An Inverse Phaseless Problem for Electrodynamic Equations in an Anisotropic Medium

Abstract

For the system of electrodynamics equations with anisotropic permittivity, the inverse problem of determining the permittivity is studied. It is supposed that the permittivity is characterized by a diagonal matrix \(\epsilon = {\text{diag}}({{\varepsilon }_{1}},{{\varepsilon }_{1}},{{\varepsilon }_{2}}),\) where \({{\varepsilon }_{1}}\) and \({{\varepsilon }_{2}}\) are positive constants everywhere outside of a bounded domain \({{{\Omega }}_{0}} \subset {{\mathbb{R}}^{3}}\). Time-periodic solutions of Maxwell’s equations related to two modes of plane waves coming from infinity and impinging on a local inhomogeneity located in \({{{\Omega }}_{0}}\) are considered. For determining functions \({{\varepsilon }_{1}}(x)\) and \({{\varepsilon }_{2}}(x),\) some information on the magnitudes of the electric strength vectors of two interfering waves is given. It is shown that this information reduces the original problem to two inverse kinematic problems with incomplete data regarding travel times of electromagnetic waves. The linearized statement for these problems is investigated. It is shown that, in the linear approximation, the problem of determining \({{\varepsilon }_{1}}(x)\) and \({{\varepsilon }_{2}}(x)\) is reduced to two X-ray tomography problems.