Fig. 2. Theoretical model of the nonlinear self-injection locking.

Model parameters for a–f: the normalized pump amplitude f = 1.6, the normalized mode-coupling parameter Γ = 0.11, the self-injection locking coefficient K₀ = 44, the locking phase ψ₀ = 0.1π. a, b Model for the conventional case where the microresonator is pumped by the laser with an optical isolator, and ζ = ξ (dark green line). c–f Model for the self-injection locked regime. The solution of Eq. (6)–(8) (thin red curves in c, e is compared with the linear tuning curve ζ = ξ (thin black lines in c, e). While tuning the laser, the actual effective detuning ζ and the intracavity power ∣a(ξ)∣² will follow red or blue lines with jumps due to the multistability of the tuning curve. The triangular nonlinear resonance curve (thick black in b) is deformed when translated from ζ frame to the detuning ξ frame (d, f) with corresponding tuning curve ζ(ξ) (c, e). The width of the locked state is larger for forward scan, but the backward scan can provide larger detuning ζ, which is crucial for the soliton generation.