Revealing the behavior of soliton build-up in a mode-locked laser

Real-time spectroscopy based on an emerging time-stretch technique can map the spectral information of optical waves into the time domain, opening several fascinating explorations of nonlinear dynamics in mode-locked lasers. However, the self-starting process of mode-locked lasers is quite sensitive to the environmental perturbation, which causes the transient behavior of laser to deviate from the true build-up process of solitons. Here, we optimize the laser system to improve its stability that suppresses the Q-switched lasing induced by the environmental perturbation. We therefore demonstrate the first observation of the entire build-up process of solitons in a mode-locked laser, revealing two possible ways to generate the temporal solitons. One way includes the dynamics of raised relaxation oscillation, quasi mode-locking stage, spectral beating behavior, and finally the stable single-soliton mode-locking. The other way contains, however, an extra transient bound-state stage before the final single-pulse mode-locking operation.


I. INTRODUCTION
Transient phenomena and dynamics are the important characteristics of numerous nonlinear systems [1]- [7]. Solitons are the localized formations in nonlinear systems [8]- [15], appearing in many nonlinear processes from fluid and biology dynamics, plasma physics to fiber lasers [16]- [19], especially in mode-locked lasers [5][11] [20]. The mode-locked fiber laser, as an ideal platform, is used for exploring new nonlinear phenomena due to its compact and low-cost configuration as well as excellent features of high stability and low noise. Solitons observed in mode-locked lasers exhibit several special behaviors such as soliton bunching and soliton bounding (i.e., the generation of soliton molecules) [21]- [23].
In the stationary state, the soliton train generated from mode-locked lasers can be described theoretically by means of the generalized nonlinear Schrödinger equation (NSE) [17] or Ginzburg-Landau equation (GLE) [11]. Despite the ultimate stability of the mode-locked pulse train, its initial self-starting process contains a rich variety of unstable phenomena which are highly stochastic and non-repetitive [24]- [28], and the theoretical modeling of these dynamical processes is beyond the NSE and GLE. While conventional technologies cannot generally measure these rapid non-repetitive processes due to the limited measurement bandwidth [1], the recently-developed time-stretch dispersive Fourier transform (TS-DFT) technique can provide an elegant way for real-time, single-shot measurements of ultrafast optical phenomena [29]- [34]. This technique helps scientists to experimentally resolve the evolution of femtosecond soliton molecules [30], the internal motion of dissipative optical soliton molecules [17] [35], and the dynamics of soliton explosions [36]- [38].
Some successful examples of using this TS-DFT technique include the measurements of rogue wave dynamics, modulation instability, and supercontinuum generation [28][32] [33].
The starting dynamics of passive mode-locking lasers has been established by a large set of experimental and theoretical investigations over the past two decades [39]- [48]. Without the emerging TS-DFT technique, the transient dynamics in mode-locked lasers had been investigated by using an oscilloscope [24] [26]. The real-time spectral evolution in the build-up process cannot be resolved, however, when no TS-DFT is used [1]. Recently, both the transient spectral and temporal dynamics were observed with the assistance of the TS-DFT technique [34] [35]. The environmental perturbation (e.g., the polarization change in laser cavity and the fluctuation of pumping strength) can cause the lasers to sustain the extra unstable stages, such as the Q-switched lasing [34] [35]. To reveal the true build-up process of solitons, we must mitigate the environmental perturbation as far as possible. This issue is not overcome yet so far, unfortunately, although the self-starting process of mode-locked lasers had been demonstrated in the previous reports [24] [26][34] [35] [49].
In this paper, the Q-switched lasing is availably suppressed by decreasing the environmental perturbation and optimizing the laser system with the assistance of polarization-insensitive carbon nanotube saturable absorber (CNT-SA). We therefore observe the entire build-up process of solitons in mode-locked lasers, for the first time to our best knowledge, in which there exist two possible ways to generate the laser solitons. One way comprises of the raised relaxation oscillation, quasi mode-locking (Q-ML) stage, spectral beating dynamics, and stable mode-locking. The other way includes, however, an extra transient bound-state stage prior to the final mode-locking operation. The numerical simulations based on the roundtrip circulating-pulse method confirm the experimental observations. Moreover, the theoretical modeling is proposed to investigate the raised relaxation oscillation during the build-up process of laser pulses. The build-up time for the birth of solitons is predicted, which agrees well with the experimental measurement. The entire build-up process can be theoretically described by means of the two-step method, as shown in Appendixes B and C. These findings highlight the importance of real-time measurements and provide some new perspectives into the ultrafast transient dynamics of nonlinear systems.

