  We develop a general Hamiltonian treatement of spontaneous four-wave mixing
in a microring resonator side-coupled to a channel waveguide. The effect of
scattering losses in the ring is included, as well as parasitic nonlinear
effects including self- and cross-phase modulation. A procedure for computing
the output of such a system for arbitrary parameters and pump states is
presented. For the limit of weak pumping an expression for the joint spectral
intensity of generated photon pairs, as well as the singles-to-coincidences
ratio, is derived.
