Mesoscale Approximation of the Electromagnetic Fields

Abstract

The point-interaction approximation (or the Foldy–Lax approximation) of the electromagnetic fields generated by a cluster of small-scaled inhomogeneities is derived in the mesoscale (i.e., mesoscopic scale) regime, that is, when the minimum distance $$\varvec{\delta }$$ between the particles is proportional to their maximum diameter $${\varvec{a}}$$ in the form $$\varvec{\delta }={\varvec{c}}_r \, {\varvec{a}}$$ with a positive constant $${\varvec{c}}_r$$ that we call the dilution parameter. The small particles are modeled by anisotropic and variable electric permittivities and/or magnetic permeabilities with possibly complex values. We provide the dominating field (the so-called Foldy–Lax field) with explicit error estimates in terms of the dilution parameter $${\varvec{c}}_r$$ uniformly in terms of the distribution of these inhomogeneities. Such approximations are key steps in different research areas as imaging and material sciences.