Dissipative Kerr cavity solitons (CSs) are self-enforcing, localized structures formed in coherently driven nonlinear cavities^[1], and they have inspired frontier applications, such as coherent optical communication^[2], astronomical spectrometer calibration^[3], low-noise microwave generation^[4], optical atomic clocks^[5], dual-comb spectroscopy^[6], and optical ranging^[7–9]. As a subset of dissipative solitons, the sustenance of CSs relies on twofold balances, i.e., the dispersion and Kerr nonlinearity, as well as the cavity loss and energy extracted from the external driving pump^[10,11]. Regarding the balances involving multiple parameters, the generation of CSs is closely associated with cavity detuning^[12,13], for which reason they have been unveiled with a rich of nonlinear dynamics, including dark pulses^[14], soliton Cherenkov radiation^[15], Raman self-frequency shift^[16], Stokes solitons^[17], Raman solitons^[18,19], soliton crystallization^[20], breathing solitons^[11], collapsed snaking^[21], and symmetry breaking^[22,23].

As a result, the underlying mechanism of these nonlinear dynamics is important for probing the comprehensive physical picture of the CS and understanding the complexity of how it transits between versatile nonlinear phenomena. To this end, it is interesting to observe the transient dynamics occurring in the vicinity of the instability regime in real time, which can intuitively manifest transient information between versatile nonlinear phenomena, such as modulation instability (MI)^[24–28], Hopf bifurcation^[29], and spatiotemporal chaos^[30]. Since most of the dynamical processes are fast-evolving and nonrepeatable, real-time observation has long been technically challenging, although significant progress has been made to single-shot technologies, e.g.,  temporal lensing^[31–34], dispersive Fourier transform (DFT, also named time-stretching)^[35-37], and high-speed optical sampling^[38]. Great efforts have been dedicated to investigating the CS phenomena, through optical buffer^[39], pulse picking^[40], and time averaging^[41]. However, enhancing understanding of the CS is still largely an unmet need.

In this work, we observe diverse CS dynamics in a passive nonlinear fiber ring cavity by leveraging a real-time DFT-based spectroscope. Being a macroscopic platform for visiting dissipative Kerr CS dynamics, the fiber resonator with a long round-trip (RT) time of $\sim 260.5\mathrm{ns}$ permits DFT spectroscopy using a large dispersion amount up to $-10.2\mathrm{ns}/\mathrm{nm}$, corresponding to a sampling spectral resolution of $\sim 1\mathrm{pm}$. Empowered by this measurement technology, a complete birth-to-annihilation process of the CS is visualized, as cavity detuning naturally sweeps in the perturbed fiber cavity specifically locked around the boundary of MI and bistable regimes. A variety of dynamics are captured, such as MI, soliton breathing, stationary CS, and its annihilation. Especially, the drift and decay dynamics of the CS are found to be associated with the collision between the solitons, evident by visible energy drop accompanied by CS breathing. Furthermore, the spectral interferograms of multiple CSs are further studied by means of field autocorrelation analysis, thereby unveiling their decay dynamics. These efforts not only demonstrate the potential of DFT-based real-time spectroscopy by revealing dynamics existing in coherently pumped passive Kerr fiber cavity but also enhance the understanding of CSs’ generation mechanism and their micro-behaviors.

Figure 1 shows the schematic diagram of the experimental setup that mainly includes a Kerr fiber cavity and a DFT measurement system used for real-time spectroscopy of the CS dynamics. The Kerr fiber cavity has a ring configuration that is made up of about a $\sim 53\mathrm{m}$ single-mode fiber (SMF), corresponding to an RT time of $\sim 260.5\mathrm{ns}$ and a free spectral range (FSR) of $\sim 3.84\mathrm{MHz}$. Two fiber optical couplers (OCs, 95/5) are inserted into the Kerr fiber cavity for different purposes. One is used to couple the pump and control (Ctrl.) beams into the cavity, while the other is employed to extract a small proportion of both clockwise- and counterclockwise-propagating signals for characterization. A polarization controller (${\mathrm{PC}}_3$) is placed inside the cavity to control the state of polarization of the bidirectionally propagating fields, which can also, to some extent, adjust the linear cavity detuning^[42]. The finesse of the cavity is estimated to be $\sim 39$ by measuring the resonance width (i.e., $\sim 98\mathrm{kHz}$; see measured resonance below). The coherent pump laser is a continuous-wave (CW) laser (NKT Photonics, KOHERAS BASIK E15) that has a linewidth of $<0.1\mathrm{kHz}$. The linewidth of the pump laser is almost 3 orders of magnitude narrower than the resonance linewidth of the cavity. The center wavelength of the pump laser is 1535 nm, with a wavelength tunable range of 1 nm through piezo tuning, which largely facilitates the subsequent laser scanning and stabilization of the Kerr fiber cavity. To flexibly control the parameter of the pump beam, the CW laser is split into two parts that, respectively, serve as coherent pump and control beams by a 90/10 OC (${\mathrm{OC}}_1$). The control beam is employed to determine and manipulate the cavity detuning^[43,44].

