  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.
