Soliton collision induced explosions in a mode-locked fibre laser

Soliton explosion refers to a striking nonlinear dynamics in dissipative systems. In this state, a dissipative soliton collapses but returns back to its original state afterwards. Yet, the origin of such exotic soliton dynamics remains elusive. Here it is revealed that soliton collision can induce soliton explosions in a mode-locked fibre laser, benefiting from synchronous measurements of the spatio-temporal intensity evolution and the real-time spectra evolution using dispersive Fourier transform. Up to seven nonlinear regimes are observed successively in the laser by increasing the pump power only, including single-pulse mode locking, standard soliton explosions, noise-like mode locking, stable double pulsing, soliton collision induced explosions, soliton molecules, and double-pulse noise-like mode locking. These experimental findings are conducive to understand complex soliton dynamics in many nonlinear dissipative systems. https://doi.org/10.1038/s42005-019-0134-8 OPEN

S oliton explosions are one of the most fascinating nonlinear dissipative phenomena in soliton dynamics. In this regime, a dissipative soliton undergoes a sudden structural collapse upon propagation. Remarkably, the exploded dissipative soliton could return back to the original state even though it experiences strong energy dissipation. Soliton explosions were firstly predicted in the framework of the complex cubic-quintic Ginzburg-Landau equation (CQGLE) 1 , emphasizing that highorder nonlinear terms are crucial to make a soliton explode. Later on, several numerical investigations have been carried out [2][3][4][5] , trying to better understand the intrinsic mechanisms involved in soliton explosions and subsequent revivals of the soliton.
Experimental observations of soliton explosions rely on technical breakthroughs to directly reveal the explosive transient evolutions that are too fast to be captured by standard measurement tools. Cundiff et al. firstly experimentally observed soliton explosions in a mode-locked Ti:sapphire laser, by virtue of a fast spectra measurement technique realized by spectrally dispersing the output of the laser across an array of six detectors and measuring the corresponding temporally resolved spectrum 6 . Later on, with the help of a novel powerful real-time spectra measurement technique called dispersive Fourier transformation (DFT) 7,8 , soliton explosions were also found experimentally in a mode-locked fibre laser 9 . The convenient access to the regime of soliton explosions via DFT has boosted the experimental investigations of such exotic dynamics in mode-locked fibre lasers recently [10][11][12] . Moreover, incoherent dissipative solitons, chaotic yet temporal localized pulses also exhibit similar explosive dynamics, as reported in a mode-locked fibre laser 13 .
Despite significant investigations on soliton explosions, a fundamental question remains elusive: why does a dissipative soliton explode and revive subsequently? For long-cavity fibre lasers (~100 m), Raman scattering was found to be related to soliton explosions 9,10 . However, for those short-cavity lasers in which Raman effect plays a negligible role 6,[11][12][13] , the origins of soliton explosions are still unclear. Although soliton explosions were numerically studied in the framework of CQGLE, considering dispersion, self-phase modulation, and high-order nonlinearity terms 1,6 , a general qualitative understanding is difficult to achieve. Indeed, it is very challenging to provide the direct physical insights behind the numerical analyses in complex nonlinear systems.
Different from soliton explosions, soliton interactions in general behave like particles. So far, soliton interactions have been discovered as ubiquitous effects accounting for various exotic nonlinear dynamics, including rogue waves [14][15][16] , Fermi-Pasta-Ulam paradox 17 , and many others 18,19 , as well as for many nonlinear phenomena, including soliton fusion 20 , soliton fission 21 , soliton annihilation 22 , and partial annihilation 23,24 . It seems a reasonable assumption that soliton explosion itself has little to do with soliton interactions. For instance, although soliton collisions were widely investigated in laser optics [25][26][27][28] , soliton explosions have not been found in these investigations.
In this work, we experimentally study explosions of dissipative solitons in a mode-locked fibre laser, and demonstrate a conceptually different type of soliton explosions induced by soliton collision. By solely increasing the pump power, at least seven  distinct nonlinear regimes are discovered successively including  stable dissipative solitons, standard soliton explosions, noise-like  mode locking, stable double dissipative solitons, soliton collision induced explosions, soliton molecules, and double-pulse noiselike mode locking. We observe soliton collision in a laser cavity, and show that it leads to soliton explosions. DFT technique is implemented to capture these transient laser dynamics. Our experimental exploration proves that soliton explosion closely relates to soliton interactions.

