Investigation of direct and inverse scattering problems for inhomogeneous medium

This paper considers the direct and inverse scattering problems for the refractive index in the case of a planar wave. Liouville´s transform is employed for transforming the Helmholtz equation in the form of the Sturm equation with a simple relationship between potential in the Sturm equation and refractive index in the Helmholtz equation. The exact and approximate solutions of the inverse and direct problems are formulated. We considered the simplest case of a plane wave incident along the normal to the 1D profile of the refractive index (without dispersion and absorption). The exact and approximate solutions are obtained for two representations of the field in an inhomogeneous layer: (i) in the form of two waves propagating locally in forward and back directions; (ii) using lost functions (the second representation results in the appearance of the well known Gel´fand-Levitan integral equation). It is shown that the first approximations corresponding to single backscattering are similar for both the representations. These results can be easily generalized for the case of an arbitrary angle of incidence of a plane wave.