On the use of the Pad\'{e}-Fourier approximation in fast evaluation of the Green's function of layered media

Efficient Green's function evaluation in layered media is a holy-grail of wave theory in general and for electromagnetics in particular. While there is a very large amount of knowledge in this context with vast literature, there are yet challenging cases such as the Green's function in thick lossy media and the Green's function at thick media with negative parameters. Here we propose a technique that can nicely tackle these issues. Our approach is based on a rational function approximation of the spectra using the Fourier-Pad\'{e} approximation that is carried out in a conformal mapped spectral plane. We show that this approach can be used in challenging scenarios such as very thick and lossy layers, materials with negative parameters such as in plasmonics, and even to approximate a dominant branch-cut contribution far from the source.