A. Build-up dynamics of solitons with beating dynamics
The time stretch technique can overcome the speed limitations of electronic digitizers [50]. The spectral interferograms are mapped into the time domain by using the TS-DFT technique. The timing data are measured via the direct detection [ Fig. 1(a)], whereas, the real-time spectra are obtained by dispersing the laser pulses over a 5 km length of dispersion-compensating fiber (DCF) prior to detection [ Fig. 1(b)]. The pulse-resolved acquisition and real-time spectral acquisition are measured simultaneously (see experimental setup in Appendix A). Comparing to the direct detection data of the laser output, its real-time spectral data is delayed in time by ~24.5 s due to the additional 5 km DCF.
The direct measurement and TS-DFT data are plotted in Figs. 1(a) and 1(b), respectively. While the direct detection gives a temporal trace of the laser output as shown in panel (I), the TS-DFT technique provides the single-shot measurement of the laser spectral information as shown in panel (II). The experimental observations reveal that there exists an evidently raised relaxation oscillation before the appearance of a stable pulse train (i.e., stable mode-locking). A representative real-time measurement is exhibited in Fig. 1(b), where the duration of the raised relaxation oscillation is ~ 4.6 ms, corresponding to ~ 1.210 5 cavity roundtrips. Before 4.32 ms, the number of cavity photons remains at the initial low value, as determined by quantum field fluctuations [51]. From this time on, the first laser spike is generated. The separation of the neighboring laser spikes is ~ 80 s. The build-up time of solitons from the beginning time of pumping process to the stable mode-locking is ~ 4.65 ms. Note that, in the experiments, we only need to measure the build-up period from the first laser spike to the stable mode-locking and, thus, we define the duration of this process as the nascent time of the laser (here the nascent time is ~0.35 ms).  Usually, the build-up process of mode-locked lasers includes the raised relaxation oscillation, Q-ML stage, beating dynamics, and finally the stable mode-locking state, as shown in Fig. 1. The experimental results show that the laser spikes are generated gradually. Figure 1(c) shows the expanded views of the 5th laser spike in Fig. 1(b). Figure 1 The time-continuous data stream shown in Fig. 1(b) evolves with a periodicity of 38.03 ns that corresponds to the cavity roundtrip time. We therefore segment the time series into intervals of 38.03 ns, achieving a two-dimensional (matrix) representation exhibited in Fig. 2(a). Here, the vertical and horizontal axes depict the information within a single roundtrip and the dynamics across consecutive roundtrips, respectively. The two-dimensional representation (also named as spatio-temporal representation) had been proposed to describe the nonlinear system with delayed feedback [52], and the analogous techniques had been used in several optical systems, such as mode-locked fiber laser [53], cavity soliton in fiber-ring resonator [54], and injection-locked semiconductor laser [55].

B. Build-up dynamics of solitons with transient bound state
Figures 1 and 2 show that the beating dynamics occurs prior to the stable mode-locking. Some of our experimental observations demonstrate, however, that a transient bound state of two solitons may appear between the two stages. A typical example is illustrated in Fig. 3 and the corresponding animation is included in Supplemental Material [61] for details. The real-time TS-DFT measurement of this unique build-up process is shown entirely in Fig. 3(a), which includes the raised relaxation oscillation, the transition region, and the final stationary single-pulse mode-locking state. The unique transition region shown in Fig. 3(a) has a duration ~3 times longer than that in Fig. 1 which is quite similar to panel (II) in Fig. 1, is the expanded view at the stable mode-locking in Fig.   3(a). Figure 3(c) is an enlarged plot that demonstrates the evolution of optical wave during this transition region in detail. The interferograms show the periodical modulation along wavelength, which is the typical result of bound-state spectrum [17] [30]. Figure 3(e) demonstrates the Fourier transform of each single-shot spectrum of the transient bound state. Obviously, the corresponding field autocorrelation with three peaks exhibits the evolution of bound state with two pulses [30].
We set the beginning time of the stationary single-pulse mode-locking state as 0 roundtrip number for the convenience of reference, as shown in Fig. 3(c). For example, the roundtrip number of -800 corresponds to the recording time 800 roundtrips prior to the stable mode-locking state.
Here  0 is the common carrier frequency.
where a 1 and a 2 are the amplitudes of the two solitons in the transient bound state, respectively, and t 1 and t 2 are the corresponding pulse widths. After the manipulation of Fourier transform, the spectral intensity I() of the optical field is given by On the basis of the experimental data shown in Fig. 3, a 1 , a 2 Fig. 3(e).

