1. Introduction

Mode-locked fiber laser is one of the most common laser used in scientific research [1–4]. Thanks to its simple structure, high stability and robustness, mode-locked fiber lasers have seen continuous development in medical and industrial applications [4]. As a typical nonlinear optical system, mode-locked laser has been a great research platform for optical solitons [5, 6], a localized optical field balanced by dispersion and nonlinear effects. In a dissipative system particularly, solitons can display a much wider range of behaviors accounted for another balance between energy gain and loss [6,7]. However, some critical dynamics in the mode-locked fiber laser, such as the formation of soliton bound states, remained unobserved due to its transient nature. Traditional measuring methodology usually fails to probe non-repetitive phenomenon and causes problems for real-time detection thereafter.

Recently, the development of dispersive Fourier transform (DFT) or time-stretch technique makes it possible to trace the real-time spectral evolution of a transient event [8–10]. By using a dispersive medium, spectral information is transferred to the time domain. The signal is slowed down so it can be captured in real time by an electronic analog-to-digital converter [11]. Thus the whole system can serve as a real-time optical spectrum analyzer (OSA). This method has been used to resolve the establishment of mode locking in a Kerr-lens Ti: sapphire laser [12]. The same research group also uncovers the soliton molecule formation and different bound state vibrations in a Kerr-lens Ti: sapphire laser [6]. The DFT technique is also used to study nonlinear dynamics in fiber systems. Grelu’s group reveals different kinds of internal pulsations of dissipative optical soliton molecules in a fiber laser, including phase drifting and vibration like dynamics [13]. Coen’s group reveals the dynamics of soliton bound states under different perturbations and provides a framework to understand soliton bound states in fiber cavities [14]. This method is also applied in the observation of transient solitons, soliton explosion and other soliton dynamics with complex interactions [15–17].

In this paper, we successfully reveal the formation dynamics of soliton bound state in a mode-locked fiber laser by using DFT method. A transient Q-switched mode-locking regime is observed in this transition period. We find two types of soliton pairs (soliton bound state and two-soliton state) in our laser system and the different evolutionary paths are analyzed and discussed. To further complete our work, we also report and analyze the dynamics from Q-switched instabilities to the stable mode locking. Although the build-up dynamics of mode locking has been reported in recent works [18, 19], our results still provide valuable information for further theoretical studies. The main difference of soliton dynamics between fiber lasers and Ti: sapphire lasers is that fiber lasers has the Q-switched instability which results from the long upper-state lifetime of Er-doped fiber. Q-switched instabilities also exist in other solid state lasers with long upper-state lifetime, such as Nd:YAG, Nd:GdVO₄ and so on [20–24]. We believe our studies have reference value for the study on these lasers. Moreover, most theoretical studies of mode-locked lasers focus on the steady state solutions rather than the dynamic evolution. We believe our results provide useful reference for future theoretical studies on the dynamic phenomenon in nonlinear optical systems.

2. Results

The laser used in our experiment is a commercial PriTel femtosecond fiber laser with a 50-MHz repetition rate. The block diagram of the laser is shown in Fig. 1(i). It contains a ring cavity and all fibers used in the laser are polarization maintaining fibers (PMF). A laser diode (LD) at 980 nm is employed as the pump source. The gain medium is a 100-cm-long Er-doped fiber (EDF). A superlattice (InGaAs) semiconductor saturable absorber mirror (SESAM) is used as the saturable absorber, whose response time is a few picoseconds. Since the SESAM works in reflection, an optical circulator is required in the cavity. The output ratio of the optical coupler (OC) is 60% in our laser. An isolator (ISO) is embedded in the cavity to guide the propagation direction of the signal light, while the pump light propagates in the opposite direction. The central wavelength of the laser is tunable by adjusting the tunable filter (TF), which is an 8 nm bandpass filter cartridge. The laser can be easily mode-locked just by increasing the pump current over its threshold.

