Inverse scattering problem for the Maxwell's equations

TL;DR

This paper introduces a new scalar integral equation approach for solving the inverse scattering problem in Maxwell's equations, enabling the reconstruction of the permittivity distribution from scattering data with stability considerations.

Contribution

It reduces the vector electromagnetic inverse problem to a scalar integral equation without symmetry assumptions and provides formulas for stable solutions.

Findings

Derived a scalar integral equation for inverse scattering

Reduced vector EM problem to scalar kernel integral equation

Proposed a method for stable solution of the ill-posed integral equation

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 {\it 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 $\ep'(x)$ in a bounded region $D\subset \R^3$ is found from the scattering amplitude $A(\beta,\alpha,k)$ known for a fixed $k=\omega\sqrt{\ep_0 \mu_0}>0$ and all $\beta,\alpha \in S^2$, where $S^2$ is the unit sphere in $\R^3$, $\ep_0$ and $\mu_0$ are constant permittivity and magnetic permeability in the exterior region $D'=\R^3 \setminus D$. The {\it novel points} in this paper include: i) A reduction of the inverse problem for {\it vector EM waves} to a {\it vector integral equation with scalar kernel} without any symmetry assumptions on the scatterer, ii) A derivation of the {\it 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.