A. Build-up process of mode-locked lasers with Q-switched lasing
Kerr-lens mode-locked laser is able to deliver the pulses and solitons [1] [30], the build-up time of lasers is more than 10 ms [1] and, even, more than 100 ms [35].
We establish a fiber laser mode-locked by NPR technique, in which the polarization change and the fluctuation of pumping strength can evidently influence the self-starting process. Figure 5(c) shows a typical build-up process of mode-locked laser. Obviously, the Q-switched lasing occurs prior to the stable mode-locking. The number of Q-switched lasing is 189 with the duration of more than 160 ms. This result is very similar to the experimental observation reported in [35].
We improve the stability of pumping strength and decrease the environmental perturbation. The build-up time of laser is shortened to be about 80 ms with 76 lines of Q-switched lasing, as shown in Fig. 5(b). We further enhance the robustness of mode-locked laser by using the hybrid saturable absorber where the single-wall carbon nanotubes together with NPR technique operate on the laser.
The experimental result is demonstrated in Fig. 5(a), where the duration of Q-switched lasing is ~3 ms and its number is 5. Every Q-switched lasing is composed of several hundreds of pulses, as shown in the left inset of Fig. 5(a).
It can be seen that, by comparing Fig. 5(a) to Figs. 5(b) and 5(c), the decrease of environmental perturbation can effectively shorten the build-up time of laser and suppress the Q-switched lasing. Fig. 5(a) shows that, therefore, the laser experiences a shorter unstable Q-switched lasing stage. Note that, within Fig. 5(a), the separation between the neighboring Q-switched lasing lines is ~700 μs.
However, the separation between the laser spikes in the raised relaxation oscillation is ~80 μs, as shown in Fig. 1(b).
When the polarization-dependent devices are removed from the laser cavity, only single-wall carbon nanotube serves as the saturable absorber. Simultaneously, we optimize the laser system and the saturable absorber. As a result, the Q-switched lasing is suppressed completely and, then, no Q-switched lasing occurs in the self-starting process of mode-locked laser. The raised relaxation oscillation is observed experimentally, as shown in Fig. 6(a) below. Therefore, the experimental results here denote the general build-up process, which can reflect the intrinsic features of mode-locked lasers.  The relaxation oscillation is a general transient behavior of laser [51]. The laser without a modelocker undergoes a damped oscillatory behavior and, finally, approaches to the stable continuous-wave (CW) operation [63]. In contrast, the modelocker-based laser experiences a raised relaxation oscillation rather than a damped behavior, as shown in Fig. 6 be from randomized to synchronized, where the phase difference  between any two neighboring modes is gradually locked to a constant value. During the raised relaxation oscillation,  is a variable rather than a constant. We therefore define the former (i.e., a variable for ) as Q-ML and the latter (i.e., a constant for ) as perfect or stable mode-locking. The transition stage from Q-ML to stable mode-locking undergoes a beating dynamics [see Fig. 2 Fig. 6(a). The first laser spike occurs at ~4.3 ms and the period of raised relaxation oscillation is ~4.6 ms. Both the theoretical prediction and experimental observation show that, although the pump power is inputted into the laser system after the turn-on time, the number of cavity photons remains at the initial low value before the first laser spike (i.e., ~4.3 ms). Therefore, no pulse appears before ~4.3 ms, as shown in Fig. 6.

C. Theoretical confirmation
The proposed modeling in Appendix B successfully simulates the relaxation oscillation process, but it cannot describe the dynamics and evolution of pulses in the stationary mode-locking. Based on the extended NSE and the roundtrip circulating-pulse method (see Appendix C), we can numerically achieve the two different evolution processes of soliton build-up. Figure 7 demonstrates two typical results of numerical simulations, revealing two different evolution ways of the soliton build-up. In one way, the single-soliton operation is directly formed via the beating dynamics, as shown in Figs.

