Universal Kerr-thermal dynamics of self-injection-locked microresonator dark pulses

Microcombs, formed in optical microresonators driven by continuous-wave lasers, are miniaturized optical frequency combs with small size, weight and power consumption. Leveraging integrated photonics and laser self-injection locking (SIL), compact and robust microcombs can be constructed via hybrid integration of a semiconductor laser with a chip-based microresonator. While the current linear SIL theory has successfully addressed the linear coupling between the laser cavity and the external microresonator, it fails to manage the complicated nonlinear processes, especially regarding to dark-pulse microcomb formation. Here, we investigate -- theoretically, numerically and experimentally -- the Kerr-thermal dynamics of a semiconductor laser self-injection-locked to an integrated silicon nitride microresonator. We unveil intriguing yet universal dark-pulse formation and switching behaviour with discrete steps, and establish a theoretical model scrutinizing the synergy of laser-microresonator mutual coupling, Kerr nonlinearity, photo-thermal effect. Numerical simulation confirms the experimental result and identifies the origins. Exploiting this unique phenomenon, we showcase an application on low-noise photonic microwave generation with phase noise purified by 23.5 dB. Our study not only add critical insight of pulse formation in laser-microresonator hybrid systems, but also enables all-passive, photonic-chip-based microwave oscillators with high spectral purity.