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