Preface ..................................................... ix
1 Origins and Manifestations of Speckle ........................ 1
1.1 General Background ...................................... 1
1.2 Intuitive Explanation of the Cause of Speckle ........... 2
1.3 Some Mathematical Preliminaries ......................... 5
2 Random Phasor Sums ........................................... 7
2.1 First and Second Moments of the Real and Imaginary
Parts of the Resultant Phasor ........................... 8
2.2 Random Walk with a Large Number of Independent Steps ... 10
2.3 Random Phasor Sum Plus a Known Phasor .................. 13
2.4 Sums of Random Phasor Sums ............................. 17
2.5 Finite Number of Equal Length Components ............... 17
2.6 Nonuniform Distribution of Phases ...................... 19
3 First-Order Statistical Properties .......................... 25
3.1 Definition of Intensity ................................ 25
3.2 First-Order Statistics of the Intensity and Phase ...... 27
3.2.1 Large Number of Random Phasors .................. 27
3.2.2 Constant Phasor plus a Random Phasor Sum ........ 30
3.2.3 Finite Number of Equal-Length Phasors ........... 34
3.3 Sums of Speckle Patterns ............................... 37
3.3.1 Sums on an Amplitude Basis ...................... 38
3.3.2 Sum of Two Independent Speckle Intensities ...... 38
3.3.3 Sum of N Independent Speckle Intensities ........ 42
3.3.4 Sums of Correlated Speckle Intensities .......... 44
3.4 Partially Polarized Speckle ............................ 47
3.5 Partially Developed Speckle ............................ 50
3.6 Speckled Speckle, or Compound Speckle Statistics ....... 53
3.6.1 Speckle Driven by a Negative Exponential
Intensity Distribution .......................... 54
3.6.2 Speckle Driven by a Gamma Intensity
Distribution .................................... 55
3.6.3 Sums of Independent Speckle Patterns Driven by
a Gamma Intensity Distribution .................. 57
4 Higher-Order Statistical Properties of Speckle .............. 59
4.1 Multivariate Gaussian Statistics ....................... 59
4.2 Application to Speckle Fields .......................... 60
4.3 Multidimensional Statistics of Speckle ................. 62
4.3.1 Joint Density Function of the Amplitudes ........ 64
4.3.2 Joint Density Function of the Phases ............ 65
4.3.3 Joint Density Function of the Intensities ....... 68
4.4 Autocorrelation Function and Power Spectrum of
Speckle ................................................ 73
4.4.1 Free-Space Propagation Geometry ................. 73
4.4.2 Imaging Geometry ................................ 80
4.4.3 Speckle Size in Depth ........................... 82
4.5 Dependence of Speckle on Scatterer Microstructure ...... 84
4.5.1 Surface vs. Volume Scattering ................... 85
4.5.2 Effect of a Finite Correlation Area of the
Scattered Wave .................................. 85
4.5.3 A Regime where Speckle Size Is Independent of
Scattering Spot Size ............................ 90
4.5.4 Relation between the Correlation Areas of the
Scattered Wave and the Surface Height
Fluctuations—Surface Scattering ................. 92
4.5.5 Dependence of Speckle Contrast on Surface
Roughness—Surface Scattering .................... 98
4.5.6 Properties of Speckle Resulting from Volume
Scattering ..................................... 102
4.6 Statistics of Integrated and Blurred Speckle .......... 105
4.6.1 Mean and Variance of Integrated Speckle ........ 106
4.6.2 Approximate Result for the Probability
Density Function of Integrated Intensity ....... 111
4.6.3 "Exact" Result for the Probability Density
Function of Integrated Intensity ............... 113
4.6.4 Integration of Partially Polarized Speckle
Patterns ....................................... 118
4.7 Statistics of Derivatives of Speckle Intensity and
Phase ................................................. 120
4.7.1 Background ..................................... 121
4.7.2 Parameters for Various Scattering Spot
Shapes ......................................... 123
4.7.3 Derivatives of Speckle Phase: Ray Directions
