An electrodynamic inverse problem in chiral media

Abstract

We consider the inverse problem of determining the electromagnetic material parameters of a body from information obtainable only at the boundary of the body; such information comes in the form of a boundary map which we assume to be known. In particular we consider the question in the case of a chiral body. In such a body, the relationship between the electromagnetic fields depends not only on the conductivity, electric permittivity and magnetic permeability of the body, but further on the chirality.We consider two problems. The first is determination of the parameters and their normal derivatives at the boundary of the body. We show that in both the chiral and non-chiral cases, such information is obtainable for all the parameters. We also show how a layer stripping algorithm may be derived to estimate the unknown parameters near the boundary in both situations. The approach is to calculate an explicit asymptotic expansion for the symbol of the boundary map which is shown to be a pseudo-differential operator; this expansion is shown in each case to determine the unknown parameters at the boundary.The second problem is that of interior determination. We show that knowledge of the boundary map determines the electromagnetic parameters in the interior under the assumption that we know the parameters to infinite order at the boundary. We rewrite Maxwell's equations as a first order perturbation of the Laplacian and construct exponentially growing solutions, and obtain the result in the spirit of complex geometrical optics.