Phase-Space Optics: Fundamentals and Applications Fundamentals and Applications

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

Prefacep. xiii
Wigner Distribution in Opticsp. 1
Introductionp. 1
Elementary Description of Optical Signals and Systemsp. 2
Impulse Response and Coherent Point-Spread Functionp. 3
Mutual Coherence Function and Cross-Spectral Densityp. 3
Some Basic Examples of Optical Signalsp. 4
Wigner Distribution and Ambiguity Functionp. 5
Definitionsp. 5
Some Basic Examples Againp. 7
Gaussian Lightp. 9
Local Frequency Spectrump. 11
Some Properties of the Wigner Distributionp. 12
Inversion Formulap. 12
Shift Covariancep. 12
Radiometric Quantitiesp. 12
Instantaneous Frequencyp. 14
Moyal's Relationshipp. 15
One-Dimensional Case and the Fractional Fourier Transformationp. 15
Fractional Fourier Transformationp. 15
Rotation in Phase Spacep. 16
Generalized Marginals-Radon Transformp. 16
Propagation of the Wigner Distributionp. 18
First-Order Optical Systems-Ray Transformation Matrixp. 18
Phase-Space Rotators-More Rotations in Phase Spacep. 19
More General Systems-Ray-Spread Functionp. 21
Geometric-Optical Systemsp. 22
Transport Equationsp. 23
Wigner Distribution Moments in First-Order Optical Systemsp. 24
Moment Invariantsp. 25
Moment Invariants for Phase-Space Rotatorsp. 26
Symplectic Moment Matrix-The Bilinear ABCD Lawp. 28
Measurement of Momentsp. 29
Coherent Signals and the Cohen Classp. 29
Multicomponent Signals-Auto-Terms and Cross-Termsp. 30
One-Dimensional Case and Some Basic Cohen Kernelsp. 32
Rotation of the Kernelp. 33
Rotated Version of the Smoothed Interferogramp. 35
Conclusionp. 40
Referencesp. 40
Ambiguity Function in Optical Imagingp. 45
Introductionp. 45
Intensity Spectrum of a Fresnel Diffraction Pattern Under Coherent Illuminationp. 47
General Formulationp. 47
Application to Simple Objectsp. 48
Contrast Transfer Functionsp. 49
Propagation through a Paraxial Optical System in Terms of AFp. 49
Propagation in Free Spacep. 49
Transmission through a Thin Objectp. 50
Propagation in a Paraxial Optical Systemp. 51
The AF in Isoplanatic (Space-Invariant) Imagingp. 52
The AF of the Image of an Incoherent Sourcep. 53
Derivation of the Zernike-Van Cittert Theorem from the Propagation of the AFp. 53
Partial Coherence Properties in the Image of an Incoherent Sourcep. 54
The Pupil-AF as a Generalization of the OTFp. 54
Phase-Space Tomographyp. 55
Another Possible Approach to AF Reconstructionp. 56
Propagation-Based Holographic Phase Retrieval from Several Imagesp. 58
Fresnel Diffraction Images as In-Line Hologramsp. 58
Application to Phase Retrieval and X-Ray Holotomographyp. 59
Conclusionp. 60
Referencesp. 60
Rotations in Phase Spacep. 63
Introductionp. 63
First-Order Optical Systems and Canonical Integral Transformsp. 64
Canonical Integral Transforms and Ray Transformation Matrix Formalismp. 64
Modified Iwasawa Decomposition of Ray Transformation Matrixp. 66
Canonical Transformations Producing Phase-Space Rotationsp. 67
Matrix and Operator Descriptionp. 67
Signal Rotatorp. 69
Fractional Fourier Transformp. 69
Gyratorp. 73
Other Phase-Space Rotatorsp. 74
Properties of the Phase-Space Rotatorsp. 74
Some Useful Relations for Phase-Space Rotatorsp. 75
Similarity to the Fractional Fourier Transformp. 76
Shift Theoremp. 77
Convolution Theoremp. 77
Scaling Theoremp. 77
Phase-Space Rotations of Selected Functionsp. 78
Eigenfunctions for Phase-Space Rotatorsp. 80
Some Relations for the Eigenfunctionsp. 80
Mode Presentation on Orbital Poincaré Spherep. 82
Optical Setups for Basic Phase-Space Rotatorsp. 84
Flexible Optical Setups for Fractional FT and Gyratorp. 85
Flexible Optical Setup for Image Rotatorp. 87
Applications of Phase-Space Rotatorsp. 88
Generalized Convolutionp. 88
Pattern Recognitionp. 90
Chirp Signal Analysisp. 94
Signal Encryptionp. 94
Mode Convertersp. 95
Beam Characterizationp. 96
Gouy Phase Accumulationp. 100
Conclusionsp. 101
Acknowledgmentsp. 102
Referencesp. 102
The Radon-Wigner Transform in Analysis, Design, and Processing of Optical Signalsp. 107
Introductionp. 107
Projections of the Wigner Distribution Function in Phase Space: The Radon-Wigner Transform (RWT)p. 108
Definition and Basic Propertiesp. 108
Optical Implementation of the RWT: The Radon-Wigner Displayp. 117
Analysis of Optical Signals and Systems by Means of the RWTp. 122
Analysis of Diffraction Phenomenap. 122
Computation of Irradiance Distribution along Different Paths in Image Spacep. 122
Parallel Optical Display of Diffraction Patternsp. 132
Inverting RWT: Phase-Space Tomographic Reconstruction of Optical Fieldsp. 134
Merit Functions of Imaging Systems in Terms of the RWTp. 138