 

Fig. 1 (i) Experimental setup of DFT technique. The output pulse of the fiber laser is stretched by 10-km dispersion compensation fiber (DCF) and its spectrum is mapped into the temporal domain. The stretched pulse is acquired by a high-speed photodetector and monitored in a real-time oscilloscope. The mode-locked fiber laser used in our experiment is a commercial PriTel femtosecond fiber laser with a ring cavity. A 980 nm laser diode (LD) is served as the pump source. All fibers in the cavity are polarization maintaining fibers (PMF), including the 100-cm-long Er-doped fiber (EDF). Terms shown in the figure are abbreviated as follows: WDM: wavelength division multiplexer, TF: tunable filter, SESAM: semiconductor saturable absorber mirror, OC: optical coupler, ISO: isolator. (ii) The evolution of soliton dynamics with the increasing of pump power. (a) Quasi-cw Q-switched bursts emerge at low pump level and produce a noise-like fluctuation under a Q-switched envelope. (b) Increased pump current stimulates transient soliton molecules. They evolve from quasi-cw Q-switched bursts and fade out rapidly. (c) If the pump power is further increased, the transient soliton explosion will convert into a stable mode-locked pulse train. (d) The transition from a single soliton operation to a soliton molecule regime has to undergo a transient period of Q-switched instabilities.

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To reveal the ultra-fast dynamics of the fiber laser, we use the simple yet powerful technique called DFT to trace the real-time spectrum. DFT is a linear wavelength-to-time mapping achieved by a spool of fiber or other dispersive elements. This method makes it possible to analyze spectral information in time domain. Thanks to the use of a real-time oscilloscope, DFT technique breaks through the restrictions about updating rate of the traditional OSA. Thus the real-time spectral analyzing can become a reality. The experimental setup is shown in the right part of Fig. 1(i). A 10-km-long DCF with a dispersion value of 1486 ps/nm is employed to stretch the ultra-fast pulse. The stretched optical pulse is then fed into a high-speed photodector (PD) with a 18-GHz bandwidth and monitored b a real-time oscilloscope (Tektronix, MSO 71604C Mixed Signal Oscilloscope). The sampling rate of the oscilloscope is set to be 25 GS/s and its maximum record length is 250 Mpts. It means the maximum recording time is 10 ms at a 25-GS/s sampling rate. The spectral resolution in our experiment is determined by the dispersion value and the sampling rate of oscilloscope. For a 1486-ps/nm dispersion and a 25-GS/s sampling rate, the spectral resolution is about 3.36 GHz. Figure 1(ii) illustrates the schematic diagrams of soliton dynamics under different pump power. With the increasing of pump current, the laser undergoes quasi-cw Q-switching, transient soliton explosion and the stable mode locking successively. If the pump power is further increased, the stable pulse train will collapse, followed by a fleeting Q-switched mode-locking period. Eventually, a stable soliton molecule will evolve from the Q-switched instabilities.

2.1 Q-switched instability

The Q-switched instability is related to relaxation oscillations of the gain medium. For lasers with upper-state lifetime longer than the cavity damping time, changes in the pump power will lead to an evident oscillation of output power, which is the so-called relaxation oscillations. This phenomenon has been observed in most solid-state lasers [24, 25]. In general, relaxation oscillations can be damped, leading back to the stable state eventually. However, for passively mode-locked lasers with a saturable absorber inside the cavity, the damping is substantially reduced so that the oscillations become persistent [25]. This phenomenon is so-called Q-switched instabilities.

Evident Q-switched bursts are observed when the pump current is over 260 mA. Figure 2(a) shows the recorded Q-switched bursts with different pump currents ranging from 260 mA to 310 mA. Q-switched envelope within this current range is usually a Gaussian-like curve. As shown in Fig. 2(b), the Q-switched period (the repetition period of Q-switched bursts) reduces from ~120$\mu s$to ~60$\mu s$with the increasing of pump current. As mentioned above, the Q-switching phenomenon is caused by the relaxation oscillation of the gain medium. When higher pump power is injected into the cavity, relaxation oscillation frequency will increase, leading to a shorter Q-switched period. Moreover, the temporal width of Q-switched bursts also shrinks from ~25 $\mu s$to ~10$\mu s$(see red line in Fig. 2(b)). It is worth nothing that the peak values under different current have been normalized in Fig. 2(a). Actually the maximum power of Q-switched burst is higher under a larger pump current.

