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Inverse scattering problem for the Maxwell's equations

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 $A(β,α,k)$ known for a fixed $k=ω\sqrt{\ep_0 μ_0}>0$ and all $β,α\in S^2$, where $S^2$ is the unit sphere in $\R^3$, $\ep_0$ and $μ_0$ 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.