Axial Point-Spread Function (PSF) and Optical Transfer Function (OTF)p. 138
Polychromatic OTFp. 143
Polychromatic Axial PSFp. 146
Design of Imaging Systems and Optical Signal Processing by Means of RWTp. 151
Optimization of Optical Systems: Achromatic Designp. 151
Controlling the Axial Response: Synthesis of Pupil Masks by RWT Inversionp. 156
Signal Processing through RWTp. 157
Acknowledgmentsp. 162
Referencesp. 162
Imaging Systems: Phase-Space Representationsp. 165
Introductionp. 165
The Product-Space Representation and Product Spectrum Representationp. 166
Optical Imaging Systemsp. 170
Bilinear Optical Systemsp. 173
Noncoherent Imaging Systemsp. 176
Tolerance to Focus Errors and to Spherical Aberrationp. 178
Phase Conjugate Platesp. 183
Referencesp. 189
Super Resolved Imaging in Wigner-Based Phase Spacep. 193
Introductionp. 193
General Definitionsp. 195
Description of SRp. 197
Code Division Multiplexingp. 200
Time Multiplexingp. 201
Polarization Multiplexingp. 202
Wavelength Multiplexingp. 203
Gray-Level Multiplexingp. 203
Description in the Phase-Space Domainp. 205
Conclusionsp. 213
Referencesp. 214
Radiometry, Wave Optics, and Spatial Coherencep. 217
Introductionp. 217
Conventional Radiometryp. 218
Lambertian Sourcesp. 221
Mutual Coherence Functionp. 221
Stationary Phase Approximationp. 224
Radiometry and Wave Opticsp. 226
Examplesp. 231
Blackbody Radiationp. 231
Noncoherent Sourcep. 232
Coherent Wave Fieldsp. 233
Quasi-Homogeneous Wave Fieldp. 234
Acknowledgmentsp. 235
Referencesp. 235
Rays and Wavesp. 237
Introductionp. 237
Small-Wavelength Limit in the Position Representation I: Geometrical Opticsp. 238
The Eikonal and Geometrical Opticsp. 239
Choosing z as the Parameterp. 242
Ray-Optical Phase Space and the Lagrange Manifoldp. 243
Small-Wavelength Limit in the Position Representation II: The Transport Equation and the Field Estimatep. 245
The Debye Series Expansionp. 245
The Transport Equation and Its Solutionp. 245
The Field Estimate and Its Problems at Causticsp. 247
Flux Lines versus Raysp. 249
Analogy with Quantum Mechanicsp. 250
Semiclassical Mechanicsp. 251
Bohmian Mechanics and the Hydrodynamic Modelp. 253
Small-Wavelength Limit in the Momentum Representationp. 254
The Helmholtz Equation in the Momentum Representationp. 254
Asymptotic Treatment and Ray Equationsp. 256
Transport Equation in the Momentum Representationp. 258
Field Estimatep. 259
Maslov's Canonical Operator Methodp. 260
Gaussian Beams and Their Sumsp. 261
Parabasal Gaussian Beamsp. 261
Sums of Gaussian Beamsp. 264
Stable Aggregates of Flexible Elementsp. 266
Derivation of the Estimatep. 266
Insensitivity to ¿p. 269
Phase-Space Interpretationp. 270
A Simple Examplep. 271
Concluding Remarksp. 275
Referencesp. 275
Self-Imaging in Phase Spacep. 279
Introductionp. 279
Phase-Space Optics Minimum Tool Kitp. 280
Self-Imaging of Paraxial Wavefrontsp. 284
The Talbot Effectp. 285
The "Walk-off" Effectp. 289
The Fractional Talbot Effectp. 290
Matrix Formulation of the Fractional Talbot Effectp. 295
Point Source Illuminationp. 298
Another Path to Self-Imagingp. 301
Self-Imaging and Incoherent Illuminationp. 302
Summaryp. 305
Referencesp. 306
Sampling and Phase Spacep. 309
Introductionp. 309
Notation and Some Initial Conceptsp. 312
The Wigner Distribution Function and Propertiesp. 312
The Linear Canonical Transform and the WDFp. 314
The Phase-Space Diagramp. 314
Harmonics and Chirps and Convolutionsp. 316
The Comb Function and Rect Functionp. 318
Comb Functionsp. 318
Rect Functionsp. 320
Finite Supportsp. 321
Band-limitedness in Fourier Domainp. 321
Band-limitedness and the LCTp. 322
Finite Space-Bandwidth Product-Compact Support in x and kp. 324
Sampling a Signalp. 325
Nyquist-Shannon Samplingp. 325
Generalized Samplingp. 328
Simulating an Optical System: Sampling at the Input and Outputp. 329
Conclusionp. 332
Referencesp. 332
Phase Space in, Ultrafast Opticsp. 337
Introductionp. 337
Phase-Space Representations for Short Optical Pulsesp. 338
Representation of Pulsed Fieldsp. 338
Pulse Ensembles and Correlation Functionsp. 340
The Time-Frequency Phase Spacep. 343
Phase-Space Representation of Paraxial Optical Systemsp. 349
Temporal Paraxiality and the Chronocyclic Phase Spacep. 353
Metrology of Short Optical Pulsesp. 357
Measurement Strategiesp. 357
Pulse Characterization Apparatuses as Linear Systemsp. 358
Phase-Space Methodsp. 361
Spectrographic Techniquesp. 362
Tomographic Techniquesp. 366
Interferornetric or Direct Techniquesp. 369
Two-Pulse Double-Slit Interferometryp. 370
Shearing Interferometryp. 374
Conclusionsp. 378
Referencesp. 379
Indexp. 385
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