[Homogénéisation du système de Maxwell 3D près des résonances et magnétisme artificiel]
Il est maintenant bien connu que l'homogénéisation d'un réseau périodique de fibres diélectriques de forte permittivité peut conduire à une perméabilité effective négative sur certaines bandes de fréquences. Cependant ce résultat basé sur l'analyse d'un résonateur en dimension deux ne pouvait être justifié sans l'hypothèse de polarisation du champ magnétique et en pratique seuls des obstacles cylindriques infinis pouvaient être considérés. Dans cette note nous proposons une extension complète au cas 3D basée sur une nouvelle notion de moyennisation du champ magnétique. Nous obtenons un problème spectral vectoriel sur la cellule unité qui décrit l'effet de résonance et conduit à un tenseur de perméabilité effective dont les parties réelles des valeurs propres changent de signe suivant la fréquence.
It is now well known 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 micro resonator problem on the section of the fibers holds merely in the case of polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this Note we propose a full 3D extension of previous asymptotic analysis based on a new averaging method for the magnetic field. We evidence a vectorial spectral problem on the periodic cell which accounts for micro-resonance effects and leads to a 3D negative effective permeability tensor. This suggests that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry.
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@article{CRMATH_2009__347_9-10_571_0,
author = {Guy Bouchitt\'e and Christophe Bourel and Didier Felbacq},
title = {Homogenization of the {3D} {Maxwell} system near resonances and artificial magnetism},
journal = {Comptes Rendus. Math\'ematique},
pages = {571--576},
publisher = {Elsevier},
volume = {347},
number = {9-10},
year = {2009},
doi = {10.1016/j.crma.2009.02.027},
language = {en},
}
TY - JOUR AU - Guy Bouchitté AU - Christophe Bourel AU - Didier Felbacq TI - Homogenization of the 3D Maxwell system near resonances and artificial magnetism JO - Comptes Rendus. Mathématique PY - 2009 SP - 571 EP - 576 VL - 347 IS - 9-10 PB - Elsevier DO - 10.1016/j.crma.2009.02.027 LA - en ID - CRMATH_2009__347_9-10_571_0 ER -
%0 Journal Article %A Guy Bouchitté %A Christophe Bourel %A Didier Felbacq %T Homogenization of the 3D Maxwell system near resonances and artificial magnetism %J Comptes Rendus. Mathématique %D 2009 %P 571-576 %V 347 %N 9-10 %I Elsevier %R 10.1016/j.crma.2009.02.027 %G en %F CRMATH_2009__347_9-10_571_0
Guy Bouchitté; Christophe Bourel; Didier Felbacq. Homogenization of the 3D Maxwell system near resonances and artificial magnetism. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 571-576. doi : 10.1016/j.crma.2009.02.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.027/
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