Analytic Solution and Spectral Analysis of the Differential Surface Admittance Electric Field Integral Equation for Spherical Geometries

Abstract

In order to enable a rigorous spectral analysis of the existing single-source differential surface admittance electric field integral equation (DSA-EFIE), a fully analytical solution for scattering at a nonmagnetic homogeneous dielectric sphere is presented. To this end, the pertinent surface unknowns and operators are expanded into vector spherical harmonics and subjected to a Galerkin method of moments (MoM). The resulting integrals are all computed analytically, the solution is proven to be fully equivalent to the reference Mie series, and the spectral properties are derived for any Sobolev testing space. Consequently, several important insights are rigorously shown in the spectral analysis. First and foremost, the DSA-EFIE provides an exact solution of Maxwell’s equations for any complex dielectric material and frequency, establishing the approach as a precise single-source boundary integral formulation. Moreover, through the application of the generalized Fourier series, closed analytical formulas are derived for all pertinent elements, leading to expressions devoid of any numerical integration or summation, thereby strengthening the method’s efficacy and accuracy. In addition, this work shows that the concatenation of the two pertinent integral operators provides a formulation with a bounded condition number in both the space of square integrable functions and the Sobolev H−1/2div space, required for a bounded EFIE Galerkin solution. Moreover, it is deduced that the inherent low-frequency breakdown of the EFIE is avoided as well. Last, the issue of internal resonances, typical for single-source formulations, persists although they are not simply those of the DSA and EFIE operator on their own. Hence, the advocated analysis approach has brought forth important insights and provides a useful avenue for analyzing alternative integral equation formulations with extended capabilities or improved properties.