Inverse Scattering Problem for the Maxwell’s Equations

Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA

^∗ Corresponding author. E-mail: ramm@math.ksu.edu

Abstract

Inverse scattering problem is discussed for the Maxwell’s equations. A reduction of the Maxwell’s system to a new Fredholm second-kind integral equation with a scalar weakly singular kernel is given for electromagnetic (EM) wave scattering. This equation allows one to derive a formula for the scattering amplitude in which only a scalar function is present. If this function is small (an assumption that validates a Born-type approximation), then formulas for the solution to the inverse problem are obtained from the scattering data: the complex permittivity ϵ′(x) in a bounded region D ⊂ R³ is found from the scattering amplitude A(β,α,k) known for a fixed k = ω √ϵ₀μ₀ >0 and all β,α ∈ S², where S² is the unit sphere in R³, ϵ₀ and μ₀ are constant permittivity and magnetic permeability in the exterior region D′ = R³\D. The novel points in this paper include: i) A reduction of the inverse problem for vector EM waves to a vector integral equation with scalar kernel without any symmetry assumptions on the scatterer, ii) A derivation of the scalar integral equation of the first kind for solving the inverse scattering problem, and iii) Presenting formulas for solving this scalar integral equation. The problem of solving this integral equation is an ill-posed one. A method for a stable solution of this problem is given.