An inverse phaseless problem for electrodynamic equations in an anisotropic medium

For the system of equations of electrodynamics which has the anisotropy of the permittivity, an inverse problem of determining the permittivity is studied. It is supposed that the permittivity is characterized by the diagonal matrix ∈ = diag (ε₁(x), ε₁(x), ε₂(x)) and ε₁ and ε₂ are positive constants anywhere outside of a bounded domain Ω₀ ⊂ ℜ³. Periodic in time solutions of the system of Maxwell’s equations related to two modes of plane waves falled down from infinity on the local non-homogeneity located in Ω₀ is considered. For determining functions ε₁(x) and ε₂(x) some information on the module of the vector of the electric strength of two interfered waves is given. It is demonstrated that this information reduces the original problem to two inverse kinematic problems with incomplete data about travel times of the electromagnetic waves. An investigation of the linearized statement for these problems is given. It is shown that in the linear approximation the problem of the determining ε₁(x) and ε₂(x) is reduced to two X-ray tomography problems.