%0 Journal Article
%T Homogenization near resonances and artificial magnetism in 3D dielectric metamaterials
%+ Institut de Mathématiques de Toulon - EA 2134 (IMATH)
%+ Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA)
%+ Laboratoire Charles Coulomb (L2C)
%A Bouchitté, Guy
%A Bourel, Christophe
%A Felbacq, Didier
%< avec comité de lecture
%@ 0003-9527
%J Archive for Rational Mechanics and Analysis
%I Springer Verlag
%V 225
%N 3
%P 1233-1277
%8 2017
%D 2017
%Z 1512.02463
%R 10.1007/s00205-017-1132-1
%K Maxwell system
%K Two-scale convergence
%K Homogenization
%K Metamaterials
%K Micro-resonators
%K Photonic crystals
%K Effective tensors
%Z 35B27, 35Q60, 35Q61, 78M35, 78M40
%Z Mathematics [math]
%Z Mathematics [math]/Analysis of PDEs [math.AP]Journal articles
%X 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.
%G English
%2 https://hal.science/hal-01240316/document
%2 https://hal.science/hal-01240316/file/BouBouFel.pdf
%2 https://hal.science/hal-01240316/file/f4.pdf
%2 https://hal.science/hal-01240316/file/J1_570.pdf
%2 https://hal.science/hal-01240316/file/J1_620.pdf
%2 https://hal.science/hal-01240316/file/J3_376.pdf
%2 https://hal.science/hal-01240316/file/J3_417.pdf
%2 https://hal.science/hal-01240316/file/J3_750.pdf
%2 https://hal.science/hal-01240316/file/muefcube310.pdf
%2 https://hal.science/hal-01240316/file/muefgeoL.pdf
%2 https://hal.science/hal-01240316/file/structure2.pdf
%L hal-01240316
%U https://hal.science/hal-01240316
%~ UNIV-TLN
%~ CNRS
%~ UNIV-LITTORAL
%~ L2C
%~ IMATH
%~ TDS-MACS
%~ LMPA-JL
%~ MIPS
%~ UNIV-MONTPELLIER
%~ ANR
%~ UM-2015-2021