Ray Paths and Frequency Dependence of Mismatched Luneburg Lenses

The recently introduced <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Mismatched Luneburg Lenses</i> 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.