0:

Λ= mλ0/dg (A3)

Equation (A1) can be rearranged as(n2S′−n1S+Λp

)×r= 0 (A4)

where p = ui+ vj+wk is a unit vector parallel to the gratingsubstrate and normal to the grating grooves at the rayintersection point P(x, y).

Because the vectors q and p are orthogonal, we can write

q=−p×r (A5)

The vector S′ defining the propagation direction of a ray afterdiffraction on the grating surface is found from Eq. (A4) in thefollowing form:

S′ =(n1S−Λp+ΓDr

)/n2 (A6)


Appendix: Diffractive Raytrace 103

Appendix: Diffractive Raytrace (cont.)

Individual components of the vector S′ are found from thefollowing equations:

L′S =

(n1LS −Λu+ΓDLr

)/n2

M′S =

(n1MS −Λv+ΓD Mr

)/n2

N ′S =

(n1NS −Λw+ΓD Nr

)/n2

(A7)

If R = xi+ yj is a radial vector at the intersection point normalto the optical axis, then for any axially symmetric diffractivestructure, vectors p and r are coplanar with vector R. Therefore,we can write

uv= Lr

Mr= x

y(A8)

Components of vector p in Cartesian coordinates can be foundby

u = xNr√x2 + y2

v = yNr√x2 + y2

w = xLr + yMr√x2 + y2

= cos(R,r

) (A9)

From Eq. (A9), it follows that sin(R,r

)= ±Nr. The factor ΓD

in Eqs. (A6) and (A7) depends on the grating type. For thetransmission grating, the factor ΓD is

ΓD =−a+√

(a)2 −b (A10)

For the reflection grating, the factor ΓD is

ΓD =−a−√

(a)2 −b (A11)

Parameters a and b in Eqs. (A10) and (A11) are defined as

a = n1 cos(r,S

)= n1 (LrLS +MrMS +NrNS) (A12)

b = (n1)2 − [(n2)2 + (Λ)2

]−2n1Λ (LSu+MSv+NSw) (A13)

In the case of diffractive lenses, the grating spacing dg (r) isa function of the local radial coordinate r. The local grating


104 Appendix: Diffractive Raytrace

Appendix: Diffractive Raytrace (cont.)

spacing dg (r) can be used during the analysis of complexmulti-element optical systems containing diffractive opticscomponents.

The explicit use of the local grating spacing dg (r) duringdiffraction raytrace allows extending the local grating theoryto optimize performance of an optical system with respect toaberrations, diffraction efficiency, and image contrast.

In the case of diffractive propagation, vectors S, r, and S′ are nolonger coplanar, and the diffracted ray does not lie in the inci-dent plane defined by the incident ray and the surface normalat the point of intersection. This is fundamentally different fromthe refraction or reflection propagation phenomenon, where theincident ray, the surface normal at the point of the ray intersec-tion, and the outgoing ray are always coplanar.

The local zone spacing of SDSs dSDS = d (r, t0) depends on boththe radial coordinate r and the step height t0 and can be foundby

dSDS (r, t0)=∂Φ(r,z)

∂r∂Φ(r,z)

∂z

t0 (A14)

The function Φ (r, z) is the analytical definition of the SDSsubstrate. When the SDS substrate is explicitly defined as arotationally symmetric even polynomial aspheric surface withN aspheric terms, vertex curvature c, and conic constant k, thelocal zone spacing as a function of the radial coordinate can bewritten as

dSDS (r, t0)= t0

crp1−(1+k)(cr)2

+2N∑

i=1iA i (r)2i−1

(A15)

The local grating parameter Λ, necessary during a diffractiveraytrace, is calculated as

ΛSDS (r, t0)= mλ0

t0

(cr√

1− (1+k) (cr)2+2

N∑i=1

iA i (r)2i−1

)


105

Equation Summary

Diffraction fundamentals:

θd ∝ λ/D ∇2U (x, y, z)+k20U (x, y, z)= 0

U (x2, y2, z2)∝Ï

1+cos(z,r12)2iλr12

exp(ikr12)U (x1, y1, z1)dx1d y1

Fresnel diffraction:

U (x2, y2, z2) ∝ exp(ikz12)iλz12

Ïexp

{ik

2z12

[(x2 − x1)2 + (y2 − y1)2

]}×U (x1, y1, z1)dx1d y1

NF∼= D2

4λLzD =

√NFλ (4Lz +NFλ) Lz =

[D2 − (NFλ)2

]4NFλ

LmT = 2md2

λL0.5m

T = LmT −d2/λ= 2(m−0.5)d2/λ

Fraunhofer diffraction:

