  Dielectric microcavities based on cylindrical and deformed cylindrical shapes
have been employed as resonators for microlasers. Such systems support spiral
resonances with finite momentum along the cylinder axis. For such modes the
boundary conditions do not separate and simple TM and TE polarization states do
not exist. We formulate a theory for the dispersion relations and polarization
properties of such resonances for an infinite dielectric rod of arbitrary
cross-section and then solve for these quantities for the case of a circular
cross-section (cylinder). Useful analytic formulas are obtained using the
eikonal (Einstein-Brillouin-Keller) method which are shown to be excellent
approximations to the exact results from the wave equation. The major finding
is that the polarization of the radiation emitted into the far-field is linear
up to a polarization critical angle (PCA) at which it changes to elliptical.
The PCA always lies between the Brewster and total-internal-reflection angles
for the dielectric, as is shown by an analysis based on the Jones matrices of
the spiraling rays.
