Degenerate four-wave mixing in triply resonant Kerr cavities

Abstract

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 of 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 ${\ensuremath{\omega}}_{0}\ensuremath{-}\ensuremath{\Delta}\ensuremath{\omega}$ can, in the presence of pump light at ${\ensuremath{\omega}}_{0}$, be converted with quantum-limited efficiency into an output shifted signal of frequency ${\ensuremath{\omega}}_{0}+\ensuremath{\Delta}\ensuremath{\omega}$, and we derive expressions for the critical input powers at which this occurs. We find the critical powers in the order of 10 mW, assuming very conservative cavity parameters (modal volumes $~10$ cubic wavelengths and quality factors $~1000$). 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.