  We study the generation of propagation invariant photons with orbital angular
momentum by spontaneous parametric down conversion (SPDC) using a Bessel-Gauss
pump beam. The angular and conditional angular spectra are calculated for an
uniaxial crystal optimized for type I SPDC with standard Gaussian pump beams.
It is shown that, as the mean value of the magnitude of the transverse wave
vector of the pump beam increases, the emission cone is deformed into two non
coaxial cones that touch each other along a line determined by the orientation
of the optical axis of the nonlinear crystal. At this location, the conditional
spectrum becomes maximal for a pair of photons, one of which is best described
by a Gaussian-like photon with a very small transverse wave vector, and the
othera Bessel-Gauss photon with a distribution of transverse wave vectors
similar in amplitude to that of the incident pump beam. A detailed analysis is
then performed of the angular momentum content of SPDC photons by the
evaluation of the corresponding transition amplitudes. As a result, we obtain
conditions for the generation of heralded single photons which are
approximately propagation invariant and have orbital angular momentum. A
discussion is given about the difficulties in the interpretation of the results
in terms of conservation of optical orbital angular momentum along the vector
normal to the crystal surface. The angular spectra and the conditional angular
spectra are successfully compared with available experimental data recently
reported in the literature.
