An Exact Solution of Mie Type for Scattering by a Multilayer Anisotropic Sphere

Abstract

The purpose of this paper is to describe methods of resolving discrepancies between experimental observations of scattering by crystalline particles and attempts to explain these observations assuming that the electrical properties of these particles can be described by the use of scalar valued functions for the permittivity, conductivity, and permeability. We can develop coupled integral equations describing the interaction of electromagnetic radiation with a heterogeneous, penetrable, dispersive, anisotropic scatterer and can use several methods of solving these integral equations. The solution of the problem of describing scattering by an anisotropic sphere can be substituted into the integral equation to check the integral equation formulation of the problem. Conditions are given for the uniqueness of the solution of the associated transmission problem. Because of the multiple propagation constants in an anisotropic material, the trivial uniqueness arguments valid for isotropic scatterers do not have a guaranteed success in understanding the more complex interaction phenomena. An exact analytical Mie-like solution has been obtained for fields induced in and scattered by an N layer sphere, where each layer has anisotropic constitutive relations. We show that as the tensor parameters change so that each layer becomes isotropic, then the distinct radial functions used in representing the electric and magnetic fields induced in the structure both converge to the same spherical Bessel and Hankel functions and all the propagation constants in each layer converge to the propagation constant k given by k² = ω²μ∈- iωμσ and the solution approaches the ordinary Mie solution for an N-layer sphere. The anisotropic sphere computer code, for the case of magnetic losses, dielectric losses, and dissipative impedance sheets, and perfectly conducting or penetrable inner cores has been validated by energy balance computations involving balancing the difference between the total energy entering the sphere minus the total energy scattered away with the sum of the surface integrals representing losses due to dissipative impedance sheets separating the layers plus the sum of the triple integrals over the layers whose values represent magnetic and dielectric losses within the anisotropic penetrable layers. Two Bessel functions with two different complex indices depending on ratios of tangential and radial magnetic properties and ratios of tangential and radial electrical properties, respectively, participate in the solution in the case of scattering by the simplest anisotropic sphere. The scattering problem is solved for the case where the scatterer consists of (i) N anisotropic dielectric layers, (ii) N of these layers separated by sheets of charge or impedances, and (iii) a perfectly conducting core surrounded by N-1 anisotropic layers.