  We propose a method for directly producing radially and azimuthally polarized
photon pairs through spontaneous parametric downconversion (SPDC). This method
constitutes a novel geometry for SPDC, in which a radially polarized
Bessel-Gauss pump beam is directed into a nonlinear crystal, with the central
propagation direction parallel to the crystal axis. The phasematching
conditions are controlled by changing the opening angle of the pump beam; as
the crystal axis cannot be tuned, we refer to this process as super-critical
phasematching. We model and plot the spatial and polarization output
distributions for Type-I and Type-II super-critical phasematching.
