Methods for 3-D vector microcavity problems involving a planar dielectric mirror

We develop and demonstrate two numerical methods for solving the class of open cavity problems which involve a curved, cylindrically symmetric conducting mirror facing a planar dielectric stack. Such dome-shaped cavities are useful due to their tight focusing of light onto the flat surface. The first method uses the Bessel wave basis. From this method evolves a two-basis method, which ultimately uses a multipole basis. Each method is developed for both the scalar field and the electromagnetic vector field and explicit ``end user'' formulas are given. All of these methods characterize the arbitrary dielectric stack mirror entirely by its 2\times2 transfer matrices for s- and p-polarization. We explain both theoretical and practical limitations to our method. Non-trivial demonstrations are given, including one of a stack-induced effect (the mixing of near-degenerate Laguerre-Gaussian modes) that may persist arbitrarily far into the paraxial limit. Cavities as large as 50 \lambda are treated, far exceeding any vectorial solutions previously reported.