The orbital angular momentum of down-converted photons

We calculate the relative amplitude of the orbital angular momentum (OAM) of entangled photon pairs from the spontaneous parametric down-conversion of a thin quadratic nonlinear crystal. The results show that the amplitude depends on both of the two Laguerre indices l, p. We also discuss the influences of the commonly used holograms and mono-mode fibres for mode analysis. We conclude that only a few dimensions can be explored from the infinite OAM modes of down-converted photon pairs.