Page 11 - Phase Space Optics Fundamentals and Applications

x Contents

8 Rays and Waves .................................... 237
8.1 Introduction .......................................... 237
8.2 Small-Wavelength Limit in the Position
Representation I: Geometrical Optics ................. 238
8.2.1 The Eikonal and Geometrical Optics ........... 239
8.2.2 Choosing z as the Parameter ................... 242
8.2.3 Ray-Optical Phase Space
and the Lagrange Manifold .................... 243
8.3 Small-Wavelength Limit in the Position
Representation II: The Transport Equation
and the Field Estimate ................................ 245
8.3.1 The Debye Series Expansion ................... 245
8.3.2 The Transport Equation and Its Solution ....... 245
8.3.3 The Field Estimate and Its Problems
at Caustics .................................... 247
8.4 Flux Lines versus Rays ............................... 249
8.5 Analogy with Quantum Mechanics ................... 250
8.5.1 Semiclassical Mechanics ....................... 251
8.5.2 Bohmian Mechanics and the Hydrodynamic
Model ......................................... 253
8.6 Small-Wavelength Limit in the Momentum
Representation ....................................... 254
8.6.1 The Helmholtz Equation in the Momentum
Representation ................................ 254
8.6.2 Asymptotic Treatment and Ray Equations ...... 256
8.6.3 Transport Equation in the Momentum
Representation ................................ 258
8.6.4 Field Estimate ................................. 259
8.7 Maslov’s Canonical Operator Method ................ 260
8.8 Gaussian Beams and Their Sums ..................... 261
8.8.1 Parabasal Gaussian Beams ..................... 261
8.8.2 Sums of Gaussian Beams ...................... 264
8.9 Stable Aggregates of Flexible Elements ............... 266
8.9.1 Derivation of the Estimate ..................... 266
8.9.2 Insensitivity to ............................... 269
8.9.3 Phase-Space Interpretation ..................... 270
8.10 A Simple Example .................................... 271
8.11 Concluding Remarks ................................. 275
References ................................................. 275

9 Self-Imaging in Phase Space ....................... 279
9.1 Introduction .......................................... 279
9.2 Phase-Space Optics Minimum Tool Kit ................ 280
9.3 Self-Imaging of Paraxial Wavefronts .................. 284