Eigenvalue algorithms and their application to photonic crystal device modelling
Eigenvalue algorithms and their application to photonic crystal device modelling
In this thesis three methods are presented which calculate the lowest eigenvalues of a set of extremely sparse generalized eigenvalue problems which arise from modelling photonic crystal (PC) structures with the Finite Element Method in 2D and 3D. These are (1) Subspace Iteration, (2) a spectral solver based on Fourier Analysis or Maximum Entropy and (3) an Implicitly Restarted Lanczos Algorithm.
Each eignevalue solver was used in a unique way to increase the efficiency of calculating the lowest few eigenvalues of a set of similar generalized eigenvalue problems. For Subspace Iteration using a low fractional accuracy and only 2 extra vectors accurate results can still be obtained with only ~ 2.2 iterations until convergence. By using Maximum Entropy or Fourier Analysis accurate density of states diagrams could be produced for propagating modes combined given a set of moments calculated from matrix vector products. A parallel implementation of this technique is presented. Modelling 3-dimensional photonic crystals with the Vector Finite Element Method leads to a large number of zero eigenvalues which do not represent physical modes. They were ‘filtered’ out by using an Implicitly Restarted Lanczos Method which selects the zero eigenvalue as a shift in the shifted QR scheme as it begins to converge.
Taking advantage of the development of a highly efficient solver for the PC problem and the use of a grid-enabled cluster, the final chapters are an initial study in exploiting our modelling capability for optimising PC structures consisting of various configurations of rods. There are three main results: (1) from an initial sample of several thousand PC structures the best ones were optimised using a simple gradient descent technique; (2) a set of canonical structures were optimised, and (3) the effect of fabrication tolerances on the properties of a PC with a triangular lattice were investigated. The optimisation increased the size of the gap-midgap ratio by over 200% in some cases. By allowing for errors in the position and radius of the rods it was shown that with current manufacturing processes potentially homogeneous band gap could be destroyed.
Beckett, D.H.
71fdbee9-30b4-4a11-accd-fee7b28f7257
2003
Beckett, D.H.
71fdbee9-30b4-4a11-accd-fee7b28f7257
Beckett, D.H.
(2003)
Eigenvalue algorithms and their application to photonic crystal device modelling.
University of Southampton, School of Engineering Sciences, Doctoral Thesis.
Record type:
Thesis
(Doctoral)
Abstract
In this thesis three methods are presented which calculate the lowest eigenvalues of a set of extremely sparse generalized eigenvalue problems which arise from modelling photonic crystal (PC) structures with the Finite Element Method in 2D and 3D. These are (1) Subspace Iteration, (2) a spectral solver based on Fourier Analysis or Maximum Entropy and (3) an Implicitly Restarted Lanczos Algorithm.
Each eignevalue solver was used in a unique way to increase the efficiency of calculating the lowest few eigenvalues of a set of similar generalized eigenvalue problems. For Subspace Iteration using a low fractional accuracy and only 2 extra vectors accurate results can still be obtained with only ~ 2.2 iterations until convergence. By using Maximum Entropy or Fourier Analysis accurate density of states diagrams could be produced for propagating modes combined given a set of moments calculated from matrix vector products. A parallel implementation of this technique is presented. Modelling 3-dimensional photonic crystals with the Vector Finite Element Method leads to a large number of zero eigenvalues which do not represent physical modes. They were ‘filtered’ out by using an Implicitly Restarted Lanczos Method which selects the zero eigenvalue as a shift in the shifted QR scheme as it begins to converge.
Taking advantage of the development of a highly efficient solver for the PC problem and the use of a grid-enabled cluster, the final chapters are an initial study in exploiting our modelling capability for optimising PC structures consisting of various configurations of rods. There are three main results: (1) from an initial sample of several thousand PC structures the best ones were optimised using a simple gradient descent technique; (2) a set of canonical structures were optimised, and (3) the effect of fabrication tolerances on the properties of a PC with a triangular lattice were investigated. The optimisation increased the size of the gap-midgap ratio by over 200% in some cases. By allowing for errors in the position and radius of the rods it was shown that with current manufacturing processes potentially homogeneous band gap could be destroyed.
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Published date: 2003
Organisations:
University of Southampton
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Local EPrints ID: 45909
URI: http://eprints.soton.ac.uk/id/eprint/45909
PURE UUID: 00a93228-76e7-47e0-929d-954e1081a434
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Date deposited: 25 Apr 2007
Last modified: 11 Dec 2021 16:29
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Author:
D.H. Beckett
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