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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. .
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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