  We demonstrate theoretical conditions for highly-efficient degenerate
four-wave mixing in triply-resonant nonlinear (Kerr) cavities. We employ a
general and accurate temporal coupled-mode analysis in which the interaction of
light in arbitrary microcavities is expressed in terms a set of coupling
coefficients that we rigorously derive from the full Maxwell equations. Using
the coupled-mode theory, we show that light consisting of an input signal of
frequency $\omega_0-\Delta \omega$ can, in the presence of pump light at
$\omega_0$, be converted with quantum-limited efficiency into an output shifted
signal of frequency $\omega_0 + \Delta \omega$, and we derive expressions for
the critical input powers at which this occurs. We find that critical powers in
the order of 10mW assuming very conservative cavity parameters (modal volumes
$\sim10$ cubic wavelengths and quality factors $\sim1000$. The standard
Manley-Rowe efficiency limits are obtained from the solution of the classical
coupled-mode equations, although we also derive them from simple
photon-counting "quantum" arguments. Finally, using a linear stability
analysis, we demonstrate that maximal conversion efficiency can be retained
even in the presence of self- and cross-phase modulation effects that generally
act to disrupt the resonance condition.
