The expansion wave concept. I. Efficient calculation of spatial Green's functions in a stratified dielectric medium

Abstract

A procedure is given to perform the inverse Fourier transformation relating a spatial Green's function to its spectral equivalent. The procedure is applied to the spectral Green's functions of the double scalar mixed-potential integral expression formulation of the electromagnetic field in a stratified dielectric medium. The extraction technique is used to annihilate every type of "problematic" behavior of the spectral Green's functions. Every annihilating function is inverse Fourier transformed analytically. It is shown that the annihilation of both the surface wave poles and the singularities at the branch point results in a set of analytical spatial functions, which are a very good approximation of the exact spatial Green's function down to relatively small lateral distances. Some very important characteristics of these functions will play a crucial role in Part II of the paper, where a new technique is introduced to model mutual coupling.

Green's Functions of Filament Sources Embedded in Stratified Dielectric Media

“…The evaluation procedure of the Green's functions always starts in the spectral domain. In this domain, several recursive techniques can be employed to calculate the required Green's functions [1][2][3][4][5].…”

“…Most of the techniques presented in the literature are approximating this asymptotic behavior using analytical functions whose inverse Fourier transform are known analytically. The approximating functions are subtracted from the spectral Green's function and their inverses are added analytically in the spatial domain [2,4,5,7].…”

“…The physical interpretation of a pole is a surface wave mode, which can be considered as an eigen solution to the unloaded dielectric layer structure. The behavior of the poles can also be analytically approximated, subtracted from the spectral functions and their inverse counterparts are analytically added in the spatial domain [2,4,5,7].…”

“…Most of the techniques presented in the literature avoid these branch points by deforming the Sommerfeld integration path from the real axis to another path in which the branch point singularities do not appear [4,7]. Since the fast variation in the spectral Green's functions are essentially dominating at large spatial values, the technique presented in [2] provides more accurate Green's functions at high spatial values. In this technique, the branch point singularity is also analytically approximated and subtracted in the spectral domain.…”

“…Using Fourier transform identities, the inverse Fourier transform is obtained analytically and added. The remaining spectral function along the real axis is a smooth and fast decaying function, which can be integrated numerically [2] over a finite interval.…”