Closed Form of the Biphoton K-Vector Spectrum for Arbitrary Spatio-Temporal Pump Modes

A closed form solution is derived for the biphoton k-vector spectrum for an arbitrary pump spatial mode. The resulting mode coefficients for the pump input that maximize the probability of biphoton detection in the far field are found. It is thus possible to include the effect of arbitrary crystal poling strucures, and optimize the resulting biphoton flux.