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Homogenization Near Resonances and Artificial Magnetism in Three Dimensional Dielectric Metamaterials

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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 three dimensional asymptotic analysis. The main ingredient is a new averaging method for characterizing the bulk effective magnetic field in the vanishing-period limit. We give evidence of 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 make explicit as a function of the frequency. As far as sign-changing permeability is 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. Part of these results have been announced in Bouchitté et al. (C R Math Acad Sci Paris 347(9–10):571–576, 2009).

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Authors and Affiliations

  1. Institut de Mathématiques, EA 2134, Université de Toulon, CS 60584, 83041, Toulon Cedex 9, France

    Guy Bouchitté

  2. Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, Univ. Littoral Côte d’Opale, EA 2597 - LMPA, 62228, Calais, France

    Christophe Bourel

  3. CNRS, 2956, Paris, France

    Christophe Bourel

  4. L2C, Université de Montpellier, 34095, Montpellier, France

    Didier Felbacq

Authors
  1. Guy Bouchitté
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  2. Christophe Bourel
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  3. Didier Felbacq
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Correspondence to Guy Bouchitté.

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Communicated by S. Serfaty

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Bouchitté, G., Bourel, C. & Felbacq, D. Homogenization Near Resonances and Artificial Magnetism in Three Dimensional Dielectric Metamaterials. Arch Rational Mech Anal 225, 1233–1277 (2017). https://doi.org/10.1007/s00205-017-1132-1

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  • DOI: https://doi.org/10.1007/s00205-017-1132-1

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