| Photonic crystals are periodic mixed dielectric media with dispersion characteristics that include optical band gaps and light localization. Photonic quasicrystals are the rotational equivalent of photonic crystals, structures that have rotational symmetry instead of translational. There are a number of numerical methods available for the determination of the optical characteristics of both classes of structure. The two most commonly used techniques are planewave expansion and finite-difference time-domain. The efficient planewave expansion method is a modal approach that requires translational symmetry within the structure being studied. The alterations required to implement it for photonic crystals with non-periodic lattice defects or photonic quasicrystals can result in significant increases in the computational requirements. The finite-difference time-domain technique is robust, placing no requirements on the symmetry of the dielectric. However, the technique can be computationally intensive and as a time based method it is optimized for modeling field propagation through the structure and not the modal solutions.;The theoretical derivation of the method, and simulation results for both a photonic crystal and two photonic quasicrystals will be presented. Finite-difference time-domain and planewave expansion results will be used as the baseline against which the efficiency and accuracy of the new method is evaluated.;This thesis proposes a new method, the Fourier-Bessel expansion method, for the determination of the wavelengths for stationary states supported by rotationally symmetric dielectric structures. The approach is a polar coordinate equivalent of the planewave expansion method where the basis of expansion is the Fourier-Bessel function. The relationship between the rotational symmetries of the states and dielectric can be used to reduce the computational requirements relative to planewave expansion while retaining its accuracy. |
