Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion

We present a theoretical analysis of optical parametric oscillations through four-wave mixing in a nonlinear high-Q whispering-gallery-mode resonator. It is shown that even a small flux of pump photons is sufficient to reach the threshold of the oscillations. We demonstrate that due to narrow bandwidth of the resonator modes as well as the high efficiency of the resonant frequency conversion the oscillator produces a stable narrow-band beat note of the pump, signal, and idler waves making an all-optical secondary frequency reference feasible.