U (x2, y2, z2)∝ exp(ikz12)iλz12

exp

[ik

(x1

2 + y12)

2z12

]×Ï

exp[

ik (x2x1 + y2 y1)2z12

]U (x1, y1, z1)dx1d y1

U (x2, y2, f )∝ exp(ik f )iλ f

exp[

ik

2 f

(x2

2 + y22)]×Ï

exp[− ik (x2x1 + y2 y1)

2 f

]U (x1, y1, z1)dx1d y1

I (r)= I0

(πD2

2λ f

)2 [J1

(πDλ f

r)/(

πDλ f

r)]2

DA∼= 2.44λ f /D

I (q)= I0

[(1− e2) qD

2

]2 [J1 (q)

q− e

J1 (eq)q

]2

U (x2, y2, f )= (ab)2 sinc2(ikax2

2 f

)sinc2

(ikay2

2 f

)Volume Bragg gratings:

ηS = [sin(υ)]2 ηP = {sin[υcos(2γ)]}2 υ= π∆nT

λ√

CRCS

CR = cos(2γ) CS = cos(2γ)− λ

ndBtan

(β−αB

2

)


106

Equation Summary

∆n = λ (s−0.5)T

√cos(2γ)

[cos(2γ)− λ

ndBtan

(β−αB

2

)]β= cos−1

(2p−12s−1

)−αB β= 180−cos−1

(2p−12s−1

)−αB

Grating equation:

n2 sinθm = n1 sinθi +mλ/dg sinθm +sinθi = mλ/dg

sinθLm = mλ/2dg

Grating properties:

∂θm

∂λ= m

dgn2 cosθm

∂θLm

∂λ= m

2dg cosθLm

= tan(ϕb)

λLb

∂θm

∂λ= n2 sinθm −n1 sinθi

λn2 cosθm= 1[

dgn1 cos(θi)m − λb

tan(ϕb)

]Free spectral range:

∆λFSR = λs

m= λl

m+1λ

dλ= |m|Ng

(λ

dλ

)max

= 2W g

λ

λ

dλ= |m| Wg

dg= Wg

λ|sinθm +sinθi|

Grating anomalies:∣∣∣∣ 1n2

(n1 sinθi + mλ

dg

)∣∣∣∣= 1 λ= dm

[sgn(m)−sin(θi)]

Blazing condition:{n1 sin(θi +ϕ)= n2 sin(θm +ϕb)n2 sin(θm)= n1 sin(θi)+mλ/dg

λb = dg

m

(n2 sin

{sin−1

[n2

n1sin(θi +ϕb)

]−ϕb

}−n1 sin(θi)

)λb = 2dg

msin(ϕb)sin(θi +ϕb) λL

b = 2dg

msin(ϕb)


107

Equation Summary

Blazed facet angle and height:

tan(ϕb)=(mλb/dg

)n1 cos(θi)−

√(n2)2 − [


)]2

φ= dg [n2 sin(θm)−n1 sin(θi)]= mλb


/{√(n2)2 − [


)]2 −n1 cos(θi)}∣∣∣∣


/[√(n2)2 − (

mλb/dg)2 −n1

]∣∣∣∣ϕp = cos−1 [

n1 sin(θi) /n2 +mλb/(n2dg

)]Scalar grating theory:

ηm =∣∣∣∣ 1dg

∫ dg

0t (x) e

i 2πλ

{[n1 cos(θi)−n2 cos(θm)] f (x)− mλ

dgx}dx

∣∣∣∣2ηm =

{sin

[mπ

(λb

λ−1

)]/[mπ

(λb

λ−1

)]}2

ηM = HIS =[sin

(M

kdp2

)/M sin

(kdp

2

)]2[sin

(kdp

2

)/kdp

2

]2

η={

sin[mπ

(λb

λ−1

)]/mπ

(λb

λ−1

)}2 [sin

(πλb

λN

)/(πλb

λN

)]2

Extended scalar theory:

ηFF ≈ ζηscalar ζ= dm

dg= 1−

[n1 sin(θi)+mλ/dg

]tan(ϕ)√

(n2)2 − (n1 sinθi +mλ/dg

)2

ζ = 1− 1√(n2)2 − (

n1 sinθi +mλ/dg)2

×[n1 sin(θi)+mλ/dg

](mλ/dg

)(∣∣∣∣n1 cos(θi)−√

(n2)2 − [n1 sin(θi)+mλ/dg

]2∣∣∣∣)

