Super-critical phasematching for photon pair generation in structured   light modes
  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. 
Rebecca Y. Saaltink, Lambert Giner, Robert W. Boyd, Ebrahim Karimi,   Jeff S. Lundeen
quant-ph physics.optics