Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts

Abstract

Axisymmetric toroidal wave fronts are pertinent to the near forward and backward scattering by objects that have rotational symmetry. For spheres, the wave field produced by such a wave front is known as glory scattering. As the wave front propagates, some porton of it becomes focused on an axis, forming a structurally unstable line caustic. A specific class of harmonic perturbations of the wave front shape is considered that leads to unfoldings of the axial caustic. When the wave front shape is perturbed to have two-fold rotational and mirror symmetry, the unfolded caustic is a four-cusped astroid curve. The three-fold symmetric perturbed wave front propagates to produce a hypocycloid caustic with three cusps. In general, perturbed wave fronts with p-fold rotational and mirror symmetry have caustics of cusped stars, with p cusps when p is odd, and 2p cusps when p is even. These wave front perturbations have applications to scattering from symmetric, slightly nonspherical, homogeneous objects such as spheroids. Wave fields are computed using a Fresnel approximation of the diffraction integral. The wave field patterns associated with astroid caustics are displayed. They have features similar to Pearcey patterns. Applications to backscattering from spheroids, distorted torii, and axicon reflectors are noted. Certain inverse problems are considered. An inequality is given for determining the magnitude of wave front perturbations needed to cause a significant change in the wave field from that of a spherical scatterer. The merging of rays as the observation direction moves across the caustic is discussed using concepts from catastrophe optics. A novel expression in polar coordinates is given for the Hessian associated with propagation.