A relation between a class of boundary value problems in a homogeneous and an inhomogeneous region

An equivalence is demonstrated between a class of rotationally symmetric ring-source-excited three-dimensional scattering problems in a homogeneous region and a related class of line-source-excited two-dimensional problems in an inhomogeneous medium with an inverse square permittivity profile. While mappings from homogeneous to inhomogeneous regions are conventional, the novelty in the present treatment is the accommodation of relatively arbitrary variations in refractive index over the obstacle surface. After a listing of various solvable problems in either category, and the formulation of an equivalence relation for two-dimensional small obstacle scattering, attention is given to the ray configurations for the primary fields and to their perturbation by interposed structures. An analytical and graphical mapping is employed to deduce the curved ray trajectories in the inhomogeneous medium directly from their straight counterparts in the homogeneous case, without the necessity of solving the differential equations for the rays. The relation between the two ray systems provides an interesting insight into analogous scattering phenomena and is used in particular for the study of reflection, caustics, diffraction by a smooth surface, and shadowing.