Effect of spatial filtering on the Schmidt decomposition of entangled photons

A simple analytic decomposition of the spatially entangled two-photon field allows us to generalize the earlier results of Law and Eberly [Phys. Rev. Lett. 92, 127903 (2004)] to more realistic experimental geometries of spontaneous parametric down-conversion. We quantify analytically how spatial filtering reduces the Schmidt number or dimensionality of the two-photon entanglement from a generated value to a generally much lower usable or detected value, for both collinear and noncollinear phase matching. We also discuss the intimate relation between the (two-photon) Schmidt number and the classical (one-photon) concept of etendue or geometric extent. (c) 2006 American Institute of Physics.