### Spie Press Book

Speckle Phenomena in Optics: Theory and Applications, Second EditionFormat | Member Price | Non-Member Price |
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Speckle, a granular structure appearing in images and diffraction patterns produced by objects that are rough on the scale of an optical wavelength, is a ubiquitous phenomenon, appearing in optics, acoustics, microwaves, and other fields. This book provides comprehensive coverage of this subject, including both the underlying statistical theory and the applications of this phenomenon. This second edition offers improvements of several topics and addition of significant amounts of new material, including discussion of: generalized random walks, speckle in the eye, polarization speckle (and the statistics of the Stokes parameters in a speckle pattern), the effects of angle and wavelength changes on speckle, the statistics of speckle from “smooth” surfaces, and a spectrometer based on speckle. Many new references are also included. As with the first edition, a multitude of areas of application are covered.

“In the mid-1960s, a young Joseph Goodman, working at the Stanford Electronics Laboratories, wrote a detailed, but unpublished, report that established the basic statistical properties of speckle. Forty years later he wrote the most comprehensive book on the subject—*Speckle Phenomena in Optics: Theory and Applications *(2007), which became an instant classic and is the definitive text book. Goodman, the master of his subject, now presents an excellent second edition of an already excellent book.”

—**Christopher Dainty, National University of Ireland, Galway, Ireland**

“This second edition of Goodman’s classic* Speckle Phenomena in Optics *tells it all. It gives a detailed description of speckle, explains the techniques for suppressing speckle, and gives several applications of speckle in imaging and metrology.”

—**James C. Wyant, University of Arizona**

“When Goodman’s* Speckle Phenomena in Optics *was published in 2007, it became my primary reference for understanding speckle, which occurs in many diverse areas such as coherent optics, radar, and ultrasound. This second edition maintains Goodman’s signature clarity.”

—**James R. Fienup, University of Rochester**

*Simulating Speckle with Mathematica®*(SPIE Vol. PM355)

