  Self-injection locking is a dynamic phenomenon representing stabilization of
the emission frequency of an oscillator with a passive cavity enabling
frequency filtered coherent feedback to the oscillator cavity. For instance,
self-injection locking of a semiconductor laser to a high-quality-factor
(high-Q) whispering gallery mode (WGM) microresonator can result in multiple
orders of magnitude reduction of the laser linewidth. The phenomenon was
broadly studied in experiments, but its detailed theoretical model allowing
improving the stabilization performance does not exist. In this paper we
develop such a theory. We introduce five parameters identifying efficiency of
the self-injection locking in an experiment, comprising back-scattering
efficiency, phase delay between the laser and the high-Q cavities, frequency
detuning between the laser and the high-Q cavities, the pump coupling
efficiency, the optical path length between the laser and the microresonator.
Our calculations show that the laser linewidth can be improved by two orders of
magnitude compared with the case of not optimal self-injection locking. We
present recommendations on the experimental realization of the optimal
self-injection locking regime. The theoretical model provides deeper
understanding of the self-injection locking and benefits multiple practical
applications of self-injection locked oscillators.
