Analytical formulas for refractive indices of a telescopic GRIN lens for aperture magnification

In this letter, we derive a new closed-form analytical formula for the refractive index of a telescopic lens based on inhomogeneous graded index (GRIN) material. The telescopic lens is composed of two cylindrical confocal GRIN lenses, one divergent and one convergent. The scope of the telescopic lens is to magnify the aperture of a flat antenna placed at the first interface of the divergent-convergent lens doublet. This telescopic GRIN lens is useful for reusing already available medium-gain arrays and bring them to a high-gain, occupying a reduced volume as compared to traditional lenses or multireflector systems. The formulation used here is based on the solution of a regularized ray-congruence integral equation recently introduced by the authors, and applied in a previous work to treat convergent cylindrical GRIN lenses. The formulation is extended here to the case of a telescopic lens leading to analytical formulas for the two inhomogenous refractive indexes that involve five independent parameters; these parameters are eventually optimized to get the maximum gain boost.