We discuss two inverse problems related to anisotropic media for Maxwell's equations. The first one is the inverse scattering problem of determining the anisotropic surface impedance of a bounded obstacle from a knowledge of electromagnetic scattered field due to incident plane waves. Such an anisotropic boundary condition can arise from surfaces covered with patterns of conducting and insulating patches. We show that the anisotropic impedance is uniquely determined if sufficient data is available, and characterize the non-uniqueness presence if a single incoming wave is used. We derive an integral equation for the surface impedance in terms of solutions of a certain interior impedance boundary value problem. These solutions can be reconstructed from far field data using the Herglotz theory underlying the Linear Sampling Method. The second problem is to obtain information about matrix index of refraction of an anisotropic media. This problem plays a special role in inverse scattering theory due to the fact that the (matrix) index of refraction is not in general uniquely determined from scattering data. Our imaging tool is a new class of eigenvalues associated with the scattering by inhomogeneous media, known as transmission eigenvalues. In this presentation we describe how transmission eigenvalues can be determined from scattering data and be used to obtain upper and lower bounds on the norm of the index of refraction. For both problems the support of the scatterer can be determined by mean of the Linear Sampling Method. Preliminary numerical results will be shown for both problems.
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