To investigate the CS dynamics under the pump control, the pump of the Kerr fiber cavity is intensity-modulated through an electro-optic modulator (EOM), and a flat-top pulse pump with a duration of $\sim 2\mathrm{ns}$ is generated. A PC (${\mathrm{PC}}_2$) is placed before the EOM to maximize the modulation depth, such that the CW background can be largely suppressed. The electric signal driving the EOM is provided by an arbitrary waveform generator (AWG, Tektronix AWG70001A). In the meantime, an external clock signal generated by an RF signal generator is employed to trigger the AWG so that the repetition rate of the pump pulse can better match the RT time of the Kerr fiber cavity. The average power of the generated 2-ns-pulse pump is amplified by a high-power Er-doped fiber amplifier (HP-EDFA, saturation power of 4.9 W), and subsequently cleaned up by a bandpass filter (BPF) before it is finally launched into the Kerr fiber cavity through ${\mathrm{OC}}_2$. Please note that, as the driving pump is provided by a narrow-linewidth laser that easily excites backward-propagating stimulated Brillouin scattering (SBS), here the scheme of using a pulse pump can dramatically decrease the possibility of optical damage by reducing the demanding average power.

The control beam is mainly employed to monitor and adjust the cavity detuning. An acousto-optic modulator (AOM) is placed in the optical path of the control beam to shift its frequency, such that we can arbitrarily vary the cavity detuning by altering the carrier frequency of the RF signal (i.e., a sinusoidal waveform). Before injecting into the Kerr fiber cavity, the control beam is also amplified to about 18 mW through an EDFA with a saturation power of 43 mW and subsequently passes through a BPF to filter out the amplified spontaneous emission (ASE) noise. To assure a stable cavity detuning, 5% of the intracavity control signal is extracted from the Kerr fiber cavity through ${\mathrm{OC}}_3$, which is then directed toward a photodetector (${\mathrm{PD}}_1$, Newport 818-BB-51F, 12.5 GHz bandwidth) and a proportional-integral-derivative (PID) controller (Stanford Research Systems SIM960) to generate a feedback signal for the CW laser. As a result, the cavity detuning can be stabilized by locking the level of the output to a setpoint designated by the PID controller. Here, two points are noteworthy: 1) the control beam must operate in a linear regime for stabilizing the cavity detuning; 2) the narrow bandwidth of the PID controller imparts a limited refresh rate for the piezo tuning of the CW laser. It is interesting to point out that, the noninstantaneous response of the PID controller can naturally scan the cavity detuning, which will be elaborated in more detail in the following.