 

Fig. 2 Q-switched instabilities before the stable mode locking. (a) Q-switched bursts under relative low pump power usually possess Gaussian-like envelopes and fixed Q-switched period. (b) As the pump power increases, the Q-switched period (blue) and temporal width (red) of the Q-switched bursts reduce gradually. This can be explained by the increased relaxation oscillations frequency under high pump power. (c) Temporal evolution of one Q-switched burst under 260 mA without dispersion. Multiple pulses coexist in the cavity with an uneven power distribution. (d) Meanwhile, we record the same Q-switched burst via insertion of a 10-km DCF (dispersion value is ~1486 ps/nm). These pulses do not present any temporal stretch compared with the data set without dispersion, indicating that the bandwidth of these fluctuations is quite narrow. Moreover, their relative timing is not altered by the dispersion, implying that all these pulses origin from a single wavelength. (e) Q-switched bursts under pump currents from 320 mA to 350 mA. Double-burst with higher power and larger temporal width can be observed in this regime. (f) The real-time evolution of a typical double-burst measured by using DFT technique. One dominant pulse explodes in power and bandwidth after the depletion of background fluctuations. (g) The Fourier transform of the real-time spectrum represents the autocorrelation of its temporal waveform. The autocorrelation traces indicate that a four-pulse soliton is created in the Q-switched burst.

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If we focus on a specific Q-switched burst, we can observe many pulses within the envelope. Figure 3(c) shows the temporal evolution of a specific Q-switched burst measured without dispersion. As illustrated, multiple pulses with different intensity coexist over the cavity and keep their relative timing until they fade away. Actually, the bandwidth of these background pulses is quite narrow. They are difficult to be stretched by a DCF with a dispersion of 1486 ps/nm. It should be noticed that the relative timing of these pulses maintain unchanged by the dispersive element, indicating that all pulses originate from a single narrow bandwidth. We name this regime quasi-cw Q-switching. Franz X. Kartner et al. analyzed the parameter ranges for quasi-cw Q-switching, indicating that a solid-state laser with long upper-state life time tends to operate in quasi-cw Q-switching under low pump power [26].

 

Fig. 3 The build-up of stable mode locking in real time. All the results are recorded at ~360 mA. (a-c) Real-time spectral evolution from Q-switched instabilities to stable mode locking is resolved using DFT method. One seed pulse outbursts from a background fluctuations, with a suddenly broadening bandwidth and a rapidly-evolving fringe pattern. A stable mode locking is established gradually at the end of the Q-switched burst. (d) The blue line shows the power evolution of the background and dominant pulses in the mode-locking build-up event. The red line illustrates the 3dB bandwidth of the dominant fluctuation. (e) We trace the central wavelength variation of the dominant pulse and observe a spectral blue shift. The central wavelength of mode-locked pulses is ~0.7 nm shorter than that of the dominant pulse before mode locking.

