1. Introduction

Compared with the Hamiltonian system, the dissipative system is more complicated in the sense that the dissipative system has the dissipative properties of continuous exchange of energy with the environment. In such a dissipative system, gain and loss play an essential role in the formation of dissipative solitons. Owing to its characteristic of energy exchange, the fiber laser can be regarded as a dissipative system and the pulses formed in such a system can be treated as dissipative solitons [1]. In addition to its excellent ability of generating ultra-short pulses which has a wide range of applications in fields from fundamental physics to industrial purposes, the mode-locked fiber laser is also known as an ideal test bed for investigating complex nonlinear dynamics. So far, various striking nonlinear phenomena have been observed in fiber lasers, such as multi-soliton patterns [2–5], vector soliton [6–9] and dissipative soliton resonance [10–13]. However, due to the lack of advanced measurement technologies, these results were not based on a real-time measurement method, which means that some important transient information may be ignored since the ultrafast nonlinear phenomena cannot be captured by the conventional measurement instruments.

Till recent years, owing to the development of real-time measurement techniques, dispersive Fourier-transform (DFT) has been regarded as one of the standard real-time measurement techniques for resolving ultrafast dynamics of solitons in laser systems [14]. Through the DFT, the spectrum of the soliton could be mapped into a temporal waveform by using dispersive element with enough group-velocity dispersion. Thus, the ultrafast spectral signals can be captured by a real-time oscilloscope with a high-speed photodetector. The DFT technique opens up the way of experimentally investigating the transient dynamics of solitons in lasers. Up to now, it has been employed to observe complex ultrafast nonlinear phenomena, including soliton explosion [15–20], build-up and transient dynamics of solitons [21–27], pulsating solitons [28–31], as well as internal dynamics and build-up of soliton molecules [32–38].

Particularly, pulsating soliton as a distinctive localized structure in dissipative systems has attracted extensive attention [39–46]. Unlike equal amplitude of pulse train generated from the laser, it is found that the amplitude of the pulsating soliton could evolve periodically or even in a more complex way along with the cavity roundtrips and the pulsating soliton exists regardless of the sign of the dispersion. In the early years, the properties of the pulsating solitons could merely be observed by the oscilloscope, from which one could only obtain the temporal intensity information of the pulsating solitons [41]. The limitation of the measuring apparatus has been broken through by the development of real-time spectroscopy techniques, i.e., DFT, which is possible to unveil the real-time spectral dynamics of pulsating solitons [28–31]. In anomalous dispersion fiber lasers, the periodic radiation dispersive waves in sync with the pulsation and the pulsating soliton with chaotic behavior were observed [28,30]. While in a normal dispersion fiber laser, Du et al. firstly demonstrated the pulsating single dissipative soliton with oscillating structure spectrum [29]. Very recently, Wang et al. reported the pure soliton pulsations and soliton explosion, which have common features such as energy oscillation, bandwidth breathing and temporal shift [31]. However, the reported dynamical behaviors of the pulsating soliton, in particular the multi-soliton pulsation, are still limited, the underlying dynamics have not been fully revealed. Therefore, it is of great interest to investigate the versatile behaviors of pulsating solitons in a fiber laser.

In this work, we firstly observe, to the best of our knowledge, the distinct dynamical diversity of pulsating solitons in a fiber laser using the DFT. We reveal the real-time spectral evolution from the regime of single-soliton pulsations to that of multi-soliton pulsations. The explosive behavior during the single-soliton pulsation is observed from weak to strong, and the rogue wave is generated. Moreover, we demonstrate particularly pulsating behaviors of the multi-soliton patterns that the energy, separation and relative phase difference of the solitons inside the molecule synchronously pulsate, while the energy pulsation of each individual in multi-soliton bunch could be in a regular or irregular way. The experimental results reveal the new transient dynamics of the pulses in laser community and pave the way towards a deeper understanding of the complex features of pulsating solitons in dissipative optical systems.

