1. Introduction

History of monstrous waves in the ocean that appear and disappear without a trace goes back hundreds of years, having been remained only as lore among sailors to prove their existence until a scientific observation verified that these waves are indeed the product of nature and sporadically observed in the deep sea environment [1]. These extreme waves, most commonly known as rogue waves (RWs), attracted a lot of research attention initially in the field of oceanography so as to comprehend and overcome their disastrous and unpredictable nature, thereby assisting human activities in the ocean [2–8]. As the RW research in the ocean was mostly stochastic and large-scaled, it was time consuming and required expensive and bulky experimental settings. Therefore, RW research in platforms other than the ocean started to draw a considerable amount of interest, opening up a possibility for a smaller-scaled study, which include optical platform as well [9–13]. In fact, optical platforms, in which light waves behave analogous to ocean waves in an incomparably faster and smaller scale, make RW research possible for greater science community. Since Solli et al. first reported optical RWs in optical-fiber-based supercontinuum generation [14], research interest in optical RWs has been propelled and spread to various other optical configurations [15]. In particular, a number of RW studies have been conducted in passively mode-locked fiber lasers (PMLFLs) in recent years [16–21]. PMLFLs are a compact, flexible, and efficient platform known to generate stable ultrashort pulses [22–28]. However, PMLFLs are, on the other hand, a complex nonlinear dissipative system, so that if forced to operate in an extraordinary condition such as in an excessively pumped cavity, they exhibit a chaotic or stochastic nature of photon dynamics, operating in quasi-mode-locked (QML) regimes [29–37]. This indicates that they may become a favorable platform for observing RWs [38].

In fact, among various types of QML regimes, the multi-soliton (MS) regime is the most probable candidate for exploring RWs, because it provides quasi-stable but well-defined pulse dynamics in contrast with the other QML regimes in which pulses are likely to behave in a form of wave packets [39]. The MS regime is a soliton complex in which multiple solitons are present in the cavity simultaneously, so that in comparison with single-soliton regime, its behavior and dynamics are substantially more nonlinear and disordered [40]. The complexity arises from the fact that the cavity can hold tens or even hundreds of solitons at the same time depending on its precise condition and configuration [41]. Individual solitons of such a large number do not necessarily remain independent in the cavity, being able to interact with one another in many cases [42]. Moreover, dispersive waves shed from soliton-soliton interaction make the pulse dynamics in the cavity even more complex [43–47]. Consequently, even slight a change in the given cavity condition may lead to the pulse dynamics in the MS regime turning into a drastically different state having extraordinary characteristics as noticed in [48]. This aspect elucidates it can serve as a promising platform for studying extreme events therein. Recently, it has been reported that soliton-collision and soliton-focusing mechanisms are mainly regarded responsible for the generation of RWs in dissipative fiber-optic systems [47,49,50]. However, due to the chaotic and random nature of RWs, analysis of the phenomenon has not been trivial, so that it still requires extensive research efforts, especially in the MS regime of PMLFLs that has yet to be investigated thoroughly as a RW-generating platform.

In this paper, we report for the first time extraordinary RWs intermittently triggered with an interval in the order of no longer than ten seconds in the MS regime of a nonlinear-polarization-rotation (NPR)-mode-locked, anomalous-dispersion fiber ring cavity. We call them “super RWs” as their intensities exceed over ten times the average intensity of all events in the cavity. With fine adjustment of the cavity saturation power, we make the laser cavity operate in an extremely narrow window of a heavily populated MS state, in which soliton interactions can grow rapidly in a cascade of interactions mediated by dispersive waves. We investigate this specific MS regime by statistically analyzing its shot-to-shot temporal evolutions in comparison with the dynamic changes observed in its optical spectrum. We eventually capture the moment of intensifying solitons mediated by accompanying dispersive radiation of a shorter wavelength (relative to the soliton wavelength) generated upon soliton-soliton interaction in the cavity, which culminates to a super RW burst. Based on systematic, spectral and spatio-temporal measurements, we verify that the gradual and abnormal increase of the Kelly sideband on account of the dispersive radiation coincides with the elongation of the histogram distribution of the pulse events towards more extreme events, and that their intermittent growth and collapse exactly match the cycle of the super RW burst. We stress that the origin of the super RW burst in the MS regime is strongly correlated not only with soliton-soliton interaction but also with the initial build-up and strong localization of accompanying dispersive radiation generated upon the soliton-soliton interaction and the cascade of their interactions with nearest neighboring solitons. We unveil and discuss the further details in the following.

