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.

Figure 1

Vernier frequency locking (VFL) of CP solitons. (a) Frequency locking of CP solitons with different repetition rates can create a pair of Vernier-like microcombs. The coupling of CP solitons can insert two types of sidebands spaced by $\mathrm{\Delta }{f}_r$ (type 1) and $\mathrm{\Delta }{\nu }_P$ (type 2). If the backscattered field is an isolated pulse, solitons are kicked into VFL when the soliton overlaps with the backscattered pulse instead of via continuous interaction as in the single-line coupling case. (b) Simulated subtracted relative delay (SRD) of the output CP solitons under different conditions. In the absence of backscattering, two solitons propagate independently with SRD being a straight line with a nonzero slope (cyan curve, color online). If only a single line is backscattered, SRD transitions smoothly into a constant delay (yellow curve). When assuming an isolated pulse is backscattered by a point scatterer, SRD has periodic jumps at a rate of $\mathrm{\Delta }{f}_r$ (blue curve). The inset shows the trajectory of the soliton and backscattered pulse; the delay jumps occur when these two pulses overlap. When assuming the backscattered soliton is stretched and distorted by adding random phases for comb lines in backscattering, CP solitons also have oscillatory motion at the rate of $\mathrm{\Delta }{f}_r$ (violet curve). (c) Relative spectral center frequency between CP solitons in VFL when assuming an isolated backscattered pulse. The relative frequency features periodic oscillations and jumps at rates of $\mathrm{\Delta }{\nu }_P$ and $\mathrm{\Delta }{f}_r$, respectively. The top panel shows the relative frequency oscillates around a frequency different from the relative frequency when backscattering the $n\mathrm{th}$ line only (yellow curve).

Figure 2

Fine structures of simulated comb lines in a VFL state. (a) Spectrum of the pump mode in a VFL state assuming a point scatterer, exhibiting sidebands spaced by $\mathrm{\Delta }{f}_r$. The right inset is an enlargement of the shaded region, while the left inset shows the enlargement of the same frequency span in an unlocked state. (b) Spectrum of the $n\mathrm{th}$ mode in the VFL state also exhibits multiple sidebands. The extinction ratio between the main tone and the sidebands is about 28 dB. The right inset is an enlargement of the shaded region, and the left inset plots the spectrum of the $n\mathrm{th}$ line when assuming multiple backscatterers, showing similar extinction ratio. (c) Extinction ratio between comb line and sidebands versus mode number in point backscatterer and multiple backscatterer cases. The extinction ratios decrease with $|\mu |$ and are close for the multiple backscatterer cases and the point backscatterer case.

Figure 3

Theoretical and simulated phase-locking dynamics in VFL. (a) The Lugiato-Lefever equation simulated (solid) and the Adler equation (dashed) predicted relative phase change for the $n\mathrm{th}$ line pair when considering the $n\mathrm{th}$ line only in the VFL. Different curves stand for phase locking starting from different initial phase difference (initial pulse delay). (b) When considering the interaction of the whole pulse in VFL, the simulated relative phase for the $n\mathrm{th}$ line pair shows periodic jumps. These jumps can be modeled by considering all the comb lines in VFL using Eqs. (3) and (4). The inset shows phase locking can be reached when excluding the $n\mathrm{th}$ line pair, but locking takes a much longer time to be established.

Figure 4

Locking bandwidth in VFL. The locking bandwidth increases linearly with the backscattering rate. The corresponding slope is close to the theoretical locking bandwidth when assuming phase locking via single-line injection locking (yellow line). Backscattering all comb-line pairs except the $n\mathrm{th}$ provides a small locking bandwidth (cyan dots). The inset shows the simulated upper and lower bounds of the locking zone assuming interaction via the whole pulse, while the solid lines are the theoretical bounds for the single-line locking case.