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

Klindworth, Dirk and Schmidt, Kersten (2014):
Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides.
In: IEEE Trans. Magn., pp. 217-220, 50, (2), ISSN 0018-9464, DOI: 10.1109/TMAG.2013.2285412, [Article]

Abstract

In this work we present a complete algorithm for the exact computation of the guided mode band structure in photonic crystal (PhC) wave-guides. In contrast to the supercell method, the used approach does not introduce any modelling error and is hence independent of the confinement of the modes. The approach is based on Dirichlet-to-Neumann (DtN) transparent boundary conditions that yield a nonlinear eigenvalue problem. For the solution of this nonlinear eigenvalue problem we present a direct technique using Chebyshev interpolation that requires a band gap calculation of the PhC in advance. For this band gap calculation we introduce as a very efficient tool a Taylor expansion of the PhC band structure. We show that our algorithm like the supercell method converges exponentially, however, its computational costs in comparison to the supercell method only increase moderately since the size of the matrix to be inverted remains constant.

Item Type: Article
Erschienen: 2014
Creators: Klindworth, Dirk and Schmidt, Kersten
Title: Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides
Language: English
Abstract:

In this work we present a complete algorithm for the exact computation of the guided mode band structure in photonic crystal (PhC) wave-guides. In contrast to the supercell method, the used approach does not introduce any modelling error and is hence independent of the confinement of the modes. The approach is based on Dirichlet-to-Neumann (DtN) transparent boundary conditions that yield a nonlinear eigenvalue problem. For the solution of this nonlinear eigenvalue problem we present a direct technique using Chebyshev interpolation that requires a band gap calculation of the PhC in advance. For this band gap calculation we introduce as a very efficient tool a Taylor expansion of the PhC band structure. We show that our algorithm like the supercell method converges exponentially, however, its computational costs in comparison to the supercell method only increase moderately since the size of the matrix to be inverted remains constant.

Journal or Publication Title: IEEE Trans. Magn.
Volume: 50
Number: 2
Uncontrolled Keywords: Chebyshev approximation;convergence of numerical methods;eigenvalues and eigenfunctions;interpolation;optical waveguides;photonic band gap;photonic crystals;Chebyshev interpolation;Dirichlet-to-Neumann transparent boundary conditions;PhC band structure;Taylor expansion;bandgap calculation;convergence;guided mode band structure;nonlinear eigenvalue problem;photonic crystal waveguides;Boundary conditions;Chebyshev approximation;Eigenvalues and eigenfunctions;Interpolation;Photonic band gap;Taylor series;Boundary conditions;eigenvalues and eigenfunctions;finite-element methods;nonlinear equations;photonic crystals (PhCs)
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 19 Nov 2018 21:32
DOI: 10.1109/TMAG.2013.2285412
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