Printed-cavity and whispering-gallery-mode resonances in discrete Luneburg lens antenna integrated with a small double-disk feed

An accurate mathematical and numerical analysis of both a Luneburg lens and a slot-fed spherical-circular printed antenna with a spherical-circular ground conductor is fully presented. The entire analysis is done in terms of the spherical vector wave function expansions in each partial domain. The feed is modelled by an horizontal magnetic dipole used as a forcing current. Applying the boundary conditions lead to the overall scattered electromagnetic field at any spatial position. The problem is cast into a coupled set of the dual-series equations for the expansion coefficients, and then to an infinite-matrix equation having favorable features by expressing the currents into the Fourier domain. This is achieved by following the Method of Analytical Regularization, which is based here on the explicit inversion of the static part of the dual-series equations thanks to the Abel integral equation properties. Such a procedure leads to a guaranteed convergence and controlled accuracy of computations in accordance with the Fredholm theorem of the Fredholm equations of the second kind. Because of its semi analytical nature, the implemented algorithm is accurate and very low in both CPU time and memory capacity consumption.