Thom's catastrophe theory classifies the generic caustics formed by families of light or sound rays — fold, cusp, swallowtail, butterfly, elliptic and hyperbolic umbilic — by their finite co-dimension, and the corresponding diffraction catastrophes give uniform wave approximations near them. Berry's 1976 review 'Waves and Thom's theorem' identified as its most important open problem the caustics that lie outside this framework: those of infinite co-dimension, exemplified by the glory (the backward-scattering enhancement from a sphere), the forward diffraction peak of semiclassical scattering, and source-point caustics, whose generic unfoldings appear to require infinitely many (possibly uncountably many) perturbation parameters. No mathematical classification scheme for such caustics, their generic unfoldings, and the associated diffraction patterns exists. The problem is to construct one, or to establish what replaces the finite-co-dimension catastrophe classification in these cases.

Background

When a family of rays (the trajectories of the short-wavelength or ray approximation to a wave equation) is focused, neighbouring rays envelope a caustic — a surface on which the ray amplitude formally diverges and ray optics fails. Away from caustics, the wave field is well described by the WKB (Wentzel–Kramers–Brillouin) or eikonal approximation, in which the wave is a slowly varying amplitude multiplying a rapidly oscillating phase given by the ray travel time; near a caustic this description must be repaired by uniform diffraction approximations, i.e. single formulas that remain valid on both sides of the caustic. René Thom's catastrophe theory classifies the stable singularities of smooth mappings: the generic caustics of co-dimension at most five are the elementary catastrophes (fold A2A_2, cusp A3A_3, swallowtail A4A_4, butterfly A5A_5, and the elliptic, hyperbolic, and parabolic umbilics), where co-dimension counts the parameters of the universal unfolding, i.e. the number of independent perturbations needed to display all nearby forms of the singularity. Berry's 1976 review 'Waves and Thom's theorem' connected this classification to wave physics: each elementary catastrophe has a canonical diffraction integral (the Airy integral near the fold, the Pearcey integral near the cusp, the elliptic-umbilic integral, etc.), and this diffraction catastrophe gives the uniform approximation of the wave field near the corresponding caustic, with diffraction corrections whose scaling is dictated by the catastrophe. This program was consolidated in the review literature as 'catastrophe optics' (Berry and Upstill, 1980). The classification, however, only covers generic caustics of finite co-dimension. Several wave-physics caustics are intrinsically non-generic and of infinite co-dimension: the glory — the bright ring and central enhancement seen in light backscattered by a spherical water droplet, arising from a toroidal, axially symmetric caustic whose uniform description requires a superposition of countably many angular (circular) harmonics; the forward diffraction peak that dominates small-angle scattering in the short-wavelength (semiclassical) regime, whose generic unfolding may involve uncountably many parameters; and caustics passing through a source point. Because their generic unfoldings appear to need infinitely many perturbation parameters, these caustics fall outside Thom's finite-co-dimension theorem, and no substitute classification exists. Related but distinct open questions raised in the same review include the structurally stable wavefront dislocation (phase singularity, i.e. a line of zero intensity where the wave phase is undefined) patterns conjectured to accompany every generic caustic, the extension to normal-mode spectra of bound systems, and the existence of a short-wave limit when caustics are dense (rough surfaces, ergodic ray fields).

Problem Statement

Construct a mathematical classification scheme, playing the role for caustics of infinite co-dimension that Thom's elementary-catastrophe classification plays for generic finite-co-dimension caustics, for the caustics that, though non-generic in Thom's sense, arise commonly in wave physics and require infinitely many unfolding parameters: the glory (the toroidal, axially symmetric backscattering caustic of a sphere, for which Berry conjectured a countably infinite unfolding in circular harmonics), the forward diffraction peak of semiclassical scattering (for which the space of generic unfoldings may be uncountable), and source-point caustics. The scheme is sought at the general level — a classification principle for infinite-co-dimension caustics as a class, exercised on the three caustics named above — and it should jointly specify their canonical forms, their generic unfoldings (here 'generic' is understood within the symmetry class that generates the caustic, such as axial/toroidal symmetry for the glory, so that it is symmetry-breaking perturbations that reduce them to finite-co-dimension catastrophes), and the corresponding uniform diffraction approximations of the wave fields near them. Included in the same objective, as its established physical instances, are the glory and forward-peak diffraction patterns themselves; excluded are the separable open questions raised in the same review (complete asymptotic series of the uniform approximations, catastrophe theory of normal-mode spectra, the short-wave limit for dense caustics, and the dislocation-pattern conjecture), which are outside the scope of this objective.

References

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