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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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