Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides

Fliss, Sonia; Klindworth, Dirk; Schmidt, Kersten

doi:10.1007/s10543-014-0521-1

Cited by 7 publications

(5 citation statements)

References 16 publications

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“…The key feature of this model is that this boundary condition is imposed on the boundary of a large rectangular domain outside of which the wave speed is constant, and allows for a numerical study of the problem. Impedance-to-impedance operators have also been used in [JLF06] when calculating periodic wave-guides with perturbations and in [FKS15] for computing modes in photonic crystal wave-guides.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

Beck¹,

Canzani²,

Marzuola³

2021

Preprint

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We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the subdomains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates on solutions of the impedance problem, which are of independent interest. In addition to the variable media setting, we also construct bounds for similar merge operators in the obstacle scattering problem.Assumption 1 (V j are non-trapping with respect to (1, 0)). There existy), satisfy 2V j (x, y) + (x − 1, y)•∇V j (x, y) ≥ c (3)for all (x, y) ∈ S, and j = 1, 2.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

Beck¹,

Canzani²,

Marzuola³

2021

Preprint

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“…The ideas were used in [9] to the study of line defects in a periodic photonic crystal; these defects have been analyzed also in [16] with the result that a line defect cannot support finite energy modes (bound states). For extensions of the numerical scheme to Robin type boundary conditions see [13].…”

Section: Literaturementioning

confidence: 99%

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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“…In this case, one asks for appropriate boundary conditions that must be imposed on the boundary of the bounded domain. This point of view leads to the construction of Dirichlet-to-Neumann maps or similar ideas [14,15,17]. Other key-words are perfectly matched layers [23] or transparent boundary conditions.…”

Section: On Radiation In Waveguides and Photonic Crystalsmentioning

confidence: 99%

Outgoing wave conditions in photonic crystals and transmission properties at interfaces

Lamacz

Schweizer

2018

ESAIM: M2AN

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We analyze the propagation of waves in unbounded photonic crystals. Waves are described by a Helmholtz equation with x-dependent coefficients, the scattering problem must be completed with a radiation condition at infinity. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new radiation condition to analyze the transmission problem where, at fixed frequency, a wave hits the interface between free space and a photonic crystal. We show that the vertical wave number of the incident wave is a conserved quantity. Together with the frequency condition for the transmitted wave, this condition leads (for appropriate photonic crystals) to the effect of negative refraction at the interface.

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Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides

Cited by 7 publications

References 16 publications

Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs

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

Outgoing wave conditions in photonic crystals and transmission properties at interfaces

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