2. Experimental setup

The dynamical diversity of pulsating solitons is observed in a passively mode-locked fiber laser with net-normal dispersion. The schematic of the generation and real-time characterization of pulsating solitons is presented in Fig.  1. As shown in Fig.  1(a), the fiber laser cavity includes a 10 m long erbium-doped fiber (EDF) with dispersion parameter of −44.5 ps/nm/km pumped by a 980 nm laser diode (LD) through a wavelength-division-multiplexer (WDM), a polarization-dependent isolator (PD-ISO), a polarization controller (PC), a carbon nanotube saturable absorber (CNT-SA) and an output coupler (OC). The other fibers in the cavity are single-mode fibers (SMF-28) with 12 m long. Therefore, in this laser, the effect of saturable absorption is realized by both nonlinear polarization rotation (NPR) effect and CNT-SA. Furthermore, the output coupler with coupling ratio of 10:90 is used to extract 10% of the laser for measurements.

 

Fig. 1. Schematic of (a) the fiber laser and (b) the real-time characterization of pulsating solitons.

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At the laser output, we measure the average optical spectrum and the radio-frequency (RF) spectrum by using an optical spectrum analyzer (OSA, Yokogawa, AQ6375B) and a RF spectrum analyzer (Advantest, R3131A), respectively, as depicted in Fig.  1(b). The autocorrelation trace of soliton is measured by an autocorrelator (Femtochrome, FR-103WS). Apart from the conventional measurements above, directly observing the spatial-temporal intensity evolution is fulfilled by a real-time oscilloscope (Tektronix, DSA71604B, 16 GHz) with a photodetector (Newport, 818-BB-35F, 12.5 GHz). On the other hand, to measure the spatial-spectral evolution of solitons, the DFT technique is adopted. In order to stretch the temporal solitons and map them into the shot-to-shot spectrum, here for pulsating non-bound-state solitons (the pulsating single-soliton, explosive pulsating soliton and pulsating soliton bunches), a ∼16 km long single-mode fiber (SMF) is used, while for the soliton molecule, a piece of dispersion compensation fiber (DCF) with dispersion amount of 2300 ps/nm is adopted. Thus, combining the photodetector and oscilloscope, the shot-to-shot spectrum can be measured in real-time with resolution of Δλ=0.29 nm and Δλ = 0.035 nm, respectively, for our DFT configuration. It is worth noting that, for the soliton molecules, because the inside soliton separation is quite large (around 106 ps in our experiment) comparing to that of soliton molecules in the anomalous dispersion regime, a large total dispersion to stretch the pulsating soliton molecules is therefore in need, which enables two individual corresponding shot-to-shot soliton spectra to overlap completely. In this case, the two superposition spectra can be regarded as a unit, ensuring the accuracy to resolve the shot-to-shot spectral dynamics of pulsating soliton molecules in the spectral domain.

3. Experimental results

3.1 Pulsating single-soliton and explosive long-period pulsating soliton

3.1.1. Pulsating single-soliton

In our experiment, by properly adjusting the PC, the fiber laser has a mode-locking threshold of 5.935 mW. Such low threshold could be attributed to the usage of the CNT-SA in the laser cavity. The dissipative soliton with typical rectangular spectrum can be firstly observed. This type of dissipative soliton has been commonly seen and fully investigated. However, the solitons with pulsating properties have not yet been completely experimentally studied. It is always inspired to explore different patterns of pulsating solitons as theoretically predicted [39,40]. Therefore, we are more interested to further reveal the dynamics of pulsating single-soliton, which is also expected to be bunched together as pulsating solitons bunch without soliton interactions or bounded together forming pulsating soliton molecule under the soliton interactions in fiber lasers.

By further adjusting the pump power to 8.43 mW and finely tuning the PC, we observed the pulsating single-soliton. Notably, the average spectrum presented on the OSA has gradient edges instead of conventional rectangular ones, as shown in Fig.  2(a). Moreover, the RF spectrum has several satellite peaks around the main peak with separation of 33.8 kHz as depicted in Fig.  2(b), indicating that the soliton is operating in pulsating regime. Then we performed the real-time measurement techniques to analyze the characteristics of the soliton in both spectral and temporal domains. Here, to confirm the accuracy of DFT, the average of 1500 roundtrips consecutive single-shot spectrum obtained through DFT is also plotted in Fig.  2(a), where the profile is consistent with that of the OSA spectrum. The 1500 continuous roundtrips evolution of shot-to-shot spectrum reconstructed through DFT is presented in Fig.  2(c). One can see that the spectrum breathes periodically. Correspondingly, the soliton energy varies sinusoidally in large amplitude with a period of ∼274 cavity roundtrips, in accordance with the modulation frequency of 33.8 kHz in RF spectrum presented in Fig.  2(b). Figure  2(d) shows the corresponding temporal evolution of the soliton. It is evident that the width and intensity of the soliton vary periodically with the cavity roundtrips, emerging pulsating properties. Note that larger pulsating amplitude and much longer pulsating period have been observed compared to those in [28–30] and the sinusoidal energy evolution is different from that in [31].

