# American Institute of Mathematical Sciences

August  2009, 3(3): 405-452. doi: 10.3934/ipi.2009.3.405

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

 1 Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Received  September 2008 Revised  April 2009 Published  July 2009

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 of 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.
Citation: B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems and Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405
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