2. Experimental arrangement

Figure 1 shows the schematic of the PMLFL based on NPR, which is, in fact, identical to the one used in [48]. The total cavity length, including 5 m of erbium-doped fiber (EDF), was ~120 m, and the net cavity dispersion was anomalous such as β₂ ~−2.95 ps². At low pump power (< ~65 mW), the cavity operated in the single-soliton regime, whereas it operated in QML regimes including the MS regime at higher pump power [48]. We measured its optical spectrum and spatio-temporal characteristics up to 837 consecutive roundtrip times using an optical spectrum analyzer (AQ-6315B, Ando/Yokogawa) and a high-speed oscilloscope having an ~80-ps resolution (DSO91204A, Agilent: 12-GHz bandwidth and 40-GS/s sampling rate) via a ultra-fast photodetector having a 45-GHz bandwidth (Model 1014, New Focus), respectively. See [48] for more details.

 

Fig. 1 Experimental setup for a passively mode-locked EDF ring laser cavity based on NPR.

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3. Experimental results

3.1. The stable MS regime

Prior to investigating extreme events out of the given laser cavity, we first locked the laser oscillation in the cavity into a typical MS regime with ~410 mW of pump power. In this regime the cavity produced ~470 solitons irregularly spaced within a single roundtrip time, in which solitons were sufficiently apart that there were no noticeable interactions among them. This regime resulted in an optical spectrum with typical Kelly sidebands and a single-spike autocorrelation trace, as shown in Figs. 2(a) and 2(b). The 3-dB bandwidth of the optical spectrum was ~4 nm, and the duration of an individual soliton pulse in full width at half maximum (FWHM) was ~3.1 ps. In Figs. 2(c) and 2(d) we also plot the spatio-temporal measurement results, showing their shot-to-shot temporal evolutions up to 837-roundtrip times, which was the maximum number of roundtrip times that our high-speed oscilloscope could continually measure. From them one can see that the temporal separations between two adjacent solitons were at least more than ~1 ns, so that solitons could hardly interact with each other in the cavity. In Fig. 2(e) we also plot a histogram representing an intensity probability density function (IPDF), counting all the individual soliton pulses generated in the cavity over the period of 837 roundtrips with respect to their intensities normalized relative to the average intensity of them all. It is noteworthy that there were no outliers of high intensity events that might form a long tail to the right, which means appearance of RWs never happened in this regime. Thus, we hereafter call this MS regime the “stable” MS regime.

 

Fig. 2 Characteristics of the stable MS regime: (a) optical spectrum; (b) autocorrelation trace; (c) spatio-temporal measurement result in a 60-ns span; spatio-temporal measurement result in a 20-ns span; (e) IPDF for the whole pulse events in the cavity.

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3.2. The breathing MS regime with an intermittent super RW burst

