Outgoing wave conditions in photonic crystals and transmission properties at interfaces

Lamacz, Agnes; Schweizer, Ben

doi:10.1051/m2an/2018026

Cited by 11 publications

(21 citation statements)

References 35 publications

(59 reference statements)

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“…We emphasize that the radiation boxes have width εL and are positioned at ±εR, while we restricted ourselves to L = R in [24] to simplify notations. The idea is now to impose (1.2) and its counterpart on the left hand side in a strong form at a finite distance; in this first attempt we demand…”

Section: An Approach Based On (12)mentioning

confidence: 99%

“…The two contributions [12,15] treat the (globally) periodic wave-guide problem and the periodic half-wave-guide problem, respectively, and they contain existence results that are based on a limiting absorption principle. A slightly different radiation condition for the locally periodic wave-guide was suggested in [24]. Regarding further results on limiting absorption principles we mention [19,30].…”

Section: Literaturementioning

confidence: 99%

“…In contrast, the analysis is essentially independent of the boundary condition on the lateral boundary (R × {0}) ∪ (R × {H}). To make a choice, we work with periodicity conditions as in [24]: values and derivatives coincide at (x 1 , 0) ∈ R × {0} and (x 1 , H) ∈ R × {H}.…”

Section: Introductionmentioning

confidence: 99%

“…It is not an easy task to formulate the radiation condition in a wave-guide. For a periodic semi-infinite wave-guide, a condition was formulated and analyzed in [15], similarly, the periodic wave-guide was analyzed in [12,19], a wave-guide with different coefficients in the two infinite directions was analyzed in [24]. The latter publication provides a the uniqueness result for the suggested radiation condition.…”

Section: Introductionmentioning

confidence: 99%

“…In the above question we avoided the precise formulation of the radiation condition -this is adequate since the different forms of radiation conditions in a periodic wave-guide are essentially equivalent. The form of the radiation condition that was suggested in [24] can be written as The condition uses the periodicity cell Y ε = (0, ε) 2 , the function {u} + R,L , which is the restriction of u (periodically extended in the vertical direction) to the domain ε(R, R + L) × ε(0, R). For large R, we hence consider the solution u on the far right.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A Bloch Wave Numerical Scheme for Scattering Problems in Periodic Wave-Guides

Dohnal¹,

Schweizer²

2018

SIAM J. Numer. Anal.

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We present a new numerical scheme to solve the Helmholtz equation in a wave-guide. We consider a medium that is bounded in the x 2 -direction, unbounded in the x 1 -direction and ε-periodic for large |x 1 |, allowing different media on the left and on the right. We suggest a new numerical method that is based on a truncation of the domain and the use of Bloch wave ansatz functions in radiation boxes. We prove the existence and a stability estimate for the infinite dimensional version of the proposed problem. The scheme is tested on several interfaces of homogeneous and periodic media and it is used to investigate the effect of negative refraction at the interface of a photonic crystal with a positive effective refractive index.

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Section: An Approach Based On (12)mentioning

confidence: 99%

Section: Literaturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Bloch Wave Numerical Scheme for Scattering Problems in Periodic Wave-Guides

Dohnal¹,

Schweizer²

2018

SIAM J. Numer. Anal.

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Periodic waveguides revisited: Radiation conditions, limiting absorption principles, and the space of bounded solutions

Kirsch,

Schweizer

2024

Math Methods in App Sciences

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We study the Helmholtz equation with periodic coefficients in a closed waveguide. A functional analytic approach is used to formulate and solve the radiation problem in a self‐contained exposition. In this context, we simplify the non‐degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied, and the radiation condition corresponding to the chosen LAP is derived; we include an example to show different LAPs lead, in general, to different solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem.

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A radiation condition arising from the limiting absorption principle for a closed full‐ or half‐waveguide problem

Kirsch

Lechleiter

2018

Math Methods in App Sciences

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new approach, which is, in our opinion, simpler and opens the possibility to treat also more general cases, such as, the scattering by waveguides in R 2 (ie, open waveguide problems, see Kirsch and Lechleiter 4 ) or the scattering by infinite cylinders in R 3 with periodic refractive indices (see Kirsch 10 ). The basic difference to all of the traditional approaches (known to the authors) is that we do not use at all the spectrum of the Floquet-Bloch transformed differential operator and its dependence on the Bloch parameter. Instead, we consider the transformed boundary value problem as a family of problems that depends singularly on 2 parameters, namely, the imaginary part ε ≥ 0 of the wave number k ¼k þ iε and the Bloch parameter α in neighbourhoods of ðε; αÞ ¼ ð0;αÞ whereα is one of the "exceptional values" or "propagative wave numbers" (in the sense of Fliss and Joly 3 ). We prove a general functional analytic theorem about the structure of the solutions of certain families of linear problems depending on 2 parameters ðε; αÞ≠ð0;αÞ and can explicitly compute the limits in the inverse Floquet-Bloch transform when ε→0. This proves the limiting absorption principle and provides an explicit form of the solution as a sum of propagating modes and a decaying part. This motivates the definition of a radiation condition, which is equivalent to the well-known formulations as, eg, in Fliss and Joly 3 (see also Joly et al 7 ) but takes a slightly different form. In our approach, we distinguish between the right-going and left-going propagation modes by the sign of the eigenvalues of an eigenvalue problem in the finite dimensional space spanned by all propagating modes corresponding to a fixed propagative wave number.Having developed the radiation condition, we can "forget" the limiting absorption principle and take the radiation condition as given. Then we are able to prove uniqueness and also existence of a solution by a direct proof, which is independent of the limiting absorption principle. The proof of existence is based again on the Floquet-Bloch transform and an abstract singular perturbation result of Colton and Kress. 14 The methods we apply are all well known and, in principle, simple enough to extend our analysis to more involved scattering problems in linear elasticity or electromagnetics (which, however, has to be done). To reduce technical difficulties, in this paper, we are however merely considering the simple Helmholtz equation in R 2 .To briefly comment on this paper's structure, the following Section 2 discusses the Floquet-Bloch transform and uses it to transform the given boundary value problem to a family of operator equations of the form ũ ε; α −K ε; α ũ ε; α ¼f ε;α in a Sobolev space depending on the parameters ε = Im k and the Bloch parameter α. In Section 3, we prove the limiting absorption principle by applying an abstract functional analytic result, while in Section 4, we formulate our radiation condition and prove uniqueness and existence in a direct way, that is, without using the limiting ab...

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Outgoing wave conditions in photonic crystals and transmission properties at interfaces

Cited by 11 publications

References 35 publications

A Bloch Wave Numerical Scheme for Scattering Problems in Periodic Wave-Guides

A Bloch Wave Numerical Scheme for Scattering Problems in Periodic Wave-Guides

Periodic waveguides revisited: Radiation conditions, limiting absorption principles, and the space of bounded solutions

A radiation condition arising from the limiting absorption principle for a closed full‐ or half‐waveguide problem

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