Waves and Thom's theorem

Abstract

Short-wave fields can be well approximated by families of trajectories. These families are dominated by their singularities, i.e. by caustics, where the density of trajectories is infinite. Thom's theorem on singularities of mappings can be rigorously applied and shows that structurally stable caustics—that is those whose topology is unaltered by ‘generic’ perturbation—can be classified as ‘elementary catastrophes’. Accurate asymptotic approximations to wave functions can be built up using the catastrophes as skeletons: to each catastrophe there corresponds a canonical diffraction function. Structurally unstable caustics can be produced by special symmetries, and the detailed form of the caustic that results from symmetry-breaking can often be determined by identifying the structurally unstable caustic as the special section of a higher-dimensional catastrophe. Sometimes it is clear that the unstable caustic is a special section of a catastrophe of infinite co-dimension; these fall outside the scope of Thom's theorem and suggest new directions for mathematical investigation. The discussion is illustrated with numerous examples from optics and quantum mechanics.