Nonsingular Laplacian Representation of the Asymptotic Part of the Layered Medium Green Function in the Mixed Potential Formulation

We consider new developments in the analytical evaluation of the near-field contribution to the matrix elements of the electric and magnetic field operators for planar conducting structures embedded in a layered medium. The method is applicable to Rao-Wilton-Glisson (RWG) basis functions supported on parallel interfaces in the layered medium. The method uses suitably constructed representations of the mixed potential formulation integral kernels in terms of two-dimensional Laplacian of auxiliary functions. Such Laplacian representations can be obtained for the asymptotic forms of the Green functions, which are being subtracted in order to regularize the behavior of the Sommerfeld-type integrals. Matrix elements resulting from these asymptotic forms, given originally as quadruple surface integrals with singular integrands, are then reduced to double contour integrals over the perimeters of the surface elements, involving simple closed-form non-singular auxiliary functions M. The new developments include: · Derivation of relations between elements of the asymptotic dyadic Green functions and the kernels of the mixed-potential representation of the fields. · Inclusion of additional terms introduced in [1], which improve convergence of the Sommerfeld integrals. These additional kernel components, related to half-line source potentials, were not included in our previous paper [1]; they constitute non-leading asymptotic contributions to the mixed-potential kernels K
^Φ
and K
^Ψ
. · Construction of two additional auxiliary functions needed to represent the above-mentioned additional terms. The resultant auxiliary functions are expressed as integrals of the previously obtained [1] functions for the leading asymptotic kernel terms. · Construction of simplified analytical expressions for the matrix elements of the asymptotic parts of the pertinent dyadic Green functions. The asymptotic matrix elements, given in terms of quadruple surface integrals with singular integrands, are subsequently converted, by using suitably constructed Laplacian representations of the Green function, to double contour integrals over the perimeters of the surface elements, with simple, non-singular, smoothly varying integrands. The line integrals can be either evaluated analytically or by means of low order numerical quadratures. We discuss the relative merits of the direct numerical and analytic evaluation if these line integrals.