 

Fig. 2. The characteristics of the single-soliton pulsation. (a) The spectrum obtained from the OSA and the average of 1500 roundtrips consecutive single-shot spectrum. (b) The RF spectrum. (c) The shot-to-shot spectra and corresponding energy evolution. (d) Pulse train.

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Furthermore, we fixed the pump power and subtly adjusted the orientation of the PC to display more characteristics of pulsating single-soliton. We find that the spectrum displayed on the OSA varies correspondingly, namely, the spectrum can vary from the relative flat top to the gradient one (see Appendix Fig.  9). Moreover, the special spectrum is related to the degree of soliton pulsation: the larger the gradient extent of the spectral edges, the stronger the modulation of pulsating soliton energy and amplitude is. Particularly, when the gradient extent of spectral edges is slight enough, the energy modulation is minimized, showing a tendency toward a stable mode-locked state. The variation of the spectral gradient extent could be attributed to the change of the pulsating states in the laser cavity. In the experiment, the pulsating states could be varied by changing the transmission of the cavity via adjusting the PC. Consequently, the characteristics of the mode-locked spectrum also vary correspondingly.

3.1.2. Explosive long-period pulsating soliton

In the process of exploring the pulsating single-soliton, especially, we also observed that the pulsating single-soliton collapsed intermittently and returned back to the original pulsating state after roundtrips of dozens to around one hundred during the evolution. This soliton nonlinear phenomenon is similar to the soliton explosion in non-pulsating soliton regime [15–19] and has been numerically predicted in pulsating soliton operation [39,40]. However, here we observed the significant different explosive pulsating soliton events. When adjusting the orientation of the PC, in addition to achieving the regularly pulsating single-soliton at the pump power of 8.98 mW, we noticed a unique average spectrum from the OSA showing a high instability: spikes and dips stochastically occur on the spectrum. The typical spectrum is presented in Fig.  3(a). Many irregular peaks in the corresponding RF spectrum can also been observed as shown in Fig.  3(b). To acquire the possible diverse characteristics of the pulse, we recorded 15000 roundtrips real-time spectra with DFT, as shown in Fig.  3(c). Obviously, during the evolution of the pulsating soliton, we can see the explosive pulsating soliton spectrum with extreme intensity peak around the 8000th roundtrip and the corresponding pulse energy suddenly raises. To clearly show the detailed explosive process of pulsating soliton, we take out 1200 roundtrips single-shot spectra from the 15000 roundtrips evolution process, as shown in Fig.  3(d). Remarkably, from the 7200th roundtrip to the 8400th roundtrip, three explosive events of the pulsating soliton corresponding to three pulsating periods can be observed, and the explosive degree strengthens gradually. Note that, comparing to the explosions of pulsating soliton around the 7550th and the 7800th roundtrips, the pulsating soliton explosion around the 8000th roundtrip is much more highly chaotic and sustains longer roundtrips (about 100 roundtrips).

 

Fig. 3. The explosive long-period pulsating soliton operation. (a) The spectrum obtained from the OSA. (b) The RF spectrum. (c) The shot-to-shot spectra and corresponding energy evolution. (d) The enlarged explosive pulsating soliton region.

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For these three different explosion events, the spectrum collapse sustains certain roundtrips and recovers to the regularly pulsating soliton. As for the strongest explosive pulsating soliton event, the spectrum broadens dramatically. To illustrate more details, one typical single-shot spectrum is drawn during the explosion at the 7982th roundtrip, as presented in the inset of Fig.  3(c). Here, the spectrum collapse can be found clearly, and the explosive spectrum has intensive peaks. Besides, we note that the RF spectrum in Fig.  3(b) has quite many satellite peaks, which also shows a slight chaotic feature. Given that the pulsating soliton behavior is along with the explosive events and long-period pulsation (∼242 roundtrips) at the same time in our experiment, we regard this unique pulsating soliton pattern as the “explosive long-period pulsating soliton”. Moreover, we investigated the extreme waves in the spectral domain by recording the maximum intensity of the single-shot spectrum. It is found that spectral rouge waves generated in such an explosive long-period pulsating soliton regime, which is shown in the Appendix B (see Fig.  10).

