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Spie Press Book

Simulating Speckle with Mathematica®
Author(s): Joseph W. Goodman
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Book Description

The speckle phenomenon is ubiquitous, occurring in all regions of the electromagnetic spectrum, as well as in both ultrasound and synthetic-aperture-radar imaging. Speckle occurs whenever radiation is reflected from a surface that is rough on the scale of a wavelength or is passed through a diffusing surface that introduces random path-length delays on the scale of a wavelength. This book is devoted to simulation of speckle phenomena using the software package Mathematica®. Various techniques for simulating speckle are discussed. Simulation topics include first-order amplitude and intensity statistics, speckle phenomena in both imaging and free-space propagation, speckle at low light levels, polarization speckle, phase vortices in speckle, and speckle metrology methods.

This book is a companion to
Speckle Phenomena in Optics: Theory and Applications, Second Edition (SPIE Vol. PM312)

Book Details

Date Published: 10 November 2022
Pages: 102
ISBN: 9781510656543
Volume: PM355

Table of Contents
SHOW Table of Contents | HIDE Table of Contents

1 Introduction
1.1 Mathematica Background
1.2 Speckle Background
1.3 Methods for Simulating Speckle

2 First-Order Statistics of Speckle Amplitude
2.1 Speckle as a Sum of Independent Random Complex Phasors
2.2 Amplitude Statistics of the Sum of Many Random Phasors with Unit Lengths and Random Phases
2.3 Amplitude Statistics of the Sum of a Large Number of Unit-Length Random Phasors and One Large Phasor
2.4 Amplitude Statistics of the Sum of a Small Number of Unit-Length Random Phasors

3 First-Order Statistics of Speckle Intensity
3.1 Intensity Statistics of the Sum of Many Random Phasors with Unit Lengths and Random Phases
3.2 Intensity Statistics of the Sum of a Large Number of Unit-Length Random Phasors and One Large Phasor
3.3 Intensity Statistics of the Sum of a Small Number of Unit-Length Random Phasors
3.4 Intensity Statistics for Sums of Independent Speckle Patterns
3.5 Intensity Statistics of Partially Developed Speckle

4 Simulation of Speckle in Optical Imaging
4.1 Generation of a Discrete Diffuser Array with Correlated Phases
4.2 Speckle in an Imaging Geometry
4.3 The Autocorrelation Function of Speckle Intensity
4.4 The Power Spectrum of Speckle Intensity
4.5 Effect of Speckle on Resolution
4.6 Speckle in Color Images

5 Simulation of Speckle in Free-Space Propagation
5.1 The Diffuser
5.2 The Fresnel Transfer Function Approach
5.3 The Fresnel Transform Approach

6 Speckle at Low Light Levels
6.1 Photocount Image of a Uniform Intensity
6.2 The Negative-Binomial Distribution
6.3 Photocount Image for a Speckle Pattern with Uniform Statistics
6.4 Photocount Image for a Diffuse Structured Object

7 Speckle Phase Vortices
7.1 Generating the Speckle Pattern
7.2 Finding the Zeros of Intensity
7.3 Phase Behavior in the Vicinity of the Zeros of Intensity

8 Polarization Speckle
8.1 The Polarization Ellipse and the Degree of Polarization
8.2 Generating the Two Polarization Components
8.3 Generating a Filtered Speckle Pattern
8.4 Generating the Polarization Ellipses
8.5 Visualizing Polarization Speckle

9 Speckle Simulation for Metrology
9.1 Measurement of In-Plane Displacement
9.2 Electronic Speckle Pattern Interferometry
9.3 Phase-Shifting Speckle Interferometry

Appendix A: Some Subtleties in Speckle Simulation With the 4f System
A.1 Effects on the Speckle Contrast
A.2 Simulation With a Smoothed Phase
A.3 Simulation With an Unsmoothed Phase

Appendix B: Some Subtleties in Dealing With Mathematica Images
B.1 Dimensions of Data Arrays and Images
B.2 Images When the Data Range Exceeds (0, 1)
B.3 Effect of Using ImageAdjust[] on an Image
B.4 Arrays With Bipolar Values
B.5 Problem Encountered When Starting With an Image




The speckle phenomenon is ubiquitous in many fields of science and technology. Speckle phenomena can be seen in many different imaging modalities, including acoustical imaging (e.g., medical ultrasound) and microwave imaging (e.g., synthetic-aperture radar imaging). This book focuses on simulating optical speckle with Mathematica, but the same methods used can in many cases be applied to other imaging modalities.

The reader may wonder why Mathematica has been chosen as the software package for this book. There are several reasons for this choice. First, and most important, Mathematica allows the interspersing of both continuous and discrete calculations under one umbrella. Second, using Mathematica, text, code, and illustrations can be included in the same document. Third, using Mathematica we can create dynamic figures, parameters of which the user can change at will. However, such manipulation cannot be performed in the printed version of the book, so we have avoided manipulable figures in what follows and replaced them by arrays of static figures. Lastly, this author loves Mathematica for its flexibility and comprehensiveness. It seems there are almost an infinite set of capabilities of the program, many of which lie hidden for the novice user but which gradually are revealed as the use of the program increases. This book has been written entirely in Mathematica. It can be read with the full program Mathematica or with the free program Wolfram Player available for download from the Wolfram site. The Mathematica files for all chapters are available online. This book is meant as a companion to the book Speckle Phenomena in Optics: Theory and Applications, 2nd Edition, published by SPIE Press. An extensive list of references can be found in that book.

Joseph W. Goodman
Stanford University

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