Resonant effects in random dielectric structures

¹ IMATH, Université du Sud Toulon-Var, 83957 La Garde cedex, France.
bouchitte@univ-tln.fr
² LMPA, Université du littoral côte d’Opale, 62228 Calais cedex, France.
bourel@univ-littoral.fr
³ LAMA, Université de Marne la Vallée, 77454 Marne la Vallée cedex 2, France.
luigi.manca@univ-mlv.fr

Received: 10 September 2013
Revised: 13 April 2014

Abstract

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.