Spherical media and geodesic lenses in geometrical optics

Abstract

This paper presents a general approach to solving the problems of inverse scattering in three-dimensional isotropic media with a spherically symmetric refractive index distribution. It is based on equivalence of the central section of an inhomogeneous medium and corresponding geodesic lens, which is a non-Euclidean surface with constant refractive index. We use this approach for solving the Luneburg inverse problem and also for the derivation and design of absolute instruments that provide perfect imaging within the frame of geometrical optics. In addition, we solve the generalized Luneburg inverse problem, which leads to the discovery of a new class of magnifying lenses.