3.2 Pulsating dual-soliton bunches

3.2.1. Irregularly pulsating dual-soliton bunch

In the following, before seeking for the pulsating soliton molecules which are expected to be formed in a laser system, we tried to explore the pulsating solitons bunch to see the possible dynamics difference between the single-soliton pulsation and multi-soliton bunch pulsation. More importantly, since the soliton molecule can be treated as a unique pattern of multi-soliton bunch when the separation between the individuals is small enough, it might be instructive to further find the pulsating soliton molecules.

As a rule, multi-soliton bunch is expected to be generated under the stronger pump power in the laser system, which has also been widely investigated. However, there is rare report on pulsating soliton bunches [31]. By increasing the pump power to 12.06 mW, the stable mode-locked dual-soliton bunch could be firstly achieved. Likewise, the corresponding spectrum with steep edges was observed. Then, we finely tuned the PC to look for the average optical spectrum with gradient edges on the OSA, as shown in Fig.  4(a). Here, note that the gradient range of the spectral edges is smaller comparing with that in the pulsating single-soliton regime. The corresponding RF spectrum is shown in Fig.  4(b), which also has several satellite peaks around the main peak with separation of 31.5 kHz, indicating that the dual-soliton bunch presents pulsating behavior. To further verify the existence of pulsating solitons bunch, temporal evolution of the laser output was measured directly by the real-time oscilloscope and shown in Fig.  4(c). It is evident that the pulsating dual-soliton bunch has been observed. To further get insight into the shot-to-shot spectral evolution of the pulsating solitons bunch, via DFT, the two pulsating solitons were stretched into corresponding real-time spectra. Instead of relatively strong periodic pulsation of spectrum in pulsating single-soliton, both two periodic spectra show a slight pulsation as illustrated in Figs.  4(e) and 4(g). Figures  4(d), 4(f) and 4(h) display the corresponding energy evolutions of solitons bunch and each individual. Particularly, it can be seen that, during the periodic evolution of soliton energy in dependence of roundtrips, irregular fluctuation happens to the periodic energy pulsation of soliton 1, while the energy of soliton 2 keeps relatively regular pulsation. For this energy pulsation difference, we suppose that one of the solitons experiences a small perturbation, leading to the variation of peak power for this soliton. Thus, the pulsating behavior changes correspondingly. In this case, the different pulsation of the two solitons could be observed. Although irregular energy pulsation takes place to the pulsating dual-soliton bunch, the pulsation frequency of the whole dual-soliton bunch with 31.5 kHz (corresponding to ∼294 roundtrips) still can be measured by the RF spectrum analyzer, as shown in Fig.  4(b).

 

Fig. 4. Irregularly pulsating dual-soliton bunch operation. (a) The spectrum obtained from the OSA. (b) The RF spectrum. (c) The pulse train. (d) The total energy evolution of the dual-soliton bunch. (e) and (g) The shot-to-shot spectra of the two solitons. (f) and (h) The energy evolutions of the two solitons.

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3.2.2. Dual-soliton bunch with regular pulsation

Then we increased the pump power to 12.63 mW and carefully tuned the PC, dual-soliton bunch with regular pulsation was achieved. The spectrum from the OSA and the RF spectrum are shown in Fig.  5(a) and 5(b), respectively. The periodic evolutions of temporal and spectral intensity are presented in Figs.  5(c), 5(e) and 5(g). Similarly, the RF spectrum in Fig.  5(b) displays multiple peaks with frequency difference of ∼35.1 kHz, which well agrees with ∼264 roundtrips pulsating period of the dual-soliton bunch in Figs.  5(c), 5(e) and 5(g). From Figs.  5(c), 5(e) and 5(g), we can clearly see that the pulsations of the dual-soliton bunch are regular and much stronger in both temporal and spectral domains comparing to those in Fig.  4. In addition, due to the increase of pump power, the pulsation period (∼264 roundtrips) is also shorter than ∼294 roundtrips pulsating period of the dual-soliton bunch with irregular pulsation. In terms of the shot-to-shot spectra, the real-time spectral characteristics are similar to those of the pulsating single-soliton (see Appendix Fig.  11). The corresponding energy evolutions of solitons bunch, soliton 1 and soliton 2 are depicted in Figs.  5(d), 5(f) and 5(h), respectively. Obviously, all three energy pulsating processes are regular and have the same period, which means the pulsating dual-soliton complex is fairly stable. Note that the observed pulsating dual-soliton bunches has a large separation over 20 ns. In fact, the pulsating dual-soliton bunch with different temporal separations could be also obtained by adjusting the PC and the pump power.

