Kerr-enhanced amplification of three-wave mixing and emergent masing regimes

Integrated optical microresonators exploiting either second-order ($\chi^{(2)}$) or third-order ($\chi^{(3)}$) nonlinearities have become key platforms for frequency conversion, low-noise microwave photonics, and quantum entanglement generation. Here, we present an analytic theory of Kerr-enhanced three-wave mixing amplification in an electro-optic microresonator with both $\chi^{(2)}$ and $\chi^{(3)}$ nonlinearities. We demonstrate that Kerr dressing hybridizes the optical sidebands, renormalizing the $\chi^{(2)}$ couplings and detunings. As a result the system exhibits gain in regions where analogous bare $\chi^{(2)}$ or $\chi^{(3)}$ amplifiers are subthreshold. Time-domain Langevin simulations confirm this threshold reduction, mapping a practical design window for experiments.