  The nonlinear optical process of spontaneous parametric down-conversion
(SPDC) is widely studied for applications in quantum information science due to
its ability to produce two photons that can be entangled in many degrees of
freedom. For applications in quantum communication, two metrics of this process
are particularly important: heralding efficiency and total joint rate. Here, we
derive expressions for both quantities for a variety of different beam
geometries and frequencies. We pay specific attention to the spectrum of both
biphotons and individual photons. We reveal the underlying mechanisms
responsible for the spectral shape and show they differ for different
geometries and frequencies. We then use these spectra to calculate heralding
efficiency and joint count rate and examine how each of these metrics changes
with different geometries, frequencies, and spectral filtering and beam
parameters. Interestingly, we find very high heralding efficiencies are
achievable for collinear geometries, while noncollinear geometries require
spectral filtering for high heralding efficiency, ultimately limiting the joint
count rate. We also find that the the spectrum is narrower in nondegenerate
cases, leading to lower joint count rates and higher heralding efficiency in
the noncollinear case. In addition to the theory, we verify selected
predictions with experimental results.
