An efficient calculation of photonic crystal band structures using Taylor expansions

Please always quote using this URN:urn:nbn:de:0296-matheon-13091
  • In this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.

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Metadaten
Author:Dirk Klindworth, Kersten Schmidt
URN:urn:nbn:de:0296-matheon-13091
Referee:Frank Schmidt
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2014/06/26
Release Date:2014/06/26
Tag:Taylor expansion; adaptivity; band structure calculation; group velocity; photonic crystals
Institute:Research Center Matheon
Technische Universität Berlin
Project:D Optics and Electronics (Electronic and photonic devices) / D26 Asymptotic analysis of the wave-propagation in realistic photonic crystal wave-guides
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N25 Eigenvalue problems
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
65-XX NUMERICAL ANALYSIS / 65Zxx Applications to physics / 65Z05 Applications to physics
78-XX OPTICS, ELECTROMAGNETIC THEORY (For quantum optics, see 81V80) / 78Mxx Basic methods / 78M10 Finite element methods
Preprint Number:1068
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