 

Fig. 5. Regularly pulsating dual-soliton bunch operation. (a) The spectrum obtained from the OSA. (b) The RF spectrum. (c) The pulse train. (d) The total energy evolution of the dual-soliton bunch. (e) and (g) The shot-to-shot spectra of the two solitons. (f) and (h) The energy evolutions of the two solitons.

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3.3 Pulsating soliton molecules

In the following, we aimed for verifying the possible existence of pulsating soliton molecules in the fiber laser which have few reports so far. Up to now, only the soliton molecules with oscillating phase difference and/or separation [32,33,36–38,47] and pulsating energy with fixed phase difference and separation [28] have been investigated. Actually, soliton molecules can be regarded as a unit when the individual solitons are bound together under their interactions. Two characteristics of soliton molecules: the spectrum with modulation and the autocorrelation trace with multiple peaks, are useful to identify whether the fiber laser operates in the bound-state regime.

We sought for the soliton molecules pulsating simultaneously in energy, phase difference and separation, based on the characteristics of pulsating solitons observed above. When fixing the pump power at 13.67 mW, we obtained pulsating soliton molecule by controlling the PC, and the results are summarized in Fig.  6. Figure  6(a) shows the spatial-spectral dynamical evolution. One can clearly see that the modulated spectra breathe periodically. For better displaying the interference fringes of the soliton molecule spectrum, we also show the spectra with a small range in Fig.  6(b). Correspondingly, the energy of the soliton molecule varies periodically, as presented in the inset of Fig.  6(a), showing a pulsating period of ∼355 roundtrips. Here, it should be noted that because the output power is too low to be transmitted in the DCF, an erbium-doped fiber amplifier (EDFA) is adopted before recording the real-time spectral dynamics via DFT. The corresponding temporal evolution of the soliton molecule is further demonstrated in Fig.  6(c). From Fig.  6(c), one can see that two pulsating solitons bound as a unit and evolves stably. Figure  6(d) shows the corresponding RF spectrum, where the frequency difference of 26.5 kHz is well consistent with the pulsating period. Both spectral and temporal characteristics demonstrate that pulsating dual-soliton molecule has been observed in our fiber laser. Further investigation shows that the separation and phase difference between the two solitons simultaneously vibrate with the roundtrips, as shown in Fig.  6(e). The soliton separation oscillates with amplitude of 1.7 ps and a period of ∼355 roundtrips. Meanwhile, the phase difference oscillates with amplitude of 0.34π and a period of ∼355 roundtrips. Figure  6(f) shows the evolution trajectory of the two interacting solitons in the phase plane, indicating that the oscillations possess stably periodic behavior. Thus, we can conclude that the distinct soliton molecule observed above pulsates not only in energy, but also in soliton separation and phase difference.

 

Fig. 6. Characteristics of periodically pulsating dual-soliton molecule. (a) The real-time spectra evolution measured with DFT, inset: the energy evolution. (b) The enlarged spectra. (c) The temporal evolution. (d) The RF spectrum. (e) The soliton separation and phase difference as a function of the roundtrips. (f) The corresponding interaction planes.

