Controlling and confining light in three dimensions is of significant interest as it represents devices in their entirety with minimal approximations. Localized modes in resonator structures are commonly modelled using the supercell plane wave expansion method, finite element method, or the finite difference time domain. However, these standard techniques are expensive in terms of computational resources when applied to three-dimensional structures. The technique presented in this thesis is a new method for solving Maxwell’s vector wave equations for localized modes in three-dimensional spherically symmetric resonator structures as well as its application to sensor configurations. The technique requires minimal implementation, provides normalized results, works for finite size and is computationally efficient. The method is such that the structures under test can be arbitrary shape, isotropic, anisotropic, lossless, or lossy.
For the structures examined, a modified basis set composed of spherical Bessel, Legendre and Fourier functions (BLF) are used to expand the electric, magnetic, and inverse relative permittivity. These expansions allow for Maxwell’s wave equations to be cast as an eigenvalue problem from which the steady state localized modes can be determined from the eigen-frequencies and eigenvectors. This work applies to a number of spherically symmetric structures. A selection of structures whose resonator properties are reported in the literature are used to compare and verify the accuracy of the use of the BLF functions as an expansion basis.