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

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Bellingham, Washington 98227-0010 USA

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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.

Printed in the United States of America.

First printing

## Introduction to the Series

Welcome to the * SPIE Field Guides*—a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the

*is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.*

**SPIE Field Guides**Each * SPIE Field Guide* addresses a major field of optical science and technology. The concept of these

*is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights, and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each*

**Field Guides***provide additional information such as related material outside the main scope of the volume, key mathematical relationships, and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.*

**Field Guide**The * SPIE Field Guides* are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new

*topics as well as what material should be added to an individual volume to make these*

**Field Guide***more useful to you. Please contact us at*

**Field Guides****fieldguides@SPIE.org**.

John E. Greivenkamp, **Series Editor**

Optical Sciences Center

The University of Arizona

## The Field Guide Series

Keep information at your fingertips with all of the titles in the Field 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

## Field Guide to Diffractive Optics

Recent advancements in microfabrication technologies as well as the development of powerful simulation tools have led to a significant expansion of diffractive optics and the commercial availability of cost-effective diffractive optical components. Instrument developers can choose from a broad range of diffractive optical elements to complement refractive and reflective components in achieving a desired control of the optical field.

Material required for understanding the diffractive phenomenon is widely dispersed throughout numerous literature sources. This * Field Guide* offers scientists and engineers a comprehensive reference in the field of diffractive optics. College students and photonics enthusiasts will broaden their knowledge and understanding of diffractive optics phenomena.

The primary objectives of this * Field Guide* are to familiarize the reader with operational principles and established terminology in the field of diffractive optics, as well as to provide 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 field distributions and graphs, providing the basis for understanding the fundamental relations and the important trends.

I would like to thank SPIE Press Manager Timothy Lamkins and Series Editor John Greivenkamp for the opportunity to write a Field Guide for one of the most fundamental physical optics phenomena, as well as SPIE Press Senior Editor Dara Burrows 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 the material while helping with proofreading the manuscript.