7(a) and 7(b). Figures 7(a) and 7(c) exhibit the spectral and temporal evolutions of pulses,
respectively. The simulation starts at an initial signal with the noise background that is illustrated as the red curve in Fig. 7(g). Obviously, the numerical results are in good agreement with the experimental observations shown in Fig. 2. When the simulation starts with another noise background field [see the black curve in Fig. 7(g)], the transient bound state can be observed in the build-up process, as shown in Fig. 7(d) (i.e., spectral evolution) and Fig. 7(f) (i.e., temporal evolution). In this way, the transient bound state with two solitons is generated at the beginning and, however, the stronger soliton gradually evolves to the stationary state and the weaker one slowly decays and ultimately vanishes via the complex dynamics. The temporal evolution of two pulses in the numerical simulation [i.e., Fig. 7(f)] is quite similar to the experimental results (i.e., Fig. 4). It is discovered that there are two ways for the build-up process of solitons in mode-locked lasers.
The entire build-up process usually includes the raised relaxation oscillation, Q-ML stage, and beating dynamics in one way (see Fig. 1 [65]. Such real-time spectroscopy technique is expected to provide new insight into a wider class of phenomena in complex nonlinear systems [30]. For more application prospects of our work, please see Supplemental Material [64].

ACKNOWLEDGMENTS
We thank X. Yao, X. Han, G. Chen, W. Li, G. Wang, and Y. Zhang for fruitful discussions.

APPENDIX B: RELAXATION OSCILLATIONS FOR MODE-LOCKED LASER
The Kerr-lens lasers cannot usually self-start [1]. In contrast, the laser here is mode-locked by the single-wall carbon nanotubes that serve as an excellent saturable absorber (i.e., modelocker) to self-start operation. The experimental setup is described in above in detail.
where  0 , 0  , and I sat are the linear limit of saturable absorption, nonsaturable loss, and saturation intensity, respectively. I represents the input optical intensity which is related to the photon number q in the laser cavity. The relationship is given as I=qchv/L, where h is Planck constant [51].
For deriving Eq. (B1), we assume that the laser is oscillating on only one cavity mode [51]. In the mode-locking regime, actually, the laser with the cavity length of ~ 7.8 m generates pulses with the spectral bandwidth of ∼8.2 nm, corresponding to about 40000 locked modes. The stable mode-locking stage occurs when phases of various longitudinal modes are synchronized such that the phase difference between any two neighboring modes is locked to a constant value [62]. Usually, pulses during the build-up process would not satisfy the ideal condition [67]. Therefore, the deviations of the phase from the ideal values had been proposed by introducing a fluctuating background [68]. During the self-starting process of the pulse laser, the modelocker can gradually drive thousands of modes from absolutely random state to the stable mode-locking state that these cavity modes have regular phase relationships. The total optical field in the laser cavity can be expressed as [69] ( where A m ,  m , and  m are the amplitude, phase, and frequency of a specific mode among 2M+1 modes, respectively. The phase difference between any two neighboring modes is defined as When the modelocker is present in the laser cavity, the laser parameters are employed in our simulations, including R p =3.7510 16  describe the evolution and interaction of solitons in the stable mode-locking stage. We use a roundtrip circulating-pulse method to simulate the behaviors of beating and stable mode-locking stages [58]. The modeling includes the Kerr effect, the group velocity dispersion of fiber, the saturable absorption of modelocker, and the saturated gain with a finite bandwidth. When the pulses encounter cavity components, we take into account their effects by multiplying the optical field by the transfer matrix of a particular component. The simulation for the roundtrip circulating-pulse method starts with an arbitrary light field with the noise background [e.g., Fig. 7(g)]. After one roundtrip circulation in the laser cavity, the obtained results are used as the input of the next round of calculation until the light field becomes self-consistent. The simulation will approach to a stable solution, which corresponds to a stable laser under certain operation condition. When the optical pulses propagate through the fiber, the extended NSE is used to simulate the dynamics and evolution of the pulses, i.e., [58] 2 2 2 2 2 2 2 2 2 . 2 Here A, β 2 , and γ represent the electric filed envelop of the pulse, the fiber dispersion, and the cubic refractive nonlinearity of the fiber, respectively. The variables t and z are the time and the propagation distance, respectively. When the pulses propagate along the SMF, the first and last terms on the right-hand side of Eq. (C1) are ignored. Ω g denotes the bandwidth of the gain spectrum.
g describes the gain function for the EDF and is given by [58] [70] 0 exp( / ), where g 0 , E p , and E s are the small-signal gain coefficient related to the doping concentration, the pulse energy, and the gain saturation energy that relies on pump power, respectively. To match the experimental conditions, we use the following parameters: g 0 =6 dB/m, Ω g =25 nm, E s =83 pJ, γ=1.8 W −1 km −1 for EDF, and γ=1 W −1 km −1 for SMF. Eq. (C1) is solved with a predictor-corrector split-step Fourier method [71].