Radial GRIN Lenses based on the Solution of a Regularized Ray Congruence Equation

We introduce a novel formulation for flat radial GRadient INdex (GRIN) lenses allowing for the optimal lens design through closed form expressions. The validity of the proposed formulation covers a very large range of GRIN lens design parameters (focal distance, lens thickness and maximum refractive index). The formulation is based on the derivation of a new form of non-linear integral equation representing the equalization of all the optical ray-path lengths, denoted as Regularized Ray Congruence (RRC) equation, and on its closed form solution. An analytical form of aperture efficiency is given for standard feed patterns. The application of the formulas presented here allows for an instantaneous design for medium/high gain antennas with controllable total aperture efficiency till 80%. The accuracy of the formulation is tested by a full-wave analysis and compared with other formulations available in the literature. We found that the new formulation proposed here significantly reduces the phase error in a wide range of the lens parameters, thus allowing for a more efficient, accurate and flexible design for GRIN lenses. As a proof of concept, a thin GRIN lens antenna for operation in E-band (60-90 GHz) is designed, demonstrating a high aperture efficiency (63%) across a wide frequency range.