Resonant effects in random dielectric structures

TL;DR

This paper develops a stochastic homogenization framework for random dielectric nanorod structures, revealing conditions for effective magnetic properties and analyzing resonance-induced singular behaviors.

Contribution

It introduces a rigorous stochastic approach to homogenize random dielectric nanorods and characterizes conditions for negative permeability and resonance effects.

Findings

Homogenization yields a deterministic dispersion law for effective permeability.

Resonances cause singular behaviors as rod density concentrates on real permittivity values.

Conditions identified for negative real part of effective permeability.

Abstract

Recently, a theory for artificial magnetism in two-dimensional photonic crystals has been developed for large wavelength using homogenization techniques. In this paper we pursue this approach within a rigorous stochastic framework: dielectric parallel nanorods are randomly disposed, each of them having, up to a large scaling factor, a random permittivity \epsilon(\omega) whose law is represented by a density on a window \Delta=[a,b]x[0,h] of the complex plane. We give precise conditions on the initial probability law (permittivity, radius and position of the rods) under which the homogenization process can be performed leading to a deterministic dispersion law for the effective permeability with possibly negative real part. Subsequently a limit analysis h->0, accounting a density law of \epsilon, which concentrates on the real axis, reveals singular behavior due to the presence of resonances in the microstructure.