Self-Injection Locking and Phase-Locked States in Microresonator-Based Optical Frequency Combs

Microresonator-based optical frequency combs have been a topic of extensive research during the last few years. Several theoretical models for the comb generation have been proposed; however, they do not comprehensively address experimental results that show a variety of independent comb generation mechanisms. Here, we present frequency-domain experiments that illuminate the transition of microcombs into phase-locked states, which show characteristics of injection locking between ensembles of comb modes. In addition, we demonstrate the existence of equidistant optical frequency combs that are phase stable but have nondeterministic phase relationships between individual comb modes.

Figure 1

(a) Optical spectra of microcombs without (1) and with (2) overlap. The amount of overlap can be controlled by changing the detuning of the pump laser with respect to the resonator mode. (b) Offset in a bunched microcomb. The dashed lines depict the position of equidistant comb modes. (c) Experimental setup to analyze and test self-injection-locking and phase-locking behavior of microcombs. $\mathrm{OSA}=\text{optical}\text{spectrum}\text{analyzer}$, $\mathrm{PLL}=\text{phase}\text{−}\text{locked}\text{loop}$, and $\mathrm{PD}=\text{photodiode}$. (d) Scheme to determine whether the frequency of a comb mode is commensurate with that of a uniformly spaced comb.

Figure 2

Injection locking of bunched comb states. (a) Measured offset of the 37th sideband of a bunched microcomb (blue crosses) showing a characteristic injection-locking behavior. The red line is a fit based on the Adler equation. The green line going through the origin of the graph is obtained from the fit and shows how the offset would tune without injection locking present. Panel (b) shows the injection-locking behavior for different comb states with different overlap between bunches. Trace (1) shows a comb state without overlap between bunches, which shows no injection locking. Trace (2) shows a state with strong overlap that leads to injection locking. Corresponding optical spectra are shown in Fig. 1. Panel (c) shows the injection-locking range as function of the microcomb spacing (tuned via launched power).

Figure 3

(a) Optical spectrum of a phase-locked microresonator comb. (b) Measurement of the offset of the $+98\mathrm{th}$ comb sideband in a phase-locked state. The mean value of the measurement is $740\mu \mathrm{Hz}\pm 3\mathrm{mHz}$. (c) Distribution of the data in panel (b). (d) Allan deviation (not normalized) of the offset of the $+37\mathrm{th}$ and $+98\mathrm{th}$ comb sideband. (e) Tuning of the 37th, 61st, and 98th sideband versus the measured comb spacing.

Figure 4

Phase measurement of microresonator modes via optical autocorrelation. (a) Experimental setup. (b),(c) Optical spectrum and autocorrelation without amplitude and phase adjustments. (d),(e) Spectrum and autocorrelation after phase adjustment. (f) Second-order dispersion of the phases of three different phase-locked states. The phases of the comb modes in all three states appear to be random and unique but are stable in time. (g),(h) Mode spacing beat note of a phase-optimized microcomb at different scales.