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

TL;DR

This paper derives a precise Foldy-Lax approximation for electromagnetic fields generated by small anisotropic or perfectly conductive inhomogeneities in the mesoscale regime, with explicit error estimates, advancing modeling in imaging and materials science.

Contribution

It introduces an improved Foldy-Lax approximation for electromagnetic fields involving anisotropic and perfectly conductive inhomogeneities, with explicit error bounds in the mesoscale regime.

Findings

Explicit error estimates for the approximation in terms of the dilution parameter.

Enhanced accuracy of the approximation for perfectly conductive inclusions.

Applicability to imaging and material science problems.

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.