  We develop a theoretical model of triply-resonant optical parametric
oscillators (OPOs) based on degenerate four-wave mixing (FWM) that includes
physics and degrees of freedom relevant to microphotonic (on-chip) device
implementations, including nonlinear loss, a general resonant mode field
structure, and mode-selective coupling to external ports. The coupled mode
theory model addresses the effect of two-photon absorption and free-carrier
absorption on parametric gain and oscillation thresholds, and ultimately on the
optimum design for an OPO. The model goes beyond a typical free-space cavity
configuration by incorporating a full modal analysis that admits distributed
modes with non-uniform field distribution, relevant to photonic microcavity
systems on chip. This leads to a generalization of the concept of nonlinear
figure of merit (NFOM) to a vector of coefficients. In addition, by considering
unconstrained signal, pump and idler resonance coupling strengths to excitation
ports, not usually available in simple cavity geometry, we show that the
efficiency-maximizing design will have unequal external Q for the three
resonances. We arrive at generalized formulas for OPO oscillation threshold
that include nonlinear absorption and free carrier lifetime, and provide a
normalized solution to the design problem in the presence of nonlinear loss in
terms of optimum choice of coupling. Based on the results, we suggest a family
of coupled-cavity systems to implement optimum FWM, where control of resonant
wavelengths can be separated from optimizing nonlinear conversion efficiency,
and where furthermore pump, signal, and idler coupling to bus waveguides can be
controlled independently, using interferometric cavity supermode coupling as an
example. Using the generalized NFOM, we address the efficiency of single and
multi-cavity geometry, as well as standing and traveling wave excitation.
