  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.