### Table of Contents

*Preface to Second Edition**Preface to First Edition***1 Origins and Manifestations of Speckle**- 1.1 General Background
- 1.2 Intuitive Explanation of the Cause of Speckle
- 1.3 Some Mathematical Preliminaries
**2 Random Phasor Sums**- 2.1 First and Second Moments of the Real and Imaginary Parts of the Resultant Phasor
- 2.2 Random Walk with a Large Number of Independent Steps
- 2.3 Random Phasor Sum Plus a Known Phasor
- 2.4 Sums of Random Phasor Sums
- 2.5 Random Phasor Sums with a Finite Number of Equal-Length Components
- 2.6 Random Phasor Sums with a Nonuniform Distribution of Phases
**3 First-Order Statistical Properties of Optical Speckle**- 3.1 Definition of Intensity
- 3.2 First-Order Statistics of the Intensity and Phase
- 3.2.1 Large number of random phasors
- 3.2.2 Constant phasor plus a random phasor sum
- 3.2.3 Finite number of equal-length phasors
- 3.2.4 Finite number of random-length phasors
- 3.2.5 Random number of random-length phasors
- 3.3 Sums of Speckle Patterns
- 3.3.1 Sums on an amplitude basis
- 3.3.2 Sum of two independent speckle intensities
- 3.3.3 Sum of
*N*independent speckle intensities - 3.3.4 Sums of correlated speckle intensities
- 3.4 Partially Developed Speckle
- 3.5 Speckled Speckle, or Compound Speckle Statistics
- 3.5.1 Speckle driven by a negative-exponential intensity distribution
- 3.5.2 Speckle driven by a gamma intensity distribution
- 3.5.3 Sums of independent speckle patterns driven by a gamma intensity distribution
**4 Higher-Order Statistical Properties of Speckle**- 4.1 Multivariate Gaussian Statistics
- 4.2 Application to Speckle Fields
- 4.3 Multidimensional Statistics of Speckle Amplitude, Phase, and Intensity
- 4.3.1 The bivariate density function
- 4.3.2 Joint density function of the amplitudes
- 4.3.3 Joint density function of the phases
- 4.3.4 Joint density function of the intensities
- 4.4 Bivariate Statistics of a Linearly Polarized Speckle Pattern
- 4.5 Speckle and Polarization
- 4.6 Statistics of the Stokes Parameters in a Fully Developed Speckle Pattern
- 4.6.1 Statistics of
*S*_{0} - 4.6.2 Statistics of
*S*_{1} - 4.6.3 Statistics of
*S*_{2} - 4.6.4 Statistics of
*S*_{3} - 4.6.5 Polarization speckle
- 4.7 Statistics of Integrated and Blurred Speckle
- 4.7.1 Mean and variance of integrated speckle
- 4.7.2 Approximate result for the probability density function of integrated intensity
- 4.7.3 "Exact" result for the probability density function of integrated intensity
- 4.7.4 Lorenz transformation
- 4.8 Statistics of Derivatives of Speckle Intensity and Phase
- 4.8.1 Background
- 4.8.2 Derivatives of speckle phase: ray directions in a speckle pattern
- 4.8.3 Derivatives of speckle intensity
- 4.8.4 Level crossings of speckle patterns
- 4.9 Zeros of Speckle Patterns: Optical Vortices
- 4.9.1 Conditions required for a zero of intensity to occur
- 4.9.2 Properties of speckle phase in the vicinity of a zero of intensity
- 4.9.3 The density of vortices in fully developed speckle
- 4.9.4 The density of vortices for fully developed speckle plus a coherent background
**5 Spatial Structure of Speckle**- 5.1 Autocorrelation Function and Power Spectrum of Speckle
- 5.1.1 Free-space propagation geometry
- 5.1.2 Imaging geometry
- 5.2 Speckle Size in Depth
- 5.3 Dependence of Speckle on Scatterer Microstructure
- 5.3.1 Surface vs. volume scattering
- 5.3.2 Effect of a finite correlation area of the scattered wave
- 5.3.3 A regime where speckle size is independent of scattering spot size
- 5.4 Effects of Surface Microstructure on the Reflected Wave
- 5.4.1 Correlation function of the reflected wave
- 5.4.2 Illumination normal to the surface
- 5.5 Effects of a Change of Illumination Angle in Free-Space Propagation
- 5.6 Effect of a Change of Wavelength in Free-Space Propagation
- 5.7 Simultaneous Changes of Illumination Angle and Wavelength
- 5.8 Speckle in a Simple Imaging System: In-Focus Case
- 5.9 Speckle in a Simple Imaging System: Out-of-Focus Cases
- 5.10 Effects of Pupil Size and rms Roughness on Speckle Contrast
- 5.11 Properties of Speckle Resulting from Volume Scattering
**6 Optical Methods for Suppressing Speckle**- 6.1 Polarization Diversity
- 6.2 Temporal Averaging with a Moving Diffuser
- 6.2.1 Background
- 6.2.2 Smooth object
- 6.2.3 Rough object
- 6.3 Wavelength and Angle Diversity
- 6.3.1 Free-space propagation: reflection geometry
- 6.3.2 Free-space propagation: transmission geometry
- 6.3.3 Imaging geometry
- 6.4 Temporal and Spatial Coherence Reduction
- 6.4.1 Coherence concepts in optics
- 6.4.2 Moving diffusers and coherence reduction
- 6.4.3 Speckle suppression by reduction of temporal coherence
