Non-iterative two-step method for solving scalar inverse 3D diffraction problem

The scalar problem of reconstruction of an unknown refractive index of an inhomogeneous solid is considered. The original boundary value problem for the Helmholtz equation with an unknown refractive index is reduced to the source-type integral equation. The solution to the inverse problem is obtained in two steps. First, the integral equation of the first kind is solved in the inhomogeneity domain using measurements of the total field outside the domain. The uniqueness of a solution to the integral equation of the first kind is proved in a special class of functions. Second, the sought-for refractive index is explicitly expressed via the found solution and the total field. The two-step method was verified by solving a test problem with a given refractive index. Procedures for refining approximate solutions were proposed and implemented. Efficiency of the proposed method was approved by comparison between the exact solution and the approximate ones.