Volume 21, Number 1, January-March 2015
|Page(s)||217 - 246|
|Published online||09 December 2014|
Resonant effects in random dielectric structures
1 IMATH, Université du Sud Toulon-Var, 83957 La Garde cedex,
2 LMPA, Université du littoral côte d’Opale, 62228 Calais cedex, France.
3 LAMA, Université de Marne la Vallée, 77454 Marne la Vallée cedex 2, France.
Revised: 13 April 2014
In [G. Bouchitté and D. Felbacq, C. R. Math. Acad. Sci. Paris 339 (2004) 377–382; D. Felbacq and G. Bouchitté, Phys. Rev. Lett. 94 (2005) 183902; D. Felbacq and G. Bouchitté, New J. Phys. 7 (2005) 159], a theory for artificial magnetism in two-dimensional photonic crystals has been developed for large wavelength using homogenization techniques. In this paper we pursue this approach within a rigorous stochastic framework: dielectric parallel nanorods are randomly disposed, each of them having, up to a large scaling factor, a random permittivity ε(ω) whose law is represented by a density on a window Δh = [a−,a+] × [0,h] of the complex plane. We give precise conditions on the initial probability law (permittivity, radius and position of the rods) under which the homogenization process can be performed leading to a deterministic dispersion law for the effective permeability with possibly negative real part. Subsequently a limit analysis h → 0, accounting a density law of ε which concentrates on the real axis, reveals singular behavior due to the presence of resonances in the microstructure.
Mathematics Subject Classification: 35B27 / 35Q60 / 35Q61 / 35R60 / 60H25 / 78M35 / 78M40
Key words: Stochastic homogenization / photonic crystals / metamaterials / micro-resonators / effective tensors / dynamical system
© EDP Sciences, SMAI 2014
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