  We present an analytic perturbation theory which extends the paraxial
approximation for a common cylindrically symmetric stable optical resonator and
incorporates the differential, polarization-dependent reflectivity of a Bragg
mirror. The degeneracy of Laguerre-Gauss modes with distinct orbital angular
momentum (OAM) and polarization, but identical transverse order N, will become
observably lifted at sufficiently small size and high finesse. The resulting
paraxial eigenmodes possess two distinct OAM components, the fractional
composition subtly depending on mirror structure.