Results
Standard soliton explosions. The laser setup is shown in Fig. 1 (see 'Methods' for details). The fibre laser was mode-locked via nonlinear polarization rotation, which could be realized by tuning the polarization controllers in the cavity. In order to study the laser dynamics in the same polarization states, we fixed the polarization controllers once mode-locking was initiated. Only the pump power was adjusted to determine its influence on the laser dynamics. Stable dissipative solitons were established as the pump power reached the mode-locking threshold (~129 mW under our experimental configuration). Figure 2a depicts the pulse spectra measured by DFT, indicating no soliton explosions. As shown in Fig. 2b, the pulse spectra measured by DFT agree well with those attained with a commercial optical spectrum analyzer (OSA), confirming the accuracy of DFT employed in the experiment. The pulse duration is 6.4 ps as measured by an autocorrelator (FR-103 XL) and the spectral width is 25 nm, indicating that the pulse is highly chirped, a distinguished feature of mode-locked normal dispersion fibre lasers 29 . Standard soliton explosions were observed when the pump power was increased to 155 mW. Figure 2c shows the real-time spectra dynamics in the regime of soliton explosions. As one can see, the stable spectra undergo drastic changes suddenly, but recover to its original state after more than~10 4 round trips (RTs). Such transitions are shown in Fig. 2d, which displays the cross-sections of the stable and chaotic spectra. The spectra of the exploded solitons broaden significantly and present several peaks, indicating that self-phase modulation could be involved in soliton explosions in this case. Soliton explosion events occurred more frequently when the pump power was further increased. One example is shown in  Remarkably, stable double pulses were observed when the pump power was increased to 188 mW. The corresponding spectra measured by DFT are also stable. There are no soliton explosions in this case. The transition from noise-like mode locking to stable double pulsing by solely pump power increasing is found for the first time here. The origin is beyond the scope of this work. The two pulses are temporally separated by 19 ns as shown in Fig. 3.
Soliton collision induced explosions. The double pulses kept stable until the pump power was boosted to 228 mW. Remarkably, a novel type of soliton explosions was observed under this pump power. The results are plotted in Fig. 4. Figure 4a shows the temporal intensity evolution of the laser outputs over consecutive RTs. As shown, two pulses (parent pulses) initially separated by 260 ps, attract each other before merging to be a 'single pulse'. Three representative cross-sections are shown in Fig. 4b, at a RT number of 5000, 10,000 and 15,000, respectively. It is beyond the resolution of the photodetector to resolve the fine structures of the exploded pulse (the 'single pulse') in the temporal domain.
However, DFT-measured spectra help to confirm and characterize soliton explosions in the spectral domain 9,11,13 . The synchronous spectra measured by DFT is presented in Fig. 4c. It shows that the spectrum of the exploded pulse is broad and chaotic. A representative cross-section at a RT number of 10,000 is shown in Fig. 4d (red). Interference patterns presenting on the spectra before the occuring of soliton explosion indicate that the two pulses are coherent. A cross-section is shown in Fig. 4d (black) at a RT number of 5000. The modulation on the spectrum is magnified in Fig. 4e. The period is 0.063 nm, corresponding to a temporal separation of 130 ps between the two pulses. Such a separation is in good agreement with the one (124 ps) shown in Fig. 4b (black), given the limited resolution of the photodetector. Moire fringes are also shown on the spectra as seen in Fig. 4c since the period of the modulation on the spectra is extremely small.