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When the PC was further adjusted slightly, the soliton molecule pulsated as a unit could be still observed. That is, the two solitons consisting of the molecule possess synchronous pulsations. However, in this case the soliton molecule did not pulsate in a periodic way. Figure  7(a) shows the real-time spectral dynamics of pulsating soliton molecule with aperiodic behaviors. The details of real-time spectral evolution with shorter spectral range are also given in Fig.  7(b). As can be seen here, the interference fringes of mode-locked spectrum present the irregular (or non-perfect periodic) patterns, indicating that the separation and phase difference between the two solitons inside the molecule irregularly evolve with roundtrip time. The whole energy evolution of the pulsating soliton molecule further demonstrated that the pulsating behavior is irregular, as depicted in the inset of Fig.  7(a). Correspondingly, the pulse train of the soliton molecule is also shown in Fig.  7(c). Here, it can be clearly seen that the two solitons pulsate as a unit, but not in a periodic manner. Moreover, because of the aperiodic pulsation of soliton molecule, the side peaks of RF spectrum with broader bandwidth could be observed under this state, as presented in Fig.  7(d). As mentioned above, the interference fringes indicate that the soliton molecule evolves when propagating in the laser cavity. Likewise, to gain insight into the dynamical characteristics of the pulsating soliton molecule from the soliton separation as well as phase difference point of view, we provide the evolution of the soliton separation and phase difference extracted from the DFT spectra in Fig.  7(e), where the aperiodically evolving (even random) trend could be observed. Moreover, the corresponding interaction plane between the two solitons is plotted in Fig.  7(f). It further proves that the pulsating behavior of the soliton molecule is not stable.

 

Fig. 7. Characteristics of aperiodically pulsating dual-soliton molecule. (a) The real-time spectra evolution measured with DFT, inset: the energy evolution. (b) The enlarged spectra. (c) The temporal evolution. (d) The RF spectrum. (e) The soliton separation and phase difference as a function of the roundtrips. (f) The corresponding interaction planes.

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Then we increased the pump power to 15.68 mW, pulsating triple-soliton molecule was achieved by carefully manipulating the PC. Figure  8 summarizes its characteristics. Figure  8(a) shows the spatial-spectral evolution. For better clarity, we show the enlarged spectra in Fig.  8(b). Obviously, the spectra with modulation pulsate periodically, like that of the dual-soliton molecule in Fig.  6. The energy evolution of the triple-soliton molecule is presented in the inset of Fig.  8(a), which also varies with a period of ∼276 roundtrips. The corresponding temporal evolution is demonstrated in Fig.  8(c). It is clear that three pulsating solitons bound as a unit and evolves stably. The pulsating period was further confirmed by the RF spectrum shown in Fig.  8(d). In addition, we analyze the temporal separation and relative phase difference of the triple-soliton molecule, as shown in Fig.  8(e). Note that the separation and phase difference between the solitons inside the molecule synchronously oscillate with the roundtrips, which are similar with those of the dual-soliton molecule in Fig.  6(e). Figure  8(f) further shows the evolution trajectory of the interacting solitons in the phase plane, indicating that the oscillations possess relatively stable periodic behavior. So far, we observed the pulsating triple-soliton molecule, similar to the pulsating dual-soliton molecule, which pulsates not only in energy, but also in soliton separation and phase difference. It should be noted that for the triple-soliton molecule, there should be two spectral interference patterns on the mode-locked spectrum because of the two temporal separations among the three solitons. However, note that the spectral resolution of DFT is about 0.035 nm. Therefore, the spectral interference pattern of the triple-soliton molecule corresponding to the larger temporal separation could not be resolved. In this case, only one pulse separation and phase difference is provided.

 

Fig. 8. Characteristics of pulsating triple-soliton molecule. (a) The real-time spectra evolution measured with DFT, inset: the energy evolution. (b) The enlarged spectra. (c) The temporal evolution. (d) The RF spectrum. (e) The soliton separation and phase difference as a function of the roundtrips. (f) The corresponding interaction planes.

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4. Discussions

In this section, we would like to further discuss the observed various dynamics of pulsating solitons shown in Section 3. Pulsating is an intermediate state between stationary stability and chaos. Various pulsating soliton states have been theoretically predicted in different parameters space based on the complex Ginzburg-Laudan equation, such as plain pulsating soliton, erupting soliton, creeping soliton, multiple-period soliton pulsations and extreme soliton pulsations [39–41,43,44]. However, there are few kinds of pulsating solitons are observed because the parameters in the experiment could not be adjusted flexibly [28–31,41]. In our experiment, in addition to the ordinary soliton pulsation, we also observed the explosive pulsating soliton. Remarkably, the explosive pulsating soliton events evolve from weak to strong, accompanying by the great broadening of the spectrum at the strongest explosive event. It may come from the fluctuation of pulse peak power. When the laser operates at pulsating regime, the gain and loss do not completely balance with each other. At certain time, the imperfectly pulsating behavior leads to the perturbation of the pulse peak power. When the peak power of the mode-locked pulse exceeds the cavity tolerant nonlinear effect (overdriven nonlinear effect), the soliton splits into small pulses with random phase difference, which shows the explosive behavior [19,44]. After some roundtrips, some small pulses dissipate and the residual pulses reshape into one. Then the next perturbation of the pulse peak power makes the pulse split and the spectrum collapse again. The degree of the explosive events depends on the peak power of the split small pulses. The higher the peak power, the broader the spectrum generates due to the self-phase modulation and cross-phase modulation effects. And it is more likely to appear rogue waves as the statistical analysis.

