Transverse distinguishability of entangled photons with arbitrarily shaped spatial near- and far-field distributions

Entangled photons generated by spontaneous parametric down conversion inside a nonlinear crystal exhibit a complex spatial photon count distribution. A quantitative description of this distribution helps with the interpretation of experiments that depend on this structure. We developed a theoretical model and an accompanying numerical calculation that includes the effects of phase matching and the crystal properties to describe a wide range of spatial effects in two-photon experiments. The numerical calculation was tested against selected analytical approximations. We furthermore performed a double-slit experiment where we measured the visibility V and the distinguishability D and obtained $D^2 + V^2 = 1.43$. The numerical model accurately predicts these experimental results.