Homogenization near resonances and artificial magnetism in 3D dielectric metamaterials

TL;DR

This paper develops a full 3D homogenization method for dielectric metamaterials, revealing how micro-resonances induce artificial magnetism and frequency-dependent permeability, potentially offering an alternative to metallic structures.

Contribution

It introduces a new averaging technique for 3D dielectric structures, deriving explicit effective permeability tensors and analyzing micro-resonances in a bounded domain.

Findings

Effective permeability tensor depends explicitly on frequency.

Micro-resonances are characterized by a vectorial spectral problem.

Dielectric inclusions can produce negative permeability without metals.

Abstract

It is now well established that the homogenization of a periodic array of parallel dielectric fibers with suitably scaled high permittivity can lead to a (possibly) negative frequency-dependent effective permeability. However this result based on a two-dimensional approach holds merely in the case of linearly polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this paper we consider a dielectric structure placed in a bounded domain of $\mathbb{R}^3$ and perform a full 3D asymptotic analysis. The main ingredient is a new averaging method for characterizing the bulk effective magnetic field in the vanishing-period limit. We evidence a vectorial spectral problem on the periodic cell which determines micro-resonances and encodes the oscillating behavior of the magnetic field from which artificial magnetism arises. At a macroscopic level we deduce an effective permeability tensor that we can be make explicit as a function of the frequency. As far as sign-changing permeability are sought after, we may foresee that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry.