  Generation of heralded single photons has recently been demonstrated using
spontaneous four-wave mixing in integrated microresonators. While the results
of coincidence measurements on the generated photon pairs from these systems
show promise for their utility in heralding applications, such measurements do
not reveal all of the effects of photon losses within the resonator. These
effects, which include a significant degradation of the heralding efficiency,
depend strongly on the relative strengths of the coupling of the ring modes to
loss modes and channel modes. We show that the common choice of critical
coupling does not optimize the rate of successfully heralded photons, and
derive the coupling condition needed to do so, as well as the condition needed
to maximize the rate of coincidence counts. Optimizing these rates has a
considerable negative effect on the heralding efficiency.