When we increase the pump power and manipulate the orientation of the PC, the transmission function of the cavity varies, and it is more likely to arouse multiple solitons. Consequently, the pulsating states change as well. Note that the separation between the two solitons in a bunch is more than 20 ns, they therefore do not have direct interactions, indicating that the pulsation for each individual could be irregular or regular, synchronous or asynchronous. However, for the pulsating soliton molecules, the separation between the two solitons inside the molecule is around 106 ps, with the soliton width of about 23 ps (see Appendix Fig.  12). The two solitons are bounded together under their direct interactions. Thus, the spectrum presents periodical modulation in soliton molecules state.

Generally, the two solitons inside the pulsating dual-soliton molecules act as a unit. By comparing the two states of the dual-soliton molecules, the periodic evolution of pulse separation and phase difference could be observed for the regularly periodic pulsation of soliton molecule. However, the irregularly periodic pulsation of soliton molecule shows the aperiodic evolution of pulse separation and phase difference. Due to the fact that the soliton pulsation is related to the gain and loss in the laser cavity, the soliton molecule possesses the same state including soliton separation and phase difference when it propagates through the EDF every roundtrip for the periodically evolving phase difference and pulse separation. Thus, the gain for this type of soliton molecule will be equilibrated, resulting in the appearance of regular pulsation. However, the soliton molecule with the aperiodic evolution experiences different states when passing through the EDF every roundtrip, leading to the irregular gain dynamics. Therefore, it is indicated that the pulsating behavior of soliton molecule significantly depends on the evolution of pulse separation and phase difference. That is to say, the locked or periodic evolution of pulse separation and phase difference is essential for achieving regular pulsation of soliton molecule.

Although we have experimentally observed these distinct diverse pulsating soliton patterns in a same fiber laser, we only manipulate the pump power and the PC, without engineering the dispersion of the cavity. In fact, as theoretically predicted in [44], dispersion has great influence on pulsating states even in normal dispersion regime. With proper normal dispersion, one can achieve the extreme soliton pulsations, which is beneficial to generating high energy pulses. Our present work is in normal dispersion regime, it is also interesting to know whether one can observe the same or more phenomena in the anomalous dispersion regime, which is an open issue and needs further experimental works to explore. Besides the spectral dynamics of the pulsating solitons in this work, we would also like to mention that it is still anticipated to further reveal the detailed temporal information of these pulsating solitons through using time-lens measurement technique [24,48,49] in the future.

5. Conclusion

In this work, we experimentally investigate the dynamical diversity of pulsating solitons from single-soliton to multi-soliton in a fiber laser by using DFT. The weak to strong explosive behaviors of pulsating soliton, as well as the rogue wave generation during explosions are observed. Moreover, we find that the pulsations of each individual in multi-soliton bunch could be regular or irregular. On the other hand, for the pulsating soliton molecule, the two solitons inside the molecule act as a unit, the simultaneous pulsations in energy, separation and relative phase difference are achieved. These findings will shed new insights into the complex nonlinear behavior of solitons in dissipative optical systems.

Appendix A: Spectral variation of the pulsating single-soliton with the PC adjustment

Here, to better display the characteristics of time-average optical spectrum displayed on the OSA when adjusting the orientation of the PC, we provide several optical spectra corresponding to different PC orientations at the same pump power, as shown in Figs.  9(a), 9(b) and 9(c). From these figures, one can see that by tuning the PC orientation, the shape of the optical spectrum varies in a certain range, from flat top to remarkable gradient edges in our experiment. Moreover, this kind of spectrum is connected with the pulsation parameters. To put it differently, the larger the gradient extent of the spectral edges, the stronger the modulation of pulsating soliton energy and amplitude is. The single-shot spectra corresponding to the maximum pulsating soliton energy for these three PC orientations through DFT are recorded in Figs.  9(d), 9(e) and 9(f). As seen from Figs.  9(d), 9(e) and 9(f), the single-shot spectra exhibit the structured profiles, where two sharp spectral peaks with oscillation structures around them locate at both edges of the spectrum. And the larger the gradient degree of the average spectrum, the stronger the spectral fringe at the edges of the single-shot spectrum is. This structured profile of the single-shot spectrum is theoretically predicted in [44]. As for the difference of the spectral fringe at the edges of the spectrum in these three cases, it comes from the increase of the soliton energy during pulsating. Therefore, the spectrum shows stronger oscillation structure at the edges.

