A General Solution for Double-Layer Gradient-Index and Geodesic Lenses with Rotational Symmetry

A double-layer lens consists of a pair of rotationally symmetric index profiles or geodesic lens shapes connected by a reflecting mirror partially covering their common periphery. Such a lens can provide a focus in each layer, and a wave travelling between the foci explores both layers. Here, we concentrate on the case with one layer being homogeneous or flat, and derive a general solution for the lens profiles by solving a Luneburg-like inverse problem with pre-specified foci inside or outside the lens, and different background indices in two layers. We demonstrate four examples of interest in ray-tracing plots. These lenses may find application in communications, sensing, and imaging from millimeter waves up to the optical bands.