ζ= 1−(mλ/dg

)2√(n2)2 − (

mλ/dg)2

(∣∣∣∣n1 −√

(n2)2 − (mλ/dg

)2∣∣∣∣)


108

Equation Summary

Rigorous analysis:

ηrig ≈ ηscalar

[1−Cm (n,θ)

λB

d

]dλ< 1√

max(n1,n2)+n1 sin(θi)

Effective medium theory:

n⊥ = n1n1√(n1)2ζg + (n2)2

(1−ζg

)n∥ =

√(n1)2ζg + (n2)2

(1−ζg

)Grating doublets:

OPD= mλb = h [n2 (λ)cos(θm)−n1 (λ)cos(θi)]d

dλ[n2 (λ)]cos(θm)− d

dλ[n1 (λ)]cos(θi)= m

OPD = mλbl = h1 [n3 (λ)cos(θm1)−n1 (λ)cos(θi1)]+h2 [n2 (λ)cos(θm2)−n3 (λ)cos(θi2)]

h1

{d



}+

h2

{d



}= m

ddλ

[n2 (λ)]h2 cos(θm2)− ddλ

[n1 (λ)]h1 cos(θi1)= m

∆OPD=∆h1 [n1 (λ)−n2 (λ)] ∆OPD=∆h1 [n1 (λ)−1]

Diffractive lens surfaces:

ΦD (λ)= 8Nkλ0/ (D0)2 φ= mλ0 ΦD (λ)=ΦD0λ

λ0

h (θi)= λ0√(n0)2 − (sinθi)2 −cosθi

t (r,θi)= h (θi) [φ (r) mod 2π] h0 = λ0/ (n0 −1)

tb = h0

2N= λ0

(n0 −1)2N∆ f D

chr =∆λ/(ΦD

0 λ0

)Field Guide to Diffractive Optics

109

Equation Summary

ψ (x, y) = 2πλ0

{[√(x− xo)2 + (y− yo)2 + (zo)2 − zo

]−

[√(x− xi)2 + (y− yi)2 + (zi)2 − zi

]}ψ (x, y)= m

K∑k=0

L∑l=0

Akl (x)k (y)l ψ (r)= mN∑

i=0A i (r)2i

Stepped diffractive surfaces:

ΦSDS (λ)=∣∣∣∣ n2 (λ)−n1 (λ)

R0

∣∣∣∣( λλ0−1

)

hSDS =

∣∣∣∣∣∣∣mλ0√

(n2)2 − [sin(θi)]2 −n1 cos(θi)

∣∣∣∣∣∣∣ hSDS =∣∣∣∣ mλ0

n2 −n1

∣∣∣∣Multi-order diffractive lenses:

hm = mλ0

n (λ0)−1mλB

n (λB)−1= kλR

n (λR)−1

Hybrid diffractive lenses:

ΦH (λ)=ΦR (λ)+ΦD (λ)=ΦR0

[1+ Dn (λ)∆λ

n0 (λ)−1

]+ΦD

0

(1+ ∆λ

λ0

)ΦR

0

ΦD0

= ∆λ [n0 (λ)−1]λ0 [n0 (λ)−n (λ)]

=− [n0 (λ)−1]λ0Dn (λ)

ΦAH =ΦD0

{1− [n0 (λ)−1]

λ0Dn (λ)

}=ΦR

0

{1− λ0Dn (λ)

[n0 (λ)−1]

}

Opto-thermal properties:

ξK =−2α ξSDS (λ)=± [ξR (λ)−ξK ]

ξR (λ)= 1

[n2 (λ)−n1 (λ)]

[dn2 (λ)

dT− dn1 (λ)

dT

]−α

ξSDS (λ)= 1

[n2 (λ)−n1 (λ)]

[dn2 (λ)

dT− dn1 (λ)

dT

]+α


110

Equation Summary

Athermalization with diffractive components:

∆Φ

Φ= ξ∆T

ΦK

ΦR=

{d

dT [n2 (λ)]− ddT [n1 (λ)]

}2α [n2 (λ)−n1 (λ)]

− 12

ΦSDSeff (λ)=−

[n2 (λ)−n1 (λ)

R0

]


111

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Bibliography

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114

Index

achromatic condition, 98achromatic hybrid structure,

98achromatic refractive

doublet, 98air-spaced grating doublet,

79Airy beams, 31Airy disk, 14, 29Airy distribution, 7Airy pattern, 13, 14amplitude division, 37amplitude filters, 33amplitude gratings, 36amplitude masks, 29, 87angular dispersion, 49angular switching, 53annular phase plates, 87aperiodically spaced

aperture (ASA), 27aperture spacing, 25apertures with central

obscuration, 20apodization, 19array beam generators, 43artificial dielectrics, 41, 67,

