Generalized Hermite-Gauss decomposition of the two-photon state produced by spontaneous parametric down-conversion

We provide a general decomposition of the two-photon state produced by spontaneous parametric down-conversion in Hermite-Gaussian modes, in the case that the pump beam is described by a Hermite-Gaussian beam of any order. We show that the spatial correlations depend explicitly on the order of the pump beam, as well as other experimental parameters. We use the decomposition to demonstrate a few interesting cases. Our results are applicable to the engineering of two-photon spatial entanglement, in particular for non-Gaussian states.