Nye J.F. Natural focusing and fine structure of light: caustics and wave dislocations
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Nye J.F. Natural focusing and fine structure of light: caustics and wave dislocations / Nye J.F. - Bristol: Institute of Physics, 1999. - 328 p. - ISBN 0-7503-0610-6.
 
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   Preface .......................................................... xi

1  Prospect .......................................................... 1


   PART 1
   CAUSTICS

2  Natural Focusing .................................................. 9
   2.1 The coffee cup caustic ........................................ 9
   2.2 The evolution of a wavefront in terms of rays ................ 12
   2.3 Form of the wavefront at a fold .............................. 16
   2.4 Caustics in the paraxial approximation ....................... 17
   2.5 Caustics at infinity ......................................... 19
   2.6 The caustic loci in angle space .............................. 22
   2.7 The caustic loci on the wavefront ............................ 24
   2.8 Examples of caustics ......................................... 25
       2.8.1 Caustics from a water surface .......................... 25
       2.8.2 Caustics from rippled glass ............................ 27
       2.8.3 Swimming pool caustics ................................. 29
       2.8.4 Rippled glass, viewed from control space ............... 30
       2.8.5 Scintillation of stars ................................. 32
       2.8.6 Reflections in rippling water; the masts of ships,
             mirages and sunsets .................................... 34
       2.8.7 Cosmic mirages—gravitational lensing ................... 39

3  Folds and Cusps in Three Dimensions .............................. 41
   3.1 Specification of the wavefront and the condition
       for a caustic ................................................ 41
       3.1.1 Caustic condition with more than two controls .......... 42
   3.2 Transformations, equivalence and Thom’s theorem .............. 43
   3.3 The fold ..................................................... 46
   3.4 The cusp ..................................................... 48
   3.5 The cusp in three dimensions ................................. 52
   3.6 Beak-to-beak and lips events ................................. 53
   3.7 Observing caustics with an optical instrument ................ 56

4  Caustics of Codimension Three .................................... 59
   4.1  The swallowtail ............................................. 59
   4.2  The shape of the swallowtail caustic ........................ 60
   4.3  General restrictions on the orientations and distortions
        of caustics ................................................. 63
   4.4  Example of a swallowtail wavefront .......................... 65
   4.5  The hyperbolic umbilic focus ................................ 66
   4.6  A water drop in a circular hole ............................. 68
   4.7  The elliptic umbilic ........................................ 74
   4.8  Caustics from an irregular water drop ....................... 77
   4.9  The wavefront surface at an umbilic point ................... 81
   4.10 Pair creation of umbilic points ............................. 84
   4.11 The wavefront surface at catastrophe points ................. 87
   Appendix: General analysis of trajectory patterns in pair
             creation ............................................... 90


   PART 2
   THE WAVELENGTH  SCALE

5  Dislocations in Scalar Wave Fields ............................... 95
   5.1  Wave dislocations and diffraction patterns .................. 95
   5.2  Examples of dislocations .................................... 95
        5.2.1 Pure screw dislocation ................................ 98
        5.2.2 Pure edge dislocation ................................. 99
   5.3  Mixed edge-screw dislocations .............................. 102
   5.4  A helical dislocation ...................................... 103
   5.5  Definitions and signs of wave dislocations ................. 104
        5.5.1 The general three-dimensional case ................... 104
        5.5.2 Two-dimensional sections ............................. 104
        5.5.3 Relation to the wave vector .......................... 106
   5.6  Evolving dislocations ...................................... 107
   5.7  Screw dislocations in laser beams .......................... 108
   5.8  Behaviour of the Airy ring system .......................... 110
   5.9  Wave dislocations and caustics ............................. 115
   5.10 Wave dislocations compared with crystal dislocations ....... 117
   5.11 Other non-optical examples of dislocations ................. 118
   5.12 Statistical questions ...................................... 119

6  Diffraction ..................................................... 123
   6.1 The diffraction integral .................................... 123
   6.2 The fold diffraction catastrophe ............................ 126
   6.3 The rainbow as an example of a fold catastrophe ............. 128
   6.4 Colours in fold catastrophes ................................ 136
   6.5 Secondary rainbow ........................................... 137
   6.6 Ellipsoidal drops ........................................... 137
   6.7 The cusp diffraction catastrophe ............................ 139
   6.8 The elliptic umbilic diffraction pattern .................... 144
   6.9 How diffraction catastrophes depend on wavelength ........... 151


