Vernier Frequency Locking in Counterpropagating Kerr Solitons

Interaction of counterpropagating (CP) solitons in a high-$Q$ microcavity can lead to phase locking of the two microcombs, endowing a Vernier-like comb pair with high mutual coherence. Here, we study numerically and analytically the soliton dynamics in this phase-locking state. Our simulations unveil that the relative group velocity of CP solitons will change periodically as opposed to being locked to a fixed value. CP solitons are kicked into the frequency-locking state instead of via continuous interaction when assuming a point backscatterer. The group-velocity modulation and periodic soliton motion induce sidebands around the comb modes. The simulated locking dynamics can be analytically modeled by a generalized Adler equation, which accounts for injection locking among multiple comb-line pairs. Our work elucidates the relevance of considering the frequency-locking dynamics when using CP solitons for dual-comb or time-of-flight-based applications.