Having reported in our pervious study [48], we could lock the laser oscillation in the given cavity in a number of different regimes having distinct characteristics by means of varying the saturation power of the cavity, i.e., by means of tweaking the PCs, while keeping the pump power constant. It is noteworthy that in cavities based on NPR even a slight alteration of the saturation power of the cavity could change the intra-cavity pulse dynamics in a drastic way [48]. With a fine adjustment of the PCs while keeping the pump power at the same level as that of the stable MS regime, i.e., at ~410 mW, we could make the cavity operate in a different-type MS regime, heavily populated with ~790 solitons within a unit roundtrip time, which was almost double the number of solitons observed in the stable MS regime. At a glance, this specific MS regime did not seem very different from the stable MS regime such that its optical spectrum still maintained the characteristic features of the MS regime [48]. However, we eventually noticed that in this specific regime, one of the Kelly sidebands was extraordinarily strong and gradually growing until it reached a maximum point, and collapsed back to its initial state, which intermittently recurred within a time scale in the order of ten seconds. Thus, we hereafter call this specific regime the “breathing” MS regime. In Fig. 3(a), we illustrate the gradual change of the optical spectrum within a unit cycle of the breathing MS regime, breaking it into five different states. [An alternative graphical illustration of Fig. 3(a) is also provided in a separate file: Visualization 1, where the optical spectra measured at different times are individually illustrated on a linear scale.] In particular, one can see that during the cycle, the utmost Kelly sideband at the shorter wavelength side relative to the soliton wavelength grew by nearly five times from State 1 to State 5 before collapsing and restarting the cycle, and that the center wavelength of the solitons and the wavelengths of the Kelly side bands drifted significantly, as shown in Fig. 3(b). The oscillation of the utmost Kelly sideband and the wavelength drifts of both solitons and dispersive waves imply that they interacted heavily in the cavity [43–51], and that the pulse dynamics going on in the cavity might be drastically different from those of the stable MS regime, although the regime still exhibited the characteristic feature of the typical MS regime without forming a wave packet [48,51]. In Fig. 3(c) we plot the IPDFs regarding State 1, State 3, and State 5 of the breathing MS regime, relying on the spatio-temporal measurements up to 837 roundtrip times. One can interestingly see that the tail of the histogram was gradually elongated with the progression of the state. In particular, at State 5 the intensity levels of extreme events could exceed twice the average intensity of all. This suggests that the higher the level of the Kelly sideband became, the more intensively the extreme events took place, possibly resulting from the escalated nonlinear interactions among the intra-cavity pulses in complex forms as discussed in the previous studies [43–51].

 

Fig. 3 Characteristics of the breathing MS regime: (a) optical spectrum; (b) zoomed-in view of the Kelly sidebands encircled by the dashed line; (c) IPDFs for State 1, State 3, and State 5.

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Moreover, we noticed that pulse interaction was becoming more intense and frequent as the state progressed from State 1 to State 5, and eventually managed to capture the most striking feature such that after the completion of every cycle of the utmost Kelly sideband’s growth, the cavity went off a burst of a super RW. In a blink of time when the growing Kelly sideband reached State 5, a super RW turned up immensely and disappeared quickly. In Fig. 4 we illustrate the spatio-temporal measurement result and the corresponding histogram of the pulse dynamics upon this burst of a super RW. While our measurement was resolution-limited in looking at even finer pulse dynamics inside the super RW than ~80 ps, one can clearly see strong localization of extreme events apparent within the time window from −5 to 0 ns, as shown in Fig. 4(a). The sudden burst emerged out of nowhere and was starting to disperse swiftly. [An alternative graphical illustration of Fig. 4(a) is also provided in a movie version available in a separate file: Visualization 2.] Looking at the histogram shown in Fig. 4(b), i.e., the IPDF for the whole pulse events during the super RW’s emergence, we could even observe super extreme events having their intensity levels exceeding more than ten times the average intensity of all. Notwithstanding, we note that our spatio-temporal measurements were resolution-limited to ~80 ps. This means that highly localized extreme events might have been measured as if they had been encompassed in a bunch within the window of the resolution limit, so that we have to bear in our mind a possibility that the real peak intensity level of the utmost extreme event comprising the super RW might have been over- or underestimated to some degree.

 

Fig. 4 Intermittent super RW burst: (a) Spatio-temporal measurement result and (b) IPDF for the whole pulse events in the cavity.

