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 nonlinear 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 high aperture efficiency (63%) across a wide frequency range.