- 6.4.4 Speckle suppression by reduction of spatial coherence
- 6.5 Use of Temporal Coherence to Destroy Spatial Coherence
- 6.6 Compounding Speckle Suppression Techniques
**7 Speckle in Certain Imaging Applications**- 7.1 Speckle in the Eye
- 7.2 Speckle in Holography
- 7.2.1 Principles of holography
- 7.2.2 Speckle suppression in holographic images
- 7.3 Speckle in Optical Coherence Tomography
- 7.3.1 Overview of the OCT imaging technique
- 7.3.2 Analysis of OCT
- 7.3.3 Speckle and speckle suppression in OCT
- 7.4 Speckle in Optical Projection Displays
- 7.4.1 Anatomies of projection displays
- 7.4.2 Speckle suppression in projection displays
- 7.4.3 Polarization diversity
- 7.4.4 A moving screen
- 7.4.5 Wavelength diversity
- 7.4.6 Angle diversity
- 7.4.7 Overdesign of the projection optics
- 7.4.8 Changing the diffuser projected onto the screen
- 7.4.9 Specially designed screens
- 7.5 Speckle in Projection Microlithography
- 7.5.1 Coherence properties of excimer lasers
- 7.5.2 Temporal speckle
- 7.5.3 From exposure fluctuations to line position fluctuations
- 7.6 Speckle in the Image of a "Smooth" Surface
- 7.6.1 Symmetry of the spectral intensity in the focal plane
- 7.6.2 Bright-field imaging
- 7.6.3 Dark-field imaging
**8 Speckle in Certain Nonimaging Applications**- 8.1 Speckle in Multimode Fibers
- 8.1.1 Modal noise in fibers
- 8.1.2 Statistics of constrained speckle
- 8.1.3 Frequency dependence of modal noise
- 8.2 Effects of Speckle on Optical Radar Performance
- 8.2.1 Spatial correlation of the speckle returned from distant targets
- 8.2.2 Speckle at low light levels
- 8.2.3 Detection statistics: direct detection
- 8.2.4 Detection statistics: heterodyne detection
- 8.2.5 Comparison of direct detection and heterodyne detection
- 8.2.6 Reduction of the effects of speckle in optical radar detection
- 8.3 A Spectrometer Based on Speckle
**9 Speckle and Metrology**- 9.1 Speckle Photography
- 9.1.1 In-plane displacement
- 9.1.2 Simulation
- 9.1.3 Properties of the spectra
*I*(_{k}*v*,_{X}*v*)_{Y} - 9.1.4 Limitations on the amount of the motion (
*x*_{0},*y*_{0}) - 9.1.5 Analysis with multiple specklegram windows
- 9.1.6 Object rotation
- 9.2 Speckle Interferometry
- 9.2.1 Systems that use photographic detection
- 9.2.2 Electronic speckle pattern interferometry (ESPI)
- 9.2.3 Speckle shearing interferometry
- 9.3 From Fringe Patterns to Phase Maps
- 9.3.1 The Fourier transform method
- 9.3.2 Phase-shifting speckle interferometry
- 9.3.3 Phase unwrapping
- 9.4 Vibration Measurement Using Speckle
- 9.5 Speckle and Surface Roughness Measurements
- 9.5.1 RMS surface height and surface covariance area from speckle contrast
- 9.5.2 RMS surface height from two-wavelength decorrelation
- 9.5.3 RMS surface height from two-angle decorrelation
- 9.5.4 Surface-height standard deviation and covariance function from measurement of the angular power spectrum
**10 Speckle in Imaging Through the Atmosphere**- 10.1 Background
- 10.2 Short- and Long-Exposure Point-Spread Functions
- 10.3 Long- and Short-Exposure Average Optical Transfer Functions
- 10.4 Statistical Properties of the Short-Exposure OTF and MTF
- 10.5 Astronomical Speckle Interferometry
- 10.5.1 Object information that is retrievable
- 10.5.2 Results of a more complete analysis of the form of the speckle transfer function
- 10.6 The Cross-Spectrum or Knox−Thompson Technique
- 10.6.1 The cross-spectrum transfer function
- 10.6.2 Recovering full object information from the cross-spectrum
- 10.7 The Bispectrum Technique
- 10.7.1 The bispectrum transfer function
- 10.7.2 Recovering full object information from the bispectrum
- 10.8 Speckle Correlography
**Appendix A Linear Transformations of Speckle Fields****Appendix B Contrast of Partially Developed Speckle Intensity and Phase****Appendix C Calculations Leading to the Statistics of Derivatives of Speckle**- C.1 The Correlation Matrix
- C.2 Joint Density Function of the Derivatives of Phase
- C.3 Joint Density Function of the Derivatives of Intensity
- C.4 Parameters for Various Scattering Spot Shapes
**Appendix D Wavelength and Angle Dependence When a Dynamic Diffuser is Projected onto a Random Screen**- D.1 Free-Space Geometry
- D.2 Imaging Geometry
**Appendix E Speckle Contrast When a Dynamic Diffuser is Projected onto a Random Screen**- E.1 Random Phase Diffusers
- E.2 Diffuser that Just Fills the Projection Optics
- E.3 Diffuser that Overfills the Projection Optics
**Appendix F Statistics of Constrained Speckle****Appendix G Sample***Mathematica*Programs for Simulating Speckle- G.1 Speckle Simulation with Free-Space Propagation
- G.2 Speckle Simulation with an Imaging Geometry
*References**Index*

