A Numerical Verification of Analytical Solution for the Mixed Fourier Transform Equation

Analysis of 2D-periodic structures is often carried out with the Method of Moments, which is capable to obtain simultaneously the phase velocity and attenuation constant of surface- or leaky-waves (i.e. eigenmodes). However, to find the eigenmodes associated with a given geometry, many samples of the impedance matrix are needed, whose evaluation can be computationally expensive. To overcome the computational burden, efficient interpolation procedures have been proposed, which are based on the decomposition of the impedance matrix into Floquet modes and the proper extraction of some of them. However, the evaluation of a given Floquet mode with the reaction integral can be difficult when the integration regions of the source and test basis functions overlap along the z-axis. In this work, we deal with this difficult case: we propose a fully analytical solution and validate it numerically. Our work paves the way to the generalization of existing fast interpolation procedures to arbitrary structures and, eventually, to the fast analysis of general 2D-periodic geometries.