Numerical realization of Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides

Klindworth, Dirk; Schmidt, Kersten; Fliss, Sonia

doi:10.1016/j.camwa.2013.03.005

Cited by 9 publications

(31 citation statements)

References 28 publications

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“…This eigenvalue problem is linear in ω 2 when fixing k ∈ B, the so called ω-formulation, and quadratic in k when fixing ω ∈ R + , the so called k-formulation. However, note that this problem is posed on the unbounded domain S. Now let us come to the spectral properties of (2.3) as shown in [5,13], which are relevant for this work. For any k ∈ B we will denote the set of frequencies ω 2 for which Bloch modes [14] in the PhC + PhC on top or in the PhC − PhC below the defect exist by σ ess (k) ⊂ R + .…”

Section: Transformation To Periodic Problem and Spectral Propertiesmentioning

confidence: 99%

“…For more details and proofs the reader is referred to [5,13]. Before we can present the spectral properties we need to introduce several function spaces.…”

Section: Transformation To Periodic Problem and Spectral Propertiesmentioning

confidence: 99%

“…If the Robin boundary condition is replaced by a Dirichlet boundary condition, as done in [5,13] for the DtN method, the problem is well posed except for a countable set of frequencies ω 2 which corresponds to the Dirichlet eigenvalues of the problem in the infinite half-strip. We will call these frequencies global Dirichlet eigenvalues.…”

Section: Definition Of the Robin-to-robin Operatorsmentioning

confidence: 99%

“…This implies that we can identify all functions of C ± n by functions of C ± 0 , and, similarly, all functions of ± n by functions of ± 0 . In [13] we introduced shift operators that allow for a rigorous notation of this identification. However, for simplicity of notation we shall refrain from introducing these shift operators in this work, keeping in mind, that such an identification is possible.…”

Section: Characterization Of the Robin-to-robin Operatorsmentioning

confidence: 99%

“…The numerical realization of these DtN transparent boundary conditions with the highorder finite element method (FEM) was shown in [13]. The numerical solution of the resulting non-linear eigenvalue problem was also addressed in [13].…”

Section: Introductionmentioning

confidence: 99%

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

2014

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The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides.

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Section: Transformation To Periodic Problem and Spectral Propertiesmentioning

confidence: 99%

“…For more details and proofs the reader is referred to [5,13]. Before we can present the spectral properties we need to introduce several function spaces.…”

Section: Transformation To Periodic Problem and Spectral Propertiesmentioning

confidence: 99%

Section: Definition Of the Robin-to-robin Operatorsmentioning

confidence: 99%

Section: Characterization Of the Robin-to-robin Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2014

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Cross‐impact of beam local wavenumbers on the efficiency of self‐adaptive artificial boundary conditions for two‐dimensional nonlinear Schrödinger equation

Trofimov

Loginova

Egorenkov

et al. 2023

Math Methods in App Sciences

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In this paper, we propose self‐adaptive artificial boundary conditions (ABCs) for the two‐dimensional (2D) time‐dependent nonlinear Schrödinger equation and studied their efficiency. This equation describes the laser pulse interaction with a semiconductor layer under laser‐induced plasma generation. Because of its nonlinearity, the interaction process is very sensitive to the appearance of the optical beam spurious reflection; therefore, highly transparent ABCs are urgently required to avoid disturbing the interaction process by an unphysical reflected wave. This can be achieved by using time‐ and spatial‐dependent local wavenumbers computed near the artificial boundaries. We claim that it is necessary to account for both wavenumbers in each of the adaptive ABCs stated on both spatial coordinates. Based on computer simulation, we demonstrate an increase in the efficiency of the adaptive ABC stated on the coordinate along which the optical pulse propagates by involving both local wavenumbers. The accuracy of the developed approach is verified using a well‐known analytical solution of the linear Schrödinger equation as well as the reference problem solution, obtained by solving the problem with zero‐value boundary conditions on the extended spatial domain. For computer modeling, the finite‐difference scheme, possessing asymptotic stability, is applied together with a two‐stage iterative process for its realization. The algorithm and peculiarities of the local wavenumber computation are discussed in detail. The applicability of the developed approach is also demonstrated via the solution of both linear and nonlinear problems. The proposed method has widespread application for solving various laser physics problems governed by the Schrödinger equation.

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High-order finite element approximation for partial differential equations

Chernov¹,

Schwab²

2014

Computers & Mathematics with Applications

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Numerical realization of Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides

Cited by 9 publications

References 28 publications

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

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

Cross‐impact of beam local wavenumbers on the efficiency of self‐adaptive artificial boundary conditions for two‐dimensional nonlinear Schrödinger equation

High-order finite element approximation for partial differential equations

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