Finesse and four-wave mixing in microresonators

TL;DR

This paper develops a comprehensive theory for the sharp variations in four-wave mixing thresholds in high-finesse Kerr microresonators, linking finesse dispersion to instability thresholds and bistability regimes.

Contribution

It provides explicit estimates for pump power differences required for sideband excitation and introduces a formal link between finesse dispersion and density of states in microresonators.

Findings

Explicit estimates for pump power differences for sideband excitation

Identification of threshold conditions for multiple sideband instabilities

Demonstration of bistability regimes without FWM in high finesse dispersion

Abstract

We elaborate a comprehensive theory of the sharp variations of the four-wave-mixing (FWM) threshold happening with the tuning of the pump frequency along the dispersive tails of the nonlinear resonance in the driven Kerr microresonators with high finesses and high finesse dispersions. Our theory leads to the explicit estimates for the difference in the pump powers required for the excitation of one sideband pair with momenta $\pm\mu$, and for the simultaneous excitation of the two neighbouring pairs with momenta $\pm\mu$, $\pm(\mu+1)$. This power difference also measures the depth of the Arnold tongues in the pump-frequency and pump-power parameter space, associated with the aforementioned power variations. A set of select pump frequencies and powers, where the instabilities of the two sideband pairs come first, forms the threshold of complexity, which is followed by a sequence of conditions specifying critical powers for the simultaneous instabilities of three, four and so on pairs. We formally link finesse dispersion and the density of states notion borrowed from the condensed matter context and report a value of the finesse dispersion and a critical mode number signalling the transition from the low- to the high-contrast tongue structure. We also demonstrate that the large finesse dispersion makes possible a surprising for the multimode resonators regime of the bistability without FWM.