Caustics of wave fronts reflected by a surface
Alexander Yampolsky, Oleksandr Fursenko
TL;DR
This work addresses how caustics of reflected wave fronts depend on the geometry of the reflecting surface. It introduces a purely geometric approach in which a virtual deformation of the mirror surface yields an explicit parameterization of the caustic, expressed as xi^*=r+(1/k^*)b with g^*_{ij}=g_{ij}-(partial_j r, a)(partial_i r, a) and B^*_{ij}=-2 cos theta B_{ij} and k^* satisfying (k^*)^2+2 cos theta (2H+k_n(a_t) tan^2 theta) k^* + 4K = 0. The main results provide closed-form equations for the two caustics for both flat and spherical incidence, together with explicit examples that illustrate the construction. The approach enables fast, exact visualization and has potential applications in geometric optics and antenna theory.
Abstract
One can often see caustic by reflection in nature, but it is rather hard to understand the way of how caustic arise and which geometric properties of a mirror surface define the geometry of the caustic. The caustic by reflection has complicated topology and much more complicated geometry. From engineering point of view, the geometry of caustic by reflection is important for antenna's theory because it can be considered as a surface of concentration of the reflected wave front. In this paper, we give purely geometric description of the caustics of a wave front (flat or spherical) after reflection from a mirror surface. The description clarifies the dependence of caustic on geometrical characteristics of a surface and allows rather simple and fast computer visualization of the caustics in dependence of location of the rays source or direction of the pencil of parallel rays.