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When we increase the pump current continuously, something interesting emerges. As illustrated in Fig. 2(e), two Q-switched bursts begin to combine, forming one burst with sequential envelopes. The temporal width and the peak power of the combined burst are usually larger than those of the single one, and its envelope becomes irregular, not Gaussian-like. If we focus on the real-time spectral evolution of the Q-switched burst, it shows a qualitatively different behavior compared to the quasi-cw Q-switched burst. Figure 2(f) shows the spectral evolution of a typical burst under the pump current of 320 mA. The first envelope is similar to the quasi-cw Q-switching, consisting of several background fluctuations, but due to the high pump power, one pulse in the background fluctuations becomes strong enough to bleach the saturable absorber. Then its intensity increases significantly within a short period. Simultaneously, high peak power stimulates self-phase modulation (SPM), resulting in a spectral broadening. More interestingly, the broadened spectra always show interferograms, implying the emergence of a transient soliton molecule in this process. A possible explanation of the emergence of transient soliton molecule is the theory of soliton energy quantization. If the pulse intensity increases beyond a limitation, the archetypal soliton will break up into several solitons with weaker peak power [27]. The temporal structure of the transient soliton molecule can be revealed by calculating the Fourier transform of its real-time spectrum. Figure 2(g) shows this the calculating result, which is actually the auto-correlation of its temporal electrical field. We can clearly observe the process that original pulse evolves into a four-soliton transient bound state. And the pulse separations can also be speculated from the auto-correlation profile. Nevertheless, it is worth nothing that the structure of these transient soliton molecules depicts great variations in the numbers and lifetimes.

2.2 The build-up of mode-locking

The criterion for Q-switched instabilities has been given by several researchers [20, 28]. If the pulse energy E_p satisfies the condition$E_p<\sqrt{E_{\text{a}}{E}_g\Delta R}$, the laser will operate in the Q-switched instabilities regime [20]. Here E_a is the saturable energy of the saturable absorber, E_g is the saturable energy of gain medium, and △R is the modulation depth of the saturable absorber. Circulating pulses with energy higher than the threshold will stimulate additional nonlinear loss in the saturable absorber, leading to an increasing damping of the relaxation oscillations [25]. The laser will enter into a stable mode locking regime under higher pump power.

In the previous section 2.1, we discussed the Q-switched instabilities before the stable mode locking. In this section, we will focus on how the mode-locked pulse train evolves from the last Q-switched burst. Figures 3(a)-3(c) illustrate the real-time spectral evolution of three starting dynamics. The zeroth roundtrip is just set at the gap between two Q-switched bursts. It is not the start time of a measurement. The 2000 roundtrips presented in Fig. 3 are part of the actual 20000 roundtrips recorded in a single-shot measurement. The acquisition of the build-up dynamics is not easy. Different from the Ti: sapphire mode-locked laser [12], there is a series of Q-switched bursts with high peak power before the stable mode locking, so we cannot acquire the build-up dynamics just by setting a trigger level in the oscilloscope. And there is no external trigger or controlled current in our measurement. The stable mode locking is initiated by increasing the pump current, not a turn-key operation. We firstly start the laser and turn up the pump current approaching to the threshold (about 360 mA for our laser). Then, we increase the current by petite until the laser enter a special regime where it switches between Q-switched instabilities and stable mode locking constantly. At this special regime, we repetitively capture the waveform until that of starting dynamics is acquired. One valid data set is obtained from multiple acquisitions. The probability to capture the starting dynamics can be enhanced by increasing the record length of the oscilloscope.

These build-up processes, though with small differences in spectrum, are enriched with common evolution features. All these starting dynamics can be divided into three phases. A quasi-cw Q-switched burst emerges firstly, followed by an explosion of transient soliton molecule. Finally, a stable mode-locked pulse train evolves from the damped soliton molecule gradually. Figure 3(d) presents the power (blue line) and bandwidth (red line) evolutions of the mode-locking build-up process. The first power peak, appearing at ~400th roundtrip, is corresponded to the power integration of all background fluctuations. Actually, the peak power of each pulses is quite small except for the dominant pulse. Their bandwidth also remains at very low level, about 0.2 nm. Then the optical power and bandwidth increase significantly during the soliton molecule explosion. The transient soliton molecule is quite unstable from our observation. Both bandwidth and pulse power undergo violent oscillations. In the end, fringe patterns fade out into a stable Gaussian-like spectrum. The optical power and bandwidth also rest at a constant value.

