Localization of high-frequency waves propagating in a locally periodic medium (original) (raw)
Related papers
Bloch wave homogenization and spectral asymptotic analysis
Journal de Mathématiques Pures et Appliquées, 1998
We consider a second-order elliptic equation in a bounded periodic heterogeneous medium and study the asymptotic behavior of its spectrum, as the structure period goes to zero. We use a new method of Eloch wave homogenization which, unlike the classical homogenization method, characterizes a renormalized limit of the spectrum, namely sequences of eigenvalues of the order of the square of. the medium period. We prove that such a renormalized limit spectrum is made of two parts: the so-called Bloch spectrum, which is explicitly defined as the spectrum of a family of limit problems, and the so-called boundary layer spectrum, which is made of limit eigenvalues corresponding to sequences of eigenvectors concentrating on the boundary of the domain. This analysis relies also on a notion of Bloch measures which can be seen as ad hoc Wigner measures in the context of semi-classical analysis. Finally, for rectangular domains made of entire periodicity cells, a variant of the Bloch wave homogenization method gives an explicit characterization of the boundary layer spectrum too. 0 Elsevier, Paris RBsuMB. -On considere une equation elliptique du deuxitme ordre dans un milieu ptriodique heterogbne borne, et on dtudie le comportement asymptotique de son spectre lorsque la p&ode tend vers zero. On utilise une nouvelle methode d'homog&&ation par ondes de Bloch qui, contrairement aux mtthodes classiques d'homogenbisation, caracterise la limite renormalisee du spectre, et plus precisement les suites de valeurs propres de I'ordre du cam? de la pbriode. On dtmontre que le spectre limite renormalise est constitut de deux parties : un spectre de Bloc/z, qui est explicitement caracterise comme le spectre d'une famille de problemes limites, et un spectre de couche limite, qui est l'ensemble des limites de suites de valeurs propres dont les vecteurs propres correspondants se concentrent sur le bord du domaine. L'analyse presentee repose sur une notion de mesures de Bloch qui peuvent &r-e vues comme des versions ad hoc des mesures de Wigner utilisees en analyse semi-classique. Enfin, pour des domaines rectangulaires constitues uniquement de cellules de p&iodicite entieres, une variante de la methode d'homogentisation par ondes de Bloch permet de donner aussi une caracterisation explicite du spectre de couche limite. 0 Elsevier, Paris
Wavepackets in Inhomogeneous Periodic Media: Propagation Through a One-Dimensional Band Crossing
Communications in Mathematical Physics, 2018
We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger's equation in one spatial dimension with a potential which is the sum of a periodic function V and a smooth function W. We assume that the period of V is much shorter than the scale of variation of W and denote the ratio of these scales by. We consider the dynamics of semiclassical wavepacket asymptotic (in the limit ↓ 0) solutions which are spectrally localized near to a crossing of two Bloch band dispersion functions of the periodic operator − 1 2 ∂ 2 z + V (z). We show that the dynamics is qualitatively different from the case where bands are well-separated: at the time the wavepacket is incident on the band crossing, a second wavepacket is 'excited' which has opposite group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a 'Landau-Zener'-type model.
arXiv (Cornell University), 2015
We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this generalized Kunz-Souillard method to almost periodic Schrödinger operators are presented. On the one hand, we show that the Schrödinger operators on ℓ 2 (Z) with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schrödinger operators on ℓ 2 (Z). More generally, for any bounded potential, one can find an arbitrarily small limit-periodic perturbation so that the resulting operator has pure point spectrum. Our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit-periodic Schrödinger operators on ℓ 2 (Z). On the other hand, we show that Schrödinger operators on ℓ 2 (Z) with arbitrarily small one-frequency quasi-periodic potential may have pure point spectrum for some phases. This was previously known only for one-frequency quasiperiodic potentials with • ∞ norm exceeding 2, namely the super-critical almost Mathieu operator with a typical frequency and phase. Moreover, this phenomenon can occur for any frequency, whereas no previous quasi-periodic potential with Liouville frequency was known that may admit eigenvalues for any phase.
Bloch-Wave Homogenization on Large Time Scales and Dispersive Effective Wave Equations
Multiscale Modeling & Simulation, 2014
We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in R n , n ∈ {1, 2, 3}. Standard homogenization theory provides, for the limit of a small periodicity length ε > 0, an effective second order wave equation that describes solutions on time intervals [0, T ]. In order to approximate solutions on large time intervals [0, T ε −2 ], one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation, and an estimate for errors between the solution of the original heterogeneous problem and the solution of the dispersive wave equation. We use Bloch-wave analysis to identify a family of relevant limit models and introduce an approach to select a well-posed effective model under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests.