in a Speckle Pattern ........................... 124
4.7.4 Derivatives of Speckle Intensity ............... 127
4.7.5 Level Crossings of Speckle Patterns ............ 130
4.8 Zeros of Speckle Patterns: Optical Vortices ........... 133
4.8.1 Conditions Required for a Zero of Intensity
to Occur ....................................... 133
4.8.2 Properties of Speckle Phase in the Vicinity
of a Zero of Intensity ......................... 133
4.8.3 The Density of Vortices in Fully Developed
Speckle ........................................ 136
4.8.4 The Density of Vortices for Fully Developed
Speckle Plus a Coherent Background ............. 138
5 Optical Methods for Suppressing Speckle .................... 141
5.1 Polarization Diversity ................................ 142
5.2 Temporal Averaging with a Moving Diffuser ............. 143
5.2.1 Background ..................................... 143
5.2.2 Smooth Object .................................. 149
5.2.3 Rough Object ................................... 151
5.3 Wavelength and Angle Diversity ........................ 153
5.3.1 Free-Space Propagation, Reflection Geometry .... 154
5.3.2 Free-Space Propagation, Transmission
Geometry ....................................... 164
5.3.3 Imaging Geometry ............................... 167
5.4 Temporal and Spatial Coherence Reduction .............. 170
5.4.1 Coherence Concepts in Optics ................... 170
5.4.2 Moving Diffusers and Coherence Reduction ....... 172
5.4.3 Speckle Suppression by Reduction of Temporal
Coherence ...................................... 175
5.4.4 Speckle Suppression by Reduction of Spatial
Coherence ...................................... 178
5.5 Use of Temporal Coherence to Destroy Spatial
Coherence ............................................. 185
5.6 Compounding Speckle Suppression Techniques ............ 186
6 Speckle in Certain Imaging Applications .................... 187
6.1 Speckle in the Eye .................................... 187
6.2 Speckle in Holography ................................. 190
6.2.1 Principles of Holography ....................... 190
6.2.2 Speckle Suppression in Holographic Images ...... 192
6.3 Speckle in Optical Coherence Tomography ............... 195
6.3.1 Overview of the OCT Imaging Technique .......... 195
6.3.2 Analysis of OCT ................................ 196
6.3.3 Speckle and Speckle Suppression in OCT ......... 200
6.4 Speckle in Optical Projection Displays ................ 203
6.4.1 Anatomies of Projection Displays ............... 204
6.4.2 Speckle Suppression in Projection Displays ..... 208
6.4.3 Polarization Diversity ......................... 208
6.4.4 A Moving Screen ................................ 209
6.4.5 Wavelength Diversity ........................... 211
6.4.6 Angle Diversity ................................ 211
6.4.7 Overdesign of the Projection Optics ............ 213
6.4.8 Changing Diffuser Projected onto the Screen .... 214
6.4.9 Specially Designed Screens ..................... 225
6.5 Speckle in Projection Microlithography ............... 228
6.5.1 Coherence Properties of Excimer Lasers ......... 228
6.5.2 Temporal Speckle ............................... 229
6.5.3 From Exposure Fluctuations to Line Position
Fluctuations ................................... 231
7 Speckle in Certain Nonimaging Applications ................. 235
7.1 Speckle in Multimode Fibers ........................... 235
7.1.1 Modal Noise in Fibers .......................... 237
7.1.2 Statistics of Constrained Speckle .............. 239
7.1.3 Frequency Dependence of Modal Noise ............ 243
7.2 Effects of Speckle on Optical Radar Performance ....... 248
7.2.1 Spatial Correlation of the Speckle Returned
from Distant Targets ........................... 250
7.2.2 Speckle at Low Light Levels .................... 252
7.2.3 Detection Statistics—Direct Detection .......... 256
7.2.4 Detection Statistics—Heterodyne Detection ...... 260
7.2.5 Comparison of Direct Detection and Heterodyne
Detection ...................................... 270
7.2.6 Reduction of the Effects of Speckle in