Yakov G. Soskind

August 2011

## Table of Contents

**Glossary of Symbols** xi

**Diffraction Fundamentals** 1

The Diffraction Phenomenon 1

Scalar Diffraction 2

Paraxial Approximation 3

**Fresnel Diffraction** 4

Fresnel Diffraction 4

Apertures with Integer Number of Fresnel Zones 5

Fresnel Zone Plates 6

Fresnel Zone Plate Properties 7

Fresnel Phase Plates 8

Comparing Fresnel Plates and Ideal Lenses 9

Efficiency of Fresnel Plates and Ideal Lenses 10

Talbot Effect 11

Fractional Talbot Distributions 12

**Fraunhofer Diffraction** 13

Fraunhofer Diffraction 13

Diffraction of Waves with Finite Sizes 14

Diffraction on Ring-Shaped Apertures 15

Energy Redistribution within Diffraction Rings 16

Diffraction on Noncircular Apertures 17

Rectangular and Diamond-Shaped Apertures 18

**Apodized Apertures** 19

Apodized Apertures 19

Apodized Apertures with Central Obscuration 20

Field Obstruction by an Opaque Semiplane 21

Apodization with Serrated Edges 22

Serrated Apertures as Apodizers 23

**Diffraction by Multiple Apertures** 24

Diffraction by Multiple Apertures 24

Effects of Aperture Spacing 25

Aperture Fill Factor 26

Aperiodically Spaced Apertures 27

**Resolution Limit in Optical Instruments** 28

Resolution Limit in Optical Instruments 28

Superresolution Phenomenon 29

Superresolution with Two-Zone Phase Masks 30

Point Spread Function Engineering 31

Adjusting Diffraction-Ring Intensity 32

Amplitude and Phase Filter Comparison 33

Vortex Phase Masks 34

Combining Amplitude and Vortex Phase Masks 35

**Diffractive Components** 36

Diffraction Gratings 36

Volume Bragg Gratings 37

Polarization Dependency of Volume Bragg Gratings 38

One-Dimensional Surface-Relief Gratings 39

GRISM Elements 40

Two-Dimensional Diffractive Structures 41

Holographic Diffusers 42

Multispot Beam Generators 43

Design of Fan-Out Elements 44

Diffractive Beam-Shaping Components 45

Digital Diffractive Optics 46

Three-Dimensional Diffractive Structures 47

**Grating Properties** 48

Grating Equation 48

Grating Properties 49

Free Spectral Range and Resolution 50

Grating Anomalies 51

Polarization Dependency of Grating Anomalies 52

Gratings as Angular Switches 53

Gratings as Optical Filters 54

Gratings as Polarizing Components 55

**Blazing Condition** 56

Blazing Condition 56

Blazed Angle Calculation 57

Optimum Blazed Profile Height 58

**Scalar Diffraction Theory of a Grating** 60

Scalar Diffraction Theory of a Grating 60

Diffraction Efficiency 61

Blaze Profile Approximation 62

**Extended Scalar Diffraction Theory** 63

Extended Scalar Diffraction Theory 63

Duty Cycle and Ghost Orders 64

Extended Scalar versus Rigorous Analysis 65

**Gratings with Subwavelength Structures** 66

Gratings with Subwavelength Structures 66

Blazed Binary Gratings 67

Relative Feature Size in the Resonant Domain 68

Effective Medium Theory 69

Scalar Diffraction Limitations and Rigorous Theory 70

**Rigorous Analysis of Transmission Gratings** 71

Analysis of Blazed Transmission Gratings 71

Polarization Dependency and Peak Efficiencies 72

Peak Efficiency of Blazed Profiles 73

Wavelength Dependency of Efficiency 74

Efficiency Changes with Incident Angle 75

Diffraction Efficiency for Small Feature Sizes 76

**Polychromatic Diffraction Efficiency** 77

Polychromatic Diffraction Efficiency 77

Monolithic Grating Doublet 78

Spaced Grating Doublet 79

Monolithic Grating Doublet with Two Profiles 80

Diffractive and Refractive Doublets: Comparison 81

**Efficiency of Spaced Grating Doublets** 82

Efficiency of Spaced Grating Doublets 82

Sensitivity to Fabrication Errors 83

Facet Width and Polarization Dependency 84

Sensitivity to Axial Components Spacing 85

Frequency Comb Formation 86

**Diffractive Components with Axial Symmetry** 87

Diffractive Components with Axial Symmetry 87

Diffractive Lens Surfaces 88

Diffractive Kinoforms 89

Binary Diffractive Lenses 90

Optical Power of a Diffractive Lens Surface 91

Diffractive Surfaces as Phase Elements 92

Stepped Diffractive Surfaces 93

Properties of Stepped Diffractive Surfaces 94

Multi-order Diffractive Lenses 95

Diffractive Lens Doublets 96

**Diffractive Surfaces in Optical Systems** 97

Diffractive Lens Surfaces in Optical Systems 97

Achromatic Hybrid Structures 98

Opto-thermal Properties of Optical Components 99

Athermalization with Diffractive Components 100

Athermalization with SDSs 101

**Appendix: Diffractive Raytrace** 102

**Equation Summary** 105

**Bibliography** 111

**Index** 114

#### Glossary of Symbols and Acronyms

ASMA

aperiodically spaced multiple apertures

**B**

base length of a PRISM

**C**_{R}

incident wave obliquity factor

**C**_{S}

diffracted wave obliquity factor

CGH

computer-generated hologram

**d**

diameter of central obscuration

**d _{B}**

Bragg plane spacing

**d _{g}**

grating period or groove spacing

**d _{i}**

step width of * i*th zone

D

aperture diameter or lateral size

**D _{A}**

Airy disk diameter

**D _{n}**

material dispersion

**D**_{0}

lens clear aperture diameter

DC

duty cycle

DDO

digital diffractive optics

DLS

diffractive lens surface

**e**

aperture obscuration

* E*(ρ, φ)

complex electric field in polar coordinates

**E**_{⊥}

electric field normal to the grating grooves

**E**_{∥}

electric field parallel to the grating grooves

**f**

focal length of a lens

${f}_{0}^{D}$

nominal focal length of a diffractive surface

FDTD

finite difference time domain

FWHM

full width at half maximum

FPP

Fresnel phase plate

FZP

Fresnel zone plate

**h**

grating profile depth or height

**h _{i}**

step height of * i*th zone

**h _{m}**

profile height of a multi-order diffractive lens

**h**_{opt}

optimum grating profile depth (height)

