Abstract
We calculate the bi-photon state generated by spontaneous parametric down conversion in a thin crystal and under collinear phase matching conditions using a pump consisting of any superposition of Laguerre-Gauss modes. The result has no restrictions on the angular or radial momenta or, in particular, on the width of the pump, signal and idler modes. We demonstrate the strong effect of the pump to signal/idler width ratio on the composition of the down-converted entangled fields. Knowledge of the pump to signal/idler width ratio is shown to be essential when calculating the maximally entangled states that can be produced using pumps with a complex spatial profile.
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