Gap localization of TE‐modes by arbitrarily weak defects

Brown, B. M.; Hoang, Vu; Plum, Michael; Radosz, Maria; Wood, Ian

doi:10.1112/jlms.12046

Cited by 4 publications

(11 citation statements)

References 40 publications

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“…An upper estimate on the number of eigenvalues created in the band gap is given in Section 5 while Section 6 provides a lower bound and combines all results to our main statement (Theorem 6.4) on the precise number of eigenvalues. Compared to the results in our previous paper [2], the assumptions we make on the band functions of the fully periodic operator are more general. The results of [2] are strengthened in the sense that we allow for multiple bands coming together at the edge of a spectral band and the analysis is refined by providing upper and lower bounds on the number of eigenvalues created by the perturbation.…”

Section: Introductionmentioning

confidence: 92%

“…Then L 0 + 1 is bijective and both L 0 and G 0 := (L 0 + 1) −1 are self-adjoint, see [2,Proposition 4.1]. L 0 corresponds to the fully periodic problem (2.1) with ε = ε 0 .…”

Section: The Operator Theoretic Formulationmentioning

confidence: 99%

“…In a previous paper [2], we studied the propagation of TE-polarized waves in two-dimensional photonic crystals that contain line defects and gave rigorous sufficient conditions which imply spectral localization in band gaps. Our results were restricted to the case where only one band function (see (3.4)) contributes to the edge of the band gap.…”

Section: Introductionmentioning

confidence: 99%

“…Additionally, all our results do not depend on the precise geometry of the perturbation, for example, the shape of the inclusions defined by the region within the periodicity cell where the perturbed material coefficients differ from the unperturbed ones. For a more detailed discussion of relevant background material, we refer to [2] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…The results of [2] are strengthened in the sense that we allow for multiple bands coming together at the edge of a spectral band and the analysis is refined by providing upper and lower bounds on the number of eigenvalues created by the perturbation. In the case of exactly one band function touching the band edge, we get the existence of gap spectrum, as in [2], but now with the additional information that there is exactly one additional eigenvalue in the gap (of the reduced problem for fixed

\overset{x}{true}

‐quasimomentum).…”

Section: Introductionmentioning

confidence: 99%

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Gap localization of multiple TE‐Modes by arbitrarily weak defects

Brown

Hoang

Plum

et al. 2020

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. Using the Floquet-Bloch theory in negative-order Sobolev spaces, we characterize the precise number of eigenvalues created by the line defect in terms of the band functions of the original periodic background medium for arbitrarily weak defects.

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Section: Introductionmentioning

confidence: 92%

“…Then L 0 + 1 is bijective and both L 0 and G 0 := (L 0 + 1) −1 are self-adjoint, see [2,Proposition 4.1]. L 0 corresponds to the fully periodic problem (2.1) with ε = ε 0 .…”

Section: The Operator Theoretic Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

\overset{x}{true}

‐quasimomentum).…”

Section: Introductionmentioning

confidence: 99%

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Gap localization of multiple TE‐Modes by arbitrarily weak defects

Brown

Hoang

Plum

et al. 2020

J. London Math. Soc.

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Subwavelength Localized Modes for Acoustic Waves in Bubbly Crystals with a Defect

Ammari¹,

Fitzpatrick²,

Hiltunen

et al. 2018

SIAM J. Appl. Math.

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The recent development of subwavelength photonic and phononic crystals shows the possibility of controlling wave propagation at deep subwavelength scales. Subwavelength bandgap phononic crystals are typically created using a periodic arrangement of subwavelength resonators, in our case small gas bubbles in a liquid. In this work, a waveguide is created by modifying the sizes of the bubbles along a line in a dilute two-dimensional bubbly crystal, thereby creating a line defect. Our aim is to prove that the line defect indeed acts as a waveguide; waves of certain frequencies will be localized to, and guided along, the line defect. The key result is an original formula for the frequencies of the defect modes. Moreover, these frequencies are numerically computed using the multipole method, which numerically illustrates our main results.Mathematics Subject Classification (MSC2000). 35R30, 35C20.

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Spectral analysis of photonic crystals made of thin rods

Holzmann

Lotoreichik

2018

Asymptotic Analysis

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In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of given frequencies. In such materials electromagnetic waves with these frequencies can not propagate; this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging to a predefined range there exists a transverse electric mode that can propagate in the medium. These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in L 2 (R 2 ). The proofs rely on perturbation theory of linear operators, Floquet-Bloch analysis, and properties of Schrödinger operators with point interactions.Here, the three-dimensional vector fields E and H are the electric and the magnetic field, respectively, ω > 0 is the frequency of the wave, ε is the relative dielectric permittivity and c > 0 stands for the Combining the statements of Theorem 3.1 and of Proposition 2.4 with the perturbation result [W, Satz 9.24 b)], we obtain the following claim on the spectrum of H r,λn . Proposition 3.3. Let 0 < λ 1 < λ 2 < · · · < λ N , let a > 0 be fixed and define η := 2π |Ω| + 1. Let the operator H r,λn be as in (1.7). Then there exist a lattice Λ and constants c 1 , . . . , c N (that appear in (1.5)) such that λ n − η − a, λ n + η + a ⊂ ρ (H r,λn ) for all sufficiently small r > 0 and all n ∈ {1, . . . , N }.

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Gap localization of TE‐modes by arbitrarily weak defects

Cited by 4 publications

References 40 publications

Gap localization of multiple TE‐Modes by arbitrarily weak defects

Gap localization of multiple TE‐Modes by arbitrarily weak defects

Subwavelength Localized Modes for Acoustic Waves in Bubbly Crystals with a Defect

Spectral analysis of photonic crystals made of thin rods

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