Two-Step Method for Solving Inverse Problem of Diffraction by an Inhomogenous Body

Abstract

The problem of reconstruction of an unknown refractive index n(x) of an inhomogeneous solid P is considered. The refractive index is assumed to be a piecewise-Hölder function. The original boundary value problem for the Helmholtz equation is reduced to the integral Lippmann-Schwinger equation. The incident wave is defined by a point source located outside P. The solution of the inverse problem is obtained in two steps. First, the “current” $J = (k^2 - k_0^2)u$ is determined in the inhomogeneity region. Second, the function k(x) is expressed via the current J and the incident wave $u_0.$ The uniqueness of the solution J to the first-kind integral equation is proved in the class of piecewise-constant functions. The two-step method is verified by solving a test problem with a given refractive index. The comparison between the approximate solutions and the exact one approved the efficiency of the proposed method.