Phase-Space Optics: Fundamentals and Applications Fundamentals and Applications

by Testorf, Markus; Hennelly, Bryan; Ojeda-Castaneda, Jorge

• ISBN13: 9780071597982
• ISBN10: 0071597980
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-09-08
• Publisher: McGraw-Hill Education

Summary

Definitive coverage of the fundamentals and potential applications of phase-space opticsPhase-space optics is a powerful new tool for understanding and applying the physics of light propagation. Written by pioneers in the field, this book begins by covering the basic properties of joint-signal representations. The book goes on to illustrate how joint-signal representations are well-suited for gaining physical insight and developing novel engineering applications.Phase Space Optics offers essential information on state-of-the-art uses of phase-space methods in optical sciences. Contributors are pioneers in their fields of expertise on the use of joint-signal representations. Application examples utilizing phase-space optics are included. .

Author Biography

Bryan Hennelly is a postdoctoral researcher at the National University of Ireland, Maynooth. He is a member of the SPIE and the OSA. .

Jorge Ojeda-Castan++eda has been teaching graduate and undergraduate courses in physics and mathematics for more than 25 years..

Markus Testorf is an assistant professor at the Thayer School of Engineering at Dartmouth College. .

Table of Contents

Preface	p. xiii
Wigner Distribution in Optics	p. 1
Introduction	p. 1
Elementary Description of Optical Signals and Systems	p. 2
Impulse Response and Coherent Point-Spread Function	p. 3
Mutual Coherence Function and Cross-Spectral Density	p. 3
Some Basic Examples of Optical Signals	p. 4
Wigner Distribution and Ambiguity Function	p. 5
Definitions	p. 5
Some Basic Examples Again	p. 7
Gaussian Light	p. 9
Local Frequency Spectrum	p. 11
Some Properties of the Wigner Distribution	p. 12
Inversion Formula	p. 12
Shift Covariance	p. 12
Radiometric Quantities	p. 12
Instantaneous Frequency	p. 14
Moyal's Relationship	p. 15
One-Dimensional Case and the Fractional Fourier Transformation	p. 15
Fractional Fourier Transformation	p. 15
Rotation in Phase Space	p. 16
Generalized Marginals-Radon Transform	p. 16
Propagation of the Wigner Distribution	p. 18
First-Order Optical Systems-Ray Transformation Matrix	p. 18
Phase-Space Rotators-More Rotations in Phase Space	p. 19
More General Systems-Ray-Spread Function	p. 21
Geometric-Optical Systems	p. 22
Transport Equations	p. 23
Wigner Distribution Moments in First-Order Optical Systems	p. 24
Moment Invariants	p. 25
Moment Invariants for Phase-Space Rotators	p. 26
Symplectic Moment Matrix-The Bilinear ABCD Law	p. 28
Measurement of Moments	p. 29
Coherent Signals and the Cohen Class	p. 29
Multicomponent Signals-Auto-Terms and Cross-Terms	p. 30
One-Dimensional Case and Some Basic Cohen Kernels	p. 32
Rotation of the Kernel	p. 33
Rotated Version of the Smoothed Interferogram	p. 35
Conclusion	p. 40
References	p. 40
Ambiguity Function in Optical Imaging	p. 45
Introduction	p. 45
Intensity Spectrum of a Fresnel Diffraction Pattern Under Coherent Illumination	p. 47
General Formulation	p. 47
Application to Simple Objects	p. 48
Contrast Transfer Functions	p. 49
Propagation through a Paraxial Optical System in Terms of AF	p. 49
Propagation in Free Space	p. 49
Transmission through a Thin Object	p. 50
Propagation in a Paraxial Optical System	p. 51
The AF in Isoplanatic (Space-Invariant) Imaging	p. 52
The AF of the Image of an Incoherent Source	p. 53
Derivation of the Zernike-Van Cittert Theorem from the Propagation of the AF	p. 53
