The Effective Permittivity and Permeability Generated by a Cluster of Moderately Contrasting Nanoparticles

TL;DR

This paper explicitly characterizes the effective electromagnetic properties of a cluster of nanoparticles with moderate contrast in permittivity and permeability, providing error estimates and a connection to Lippmann-Schwinger equations.

Contribution

It introduces a novel explicit characterization of effective permittivity and permeability for nanoparticle clusters with anisotropic properties, including error bounds and a link to integral equations.

Findings

Effective permittivity and permeability are expressed via polarization tensors.

The approximation error is inversely proportional to the dilution parameter.

The Lippmann-Schwinger operator is proven invertible in Hölder spaces.

Abstract

In a $3D$ bounded and $C^{1, \alpha}$-smooth domain $\Omega$, $\alpha \in (0, 1)$, we distribute a cluster of nanoparticles enjoying moderately contrasting relative permittivity and permeability which can be anisotropic. We show that the effective permittivity and permeability generated by such cluster is explicitly characterized by the corresponding electric and magnetic polarization tensors of the fixed shape. The error of the approximation of the scattered fields corresponding to the cluster and the effective medium is inversely proportional to the dilution parameter $c_r:=\frac{\delta}{a}$ where $a$ is the maximum diameter of the nanoparticles and $\delta$ the minimum distance between them. The constant of the proportionality is given in terms of a priori bounds on the cluster of nanoparticles (i.e. upper and lower bounds on their permittivity and permeability parameters, upper bound on the dilution parameter $c_r$, the used incident frequency and the domain $\Omega$). A key point of the analysis is to show that the Foldy-Lax field appearing in the meso-scale approximation, derived in \cite{AM}, is a discrete form of a (continuous) system of Lippmann-Schwinger equations with a related effective permittivity and permeability contrasts. To derive this, we prove that the Lippmann-Schwinger operator, for the Maxwell system, is invertible in the H\"{o}lder spaces. As a by-product, this shows a H\"{o}lder regularity property of the electromagnetic fields up to the boundary of the inhomogeneity.