The Foldy--Lax Approximation for the Full Electromagnetic Scattering by Small Conductive Bodies of Arbitrary Shapes

Abstract

We deal with the electromagnetic wave propagation in the harmonic regime. We derive the Foldy--Lax approximation of the scattered fields generated by a cluster of small conductive inhomogeneities of arbitrary shapes and arbitrarily distributed in a bounded domain $\Omega$ of $\mathbb{R}^3$. The dominating term in the approximation of the electric fields (respectively, the magnetic fields) is a combination of both electric and magnetic dipoles through scattering coefficients which are solutions of an algebraic system, i.e., the Foldy system. This algebraic system describes the multiple electromagnetic interactions between the small conductivities. This approximation is valid under a sufficient but general condition on the number of such inhomogeneities $m$, their maximum radii $\epsilon$, and the minimum distances between them, $\delta$, of the form $(\ln m)^{\frac{1}{3}}\frac{\epsilon}{\delta} \leq C, $ where $C$ is a constant depending only on the Lipschitz characters of the scaled inhomogeneities. In addition, we provide explicit error estimates of this approximation in terms of the aforementioned parameters, $m, \epsilon, \delta$ but also the used frequencies $k$ under the Rayleigh regime. Both the far fields and the near fields (stated at a distance $\delta$ to the cluster) are estimated. In particular, for a moderate number of small inhomogeneities $m$, the derived expansions are valid in the mesoscale regime where $\delta \sim \epsilon$.