Heralding pure single photons: A comparison between counterpropagating and copropagating twin photons

We investigate different strategies suitable to generate pure heralded single photons through spontaneous parametric down-conversion, comparing the counterpropagating geometry with more conventional copropagating configurations which enhance the purity of the heralded photon state through the technique of group-velocity matching. We derive general results for the Schmidt number associated to the temporal modes, which provide a quantitative estimate of the purity. An analysis of the correlation of twin photons in the temporal domain provides a more physical view of the mechanisms that permit to eliminate the temporal entanglement of the state and to generate high-purity heralded photons. The efficiency of the various strategies and the individual properties of the heralded photons thereby generated are then compared.