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Register a new userElectromagnetic wave propagation in media consisting of dispersive metamaterials
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We establish the well-posedness, the finite speed propagation, and a regularity result for Maxwells equations in media consisting of dispersive (frequency dependent) metamaterials. Two typical examples for such metamaterials are materials obeying Drudes and Lorentz models. The causality and the passivity are the two main assumptions and play a crucial role in the analysis. It is worth noting that by contrast the well-posedness in the frequency domain is not ensured in general. We also provide some numerical experiments using the Drudes model to illustrate its dispersive behaviour.
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Surveys on wave propagation in dispersive media have been limited since the pioneering work of Sommerfeld [Ann. Phys. 349, 177 (1914)] by the presence of branches in the integral expression of the wave function. In this article, a method is proposed to eliminate these critical branches and hence to establish a modal expansion of the time-dependent wave function. The different components of the transient waves are physically interpreted as the contributions of distinct sets of modes and characterized accordingly. Then, the modal expansion is used to derive a modified analytical expression of the Sommerfeld precursor improving significantly the description of the amplitude and the oscillating period up to the arrival of the Brillouin precursor. The proposed method and results apply to all waves governed by the Helmholtz equations.
We consider quantum walks with position dependent coin on 1D lattice $mathbb{Z}$. The dispersive estimate $|U^tP_c u_0|_{l^infty}lesssim (1+|t|)^{-1/3} |u_0|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_c$ is the projection to the continuous spectrum. This is an analogous result for Schrodinger operators and discrete Schrodinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.
We study the behavior of wave propagation in materials for which not all of the principle elements of the permeability and permittivity tensors have the same sign. We find that a wide variety of effects can be realized in such media, including negative refraction, near-field focusing and high impedance surface reflection. In particular a bi-layer of these materials can transfer a field distribution from one side to the other, including near-fields, without requiring internal exponentially growing waves.
The extreme magnetoelectric medium (EME medium) is defined in terms of two medium dyadics, $alpha$, producing electric polarization by the magnetic field and $beta$, producing magnetic polarization by the electric field. Plane-wave propagation of time-harmonic fields of fixed finite frequency in the EME medium is studied. It is shown that (if $omega eq 0$) the dispersion equation has a cubic and homogeneous form, whence the wave vector $k$ is either zero or has arbitrary magnitude. In many cases there is no dispersion equation (NDE medium) to restrict the wave vector in an EME medium. Attention is paid to the case where the two medium dyadics have the same set of eigenvectors. In such a case the $k$ vector is restricted to three eigenplanes defined by the medium dyadics. The emergence of such a result is demonstrated by considering a more regular medium, and taking the limit of zero permittivity and permeability. The special case of uniaxial EME medium is studied in detail. It is shown that an interface of a uniaxial EME medium appears as a DB boundary when the axis of the medium is normal to the interface. More in general, EME media display interesting wave effects that can potentially be realized through metasurface engineering.
In arXiv:1201.4067 and arXiv:1611.08030, Eyink and Shi and Chibbaro et al., respectively, formally derived an infinite, coupled hierarchy of equations for the spectral correlation functions of a system of weakly interacting nonlinear dispersive waves with random phases in the standard kinetic limit. Analogously to the relationship between the Boltzmann hierarchy and Boltzmann equation, this spectral hierarchy admits a special class of factorized solutions, where each factor is a solution to the wave kinetic equation (WKE). A question left open by these works and highly relevant for the mathematical derivation of the WKE is whether solutions of the spectral hierarchy are unique, in particular whether factorized initial data necessarily lead to factorized solutions. In this article, we affirmatively answer this question in the case of 4-wave interactions by showing, for the first time, that this spectral hierarchy is well-posed in an appropriate function space. Our proof draws on work of Chen and Pavlovi{c} for the Gross-Pitaevskii hierarchy in quantum many-body theory and of Germain et al. for the well-posedness of the WKE.
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