Geodesic
Resonant effects in random dielectric structures
Bouchitté, Guy1  ; Bourel, Christophe2  ; Manca, Luigi3 
1 IMATH, Université du Sud Toulon-Var, 83957 La Garde cedex, France.
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.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 217-246

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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 Δ h =[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.

Reçu le :
DOI : 10.1051/cocv/2014026
Classification : 35B27, 35Q60, 35Q61, 35R60, 60H25, 78M35, 78M40
Keywords: Stochastic homogenization, photonic crystals, metamaterials, micro-resonators, effective tensors, dynamical system

Bouchitté, Guy 1 ; Bourel, Christophe 2 ; Manca, Luigi 3

1 IMATH, Université du Sud Toulon-Var, 83957 La Garde cedex, France.
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. 
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     title = {Resonant effects in random dielectric structures},
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Bouchitté, Guy; Bourel, Christophe; Manca, Luigi. Resonant effects in random dielectric structures. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 217-246. doi: 10.1051/cocv/2014026

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