   PART 3
   HIGHER CATASTROPHES, NETWORKS AND STATISTICS

7  Higher Caustics as Organizing Centres ........................... 157
   7.1  Higher catastrophes ........................................ 157
   7.2  The butterfly A5 and higher cuspoids ....................... 158
   7.3  The parabolic umbilic ...................................... 161
   7.4  Theory for a tilted water drop ............................. 165
   7.5  Analysis for the parabolic umbilic ......................... 169
   7.6  Forward look to higher catastrophes ........................ 172
   7.7  The symbolic umbilic E6 .................................... 172
   7.8  Using gratings to shape wavefronts to order ................ 178
   7.9  Producing D6+ .............................................. 184
   7.10 Geometry of D6+ ............................................ 185
   7.11 The catastrophe D6- ........................................ 191

8  The Higher Catastrophe X9 ....................................... 193
   8.1  The relevance of X9 to optics .............................. 193
   8.2  Fourfold symmetric caustics ................................ 194
   8.3  Twofold symmetry ........................................... 197
   8.4  Threefold symmetry; the problem of the triple junction ..... 202
   8.5  Analytical simulation of a triple junction ................. 205
   8.6  Other unfoldings ........................................... 211
   8.7  The effect of gravity on the shape of a horizontal
        water drop ................................................. 214
   8.8  Non-paraxiality ............................................ 216
   8.9  Gravitational lensing ...................................... 219
   8.10 The distortion of images of extended sources ............... 225
   Appendix ........................................................ 235

9  Network Patterns of Catastrophes ................................ 237
   9.1 A two-wave model ............................................ 237
   9.2 Triple junctions ............................................ 241
   9.3 Networks on a rectangular lattice ........................... 243

10 Statistics of Caustics and Twinkling ............................ 249
   10.1 Two different kinds of statistical question ................ 249
   10.2 Statistics of geometrical focusing in two dimensions ....... 250
   10.3 Statistics of geometrical focusing in three dimensions ..... 252
   10.4 Fluctuations of intensity .................................. 253
       10.4.1 Wave propagation in two dimensions ................... 257
       10.4.2 Wave propagation in three dimensions ................. 258
   10.5 Application of the theory .................................. 259

11 Caustics within a General Refracting Medium ..................... 261
   11.1 Principles for a local treatment without the remote
        wavefront .................................................. 261
   11.2 Specification of the wave field ............................ 263
   11.3 Construction of a phase function and a potential ........... 264
   11.4 Two reservations and their resolution ...................... 266
   11.5 Conclusion ................................................. 267


   PART 4
   POLARIZATION SINGULARITIES

12 Singularities in Paraxial Electromagnetic Waves ................. 271
   12.1 The polarization ellipse ................................... 271
   12.2 Lines of circular polarization (C lines) ................... 273
   12.3 C line as a dislocation .................................... 277
   12.4 Observation of C lines (microwaves and optics) ............. 278
   12.5 Surfaces of linear polarization (S surfaces) ............... 280
   12.6 Unfolding a perfect interference fringe .................... 281
   12.7 The magnetic field ......................................... 285
   12.8 Time dependence and disclinations .......................... 286
   12.9 Polarization ellipses in the tides ......................... 290

13 Singularities in Waves Travelling in Many Directions ............ 293
   13.1 The polarization ellipse ................................... 293
   13.2 Lines of true circular polarization ........................ 295
   13.3 Lines of true linear polarization .......................... 296
   13.4 Defining the handedness .................................... 298
   13.5 Why choose the Pancharatnam phase difference? .............. 301
   13.6 The effect of symmetry ..................................... 303

   Retrospect ...................................................... 307

   References ...................................................... 309

   Name Index ...................................................... 319

   Subject Index ................................................... 323


Вверх Nye J.F. Natural focusing and fine structure of light: caustics and wave dislocations / Nye J.F. - Bristol: Institute of Physics, 1999. - 328 p. - ISBN 0-7503-0610-6.

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