Resonant effects in random dielectric structures

Abstract

In [3,11,14], 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 ∆ 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.1 arXiv:1305.7129v1 [math-ph]

Homogenization Near Resonances and Artificial Magnetism in Three Dimensional Dielectric Metamaterials

“…It depends only on the geometry of Σ and on the permittivity in the matrix surrounding inclusions. The computation of eff looks similar as the one used in the twodimensional case (see [5,9]) where the classical ingredients of homogenization theory for Neumann problems with holes can be recognized. The entries of the tensor eff are given for (k , l ) ∈ {1, 2, 3} 2 by eff…”

“…Here r is a complex parameter such that: 4) while the scaling factor 1/η 2 is responsible of a high contrast becoming larger and larger as the period parameter η of the structure decreases to zero. The choice of this scaling is not new (see [5,9,12]). It ensures that the optical thickness of the inclusions remain constant and therefore the Mie resonances of each dielectric inclusion appear at frequencies which are independent of η (see [23]).…”

“…The dependence of permeability tensor µ eff with respect to ω is ruled by the internal resonances of the composite structure which are responsible for the magnetic activity. The description of the underlying spectral problem is quite involved due to the fact that strong oscillations of the microscopic magnetic field H 0 (x , ·) are allowed not only in Σ (in a similar way as in the 2D case [5,9,12]), but also in the surrounding matrix. A nice way to circumvent this difficulty consists in looking at the curl of the magnetic field which accounts for the magnetic activity.…”

“…The first mathematical study of this kind of dielectric metamaterials was made in the particular case where the structure is invariant in one direction (see [5,12] and [9] for a generalization to the random case). In these papers the inclusions are infinite cylinders and the incident wave is polarized with a magnetic field parallel to the axis of the cylinders.…”

The electromagnetic waves generated by a cluster of nanoparticles with high refractive indices

“…As far as the effective medium theory is concerned, few results are known for the electromagnetism with highly contrasting or negative permittivity or permeability. We can cite [1, 10–12, 16–18, 27, 31, 32] who assume periodicity and derive the equivalent coefficients, via the homogenization theory, for dielectric nanoparticles. The results provided in the current work are a contribution to fill in this gap.…”