  In this paper we consider geometrical two-photon optics of Bessel-Gaussian
modes generated in spontaneous parametric down-conversion of a Gaussian pump
beam. We provide a general theoretical expression for the orbital angular
momentum (OAM) spectrum and Schmidt number in this basis and show how this may
be varied by control over the radial degree of freedom, a continuous parameter
in Bessel-Gaussian modes. As a test we first implement a back-projection
technique to classically predict, by experiment, the quantum correlations for
Bessel-Gaussian modes produced by three holographic masks, a blazed axicon,
binary axicon and a binary Bessel function. We then proceed to test the theory
on the down-converted photons using the binary Bessel mask. We experimentally
quantify the number of usable OAM modes and confirm the theoretical prediction
of a flattening in the OAM spectrum and a concomitant increase in the OAM
bandwidth. The results have implications for the control of dimensionality in
quantum states.
