Ray Paths and Frequency Dependence of Mismatched Luneburg Lenses

The recently introduced mismatched Luneburg lenses (MLLs) are studied with an in-depth analysis regarding their frequency dependence. This article proposes an analytical expression of the ray paths inside such lenses, which is then used to derive a radiation integral. The effect of the mismatch appears to be frequency dependent, and an expression for the optimal homogeneous lens is given as a function of frequency. The previously proposed law linking the refractive index at the lens center and the lens edge is now modified into a generalized rule, taking into account that frequency dependence.