On the spectrum of waveguides in planar photonic bandgap structures

Key wordsWe 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 three-dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.

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“…We remark that in the case of planar wave‐guides, where

k \in [- π, π]

, the finiteness of Σ follows also from the analyticity of the band functions

λ_{s} (k)

, and the Thomas argument excluding constant band functions (see, e.g. ).…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…In the case of a line defect in two dimensions the analysis involves studying the band functions arising from the Floquet‐Bloch decomposition as functions of one complex variable in which they are analytic. This was of great help in . In the three dimensional situation this analyticity is no longer at hand.…”

Section: Introductionmentioning

confidence: 97%

Spectrum created by line defects in periodic structures

Brown

Hoang

Plum

et al. 2014

Mathematische Nachrichten

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“…Brown et al . showed weak gap localization in for a periodic Helmholtz‐type operator corresponding to TM‐mode polarization. We also refer to the paper of Parzygnat et al .…”

Section: Introductionmentioning

confidence: 99%

Gap localization of TE‐modes by arbitrarily weak defects

Brown

Hoang

Plum

et al. 2017

J. London Math. Soc.

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This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. It is shown that arbitrarily weak perturbations introduce spectrum into the spectral gaps of the background operator.

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“…This statement, known as the Bethe Sommerfeld conjecture is fully demonstrated in [49,50] for the periodic Schrödinger operator but is still partially open for Maxwell equations (see [58]). For the localization effect, [15,16,1,33,37,45,8,9] several papers exhibit situations where a compact (resp. lineic) perturbation of a periodic medium give rise to localized (resp.…”

Section: Introductionmentioning

confidence: 99%

“…It seems that the first results concern strong material perturbations : for local perturbation [15,16] and for lineic perturbation [33,37]. There exist fewer results about weak material perturbations : [8,9] deal with 2D lineic perturbations. Finally, geometrical perturbations are considered in [40,45], where the geometrical domain under investigation is exactly the same as ours but with homogeneous Dirichlet boundary conditions on the boundary of the ladder.…”

Section: Introductionmentioning

confidence: 99%

Trapped modes in thin and infinite ladder like domains. Part 1: Existence results

Delourme

Fliss

Joly

et al. 2017

ASY

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The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter ε > 0) whose limit (as ε tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the ε tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that ε is small enough. Numerical experiments illustrate the theoretical results.

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On the spectrum of waveguides in planar photonic bandgap structures

Cited by 5 publications

References 34 publications

Spectrum created by line defects in periodic structures

Spectrum created by line defects in periodic structures

Gap localization of TE‐modes by arbitrarily weak defects

Trapped modes in thin and infinite ladder like domains. Part 1: Existence results

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