### Spie Press Book

Simulating Speckle with*Mathematica*

^{®}

Format | Member Price | Non-Member Price |
---|---|---|

*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)

Pages: 102

ISBN: 9781510656543

Volume: PM355

**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 4***f*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
**Acknowledgements****References**

**Preface**

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

**© SPIE.**Terms of Use