Reflecting Luneburg Lens: Analytical Solution and Applications

This paper presents the closed-form solution for a new type of lens beamformer, which consists of two stacked parallel plate waveguides (PPWs) of circular shape. The rays launched by a source at the rim of the bottom PPW are collimated in the top PPW after being reflected at a cylindrical metallic wall that surrounds both layers. Since it reproduces the behavior of the Luneburg lens for reflected rays, this lens is hereinafter referred to as Reflecting Luneburg lens (RLL). The refractive index profile for the azimuthally symmetric circular domain is found by solving the non-linear integral equation of ray-congruence through a truncated Abel transform method. The validity of the derived exact formula is verified by full-wave simulations, practical implementation and possible applications of RLLs in antenna systems are also briefly discussed.