Outgoing wave conditions in photonic crystals and transmission properties at interfaces

We analyze the propagation of waves in unbounded photonic crystals, the waves are described by a Helmholtz equation with $x$-dependent coefficients. The scattering problem must be completed with a radiation condition at infinity, which was not available for $x$-dependent coefficients. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a (weak) uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new radiation condition to analyze the transmission problem where, at fixed frequency, a wave hits the interface between free space and a photonic crystal. We derive that the vertical wave number of the incident wave is a conserved quantity. Together with the frequency condition for the transmitted wave, this condition leads (for appropriate photonic crystals) to the effect of negative refraction at the interface

Moment bounds for the corrector in stochastic homogenization of a percolation model

We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on $\mathbb{Z}^d$, $d>2$. The model is obtained from the classical $\{0,1\}$-Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result in [GO1], where uniformly elliptic conductances are treated, to the degenerate case. With regard to the associated random conductance model, we obtain as a side result that the corrector not only grows sublinearly, but slower than any polynomial rate. Our argument combines a quantification of ergodicity by means of a Spectral Gap on Glauber dynamics with regularity estimates on the gradient of the elliptic Green's function

Dispersive effective models for waves in heterogeneous media

We study the long time behavior of waves in a strongly heterogeneous medium, starting from the one-dimensional scalar wave equation with variable coefficients. We assume that the coefficients are periodic with period ɛ and ɛ > 0 is a small length parameter. Our main result is the rigorous derivation of two different dispersive models. The first is a fourth-order equation with constant coefficients including powers of ɛ . In the second model, the ɛ-dependence is completely avoided by considering a third-order linearized Korteweg-de-Vries equation. Our result is that both simplified models describe the long time behavior well. An essential tool in our analysis is an adaption operator which modifies smooth functions according to the periodic structure of the medium

Dispersive homogenized models and coefficient formulas for waves in general periodic media

We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix $a^\varepsilon$ that is periodic with characteristic length scale $\varepsilon$; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions $u^\varepsilon$ well by a non-dispersive wave equation on fixed time intervals $(0,T)$. Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive equation with homogeneous coefficients such that its solutions $w^\varepsilon$ describe $u^\varepsilon$ well on time intervals $(0,T\varepsilon^{-2})$. More precisely, we provide a norm and uniform error estimates of the form $\| u^\varepsilon(t) - w^\varepsilon(t) \| \le C\varepsilon$ for $t\in (0,T\varepsilon^{-2})$. They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an $\varepsilon$-independent equation of third order that describes dispersion along rays and we present numerical examples.Comment: 28 pages, 7 figure

Outgoing wave conditions in photonic crystals and transmission properties at interfaces

We analyze the propagation of waves in unbounded photonic crystals, the waves are described by a Helmholtz equation with x-dependent coefficients. The scattering problem must be completed with a radiation condition at infinity, which was not available for x-dependent coefficients. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a (weak) uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new radiation condition to analyze the transmission problem where, at fixed frequency, a wave hits the interface between free space and a photonic crystal. We derive that the vertical wave number of the incident wave is a conserved quantity. Together with the frequency condition for the transmitted wave, this condition leads (for appropriate photonic crystals) to the effect of negative refraction at the interface

A negative index meta-material for Maxwell´s equations

We derive the homogenization limit for time harmonic Maxwell's equations in a periodic geometry with periodicity length η > 0. The considered meta-material has a singular sub-structure: the permittivity coefficient in the inclusions scales like η⁻² and a part of the substructure (corresponding to wires in the related experiments) occupies only a volume fraction of order η²; the fact that the wires are connected across the periodicity cells leads to contributions in the effective system. In the limit η → 0, we obtain a standard Maxwell system with a frequency dependent effective permeability μ^eff (ω) and a frequency independent effective permittivity ε^eff. Our formulas for these coefficients show that both coefficients can have a negative real part, the meta-material can act like a negative index material. The magnetic activity μ^eff≠1 is obtained through dielectric resonances as in previous publications. The wires are thin enough to be magnetically invisible, but, due to their connectedness property, they contribute to the effective permittivity. This contribution can be negative due to a negative permittivity in the wires