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4. Discussion

From the experiments conducted and discussed in the preceding section, we have figured out that the extreme events could be triggered and intensified inside the cavity via going through a cascade of vivid pulse interactions without having any external modification or adjustment of the cavity parameters. In contrast with the case of the stable MS regime, where the intra-cavity solitons hardly interacted with one another, in the breathing MS regime the substantial growth of the soliton population inside the cavity, which subsequently reduced physical spacing among the intra-cavity solitons, exponentially escalated the probability of interactions among them directly or indirectly via dispersive waves. We stress that the role of dispersive waves cannot be underestimated, because they can readily be generated upon soliton-soliton interaction [44], and can also lead not only to soliton fission [45] and but also to soliton fusion [46] upon interaction with solitons, depending on their phase mismatch [43,47]. Thus, one can think of a plausible scenario for the intermittent burst of a super RW based on our spatio-temporal measurement result as the following: Initially, some of the solitons generated in a place where their population density was most immensely high, might start to interact with the nearest neighboring ones, resulting in the higher interactions among them and also producing the more intense dispersive waves upon their interactions [51–56]. The combination of these probabilistically initiated soliton interactions and subsequently accompanying dispersive waves in their vicinity as a catalyst, in turn, triggered another extreme event and so on, eventually leading to an avalanche of interactions resulting in a super extreme event with substantially higher intensity. In fact, the increasing trend in the extreme events observed in the histograms shown in Fig. 3(c) was completely in accord with the growing trend in the utmost Kelly sideband shown in Fig. 3(a), which supports our aforementioned scenario. In other words, once the build-up process driven by the initial soliton interactions had passed a certain threshold point, it became unstoppable, thereby going off a super RW burst and eventually being settled down into its initial state. Once there was the burst of a super RW in the cavity, it could sweep as much stored energy in the cavity as possible and subsequently dispersed into the calmed-down state, i.e., the initial state of the breathing MS regime, because it was by no means a form of wave that the cavity could stably support in the given condition and configuration. The settlement into the initial state of the breathing MS regime means that the cavity again returned to a state of excessively populated and irregularly distributed solitons within the fixed length of the cavity. Thus, this circumstance must still bear a very high chance of probabilistic initiation of casual soliton interactions. Therefore, the whole process of the burst of a super RW could again be triggered probabilistically, thereby being intermittent fully depending on the initial formation of the excessive number of solitons as well as the ambient conditions. In addition, we note that the burst of a super RW was a completely independent, stochastic event turning up out of nowhere, since one can see in Fig. 4(a) that even when it was happening in the middle of the cavity, it seemed as if all the intra-cavity solitons placed distant to it except for the very nearby ones had not known what was going on out there until it physically reached them.

In Fig. 5 we illustrate the spatio-temporal measurement result at the very specific moment of the burst of a super RW: Fig. 5(a) depicts it in the original color scale whereas Fig. 5(b) depicts it in the adjusted color scale such that the background events with relative intensities below 0.02 were deliberately fixed to a dark color for visual aids. In Fig. 5(a), one can see that there had been notable soliton fission events at roundtrip time of ~360 and at roundtrip time of ~180 (see the locations pointed out by arrows) just before the most extreme events occurred. Since the intensity levels of the solitons were much lower than those of the most extreme events, they were depicted in too dark colors in Fig. 5(a), so that it is very hard to resolve the detailed dynamics around the soliton fission events. Thus, in Fig. 5(b) we adjusted their brightness along with fixing the background events with relative intensities below 0.02 to a dark color. In particular, the area encircled by the dashed line indicates the strong localization of the moderate extreme events right before the super RW burst. Interestingly, this new graphic reveals that the soliton fission events appeared to be caused by the interaction with the dispersive radiation started from the soliton-soliton interaction positioned on their right side, which also indicates that the wavelength of the dispersive radiation should be considerably shorter than that of the solitons. As reported in [45], such dispersive waves can surely cause soliton fission upon collision. In particular, the soliton fission event at roundtrip time of ~180 was so significant that it triggered an avalanche of dispersive radiation and subsequent soliton fission roundtrip after roundtrip, eventually leading to the burst of a super RW. We note that in this situation the excessive number of solitons along with the escalated background dispersive radiation might also lead to soliton fusion events as discussed in [45], which would drastically increase the probability of extreme events. Such extreme soliton fission and fusion events would in turn produce a substantial amount of dispersive radiation, which should be signified by extraordinary growth of the utmost Kelly side band. We note that the traces of the radiation initially dispersed from the soliton-soliton interaction in the dashed-circle area are asymptotically running in a slanted way towards the top-left corner, which suggests that they were travelling significantly faster than the background solitons by ~5.8 ps per roundtrip. In fact, this speed difference matches well with the wavelength difference between the main solitons centered at ~1552.5 nm and the growing Kelly sideband positioned at ~1550 nm, considering that the net cavity dispersion was given by β₂ ~−2.95 ps². With these intense accompanying dispersive waves as a catalyst, an avalanche of extreme events started to occur, finally leading to strongly localized extreme events, i.e., the burst of a super RW, sweeping as much stored energy in the cavity as possible. It is noteworthy that the saturation power of the cavity remained unchanged throughout the entire cycle of the super RW burst, so that this extraordinary wave could not survive long in the cavity, and thus dispersed quickly. After all the energy from the super RW had been released and spread into the cavity upon its collapse, the cavity state became to return to its initial state to start the cycle again. We note that since the high-speed oscilloscope used in our experiment could only trace pulses up to 837 round-trip times, we were not able to record the whole event continuously to the end of its cycle. However, we could observe that the super RW swiftly dispersed to the background level, and the cycle intermittently restarted. Upon the disappearance of the super RW, some extreme waves started to build up among a new set of solitons in the cavity. Since they were irregularly distributed in the cavity and changing with time, it was hard to tell whether they were from the same set as the previous one or from another soliton set.

