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

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. Keywords directional wave-field decomposition, wave-splitting, anisotropy, electromagnetic system's matrix, generalized eigenvalue problem, algebraic Riccati operator equation, generalized vertical wave number.where J e = external electric current density [A/m 2 ] and K e = external magnetic current density [V/m 2 ]. The external currents are applied, prescribed, sources. The causality of the field is taken into account by requiring that all field quantities are bounded functions of B. L. G. Jonsson

Asymptotic wave-splitting in anisotropic linear acoustics

“…Wave-splitting methods have been implemented in several physically different contexts and for a range of constitutive relations: wave-splitting for wave equations [32,34]. The electromagnetic equations are wave-field decomposed both for isotropic [22,5,24], anisotropic lossless (the spectral theoretical approach) [16] and wave-splitting has been extended to the homogeneous lossless stratified bi-anisotropic case [26,20]. Wave-splitting methods have been applied to linear-elastodynamic equations as for propagation on beams see e.g., [15] as well as in the half-space in homogeneous stratified anisotropic media [9] and up-/down symmetric media [13].…”