Another interesting observation is that we glanced a blue-shift of the mode-locked spectrum compared to the fluctuation before the mode locking. As shown in Fig. 3(e), the spectrum of mode-locked pulses shifts to a shorter wavelength by ~0.7 nm compared to the initial dominant pulse. Similar to the power and bandwidth evolutions, the central wavelength also undergoes vibrations before mode locking. We believe that the frequency shift is related to the increasing pulse power, but the specific mechanism remains unclear. A possible explanation of this spectral blue shift is the gain saturation in the cavity, as mentioned in [29]. It could also be a by-product of spectral broadening caused by SPM, since the frequency shift is quite small (just ~0.7 nm).

2.3 The starting dynamics of soliton molecules

If we continue increasing the pump current after the stable mode locking, the soliton spectrum will broaden gradually due to the enhanced SPM, as shown in Fig. 4(a). Meanwhile, a cw component emerges on the soliton spectrum and grows significantly. It is worth nothing that the cw component is located at the center of the mode-locked spectrum, different from those symmetrically-located Kelly sidebands [30–33]. Moreover, we find that the wavelength of the cw component is the same as that of the quasi-cw Q-switching before the stable mode-locking regime. Similar cw component is also reported in [18]. We think this cw component is just one of the longitudinal modes in the fiber cavity. When the pump power is increased beyond a critical level, the excessive gain cannot be depleted by the mode-locked pulse. Then the excessive gain supports an independent cw or quasi-cw fluctuation. Meantime, a strong quasi-cw fluctuation (~20 dB higher than the mode-locked spectrum) could make the system unstable. In a mode-locked fiber laser, a strong disturbance tends to stimulate relaxation oscillations. Eventually, the laser transforms into a Q-switched instability regime.

 

Fig. 4 The formation dynamics of soliton molecule above 460 mA. (a) The optical spectral evolution recorded by an OSA before entering soliton molecule regime. A cw sideband emerges at high pump power. (b) The recorded transient regime from single pulse regime to the soliton bound state over a time-window of 4 s, in which a ~1.2 s Q-switched mode-locking regime is illustrated. The results are measured by time-stretch technique, so the relative low peak power of single pulse operation mainly results from its large bandwidth. (c) Experimental real-time spectral evolution of the formation of a soliton molecule from the Q-switched mode-locking burst. (d) The Fourier transform of the measured real-time spectrum, representing the autocorrelation of its temporal waveform. (e) The autocorrelation of the 1500^th roundtrip. The inset shows the position relation of the three solitons. (f) The separation and relative phase as a function of roundtrips over roundtrips from 2000 to 5000, illustrating a decaying fluctuation before entering the stable bound state.

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The time duration of the Q-switched instabilities usually sustains hundreds of microseconds. Sometimes it can even be longer than 1 s. As shown in Fig. 4(b), the single pulse operation and the soliton molecule regime are divided by ~1.2-s Q-switched instabilities. Note that the results shown in Fig. 4(b) are measured by DFT technique. Since the bandwidth at the end of single pulse operation is much larger than the bound state regime, the time-stretched results show a lower peak power in single pulse regime.

Based on the soliton energy quantization theory [27], the large cavity energy corresponds to a multi-soliton operation. After a transient Q-switched mode-locking regime, a stable soliton bound state is achieved eventually. The Q-switched instability works as an energy redistribution process. The transient solitons emerging in this regime can be regarded as lots of unsuccessful attempts to stable soliton molecules.

Bound state properties are determined by the parameters of optical systems. This topic has been theoretically studied by several groups based on complex Ginzburg-Landau equation (CGLE) or nonlinear Schrodinger equation (NLSE) [27, 34–36]. In general, soliton molecules tend to settle on the holes of the potential energy surface. One optical system can support bound state with different separations, meaning that several energy holes exist on the potential energy surface. This has been theoretically predicted [37] and experimentally observed by previous researchers [14]. However, which bound state will emerge in a particular experiment depends on initial conditions.

In our experiment, we observed two types of stable soliton pairs. One is a bound states with a separation of ~50 ps and the other is a two-soliton state with a separation of ~4 ns. Their starting dynamics are recorded and analyzed by using DFT method. The minimum threshold current of the stable soliton molecule observed in our experiment is ~460 mA, though such threshold varies slightly in different acquisitions. The following real-time evolutions are acquired above the bound state threshold.

