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 Elerment 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.