Partial Coherence Properties in the Image of an Incoherent Source	p. 54
The Pupil-AF as a Generalization of the OTF	p. 54
Phase-Space Tomography	p. 55
Another Possible Approach to AF Reconstruction	p. 56
Propagation-Based Holographic Phase Retrieval from Several Images	p. 58
Fresnel Diffraction Images as In-Line Holograms	p. 58
Application to Phase Retrieval and X-Ray Holotomography	p. 59
Conclusion	p. 60
References	p. 60
Rotations in Phase Space	p. 63
Introduction	p. 63
First-Order Optical Systems and Canonical Integral Transforms	p. 64
Canonical Integral Transforms and Ray Transformation Matrix Formalism	p. 64
Modified Iwasawa Decomposition of Ray Transformation Matrix	p. 66
Canonical Transformations Producing Phase-Space Rotations	p. 67
Matrix and Operator Description	p. 67
Signal Rotator	p. 69
Fractional Fourier Transform	p. 69
Gyrator	p. 73
Other Phase-Space Rotators	p. 74
Properties of the Phase-Space Rotators	p. 74
Some Useful Relations for Phase-Space Rotators	p. 75
Similarity to the Fractional Fourier Transform	p. 76
Shift Theorem	p. 77
Convolution Theorem	p. 77
Scaling Theorem	p. 77
Phase-Space Rotations of Selected Functions	p. 78
Eigenfunctions for Phase-Space Rotators	p. 80
Some Relations for the Eigenfunctions	p. 80
Mode Presentation on Orbital Poincaré Sphere	p. 82
Optical Setups for Basic Phase-Space Rotators	p. 84
Flexible Optical Setups for Fractional FT and Gyrator	p. 85
Flexible Optical Setup for Image Rotator	p. 87
Applications of Phase-Space Rotators	p. 88
Generalized Convolution	p. 88
Pattern Recognition	p. 90
Chirp Signal Analysis	p. 94
Signal Encryption	p. 94
Mode Converters	p. 95
Beam Characterization	p. 96
Gouy Phase Accumulation	p. 100
Conclusions	p. 101
Acknowledgments	p. 102
References	p. 102
The Radon-Wigner Transform in Analysis, Design, and Processing of Optical Signals	p. 107
Introduction	p. 107
Projections of the Wigner Distribution Function in Phase Space: The Radon-Wigner Transform (RWT)	p. 108
Definition and Basic Properties	p. 108
Optical Implementation of the RWT: The Radon-Wigner Display	p. 117
Analysis of Optical Signals and Systems by Means of the RWT	p. 122
Analysis of Diffraction Phenomena	p. 122
Computation of Irradiance Distribution along Different Paths in Image Space	p. 122
Parallel Optical Display of Diffraction Patterns	p. 132
Inverting RWT: Phase-Space Tomographic Reconstruction of Optical Fields	p. 134
Merit Functions of Imaging Systems in Terms of the RWT	p. 138
Axial Point-Spread Function (PSF) and Optical Transfer Function (OTF)	p. 138
Polychromatic OTF	p. 143
Polychromatic Axial PSF	p. 146
Design of Imaging Systems and Optical Signal Processing by Means of RWT	p. 151
Optimization of Optical Systems: Achromatic Design	p. 151
Controlling the Axial Response: Synthesis of Pupil Masks by RWT Inversion	p. 156
Signal Processing through RWT	p. 157
Acknowledgments	p. 162
References	p. 162
Imaging Systems: Phase-Space Representations	p. 165
Introduction	p. 165
The Product-Space Representation and Product Spectrum Representation	p. 166
Optical Imaging Systems	p. 170
Bilinear Optical Systems	p. 173
Noncoherent Imaging Systems	p. 176
Tolerance to Focus Errors and to Spherical Aberration	p. 178
Phase Conjugate Plates	p. 183
References	p. 189
Super Resolved Imaging in Wigner-Based Phase Space	p. 193
Introduction	p. 193
General Definitions	p. 195
Description of SR	p. 197
Code Division Multiplexing	p. 200
Time Multiplexing	p. 201
Polarization Multiplexing	p. 202
Wavelength Multiplexing	p. 203
Gray-Level Multiplexing	p. 203
Description in the Phase-Space Domain	p. 205
Conclusions	p. 213
References	p. 214
