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