69athermal achromat, 100athermal condition, 100athermal lens, 100athermalization, 94autocollimation, 48, 49axial chromatic aberration,

91

band diagram, 47band-pass filter, 54beam obscuration, 15bidirectional propagation, 87

binary diffractive lenses, 90binary level, 88binary phase gratings, 44,

53, 62binary surface structures,

69, 88blazed binary grating, 67blazed facet angle, 56, 57, 88blazed gratings, 56blazed transmission

gratings, 71blazing condition, 40, 49, 56blazing wavelength, 56Bloch states, 47Bragg phenomenon, 47Bragg plane, 37Brillouin zones, 47broadband blazing, 77, 78broadband diffraction

efficiency, 83, 84

central core, 15central obscuration, 5, 16classical mount, 48coefficient of thermal

expansion (CTE), 99coherent illumination, 28complex amplitude, 2computer-generated

hologram, 46conical mount, 48contrast reduction, 17convolution, 28coupled wave analysis, 70cut-off orders, 51

Dammann grating, 44, 53design wavelength, 89, 94diffraction, 1


115

Index

diffraction efficiency, 10, 61diffraction gratings, 3, 36diffraction-limited (lens

performance), 13diffraction rings, 32diffractive beam-shaping

components, 45diffractive doublet, 77diffractive homogenizers, 42diffractive kinoforms, 88, 89diffractive lens doublet, 96diffractive lens surface

(DLS), 87, 88diffractive optical power, 81diffractive phase cells, 43diffractive polarizer, 55diffractive singlet, 77, 78digital diffractive optical

element, 45, 46doughnut-shaped field, 35duty cycle, 26, 63, 64

echelle, 95effective medium theory, 67,

69effective optical power, 101effective refractive index, 41efficiency angular shift, 76elliptical distribution, 17etalon effect, 54etching, 89evanescent orders, 51extended depth of field, 31extended object, 28extended scalar diffraction

theory, 63, 65

facet angle, 39facet width, 84

fan-out elements, 43far field, 3, 14far-field shaping

components, 42, 43, 45fill factor, 26, 63finite difference method, 70finite difference time

domain (FDTD), 41, 70fluorescence depiction

microscopy, 34focus, 1form birefringence, 69, 73Fourier grating, 43Fourier transform lens, 45fractional Talbot

distributions, 11, 12Fraunhofer approximation,

3Fraunhofer diffraction, 3, 14free spectral range, 50frequency comb, 86Fresnel approximation, 3Fresnel diffraction integral,

3Fresnel lens, 89Fresnel phase plate (FPP), 8Fresnel reflections, 63, 70Fresnel zone number, 4Fresnel zone plate (FZP), 6Fresnel zones, 4

Gaussian lens formula, 92ghost orders, 64grating doublet, 77, 78grating equation, 48grating parameter, 102grating resolution, 50grayscale apodizer, 23grazing incidence, 49


116

Index

grazing mount, 49GRISM, 40groove spacing, 39

Helmholtz equation, 2holographic diffuser, 42holographic optical element,

46holographic recording, 42Huygens-Fresnel principle,

2, 3hybrid achromat, 98hybrid components, 97hybrid structures, 88, 97

ideal lens, 9incoherent illumination, 28incident plane, 52, 104integral method, 70interference, 37isofrequency diagram, 47

kinoform, 43, 88Kirchhoff’s diffraction

integral, 3

Lambertian scatter, 42lamellar grating, 39lateral chromatic

aberration, 91lattice constant, 47lens transfer function, 13linear dispersion, 49linear gratings, 36lithographic techniques, 62Littrow mounting, 48, 56local grating theory, 104local groove spacing, 102longitudinal chromatic

aberration, 91

material dispersion, 91Maxwell’s equations, 2, 66modal analysis, 70monolithic grating doublet,

80multi-order diffractve

lenses, 95multiple apertures, 25multispot beam generator,

43

near-field shapingcomponents, 45

negative refraction, 47

observation point, 4opaque semiplane, 21optical path difference

(OPD), 30, 58, 78, 80optical power, 88, 89, 91, 93,

94optical tweezers, 34optimum profile height, 58,

59optimum zone height, 89opto-thermal coefficient

(OTC), 99

P-polarized (light), 37, 38, 52paraxial approximation, 3passing-off orders, 51passive facet angle, 56, 59pattern distortion, 17peak diffraction efficiency,