**h**_{SDS}

step height of SDS

HOE

holographic optical element

$\overrightarrow{\mathit{i}}$

unit vector codirectional with * x* axis

* I*(

*)*

**r**radial intensity distribution

IR

infrared

$\overrightarrow{\mathit{j}}$

unit vector codirectional with * y* axis

**J**_{0}(ρ)

Bessel function of the first kind of the zero order

**J**_{1}(ρ)

Bessel function of the first kind of the first order

$\overrightarrow{\mathit{k}}$

unit vector codirectional with * z* axis

**k**(

*,*

**x***)*

**y**wave vector of the propagating wavefront

**k**_{0}

wavenumber

**L**

observation distance

${L}_{\mathrm{T}}^{m}$

Talbot distance of order **m**

LED

light-emitting diode

**m**

diffraction order

**n**

refractive index of optical material

**n**_{1}

refractive index before optical interface

**n**_{2}

refractive index after optical interface

**n _{d}**

refractive index of diffraction grating layer

**n _{p}**

refractive index of prism material

${n}_{\perp}$

effective index for the electric field ${\mathit{E}}_{\perp}$

${n}_{\parallel}$

effective index for the electric field ${\mathit{E}}_{\parallel}$

**N**

number of binary levels

**N _{F}**

Fresnel zone number

**N _{g}**

number of grating (groove) facets

**N _{k}**

number of kinoform zones

OPD

optical path difference

PSF

point spread function

$\overrightarrow{\mathit{q}}$

vector orthogonal to the grating plane of symmetry at the point of intersection

**Q**

grating “thickness” parameter

$\overrightarrow{\mathit{r}}$

vector normal to the grating surface at the incoming ray intersection point

${\overrightarrow{\mathit{r}}}_{12}$

vector connecting points in two lateral planes

**R**_{0}

substrate radius of curvature

$\overrightarrow{\mathit{S}}$

propagation direction vector before diffractive surface

$\overrightarrow{{\mathit{S}}^{\text{'}}}$

propagation direction vector after diffractive surface

SDS

stepped diffractive surface

SPDT

single point diamond turning

**t**

axial spacing of a grating doublet

**T**

effective thickness of volume phase grating

**t _{b}**

thickness of a binary lens level

**T _{g}**

geometrical transmission pattern of a facet

TE

transverse electric

TM

transverse magnetic

* U*(

*,*

**x***,*

**y***)*

**z**complex field amplitude

UV

ultraviolet

VBG

volume Bragg grating

VLSI

very large-scale integration

* W*(

*,*

**x***)*

**y**propagating wavefront

**W _{g}**

grating width

$\overrightarrow{\mathit{z}}$

vector normal to the two reference planes

α

coefficient of thermal expansion

α_{B}

angle of the incident light after refraction into the volume phase medium

α_{d}

deflection angle

β

diffracted angle inside the volume phase medium

γ

angle between the Bragg planes and the incident light

δ

minimum feature size of the diffractive component

ε

grating minor (secondary) facet angle

ζ

fill factor of radiation

η

normalized diffraction efficiency

η_{m}

diffraction efficiency in * m*th diffraction order

η_{M}

diffraction efficiency of a grating with * M* facets

η_{P}

diffraction efficiency of * P*-polarized light

η_{S}

diffraction efficiency of * S*-polarized light

θ_{d}

diffraction angle

θ_{i}

angle of incidence

θ_{m}

* m*th order diffraction angle

${\text{\theta}}_{\mathrm{m}}^{L}$

* m*th order diffraction angle in Littrow mount

${\text{\theta}}_{m}^{{\text{\lambda}}_{l}}$

* m*th order diffraction angle of the wavelength λ

_{l}

${\text{\theta}}_{m}^{{\text{\lambda}}_{s}}$

* m*th order diffraction angle of the wavelength λ

_{s}

θ_{φ}

incidence angle with respect to the grating facet

λ

wavelength of light

λ_{b}

blazing wavelength

${\text{\lambda}}_{b}^{L}$

blazing wavelength in Littrow configuration (mount)

λ_{s}

the shortest wavelength within the spectral range

λ_{l}

the longest wavelength within the spectral range

λ_{0}

design wavelength

ξ

opto-thermal coefficient of a surface

ρ

radial coordinate

υ

volume phase grating parameter

φ

grating primary facet angle

φ_{b}

grating facet blaze angle

φ_{i}

facet angle of **i**^{th} kinoform zone

φ_{p}

grating passive facet angle

φ_{max}

maximum value of facet blazed angle

ϕ

optical path difference

Φ^{AH}

optical power of an achromatic hybrid

Φ^{D}

optical power of a diffractive surface or lens

Φ^{H}

optical power of a hybrid surface or lens

Φ^{R}

optical power of a refractive surface or lens

Φ^{SDS}

optical power of SDS

${\text{\Phi}}_{\mathrm{eff}}^{\mathrm{SDS}}$

effective optical power of SDS

ψ(* r*)

radial phase profile of a diffractive surface

ψ(* x*,

*)*

**y**phase profile of a diffractive surface

Δ**f**_{chr}

axial or longitudinal chromatic aberration

Δ**H**_{chr}

lateral or transverse chromatic aberration

Δλ

spectral bandwidth

Δλ_{FSR}

free spectral range of a grating

Δ**n**

refractive index modulation

Λ

grating parameter