## Preface to Second Edition

To explain the origins of this second edition of *Speckle Phenomena in Optics*, it is first helpful
to trace the history of the first edition. The book was originally published by Roberts &
Company Publishers (a small startup publisher) in 2007. Roberts & Company was subsequently
acquired by MacMillan Publishing Company. The book continued to be published
by MacMillan under the Roberts & Company label. In 2019, MacMillan decided not to
continue marketing and selling the book, and returned the copyright to me, the author.
In the meantime, over a period of three or four years, I had been adding material to the
manuscript and making certain revisions to the original manuscript in preparation for an
improved second edition. In this Preface, I outline some of the improvements that the
reader will find in the second edition.

The first set of improvements I would call "stylistic." A reviewer of the first edition pointed out that some equations (especially those with exponents) had been set in a typeface that was too small to read without a magnifying glass. All equations with this problem have been modified to improve their readability. A second stylistic problem was that in many figures, the curves representing results of the analyses had been drawn with a point size that was too small, so that they appeared too faint in the figures, at least too faint for my taste. This error has been corrected throughout.

In addition to these stylistic changes, considerable material has been added to this new edition:

• A new Section 3.2.4 considers a random walk with random-length phasors, introducing different models for the length statistics;

• A new Section 3.2.5 considers a random walk with a random number of random-length phasors;

• Sections 4.1 through 4.3 have been generalized so that they can be applied to the subject of polarization speckle in later sections;

• In Chapter 4, entirely new Sections 4.5 and 4.6 have been added covering the subject of polarization in speckle patterns, including statistics of the Stokes parameters and polarization speckle;

• Sections 5.5, 5.6, and 5.7 are new simplified explanations of the effects of a change of angle-of-illumination and change of wavelength on the spatial structure of speckle;

• Sections 5.8 and 5.9 are new simplified explanations of the spatial structure of speckle in an in-focus imaging system and an out-of-focus imaging system;

• Section 6.1 in the original edition contained an incorrect discussion of speckle in the eye; this discussion has been corrected in the second edition;

• Section 7.6 is an entirely new discussion of the characteristics of speckle produced by "smooth" surfaces;

• Section 8.3 is a new section discussing how to create a spectrometer that measures the spectrum of a source from the speckle it produces; and

• A number of new references have been introduced in the second edition.

The reader may note that some of the material presented in Chapter 4 of the first edition has been moved to a new Chapter 5 entitled "Spatial Structure of Speckle" in the second edition, a move that seemed logical when coupled with the several new sections introduced in that chapter.

I would like to thank Prof. Mitsuo Takeda and Prof. Wei Wang for educating me on the subject of polarization speckle. I would like to extend a heartfelt thanks to two anonymous reviewers whose comments were extremely helpful in preparing the final version of the manuscript. Again I thank my wife, Honmai, for tolerating my many hours at the computer working on this new edition. Finally, I thank Ms. Dara Burrows of SPIE for her extremely careful copy editing of the manuscript; her suggestions improved the book significantly.

**Joseph W. Goodman**

November 2019

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