Resonant effects in random dielectric structures

Bouchitté, Guy; Bourel, Christophe; Manca, Luigi

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 217-246.

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 $\epsilon \left(\omega \right)$ whose law is represented by a density on a window $\Delta {}_h=[a^{-},a^{+}]\times [0,h]$ 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 $h\to 0$, accounting a density law of $\epsilon$ which concentrates on the real axis, reveals singular behavior due to the presence of resonances in the microstructure.

Classification : 35B27, 35Q60, 35Q61, 35R60, 60H25, 78M35, 78M40
Mots clés : Stochastic homogenization, photonic crystals, metamaterials, micro-resonators, effective tensors, dynamical system

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