Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients

Filonov, N.

doi:10.1007/s00220-003-0904-7

Cited by 32 publications

(27 citation statements)

References 11 publications

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“…we must prove that a band gap of the unperturbed problem (modelled by −ε −1 0 ) contains spectrum of −ε −1 induced by the perturbation. We first note that existence of gaps in the spectrum of some problems with periodic background media was proved in [2][3][4] and in [5] for the full Maxwell case. Using layer potential techniques, this question has been studied in [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

On the spectrum of waveguides in planar photonic bandgap structures

et al. 2015

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We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a twodimensional periodic medium; the defect is infinitely extended and aligned with one of the coordinate axes. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that guided mode spectrum can be created by arbitrarily 'small' perturbations. Secondly, we show that, after performing a Floquet decomposition in the axial direction of the waveguide, for any fixed value of the quasi-momentum k x , the perturbation generates at most finitely many new eigenvalues inside the gap.

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Section: Introductionmentioning

confidence: 99%

On the spectrum of waveguides in planar photonic bandgap structures

et al. 2015

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“…It should be noted that the known examples of a gap opening in the continuous spectrum pertain precisely to the setting of elliptic systems with periodic coefficients in R n , as described in the preceding paragraph (see the papers [23], [26]- [30] and the survey [31]), and the result is achieved with the help of the method of variation of periodic coefficients. Below, a gap in the continuous spectrum of the operator of problem (1.4)-(1.6) will be constructed exclusively by choosing the form of the cell; i.e., the matrix A 0 of elastic modules and the density ρ may be thought of as constants.…”

Section: Preliminary Description Of the Resultsmentioning

confidence: 99%

Gap opening in the essential spectrum of the elasticity theory problem in a periodic half-layer

Назаров

2010

St. Petersburg Math. J.

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Abstract. Rayleigh waves are studied in an elastic half-layer with a periodic end and rigidly clamped faces. It is established that the essential spectrum of the corresponding problem of elasticity theory has a band structure, and an example of a waveguide is presented in which a gap opens in the essential spectrum; i.e., an interval arises that contains points of an at most discrete spectrum. §1. Introduction 1. Preamble. In the case of a homogeneous isotropic elastic half-space, surface waves were discovered by Lord Rayleigh [1], and since then many investigations devoted to similar effects have appeared (see a survey of modern literature in [2], and also the paper [3], which is absent in [2]). A Rayleigh wave is a plane wave of the forma vector-valued factor U (z) decaying exponentially as z = x 3 → −∞. The arising of such waves explains specific wave processes in elastic bodies.The wave number k ∈ R + = [0, +∞) determines a frequency cutoff ω † (k) above which, i.e., for ω ≥ ω † (k), the wave (1.1) exists necessarily. In the present paper we deal with a problem related to a similar phenomenon. Namely, we study an elastic, but not necessarily homogeneous and isotropic cushion Ω 0 having the form of a half-layer with a periodic end and rigidly clamped side faces (see Figure 1, where the clamped surface is shadowed). Some Rayleigh waves decaying exponentially as z → −∞ can propagate along the end of the cushion, and if Ω 0 is a cylinder ∆ × R, then we have a single cutoff ω † > 0; i.e., the corresponding operator of the elasticity theory system acquires a continuous spectrum [ω † , +∞). Our main goal in this paper is to show that, in the periodic case, a gap can open in the essential spectrum; i.e., an interval can exist the ends of which belong to the continuous spectrum, but inside which only points of the discrete spectrum may occur. Some of the results were announced earlier in [4].

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“…Another approach is creating very high contrast media, which succeeds in creating spectral gaps for periodic Maxwell operators [141][142][143]145] as well as for Laplace-Beltrami operators on abelian coverings of compact Riemannian manifolds [197, and references therein; 331]. A two-scale homogenization approach has been developed in [407].…”

Section: Gap Creationmentioning

confidence: 99%

Untitled

2016

Bull. Amer. Math. Soc.

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Abstract. The article surveys the main topics, techniques, and results of the theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which significantly influence most spectral features of such operators.The approaches described are applicable not only to the standard model example of Schrödinger operator with periodic electric potential −Δ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases.Important applications are mentioned. However, due to the size restrictions, they are not dealt with in detail.

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Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients

Cited by 32 publications

References 11 publications

On the spectrum of waveguides in planar photonic bandgap structures

On the spectrum of waveguides in planar photonic bandgap structures

Gap opening in the essential spectrum of the elasticity theory problem in a periodic half-layer

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