To investigate the details of the soliton collision process, Fig. 4c is magnified from the RT number of 8500 to 9000, as depicted in Fig. 5a. Three representative cross-sections are shown in Fig. 5b, at a RT number of 8600, 8850, and 8900, respectively. Note that the related temporal intensity measurements shown in Fig. 4a fail to resolve the temporal dynamics due to limited resolution (33 ps). Fortunately, field autocorrelation traces shed light on the temporal dynamics. A field autocorrelation trace can be obtained through the Fourier transform of each single-shot spectrum using the Wiener-Khinchin theorem. This method has been used to probe the evolving soliton separation within soliton molecules 24,31,32 and reveals the ordering of incoherent dissipative solitons 13 . If the number of pulses is n then the corresponding field autocorrelation trace gives 2n-1 peaks. The Fourier transforms of the spectra (Fig. 5a) yielded the field autocorrelation traces shown in Fig. 5c. The field autocorrelation traces reveal that the two pulses attract each other before explosion. Well-defined interference patterns are present on the spectra from the RT number of 8500 to~8800. An example is shown in Fig. 5b at a RT number of 8600. The patterns become unstable between the RTs of~8800 and 8900, but modulations are still present on the spectra. An example is shown in Fig. 5b at a RT number of 8850. The corresponding field autocorrelation traces show noisy structures. This indicates that fine structures appear on the two pulses. Finally, the spectra become chaotic starting from the RT number of 8900 (Fig. 5a). A cross-section is shown in Fig. 5b at that round trip. This means explosions of the two pulses. The corresponding autocorrelation traces also indicate that the two pulses with fine structures attract each other to form a single complex. The energy evolution gives additional insights on the dynamics, as shown in Fig. 5a (the white solid line). The energy evolution of the whole process is shown in Fig. 4a. The energy suddenly increases once soliton explosion takes place around the RT number of 8900 (Fig. 5a). The energy decreases afterwards due to two dissipative processes. On the one hand, the spectra of the exploded solitons are very broad, hence the limited gain bandwidth of erbium-doped fibre (EDF) (gain filtering) dissipates the pulse energy and gives rise to narrower spectra as shown in Fig. 5a after the RT number of 8900. On the other hand, soliton explosion means a soliton explodes to pieces in the temporal domain, and therefore the weak components suffer from losses from the saturable absorber (nonlinear polarization rotation) which imposes large loss on the weak pulses. Such intensity-dependent loss is crucial for the build-up of a dissipative soliton in a mode-locked fibre laser, which dissipatives the weak A second pulse suddenly appears around a RT number of 13,000, as shown in Fig. 4a (dashed box). A magnified version is exhibited in the inset of Fig. 4a (top right). More detailed investigations are provided in Fig. 6a, b which are the close-up versions of Fig. 4a, c between the RT number 12,000 and 14,500. The gradual growth of the second pulse is depicted in Fig. 6b before a RT number of 13,000. Since the energy of the exploded pulse decreases (Fig. 4a, the white line) due to dissipative processes as discussed above, the remaining gain of the laser is transferred to a new pulse (the second pulse). This process illustrates the revival of the double solitons. Figure 6a depicts that there are mainly three types of spectra in the figure (denoted as '1', '2', and '3'). The spectra are broader in the beginning (RTs 12,000 to~13,000, stage 1), but become narrower later (RTs 13,000 to 14,000, stage 2). This could be understood from the temporal dynamics as shown in Fig. 6b. In stage 1, there are one strong pulse and a growing weak pulse, while there are two pulses with equal energies in stage 2 (the weak pulse grows up). Most of the energy is owned by the strong pulse, therefore self-phase modulation is stronger in stage 1, giving rise to broader spectra. Typical cross-sections are shown in Fig. 6c, d, at RT numbers of 12,900, 13,500 and 14,500, respectively.
Although there are two pulses between the~15,000 and 25,000 RTs in Fig. 4c, no spectral interference patterns are observed. A cross-section is shown in Fig. 4d (blue), at a RT number of 15,000. This is because the spectral patterns are too dense to be resolved by DFT which has a resolution of 0.025 nm (see 'Methods'). Such a spectral resolution cannot resolve the spectral interference patterns of double pulses with a separation larger than 326 ps.
Further increasing the pump power resulted in the generation of bound solitons which are usually termed as soliton molecules 33,34 . Various internal dynamics of soliton molecules have been experimentally revealed using DFT to measure the evolving spectra patterns of soliton molecules 31,32,35 . A small portion of the spectra is shown in Fig. 7a. Plotting the whole spectra makes the patterns invisible because the period of the patterns is extremely small (0.07 nm) compared to the whole spectra (25 nm). An example of the whole spectra measured by DFT is displayed in Fig. 7b in which the period of the pattern is not visible. The period becomes visible as depicted in Fig. 7c, by magnifying the spectrum from 1576 to 1578 nm in Fig. 7b. Although the spectra changes in Fig. 7a, the period of the patterns is fixed at 0.07 nm, which is also confirmed by the field autocorrelation traces. The field autocorrelation traces in Fig. 7d, show that the temporal separation is fixed at 120 ps. Therefore, the evolving spectra patterns imply that the phase is changing. Similar dynamics have been reported elsewhere 31,35 . Interestingly, the soliton molecule transferred to stable double solitons separated by 19 ns, exactly the same state shown in Fig. 3, by increasing the pump power. Increasing the pump power from this state, the two coherent pulses became double noise-like pulses, but their separation remained fixed at 19 ns. Further investigation is limited by the available pump power from the laser diode.