73phase delay, 88phase filters, 33phase gratings, 36phase mask, 28, 29phase profile, 92


117

Index

photonic bandgap, 47photonic crystal, 41, 47photoresist, 39point spread function (PSF),

13point spread function

engineering, 31polarization anisotropy, 52,

75polarization dependency, 52,

72, 84polarization extinction ratio,

55polychromatic diffraction

efficiency, 77, 81, 85polychromatic efficiency

modulation, 85polychromatic point spread

function, 81propagating orders, 51pupil filter, 29, 33pupil mask, 29

Rayleigh anomalies, 52Rayleigh resolution

criterion, 28rectangular aperture, 18reflection, 36, 87refractive achromat, 98refractive doublet, 81refractive optical power, 81relative diffraction

efficiency, 61relative feature size, 68resolution, 1, 15resonant domain, 67, 68rigorous diffraction analysis,

65, 66, 70

S-polarized (light), 37, 38, 52

sagittal plane, 52scalar diffraction theory, 2,

60, 70scalar domain, 60, 73serrated aperture, 23serrated edge, 22shadowing effect, 63, 93single-point diamond

turning (SPDT), 93sinusoidal grating, 39Snell’s Law, 48soft-edge aperture, 19soft-edge apodizer, 22spaced grating doublet, 79,

82Sparrow resolution

criterion, 28speckle, 45spectral bandwidth, 91spectral shift, 74spurious diffraction orders,

96staircase lenses, 93star test, 13stepped diffractive surface

(SDS), 87, 93Strehl ratio, 29, 30structural cutoff, 67supercollimation, 47superprism effect, 47superresolved PSF, 29, 35surface relief, 36

Talbot distance, 11Talbot effect, 11Talbot image, 11Talbot plane, 11tangential plane, 52TE-polarized (light), 37, 52


118

Index

thin-element approximation,60

three-dimensionaldiffractive structures, 47

threshold condition, 51TM-polarized (light), 37, 52topological charge, 34Toraldo di Francia, 29transfer etching, 39transmission, 36, 87transverse chromatic

aberration, 91triangular grooves, 39two-dimensional diffractive

structures, 41

very large-scale integration(VLSI), 90

volume Bragg gratings(VBG), 37, 47

volumetric anisotropy, 47vortex phase masks, 31, 34

wave equation, 60wave vector diagrams, 47wavenumber, 2Wood anomalies, 51

zero deflection, 40zero-order gratings, 69


Yakov G. Soskind is Principal Sys-tems Engineer for DHPC Technolo-gies, Inc., in Woodbridge, NJ. He pro-vides innovative solutions and tech-nical expertise to customers in theareas of laser optics, electro-opticalsensors, photonics instrumentation,and system-level integration.

Dr. Soskind is a recognized expert inthe fields of optical design, diffrac-tive optics, laser systems, and illumi-nation. For more than 30 years, hehas contributed to the fields of electro-

optical and photonics engineering, successfully reducing topractice numerous innovative solutions in the form of fiberoptics and photonics devices, diffractive structures, laser optics,imaging, and illumination devices. His innovative work in thefield of diffractive optics has led to the development of noveltypes of diffractive structures that have enhanced diffractionefficiency and are covered in several issued patents.

Dr. Soskind has been awarded more than 20 domestic andinternational patents and has authored and co-authored severalpublications and conference presentations. He also serves onthe technical committees of two international conferences.

SPIE Field Guides The aim of each SPIE Field Guide is to distill a major fi eld of optical science or technology into a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena.

Written for you—the practicing engineer or scientist—each fi eld guide includes the key defi nitions, equations, illustrations, application examples, design considerations, methods, and tips that you need in the lab and in the fi eld.

John E. GreivenkampSeries Editor

P.O. Box 10Bellingham, WA 98227-0010ISBN: 9780819486905SPIE Vol. No.: FG21

Diffractive OpticsYakov G. Soskind

Recent advancements in microfabrication technologies and the development of powerful simulation tools have led to a signifi cant expansion of diffractive optics and diffractive optical components. Instrument developers can choose from a broad range of diffractive optics elements to complement refractive and refl ective components in achieving a desired control of the optical fi eld. This Field Guide provides the operational principles and established terminology of diffractive optics as well as a comprehensive overview of the main types of diffractive optics components. An emphasis is placed on the qualitative explanation of the diffraction phenomenon by the use of fi eld distributions and graphs, providing the basis for understanding the fundamental relations and important trends.

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