Spatial and spectral properties of entangled photons from spontaneous parametric down-conversion with a focused pump

The spatial and spectral properties of entangled photons from spontaneous parametric down-conversion (SPDC) with a focused pump are investigated with theoretical analysis and numerical simulation. Here, the spatial and spectral properties of down-converted photons are fully calculated without using the transverse momentum conservation assumption. We have obtained the spatial and spectral properties of SPDC photons under focused pumping in various SPDC configurations, including type-I, type-II for both positive and negative uniaxial crystals. Our result not only helps to understand the effect of pump focusing in SPDC better but also can find use in application of SPDC in realizing quantum information protocols.