Phototonic crystal modelling using finite element analysis

Abstract

This thesis presents an efficient finite element method for computing spectra of photonic band gap materials. These periodic dielectric crystals exhibit a photonic band gap analogous to the electronic band gap present in semiconducters. Photons in the frequency range of the band gap are completely excluded so that atoms within such materials are unable to spontaneously absorb and re-emit light in this region. Photonic band gap devices offer enormous potential in the development of highly efficient narrow band lasers, integrated optical computing and high-speed optical communication networks, particularly in the production of purely optical circuits for dense wavelength division multiplexing.

Computational modelling of photonic band gap devices has traditionally been approached using plane wave expansion techniques.  These have the disadvantages of being expensive in terms of computational and memory. By contrast, the finite element method is considerably more efficient since the eigensystem matrices are very sparse and the discontinuous dielectric constant is handled in real space.  We have developed finite element software and used it to compute the band structure for a variety of common photonic crystal structures along with more novel structures such as the 12-fold symmetric quasicrystal.

We compare our results with those obtained from other sources, including plane wave expansion techniques, finite difference methods and experimental data. The performance of the algorithm is analysed in terms of memory and computational cost confirming the O(n) problem scaling.  Photonic band gap device optimisation is performed via multi-dimensional minimisation algorithms and the analysis of a canonical set of lattice arrangements.  High-performance grid-enabled compute resources were utilised due to the computationally intensive nature of the process.  This research has ascertained those crystal structures that produce the largest, most robust photonic band gaps with the best being the simple triangular lattice with a high filling fraction.

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