Photonic Metamaterials: From Random to Periodic, 2007
We have measured pulsed microwave transmission through quasi-1D samples with lengths up to three localization lengths. For times approaching four times the diffusion time τD, transmission is diffusive in accord with the self-consistent theory of localization for the renormalized diffusion coefficient in space and frequency, D(z, Ω). For longer times, the transmission decay rate first agrees with and later falls increasingly below the self-consistent theory. Beyond the Heisenberg time, the decay rate approaches the predictions of a dynamic single parameter scaling model which reflects the decay of long-lived localized modes and converges to the results of 1D simulations.
On the spectral asymptotics of waves in periodic media with Dirichlet or Neumann exclusions
The Quarterly Journal of Mechanics and Applied Mathematics
Summary We consider homogenization of the scalar wave equation in periodic media at finite wavenumbers and frequencies, with the focus on continua characterized by: (a) arbitrary Bravais lattice in mathbbRd\mathbb{R}^dmathbbRd, dgeqslant2d \geqslant 2dgeqslant2, and (b) exclusions, that is, ‘voids’ that are subject to homogeneous (Neumann or Dirichlet) boundary conditions. Making use of the Bloch-wave expansion, we pursue this goal via asymptotic ansatz featuring the ‘spectral distance’ from a given wavenumber-eigenfrequency pair (situated anywhere within the first Brillouin zone) as the perturbation parameter. We then introduce the effective wave motion via projection(s) of the scalar wavefield onto the Bloch eigenfunction(s) for the unit cell of periodicity, evaluated at the origin of a spectral neighborhood. For generality, we account for the presence of the source term in the wave equation and we consider—at a given wavenumber—generic cases of isolated, repeated, and nearby eigenvalues. In this way, we obtain ...
Journal of Mathematical Analysis and Applications, 2017
We consider wavepackets composed of two modulated carrier Bloch waves with opposite group velocities in the one dimensional periodic Nonlinear Schrödinger/Gross-Pitaevskii equation. These can be approximated by first order coupled mode equations (CMEs) for the two slowly varying envelopes. Under a suitably selected periodic perturbation of the periodic structure the CMEs possess a spectral gap of the corresponding spatial operator and allow families of exponentially localized solitary waves parametrized by velocity. This leads to a family of approximate solitary waves in the periodic nonlinear Schrödinger equation. Besides a formal derivation of the CMEs a rigorous justification of the approximation and an error estimate in the supremum norm are provided. Several numerical tests corroborate the analysis.
Bloch-wave homogenization for spectral asymptotic analysis of the periodic Maxwell operator
Waves in Random and Complex Media, 2007
This paper is devoted to the asymptotic behaviour of the spectrum of the three-dimensional Maxwell operator in a bounded periodic heterogeneous dielectric medium T = [−T, T ] 3 , T > 0, as the structure period η, such that η −1 T is a positive integer, tends to 0. The domain T is extended periodically to the whole of R 3 , so that the original operator is understood as acting in a space of T-periodic functions. We use the so-called Bloch wave homogenisation technique which, unlike the classical homogenisation method, is capable of characterising a renormalised limit of the spectrum (called the Bloch spectrum). The related procedure is concerned with sequences of eigenvalues Λη of the order of the square of the medium period, which correspond to the oscillations of high-frequencies of order η −1. The Bloch-wave description is obtained via the notion of two-scale convergence for bounded self-adjoint operators, and a proof of the "completeness" of the limiting spectrum is provided.
Applicable Analysis, 2021
We pursue a low-wavenumber, second-order homogenized solution of the timeharmonic wave equation in periodic media with a source term whose frequency resides inside a band gap. Considering the wave motion in an unbounded medium R d (d 1), we first use the Bloch transform to formulate an equivalent variational problem in a bounded domain. By investigating the source term's projection onto certain periodic functions, the second-order model can then be derived via asymptotic expansion of the Bloch eigenfunction and the germane dispersion relationship. We establish the convergence of the second-order homogenized solution, and we include numerical examples to illustrate the convergence result.
Localization for a class of one dimensional quasi-periodic Schrödinger operators
Communications in Mathematical Physics, 1990
We prove for small ε and α satisfying a certain Diophantine condition the operator H =-ε 2 A +-cos 2πO'α + θ) j € Z 2π has pure point spectrum for almost all θ. A similar result is established at low energy for H =-^-K 2 (cos2πx-f cos2π(αx + 0)) provided K is sufficiently large.