 

Fig. 5 Burst of a super RW mediated by dispersive waves: Spatio-temporal measurement result (a) in the original color scale and (b) in the adjusted color scale such that the background events with relative intensities below 0.02 were deliberately fixed to a dark color for visual aids. In (a), the arrows point out the soliton fission events triggered by dispersive waves.

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We stress another compelling point from our experimental observation, which was the fact that a unit cycle of a super RW’s emergence was in the order of no longer than ten seconds although it was not exactly periodic but intermittent. While we do not completely rule out that this type of time scale might be related with thermal relaxation of the cavity or the ambient environment [57,58], we suspect this possibility is quite unlikely, because the experiment was conducted in fully temperature-controlled environment. Since we clearly observed from the spatio-temporal measurement that the burst of a super RW was initiated by fully stochastic events, we reckon that the time scale for the unit cycle of a super RW burst was probabilistically determined, depending on the population density of solitons. In fact, we have noticed in our previous numerical study reported in [42] that there can exist a transitional or intermediate state of multiple pulses that cannot be categorized into the three constitutive QML regimes, i.e., the noise-like pulse, symbiotic or MS regimes, because it does not clearly exhibit a quasi-stationary feature. In fact, such a non-quasi-stationary feature resembles the intermittent change of the complex pulse dynamics observed here. Whilst a more thorough theoretical and numerical study on this matter should follow in the future study, the experimental results and evidences presented here provide a physical insight that the dispersive-wave-mediated soliton interaction plays a significant role in triggering a super RW in the breathing MS regime.

5. Conclusion

We have experimentally observed and analyzed the intermittent burst of a super RW in the MS regime of an anomalous-dispersion fiber ring cavity mode-locked by NPR. We showed that adjusting the cavity to hold a greater number of solitons drastically increased the natural probability of interactions among neighboring solitons, which, in turn, stimulates dispersive waves to take place. Combination of these probabilistically initiated soliton interactions and accompanying dispersive waves in their vicinity functioning as a catalyst, eventually led to an avalanche of nonlinear interactions, such as soliton fission, soliton fusion, and cascade dispersive radiation, thereby resulting in super extreme events with even higher intensities, i.e., the burst of a super RW. We verified all these extreme events through spatio-temporal measurement and analysis, and also confirmed the exact match between the interval of the utmost Kelly sideband’s growth and collapse and the cycle of the burst of a super RW. This feature was also clearly reflected in the IPDF for the whole pulse events in the cavity. The intermittent recurrence of the burst of a super RW in the breathing MS regime manifests that even though the heavier soliton population might naturally push the cavity to triggering more and more pulse interactions and thereby the more extreme events, the pulse dynamics in the cavity kept attempting to go back to the initial state with the lowest order of pulse interactions, because such solitons with excessive intensities inevitably result in even faster break-up and collapse [47]. Our experimental work presented here should provide richer perspective on the study of RWs not only in the field of optics but also in other disciplines [59–61].