  We calculated the Fresnel paraxial propagator in a birefringent plate having
topological charge $q$ at its center, named "$q$-plate". We studied the change
of the beam transverse profile when it traverses the plate. An analytical
closed form of the beam profile propagating in the "$q$-plate" can be found for
many important specific input beam profiles. We paid particular attention to
the plate having a topological unit charge and we found that if small losses
due to reflection, absorption and scattering are neglected, the plate can
convert the photon spin into orbital angular momentum with up to 100%
efficiency, provided the thickness of the plate is less than the Rayleigh range
of the incident beam.
