Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations

TL;DR

This paper presents a constructive method for wave-field decomposition in anisotropic media based on Maxwell's equations, involving spectral analysis and operator integrals, extending the Dirichlet-to-Neumann map concept.

Contribution

It introduces a novel constructive proof for wave-splitting in anisotropic media using operator theory and spectral analysis, generalizing impedance mapping.

Findings

Existence of directional wave-field decomposition proved.

Construction of a splitting matrix via Dunford-Taylor integral.

Extension of Dirichlet-to-Neumann map to anisotropic media.

Abstract

The equations for the electromagnetic field in an anisotropic media are written in a form containing only the transverse field components relative to a half plane boundary. The operator corresponding to this formulation is the electromagnetic system's matrix. A constructive proof of the existence of directional wave-field decomposition with respect to the normal of the boundary is presented. In the process of defining the wave-field decomposition (wave-splitting), the resolvent set of the time-Laplace representation of the system's matrix is analyzed. This set is shown to contain a strip around the imaginary axis. We construct a splitting matrix as a Dunford-Taylor type integral over the resolvent of the unbounded operator defined by the electromagnetic system's matrix. The splitting matrix commutes with the system's matrix and the decomposition is obtained via a generalized eigenvalue-eigenvector procedure. The decomposition is expressed in terms of components of the splitting matrix. The constructive solution to the question on the existence of a decomposition also generates an impedance mapping solution to an algebraic Riccati operator equation. This solution is the electromagnetic generalization in an anisotropic media of a Dirichlet-to-Neumann map.