Determination of Permittivity from the Modulus of the Electric Strength of a High-Frequency Electromagnetic Field

Abstract

For an unmagnetized nonconducting medium, a system of electrodynamic equations corresponding to time-periodic oscillations is considered. For this system, we study the problem of determining the permittivity \(\varepsilon \) from a given electric field magnitude in the electromagnetic field resulting from the interference of two fields produced by external point current sources. It is assumed that the permittivity differs from a given positive constant \({{\varepsilon }_{0}}\) only inside a compact domain \({{\Omega }_{0}} \subset {{\mathbb{R}}^{3}}\) and the electric field magnitude is given for all frequencies, starting at a fixed one \({{\omega }_{0}}\), on the boundary \(S\) of a domain \(\Omega \) surrounding \({{\Omega }_{0}}\). By analyzing an asymptotic expansion of the solution to a direct problem at high frequencies, it is shown that, with the use of given information, the original problem is reduced to the well-known inverse kinematic problem of determining the refraction coefficient inside \(\Omega \) from the travel times of an electromagnetic wave propagating between boundary points of this domain. This reduction opens up the possibility of obtaining a constructive solution of the original problem.