To visualize the evolutionary behavior of the CS in real time, a DFT spectroscopy system composed primarily of five dispersion compensating modules (${\mathrm{DCM}}_{1–5}$, OFS DCM-120-C, about $-2\mathrm{ns}/\mathrm{nm}$ for each, a total dispersion of about $-10.2\mathrm{ns}/\mathrm{nm}$) and four C-band EDFAs (i.e., ${\mathrm{EDFA}}_{2–5}$). After the DCMs, the intracavity signal with a broad spectrum is largely stretched, such that the spectral information can be mapped onto the temporal pulse waveform, which is typically tens of nanoseconds in pulse width and can directly be read out through temporal sampling. To verify the large time stretching, the MI signal generated by this Kerr fiber cavity is injected into the DFT system. The time-stretched MI signal, displayed below the DFT system diagram on the bottom panel of Fig. 1, covers about 70% of the pump pulse period, showing an effectively large time stretching as well as the ability of real-time spectral detection. The data acquisition is implemented with a combination of high-speed ${\mathrm{PD}}_2$ (Newport 818-BB-51F, 12.5 GHz bandwidth) and a real-time oscilloscope (Teledyne LeCroy SDA 820Zi-A, 20 GHz bandwidth). It should be pointed out that, before feeding ${\mathrm{PD}}_2$, the extracted signal passes through a notch filter (NF, 0.8-dB band-stop bandwidth of $\sim 0.5\mathrm{nm}$) for removing the narrow-band driving pump and thus enhancing the signal-to-noise ratio (SNR) of the intracavity signal. Otherwise, the PD can be easily saturated by the residual pump, which will eventually prevent the visualization of the CS dynamics. In comparison with the typical configuration of the DFT system amplified by the distributed Raman gain^[45], the present system relying on lumped amplification with EDFAs has to carefully optimize the pump power of EDFAs to suppress the accumulated nonlinear effects.

Before determining the cavity detuning, we first visit a master diagram of the mean-field Lugiato–Lefever equation (LLE) in the parameter space (${\delta }_0$, $P_{\text{in}}$)^[46,47], as shown in Fig. 2(a). According to the analytical analysis (see the detailed description in Sec. S1, Supplementary Materials), we can identify the MI (hatched) and bistable (shaded) regimes, which are defined by the boundary line ($P_{\omega }={\delta }_0/2\gamma L$) that separates the domains with unstable and stable CW solutions in the bistable regime. Thanks to the analytical inspiration, we are able to investigate the bifurcation by manipulating the pump–cavity parameter in the vicinity of the boundary line, such that distinguishable dynamical behaviors are accessible. By tuning the frequency shift and simultaneously offsetting the resonance by $\sim 0.728\mathrm{rad}$, the cavity detuning ${\delta }_0$ is locked at around 0.824 rad for a certain setpoint level [Fig. 2(b)] and can be mapped to the point (${\delta }_0\approx 0.824$, $P_{\text{in}}\approx 6$) located close to the boundary line, as indicated in Fig. 2(a).

Subsequently, we scan the cavity detuning to access the dynamics of the CS. Instead of directly tuning the frequency of the CW laser, we here employ a PID stabilization scheme^[26], i.e., configuring the Kerr fiber cavity with a PID controller when the initial locking state is externally perturbed by an abrupt change of the cavity detuning. In this scenario, the PID controller’s noninstantaneous response to the environmental perturbation from the CW laser or the fiber cavity creates a natural sweep of the cavity detuning ${\delta }_0$, such that it produces a complete process of the birth-to-annihilation dynamics of the CS. As described in Fig. 2(c), an environmental perturbation that changes the cavity detuning ${\delta }_0$ can drive the operation state [denoted by ① in Figs. 2(a) and 2(c)] to the regime with the unstable homogeneous CW (i.e., upper branch solution) and admit the onset of MI responsible for a spontaneous excitation of the CS. Then, with the gradual increase of the cavity detuning ${\delta }_0$ by recovering to the locking level, the state (denoted by ②) begins to enter the regime with CS solution and is expected to exhibit oscillating behavior in the nascent stage because of a Hopf bifurcation^[46,48]. For a large enough cavity detuning, the state behaving with a limit cycle can no longer be sustained and turns to a chaotic transient state^[46], which triggers the annihilation dynamics of the CS (denoted by ③). To identify such a dynamical evolution from state ① to state ③, the numerical simulation is performed for the cavity detuning ${\delta }_0$ varying from 0.47 to 1.17 rad (see more details provided in Sec. S2, Supplementary Materials). Figure 2(d) shows the birth-to-annihilation dynamics of the CS, wherein the cavity experiences MI, breather, stationary CS, and the annihilation of the CS in sequence, which partially validates the proposed scheme of scanning the cavity tuning for exciting the CS’s dynamics.