Additionally, it should be noted that these three average spectra with diverse gradient extent edges correspond to the different pulsating solitons with different energy modulations, respectively. In other words, in our experiment, we find that the pulsating soliton energy modulation varies according to the gradient extent of average spectrum. When the energy modulation becomes larger, the structured spectrum corresponding to the maximum pulsating soliton energy varies, from peaks appearing at the single-shot spectral edges without remarkable fringes to sharp peaks with deep fringes.

 

Fig. 9. Optical spectra at different PC orientations. (a)–(c) Average spectra obtained from OSA. (d)–(f) Shot-to-shot spectra corresponding to the maximum pulsating soliton energy.

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Appendix B: Rogue wave generation in explosive long-period pulsating soliton regime

Rogue wave has been observed in different operation regimes in fiber lasers, like in noise-like pulse and chaotic multi-pulse cases [50–52]. It seems that all these complex nonlinear phenomena to some extent share a common characteristic: solitons with chaotic behaviors. Since the explosive long-period pulsating soliton in our experiment has remarkable chaotic feature and the extreme spectral events appear, we perform statistical measurements to analyze the link between the highly chaotic pulsating soliton and generation of rogue wave. Note that, instead of studying the temporal solitons which are expected to generate the rogue wave, due to our temporal resolution (80 ps) is not enough to resolve the timing between the splitting pulses from a soliton when explosion of pulsating solitons takes place, we investigate the extreme waves in the spectral domain by recording the maximum intensity of the single-shot spectrum. To display the histogram of the statistical distribution of the maximum spectrum intensity in the explosive pulsating soliton regime, 100000 trace samples are recorded by the oscilloscope. The results are shown in Fig.  10. From Fig.  10, we can see that the significant wave height (SWH) is 0.0171V. In these pulse amplitude events, the vast majority of them are concentrated in a lower intensity. The largest intensity of extreme spectral wave is more than 6 times of the SWH. Moreover, as shown in Fig.  3(c), the chaotic spectral fringes happen around the maximum pulsating soliton energy. Therefore, we believe that rogue waves generated in such an explosive long-period pulsating soliton regime.

 

Fig. 10. Statistical distribution histogram of the amplitude fluctuations in explosive long-period pulsating soliton regime.

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Appendix C: Typical single-shot spectrum of dual-soliton bunch with regular pulsation

To better show the characteristics of the single-shot spectrum of dual-soliton bunch with regular pulsation, three typical real-time spectra are taken within one pulsation period, as shown in Fig.  11. One can observe that at the positions of the 335th and 365th roundtrips, the sharp peaks appear at the edges of the spectrum and gradually become stronger. However, at the 454th roundtrip, the spectrum has parabola-like top. These spectral characteristics are quite similar to those of pulsating single-soliton in the inset of Fig.  2(c).

 

Fig. 11. Single-shot spectra of dual-soliton bunch with regular pulsation.

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Appendix D: The spectrum displayed on the OSA and the autocorrelation trace of the periodically pulsating soliton molecule

Figure  12 shows the spectrum presented on the OSA and the corresponding autocorrelation trace of the periodically pulsating soliton molecule. As can be seen from Fig.  12(a), the spectrum shows evident fringes, which is the typical characteristic of a soliton molecule. The corresponding autocorrelation trace depicted in Fig.  12(b) further confirms that the soliton molecule consists of 2 identical solitons. Moreover, the temporal separation of 106.5 ps coincides with the 0.08 nm modulation spacing on the spectrum.

 

Fig. 12. (a) The spectrum displayed on the OSA. (b) The autocorrelation trace of the periodically pulsating soliton molecule.

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Funding

National Natural Science Foundation of China (61875058, 11874018, 61875242, 61378036); Science and Technology Program of Guangzhou (201607010245).

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