Foldy-Lax approximation of the electromagnetic fields generated by anisotropic inhomogeneities in the mesoscale regime with complements for the perfectly conducting case

Abstract

The Foldy-Lax (or the point-interaction) approximation of the electromagnetic fields generated by a cluster of small scaled inhomogeneities is derived in the mesoscale regime, i.e. when the minimum distance $\delta$ between the particles is proportional to their maximum radi $a$ in the form $\delta=c_r \; a$ with a positive constant $c_r$ that we call the dilution parameter. We consider two types of families of inhomogeneities. In the first one, the small particles are modeled by anisotropic electric permittivities and/or magnetic permeabilities with possibly complex values. In the second one, they are given as perfectly conductive inclusions. In both the cases, we provide the dominating field (the so-called Foldy-Lax field) with explicit error estimates in terms of the dilution parameter $c_r$. In the case of perfectly conductive inclusions, the results provided here improve sharply the ones derived recently in \cite{AB-SM:MMS2019}. Such approximations are key steps in different research areas as imaging and material sciences.