The plane wave expansion (PWM) technique applied to Maxwell’s wave equations provides researchers with a
supply of information regarding the optical properties of dielectric structures. The technique is well suited for
structures that display a linear periodicity. When the focus is directed towards optical resonators and structures that
lack linear periodicity the eigen-process can easily exceed computational resources and time constraints. In the case
of dielectric structures which display cylindrical or spherical symmetry, a coordinate system specific set of basis
functions have been employed to cast Maxwell’s wave equations into an eigen-matrix formulation from which the
resonator states associated with the dielectric profile can be obtained. As for PWM, the inverse of the dielectric and
field components are expanded in the basis functions (Fourier-Fourier-Bessel, FFB, in cylindrical and Fourier-
Bessel-Legendre, BLF, in spherical) and orthogonality is employed to form the matrix expressions. The theoretical
development details will be presented indicating how certain mathematical complications in the process have been
overcome and how the eigen-matrix can be tuned to a specific mode type. The similarities and differences in PWM,
FFB and BLF are presented. In the case of structures possessing axial cylindrical symmetry, the inclusion of the z
axis component of propagation constant makes the technique applicable to photonic crystal fibers and other
waveguide structures. Computational results will be presented for a number of different dielectric geometries
including Bragg ring resonators, cylindrical space slot channel waveguides and bottle resonators. Steps to further
enhance the computation process will be reported.
For spherically symmetric dielectric structures, a basis set composed of Bessel, Legendre and Fourier functions, BLF, are
used to cast Maxwell's wave equations into an eigenvalue problem from which the localized modes can be determined.
The steps leading to the eigenmatrix are reviewed and techniques used to reduce the order of matrix and tune the
computations for particular mode types are detailed. The BLF basis functions are used to expand the electric and
magnetic fields as well as the inverse relative dielectric profile. Similar to the common plane wave expansion technique,
the BLF matrix returns the eigen-frequencies and eigenvectors, but in BLF only steady states, non-propagated, are
obtained. The technique is first applied to a air filled spherical structure with perfectly conducting outer surface and then
to a spherical microsphere located in air. Results are compared published values were possible.