Optical Radar Detection ........................ 273
8 Speckle and Metrology ...................................... 275
8.1 Speckle Photography ................................... 275
8.1.1 In-Plane Displacement .......................... 277
8.1.2 Simulation ..................................... 279
8.1.3 Properties of the Spectra  k(vX,vY) ............ 281
8.1.4 Limitations on the Size of the Motion
(x0,y0) ........................................ 284
8.1.5 Analysis with Multiple Specklegram Windows ..... 285
8.1.6 Object Rotation ................................ 286
8.2 Speckle Interferometry ................................ 287
8.2.1 Systems That Use Photographic Detection ........ 287
8.2.2 Electronic Speckle Pattern Interferometry
(ESPI) ......................................... 291
8.2.3 Speckle Shearing Interferometry ................ 294
8.3 From Fringe Patterns to Phase Maps .................... 296
8.3.1 The Fourier Transform Method ................... 297
8.3.2 Phase-Shifting Speckle Interferometry .......... 298
8.3.3 Phase Unwrapping ............................... 299
8.4 Vibration Measurement Using Speckle ................... 301
8.5 Speckle and Surface Roughness Measurements ............ 305
8.5.1 RMS Surface Height and Surface Covariance
Area from Speckle Contrast ..................... 305
8.5.2 RMS Surface Height from Two-Wavelength
Decorrelation .................................. 306
8.5.3 RMS Surface Height from Two-Angle
Decorrelation .................................. 307
8.5.4 Information from Measurement of Angular Power
Spectrum ....................................... 308
9 Speckle in Imaging Through the Atmosphere .................. 311
9.1 Background ............................................ 311
9.1.1 Refractive Index Fluctuations in the
Atmosphere ..................................... 311
9.2 Point-Spread Functions ................................ 313
9.3 Average Optical Transfer Functions .................... 315
9.4 Statistical Properties of the Short-Exposure OTF and
MTF ................................................... 316
9.5 Astronomical Speckle Interferometry ................... 322
9.5.1 Object Information that Is Retrievable ......... 322
9.5.2 Results of a More Complete Analysis ............ 325
9.6 The Cross-Spectrum or Knox-Thompson Technique ......... 327
9.6.1 The Cross-Spectrum Transfer Function ........... 327
9.6.2 Recovering Full Object Information from the
Cross-Spectrum ................................. 329
9.7 The Bispectrum Technique .............................. 331
9.7.1 The Bispectrum Transfer Function ............... 331
9.7.2 Recovering Full Object Information from the
Bispectrum ..................................... 332
9.8 Speckle Correlography ................................. 333
A Linear Transformations of Speckle Fields ................... 337
В Contrast of Partially Developed Speckle .................... 341
С Statistics of Derivatives of Speckle ....................... 345
C.l The Correlation Matrix ................................ 345
C.2 Joint Density Function of the Derivatives of Phase .... 348
C.3 Joint Density Function of the Derivatives of
Intensity ............................................. 349
D Wavelength and Angle Dependence ............................ 351
D.l Free-Space Geometry ................................... 351
D.2 Imaging Geometry ...................................... 355
E Speckle Contrast with a Projected Diffuser ................. 359
E.l Random Phase Diffusers ................................ 359
E.2 Diffuser that Just Fills the Projection Optics ........ 362
E.3 Diffuser that Overfills the Projection Optics ......... 362
F Statistics of Constrained Speckle .......................... 365
G Sample Mathematica Programs for Simulating Speckle ......... 369
G.l Speckle Simulation With Free Space Propagation ........ 369
G.2 Speckle Simulation With an Imaging Geometry ........... 369
Bibliography ............................................... 373
Index ...................................................... 383
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