  High flux of hyperentangled photons entails collecting the two-photon
emission over relatively wide extent in frequency and transverse space within
which the photon pairs are simultaneously entangled in multiple degrees of
freedom. In this paper, we present a numerical approach to determining the
spatial-spectral relative-phase and time-delay maps of hyperentangled photons
all over the spontaneous parametric down conversion (SPDC) emission cone. We
consider the hyperentangled-photons produced by superimposing noncollinear SPDC
emissions of two crossed and coherently-pumped nonlinear crystals. We adopt a
vectorial representation for all parameters of concern. This enables us to
study special settings such as the self-compensation via oblique pump
incidence. While rigorous quantum treatment of SPDC emission requires Gaussian
state representation, in low-gain regime (like the case of the study), it is
well approximated to the first order to superposition of vacuum and two-photon
states. The relative phase and time-delay maps are then calculated between the
two-photon wavepackets created along symmetrical locations of the crystals.
Assuming monochromatic plane-wave pump field, the mutual signal-idler relations
like energy conservation and transverse-momentum conservation define well one
of the two-photon with reference to its conjugate. The weaker conservation of
longitudinal momentum (due to relatively thin crystals) allows two-photon
emission directions coplanar with the pump beam while spreading around the
perfect phase-matching direction. While prior works often adopt first-order
approximation, it is shown that the relative-phase map is a very well
approximated to a quadratic function in the polar angle of the two-photon
emission while negligibly varying with the azimuthal angle.
