Fig. 1. Schematic diagram of the experimental setup. The experimental setup mainly includes two parts, i.e., the Kerr cavity (top) and the DFT system (bottom). The Kerr cavity has a typical configuration, and it is coherently pumped. The DFT system has a total group delay dispersion (GDD) of ∼12642 ps², provided by five pieces of a dispersion compensating module (DCM), i.e., −10.2 ns/nm @ 1555 nm, corresponding to a sampling resolution of ∼1 pm by accounting for a digitizer with an acquisition rate of 80 GS/s. The inset of the bottom panel shows the performance of time stretching of the DFT system. CW, continuous-wave laser; OC, optical coupler; PC, polarization controller; AOM, acousto-optic modulator; EOM, electro-optic modulator; AWG, arbitrary waveform generator; PID, proportional-integral-derivative controller; EDFA, Er-doped fiber amplifier; HP-EDFA, high-power Er-doped fiber amplifier; BPF, bandpass filter; NF, notch filter; PD, photodetector; SMF, single-mode fiber; ISO, isolator; DCM, dispersion compensating module; MI, modulation instability.

Fig. 2. Illustration of birth-and-annihilation dynamics of the CS. (a) Different regimes of the mean-field LLE in the parameter space as functions of cavity detuning and injected pump power. At an injected power P_in of ∼6 W in this case, the cavity detuning (blue dot) is near the boundary between MI (hatched) and bistable (shaded) regimes, defined by P_ω = δ₀/2γL, i.e., the dashed red line. (b) Resonances with respect to the pump and control beams. The offset between the resonance peaks of the pump and control beams indicates a locked cavity detuning of 0.824 rad when the control beam is fixed at a setpoint level of ∼7 mV, marked by the dashed blue line in (a). The full width at half-maximum (FWHM) of the resonance is ∼0.16 rad, yielding a finesse of ∼39 (i.e., 2π over 0.16). (c) Natural sweeping of the cavity detuning in response to the perturbation. Three states, i.e., MI, CS, and its annihilation that exhibit different dynamics [marked by ①, ②, and ③, also designated in (a)], are investigated at different cavity detunings, i.e., δ₀ = 0.47, 1, 1.17 rad. (d) Simulated spectral (top) and temporal (bottom) evolutions of the intracavity field when the perturbation is applied. More details about numerical simulation are provided in Sec. S2, Supplementary Materials.

Fig. 3. Real-time measurement of the birth-to-annihilation process of the CS. (a), (b) Oscilloscopic trace of the birth-to-annihilation process recorded by the DFT system and the corresponding spectral evolution after data processing (see the detailed description in Sec. S3, Supplementary Materials). When the cavity is locked at a cavity detuning of ∼0.824 rad, its response to the external perturbation may cause a natural sweep of cavity detuning [illustrated in Fig. 2(c)], which produces a complete birth-to-annihilation process. (c) Typical optical spectra of the three different states over the birth-to-annihilation process, i.e., MI (left), CS (middle), and the annihilation of the CS (right), as indicated in (b).

Fig. 4. Breathing dynamics of the CS. (a) Spectral evolution (top) and corresponding normalized energy evolution (bottom) of the breathing CS in the experiment. The energy here is calculated by intensity integration for each RT. The black arrows indicate spectral breathing. (b) Simulated spectral and energy evolutions. (c) Corresponding temporal evolution in the simulation. (d) Snapshots of the temporal evolution, i.e., RTs of 19, 22, and 29. Two CSs collide and one soliton is eventually decayed. The key parameters used in the simulation are provided in Sec. S2, Supplementary Materials.

Fig. 5. Spectral-temporal dynamics of multiple CSs. (a) Spectral evolution of multiple CSs (left panel) and snapshots before and after CS decay (right panel). Inset is the energy variation from RTs 100 to 160. (b) Corresponding field autocorrelation evolution of the spectral evolution [left panel of (a)] from RTs 120 to 160. (c) Simulated temporal evolution of multiple CSs.