Figure 4(c) traces the formation of a stable bound state with a separation of ~50 ps. Same with the initiating of the single soliton operation, we stimulate the soliton bound state just by increasing the pump power. We record a data set over 10 ms (~500000 roundtrips), in which 5000 roundtrips are shown in Fig. 4(c). We set the zeroth roundtrip at the gap before the formation dynamics. As shown, a spectrum with superimposed fringe periodicities emerges within the Q-switched burst, which implies an arise of a triplet state. This can be verified by the temporal autocorrelation shown in Fig. 4(d). A transient triplet is created in the Q-switched burst before evolving to a stable doublet state. Figure 4(e) shows the autocorrelation of the 1500^th roundtrip, indicating the position of the triplet state. During the following 1000 roundtrips, the spectrum range narrows from ~8 nm to ~2 nm and two pulses of the triplet bound state draw each other gradually. At the end of the Q-switched burst, the triplet state decays into a doublet state. Their relative phase and binding separation experience a fluctuation before reaching a locked relative phase and a constant separation of ~53 ps (see Fig. 4(f)). The bound state formation takes several thousand roundtrips, similar to the booting duration of the single soliton, since the separation of the exploded transient soliton molecule is close to that of the final state. If the initial soliton explosion has a larger separation, the spectral evolution will exhibit different behavior, as shown in Fig. 5 and Fig. 6.

 

Fig. 5 Formation dynamics of another soliton bound state above 460 mA. (a) The data set recorded after a dispersive element. The formation process takes about 6 ms to achieve a stable bound state. (b) Real-time spectral evolution at the beginning of bound state. (c) The synchronously recorded data set without dispersion shows the starting dynamics in temporal domain. Two solitons with ~700 ps separation emerge simultaneously in the cavity. (d) Spectral evolution from ~3.8 ms to ~6.8 ms. In this period, the two solitons are close enough so that the spectral patterns can be resolved by our oscilloscope. (e) Fourier transform of the real-time spectrum shown in (d). It represents the autocorrelation of the temporal waveform. The turn-back trace is caused by the under-sampling of the spectral information. Actually, the soliton separation is decreasing monotonously, until achieving a stable state at ~50 ps separation.

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Fig. 6 The formation of large separation bound state above 460 mA. (a) Real-time evolution of the starting dynamics of two solitons after time-stretch. (b) The synchronously measured temporal evolution. (c) The separation variation in the whole record range of ~8 ms, showing a slight decreasing tendency. (d) The spectra of single soliton operation (under 360 mA) and two soliton pairs (under 465 mA).

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Figure 5 shows a bound state formation event with an initial separation of 700 ps. The whole formation process takes about 6 ms, tens of times longer than the event shown in Fig. 4. The real-time spectral evolution at the initial period is similar to the build-up of single soliton (see Fig. 5(b)), but it is actually two solitons at a separation of ~700 ps that explode synchronously from the Q-switched burst. Due to its large separation, the spectral fringe patterns are too fine to be recognized. The rise of the amplitude from ~4 ms to ~6 ms in Fig. 5(a) means that the two solitons are close enough so that the spectral patterns can be recognized by our oscilloscope. The real-time spectral evolution of this period is shown in Fig. 5(d). An interferogram appears gradually in the Gaussian-like spectrum, implying the solitons draw each other continuously. Figure 5(e) shows how the two solitons transform into a stable bound state. The turn-back trace is caused by the under-sampled real-time spectrum. If the oscilloscope has ultra-high resolution, we would observe an evolving trace with monotone decreasing separation until achieving a stable state at ~50 ps separation.

In our experiments, we also observed bound state vibrations of the 50-ps-separation soliton molecules, as noticed in G. Herink’s work [6]. The typical period of the vibrations is about 400 roundtrips. This soliton vibration can be prepared by decreasing the pump current approaching to the bound state threshold (~460mA). Recently, soliton molecule vibrations in a fiber laser has been experimentally observed and theoretically analyzed [13], so observations about that will not be discussed in this paper.

