By Yu.A. Kravtsov
Caustics, Catastrophes and Wave Fields in a feeling keeps the therapy of the sooner quantity 6 "Geometrical Optics of Inhomogeneous Media" through analysing caustics and their fields at the foundation of recent disaster idea. the current quantity covers neighborhood and uniform caustic asymptotic expansions: The Lewis-Kravtsov approach to average features, Maslov's approach to canonical operators , Orlov's approach to interference integrals, in addition to their transformations for penumbra, space-time, random and different different types of caustics. the entire tools are amply illustrated by way of labored difficulties pertaining to appropriate wave-field functions.
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Extra resources for Caustics, Catastrophes and Wave Fields
9. These caustics have codimension m = 3 and corank 1 = 2. Caustics of higher codimension with m > 3 may be illustrated only by sections with planes. By way of example, Fig. lOa shows a three-dimensional section of an As type caustic (m = 4, 1 = 1) called the butterfly, and the diagram at (b) shows two-dimensional cross sections of this bifurcation set. For all caustics the maximum number of ray is dictated by codimension and equals m+1. Fig. 7. Bifurcation set of the swallowtail (type A4 , m = 3, I = I) Fig.
Indeed, in two dimensions caustics unambiguously dictate the course of rays. This fact has engendered the known property of a homogeneous two-dimensional space where the phase fronts are evolvments of caustics and the caustics are evolutes of phase fronts. We cannot exclude solving the problem of reconstruction of rays by the form of caustics with the formalism of catastrophe theory. Moreover, one may think of a law used to transform a given caustic to the normal form (to be discussed in Chap. 3) as the transformation of the family of rays.
Therefore, in what follows we shall stick to the latter. By way of illustration we estimate the width of a caustic zone near a nonsingular segment of a caustic. 17-19] (also see Sect. 10) where v is the distance normal to the caustic, nc the refractive index at the caustic, K re1 the relative curvature of ray and caustic that in a two-dimensional problem is combined as KR ± Kc of the ray curvature KR and the caustic curvature Kc with the plus sign for the ray and caustic lying on either side of a common tangent.