Analytic Solution of Surface–Volume–Surface Electric Field Integral Equation on Dielectric Sphere and Analysis of Its Spectral Properties

Abstract

The exact solution of the surface–volume–surface electric field integral equation (SVS-EFIE) is presented for the problem of radiation in the vicinity of the homogeneous dielectric sphere. The solution is obtained via the Galerkin method of moments (MoM) utilizing rotational and irrotational complete sets of orthogonal vector spherical harmonics as basis and test functions according to the Helmholtz decomposition. In the case of radial or tangential electric dipole radiation, the electric field throughout the sphere evaluated via the analytic MoM solution of the SVS-EFIE is compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. The exact solution confirms and validates the rigorous nature of the SVS-EFIE formulation. It also reveals the spectral properties of its individual operators, their products, and their linear combination, as well as the spectrum of the MoM impedance matrix. It is shown that, upon choosing basis and test functions in the Sobolev space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{1/2}(S)$ </tex-math></inline-formula> and performing testing inner products evaluation in the space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{-1/2}(S)$ </tex-math></inline-formula> , with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> being the surface of the sphere, the MoM impedance matrix features bounded condition number with increasing order of discretization similar with analogous exact MoM solution of the surface EFIE (S-EFIE) on the perfectly electrically conducting (PEC) sphere.