In the experiment, we verify the birth-to-annihilation process of the CS by measuring the RT evolving optical spectrum of the intracavity signal under external perturbation. To facilitate a natural cavity detuning for ${\delta }_0\approx 0.824\mathrm{rad}$ and $P_{\mathrm{in}}\approx 6\mathrm{W}$, we tune the settings (i.e., a proportional gain of 1.4 and integral parameter of 100) of the PID controller slightly away from its optimal condition. By this means, the birth-to-annihilation process of the CS can be repeatedly excited, such that versatile spectral dynamics of the CS can be observed by the real-time spectroscopy measurement. Figure 3(a) shows the raw oscilloscopic train of the DFT signal, which intuitively reveals the evolution of the intracavity signal responding to the external perturbation that produces the scanning of the cavity detuning. After data processing (see the detailed description in Sec. S3, Supplementary Materials), the evolution landscape of the optical spectra of the intracavity signal is shown in Fig. 3(b), wherein the evolutionary characteristics are well consistent with the numerical prediction [top panel of Fig. 2(d)]. To be more specific, the external perturbation to the locked Kerr fiber cavity first initiates the MI, which results in a dramatic spectral broadening in the early stage, as a higher parametric gain is available in the undepleted pump approximation^[49]. After the MI evolution, in analogy with the numerical prediction, the MI triggers the generation of CSs, and the intracavity signal undergoes spectral breathing and subsequently tends to be stable, eventually giving rise to annihilation. The real-time spectroscopy manifests the typical spectrum features of the MI, CS, and its annihilation, as shown in Fig. 3(c).

The above numerical and experimental studies reveal the observation of breathing CSs—one of the interesting dynamics of CSs^[11,50]. To gain a deeper understanding of the underlying mechanism of the CS’s dynamics, we here explore more details of the breathing CS from both the experimental and numerical perspectives. The top panel of Fig. 4(a) shows the closeup of the spectral evolution of the breathing CS, extracted from Fig. 3(b) (i.e., from RT 25 to 65). As indicated by the corresponding integrated energy provided in the bottom panel of Fig. 4(a), there exists a visible drop of the energy over the breathing process of the CS. To probe the underlying physics, we perform numerical simulation and confirm an analogous spectral evolution with similar variation characteristics of the energy, as shown in Fig. 4(b). By carefully analyzing the numerical results over the complete parameter space, we find that the collision between the solitons and the subsequent annihilation of the CS can have caused energy reduction, as evident by Fig. 4(c). After the collision, the remaining soliton drifts and undergoes a quick stabilization of the relative motion, as shown in Fig. 4(d). Similar dynamics of the dissipative solitons have also been reported in Kerr microresonators^[41,51] and mode-locked lasers^[52–54].

In addition to breathing dynamics of the CS mediated with collision decay, we also access different birth-to-annihilation dynamics with the excitation of multiple CSs through unstable MI. Figure 5(a) shows a spectral evolution pattern that involves typical spectral fringes as a signature of bound solitons^[55]. There is a visible change of the modulated structure in frequency and accompanies a step-like variation of the energy during this process [inset of Fig. 5(a)]. The two different states of spectral interference, respectively portrayed by the optical spectra at RTs of 124 and 156 [right panel of Fig. 5(a)], imply that the multiple CSs are decomposed into another temporal pattern of CSs. For intuitive understanding, we perform a field autocorrelation analysis by applying the Fourier transform to the optical spectrum of each RT^[55]. As shown in Fig. 5(b), a decaying CS at a delay time of $\sim 40\mathrm{ps}$ is observed. Thus, the survived bound CSs with a much closer temporal separation give rise to a modulated spectrum with larger wavelength spacing. Such dynamics of multiple CSs decay can be well reproduced in the numerical simulation, as shown in Fig. 5(c).

To summarize, we have investigated the birth-to-annihilation dynamics of dissipative Kerr CSs by both real-time spectral observation using DFT spectroscopy technology and numerical modeling based on LLE. Empowered by the high-resolution real-time spectroscopy system, a complete birth-to-annihilation process of the CS covering a variety of dynamics was observed, like MI, soliton breathing, stationary CS, and its annihilation, as the cavity detuning naturally swept in the perturbed Kerr fiber cavity that was specifically stabilized at the boundary of the MI and bistable regimes. Especially, different mechanisms of CS’s annihilation have been revealed. One is caused by the collision between the solitons in the process of soliton breathing. The other one is natural decay, which has been investigated by means of field autocorrelation analysis based on the spectral interferograms of multiple CSs. Our findings of rich CS dynamics may shed new light on the understanding of the soliton localized structures in other physical systems that obey the LLE, such as the microresonators and ring quantum cascade lasers.