Our results show the rich nonlinear dynamics embedded in mode-locked fibre lasers. Seven distinct nonlinear regimes are observed by solely increasing the pump power. Dissipative solitons are established above the mode-locking threshold. Dissipative solitons explode as the pump power is controlled a little bit above the mode-locking threshold, and the exploded solitons evolve to the original states after~10 4 RTs in the cavity. The life time of an exploded soliton depends upon the pump power. A higher pump power gives rise to a longer time in which the exploded solitons circulate in the cavity. Noise-like modelocking is observed when the pump power is increased. As the pump power is further increased, stable double solitons are formed. As the laser gain is increased even further, double solitons become unstable. Soliton collision occurs in the fibre laser cavity, and it induces soliton explosions. Soliton molecules are formed by slightly increasing the pump power, without soliton collision or explosion. Finally, increasing the pump power makes the soliton molecule transfer to double noise-like pulses.  Fig. 7 The dynamics of soliton molecules. a The evolving spectra patterns measured by dispersive Fourier transformation. Note that only a small portion of the spectra is shown as plotting the whole spectra causes the patterns invisible. b An example of the whole spectrum shows very dense peaks. c The period is visible by magnifying the spectrum from 1576 to 1578 nm in b. d The field autocorrelation traces calculated from the whole spectra evolution show that the separation between the solitons is stable (120 ps). e A cross-section in d Discussion Soliton interactions can be divided to short-and long-range interactions depending on the separation between solitons. Direct soliton-soliton interaction accounts for short-range interaction when solitons are separated by several times their width [36][37][38] . Long-range interactions are mediated by different mechanisms, including dispersive waves [38][39][40] , acoustic effects [41][42][43] , and gain depletion and recovery 44,45 . The attractive interactions shown in Fig. 4a refer to long-range interactions as the two solitons are initially separated by 260 ps which is forty times the pulse width; the separation is almost one hundred times the width at the RT number 15,000 (~600 ps). Dispersive waves can be neglected here as they generally do not present in mode-locked normal-dispersion fibre lasers. Acoustic effects make two solitons separated by 260 ps repel each other to a stable separation (510 ps) 42 . However attractive interactions are observed in Fig. 4a, implying that the acoustic effects can also be excluded here. The attractive soliton interactions could arise from gain depletion and recovery. A leading soliton depletes the gain of a laser, resulting in less gain for a tailing soliton until it is recovered to the value before the leading soliton. Such gain dynamics makes two solitons attract (repel) each other if their separation is shorter (longer) than the recovery time 45 . Here, the initial 260-ps separation between the two solitons is much shorter than the recovery time (~ns) 44,45 , resulting in attractive interactions. Note that the recovery time here is different from the standard gain recovery time of EDF which has a typical value of ms 46 .
Finally, we believe that soliton collision induced explosions could also prevail in various laser systems and beyond. In addition, the pieces shed by exploded solitons could relate to turbulence, constituting an excellent platform of wave turbulence investigation. We therefore expect our results to pave the way for extensive investigations on these significant nonlinear phenomena.

Methods
The laser setup. The total cavity length is 10.5 m, consisting of dispersioncompensating fibre (DCF), single-mode fibre, and erbium-doped fibre (EDF), with the corresponding group-velocity dispersion (GVD) of 65.0, 62.5, and -22.8 ps 2 / km, respectively. The net dispersion of the laser is normal. As shown in Fig. 1, the output of the laser was split into two ports by an optical coupler. One port (undispersed) was used to measure the evolution of the instantaneous intensity I(t). The other port was stretched in time domain by using a long dispersive fibre (~11 km in our experiments), in order to imitate its pulse spectrum in real-time according to the stretched temporal waveform via DFT 7,47 . The two identical photodetectors (PD1, PD2) have a bandwidth of 50 GHz, and the signals were measured by a real-time oscilloscope (33 GHz,Agilent). Note that by measuring the delay between the two photodetectors (53.651 µs), synchronous measurements of the spectral and temporal intensities of the output pulses can be realized. In particular, DFT facilitates direct measurements of round-trip resolved spectra and accordingly enables us to reveal various fast laser dynamics 8,13,31,32,35,[47][48][49] .
The spectral resolution of DFT. The spectral resolution (Δλ) of DFT technique is determined by Δλ = Δt/DL. Δt is the response time of the system (30 ps), while D and L are the dispersion parameter and the length of the fibre used (DL = 1200 ps nm −1 ). The resulting resolution of DFT is 0.025 nm in our experiment. Time to wavelength conversion can be calculated approximately by multiplication of pulse spectral width and fibre dispersion (DL). The stretched pulses have a duration around 30 ns in our experiments.
Reporting summary. Further information on experimental design is available in the Nature Research Reporting Summary linked to this article.

Data availability
The data that support the findings of this study are available from the corresponding author on request.