Analytic Theory for Parametric Amplification in High-Q Micro-Ring Resonators

Optical parametric amplifiers (OPAs) are optical amplifiers based on four-wave mixing processes, showing great potential for applications in communications, optical signal processing, quantum optics, etc. In recent years, significant progress has been made to integrated OPAs based on highly nonlinear micro-ring resonators (MRRs), benefiting from the greatly enhanced optical-matter interactions. Notable parametric gain becomes available at unprecedented low power levels, allowing for example the source-integrated coherent optical frequency combs. Therefore, an analytical formula for OPAs in high-quality (Q) MRRs is of great importance in the design of optical parametric devices. Analytical theory for OPAs in high-Q MRRs in the high pump power scenario remains elusive, where intensity-dependent nonlinear phase that brings significant parametric gains cannot be ignored. In this work, analytical formulas for the parametric gain and conversion efficiency (CE) of a high-Q MRR with high-pump excitation are derived. We show the interplay between parametric gain and field enhancement: the field enhancement of the signal and idler wave can be greatly boosted due to the compensation of the round-trip loss by the parametric gain, which in turn leads to increased field enhancement as well as greater parametric gain and CE. Our theory agrees well with numerical and experimental results.