Canonical Resonant Four-Wave-Mixing in Photonic Crystal Cavities : tuning, tolerances and scaling

We refer to Canonical Resonant Four-Wave-Mixing when only four (three in the degenerate case) modes of the cavity are able to interact through this parametric process; thereby inhibiting competing parametric processes. This condition is achieved in the recently introduced photonic crystal parametric oscillator, Marty et al. Nat. Photonics, 15 , 53 (2021). Here, we discuss extensively the properties of this parametric source and compare the design with other geometries of photonic crystal resonators. Based on a statistical study over more than 100 resonators and 10 parametric oscillators, robustness against fabrication tolerances is assessed, performances are evaluated in terms of average values and their dispersion, and the dependence on the main parameters is shown to follow the theoretical scaling. The lowest pump power at threshold is \approx 40 \muW and we show the existence of a minimum value of the cavity photon lifetime as a condition for parametric oscillation, which is related to three photon absorption.