Radiometry, Wave Optics, and Spatial Coherence	p. 217
Introduction	p. 217
Conventional Radiometry	p. 218
Lambertian Sources	p. 221
Mutual Coherence Function	p. 221
Stationary Phase Approximation	p. 224
Radiometry and Wave Optics	p. 226
Examples	p. 231
Blackbody Radiation	p. 231
Noncoherent Source	p. 232
Coherent Wave Fields	p. 233
Quasi-Homogeneous Wave Field	p. 234
Acknowledgments	p. 235
References	p. 235
Rays and Waves	p. 237
Introduction	p. 237
Small-Wavelength Limit in the Position Representation I: Geometrical Optics	p. 238
The Eikonal and Geometrical Optics	p. 239
Choosing z as the Parameter	p. 242
Ray-Optical Phase Space and the Lagrange Manifold	p. 243
Small-Wavelength Limit in the Position Representation II: The Transport Equation and the Field Estimate	p. 245
The Debye Series Expansion	p. 245
The Transport Equation and Its Solution	p. 245
The Field Estimate and Its Problems at Caustics	p. 247
Flux Lines versus Rays	p. 249
Analogy with Quantum Mechanics	p. 250
Semiclassical Mechanics	p. 251
Bohmian Mechanics and the Hydrodynamic Model	p. 253
Small-Wavelength Limit in the Momentum Representation	p. 254
The Helmholtz Equation in the Momentum Representation	p. 254
Asymptotic Treatment and Ray Equations	p. 256
Transport Equation in the Momentum Representation	p. 258
Field Estimate	p. 259
Maslov's Canonical Operator Method	p. 260
Gaussian Beams and Their Sums	p. 261
Parabasal Gaussian Beams	p. 261
Sums of Gaussian Beams	p. 264
Stable Aggregates of Flexible Elements	p. 266
Derivation of the Estimate	p. 266
Insensitivity to ¿	p. 269
Phase-Space Interpretation	p. 270
A Simple Example	p. 271
Concluding Remarks	p. 275
References	p. 275
Self-Imaging in Phase Space	p. 279
Introduction	p. 279
Phase-Space Optics Minimum Tool Kit	p. 280
Self-Imaging of Paraxial Wavefronts	p. 284
The Talbot Effect	p. 285
The "Walk-off" Effect	p. 289
The Fractional Talbot Effect	p. 290
Matrix Formulation of the Fractional Talbot Effect	p. 295
Point Source Illumination	p. 298
Another Path to Self-Imaging	p. 301
Self-Imaging and Incoherent Illumination	p. 302
Summary	p. 305
References	p. 306
Sampling and Phase Space	p. 309
Introduction	p. 309
Notation and Some Initial Concepts	p. 312
The Wigner Distribution Function and Properties	p. 312
The Linear Canonical Transform and the WDF	p. 314
The Phase-Space Diagram	p. 314
Harmonics and Chirps and Convolutions	p. 316
The Comb Function and Rect Function	p. 318
Comb Functions	p. 318
Rect Functions	p. 320
Finite Supports	p. 321
Band-limitedness in Fourier Domain	p. 321
Band-limitedness and the LCT	p. 322
Finite Space-Bandwidth Product-Compact Support in x and k	p. 324
Sampling a Signal	p. 325
Nyquist-Shannon Sampling	p. 325
Generalized Sampling	p. 328
Simulating an Optical System: Sampling at the Input and Output	p. 329
Conclusion	p. 332
References	p. 332
Phase Space in, Ultrafast Optics	p. 337
Introduction	p. 337
Phase-Space Representations for Short Optical Pulses	p. 338
Representation of Pulsed Fields	p. 338
Pulse Ensembles and Correlation Functions	p. 340
The Time-Frequency Phase Space	p. 343
Phase-Space Representation of Paraxial Optical Systems	p. 349
Temporal Paraxiality and the Chronocyclic Phase Space	p. 353
Metrology of Short Optical Pulses	p. 357
Measurement Strategies	p. 357
Pulse Characterization Apparatuses as Linear Systems	p. 358
Phase-Space Methods	p. 361
Spectrographic Techniques	p. 362
Tomographic Techniques	p. 366
Interferornetric or Direct Techniques	p. 369
Two-Pulse Double-Slit Interferometry	p. 370
Shearing Interferometry	p. 374
Conclusions	p. 378
References	p. 379
Index	p. 385
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