If the initial pulse separation is several nanoseconds, the laser will evolve into a two-soliton state with separation of ~4 ns. Figure 6(a) shows the booting of the two-soliton state by using DFT technique. Two mode-locked pulses at an initial separation of ~7.4 ns evolve from the noise fluctuations simultaneously (see Fig. 6(b)). The initial pulse separation shows randomness in different booting events. However, all these pulse pairs will finally evolve to a stable state at ~4 ns separation. The evolving time is much longer than the measurement time, so we cannot record the whole evolution process. Figure 6(c) show the separation variation and the decreasing tendency is apparent. During the initial period, the pulse separation reduces by ~16.5 ps per millisecond. If the declining rate remains constant, it will take over 200 ms to achieve the stable state at a separation of 4 ns.

In general, the two-soliton state can hardly be regarded as a soliton bound state. Since its separation is few orders of magnitude larger than the pulse width, interaction between the two solitons must be negligible. A few years ago, researchers in New Zealand observed ultra-weak long-range interactions of solitons in a fiber cavity [38, 39]. The interaction was explained as a result of acoustic waves stimulated by the propagating solitons through electrostriction. In their experiment, the interaction can last hundreds of seconds with a soliton separation of several nanoseconds. However, soliton behaviors are different in our experiment. Solitons evolve into a stable state very quickly (less than 1 second) after Q-switched instabilities. Then the solitons settle at a constant separation without variations over time. There should be other mechanisms that support such a two-soliton state with a large but fixed separation. Perhaps, there is some quasi-linear periodic wave propagating in the cavity, with a large period and small amplitude. Then each soliton is pinned to a local potential minimum created by the periodic wave, independently from the others. However, if the hypothetical quasi-linear wave is generated by the radiation from the solitons, we may regard it as a sort of soliton interactions.

Figure 6(d) shows the spectra of single soliton and two types of soliton pairs. The spectrum of the two-soliton state with a 4-ns separation is almost the same as the single pulse regime, except for its higher intensity. The spectral intensity of bound state 2 (with a 4-ns separation) is about two times higher than that of the single pulse. It is reasonable because two solitons circulate in the cavity. The spectrum of the bound state with a 50-ps separation has evident fringe patterns under a Gaussian-like envelope. From the spectrum, we can easily recognize which state emerges in the fiber cavity. However, we are still unable to predict which state will arise in a particular experiment. And it is a bit difficult to give a certain probability of the two bound states because it varies greatly in different experiments. The temperature, vibrations and working time can influence the laser behavior significantly. However, we can give a quantitative estimate for which state the different initial separations will lead to. In our experiments, the maximum initial separation of 50-ps bound states is ~1.6 ns and the minimum initial separation of 4-ns bound states is 2.8 ns. We can speculate that the critical initial separation is around 2 ns. This conforms to our intuition that the solitons will settle on a separation close to their initial state.

3. Conclusion

By means of DFT technique, we resolve the ultra-fast soliton dynamics in a passively mode-locked fiber laser. Since the upper-state lifetime of EDF is much longer than Ti: sapphire mode-locked laser, the soliton dynamics in fiber lasers show different behaviors. We find two types of Q-switched instabilities before stable mode locking and analyze how a stable mode-locked pulse train evolves from the Q-switched burst. With the increase of pump power, soliton splitting is inevitable due to the soliton energy quantization. Interestingly, we find the energy distribution is realized by a transient Q-switched instability. The soliton bound state evolves from a Q-switched burst rather than the single soliton train. Additionally, we find the soliton separation and evolution paths are highly dependent on the initial condition. Two types of soliton pairs (soliton bound state and two-soliton state) are observed and the typical forming dynamics are recorded and analyzed. These findings reveal the overall insight of solitons dynamics in a mode-locked fiber laser. Our results also provide valuable materials for theoretical studies on the dynamic phenomenon in nonlinear optical systems.