Reflecting Luneburg Lenses

This article presents the exact closed-form solution for a new planar lens, hereinafter referred to as reflecting Luneburg lens (RLL). The proposed structure consists of two stacked parallel-plate waveguides of circular shape. The rays generated by a point source located at the periphery of the bottom waveguide propagate along curvilinear paths, whose trajectories result from a variable refractive index profile with azimuthal symmetry. Then, these rays encounter a reflecting boundary and emerge all parallel in the upper uniform waveguide. The behavior of this lens resembles that of a flat Luneburg lens, with the fundamental difference that it works in reflection. The exact refractive index profile is found by solving the nonlinear integral equation of ray congruence through a truncated Abel transform method. The concept is numerically verified through different implementations of the effective refractive index profile, including a metasurface-based implementation. The proposed lens triggers new possibilities that the normal flat Luneburg lens does not offer, and it is applicable in a large variety of microwave, terahertz, and optical devices.