Archive for Rational Mechanics and Analysis

, Volume 225, Issue 3, pp 1233–1277 | Cite as

Homogenization Near Resonances and Artificial Magnetism in Three Dimensional Dielectric Metamaterials

  • Guy Bouchitté
  • Christophe Bourel
  • Didier Felbacq


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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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Guy Bouchitté
    • 1
  • Christophe Bourel
    • 2
    • 3
  • Didier Felbacq
    • 4
  1. 1.Institut de Mathématiques, EA 2134Université de ToulonToulon Cedex 9France
  2. 2.Laboratoire de Mathématiques Pures et Appliquées Joseph LiouvilleUniv. Littoral Côte d’OpaleCalaisFrance
  3. 3.CNRSParisFrance
  4. 4.L2CUniversité de MontpellierMontpellierFrance

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