Rogue waves generation via nonlinear soliton collision in multiple-soliton state of a mode-locked fiber laser

We report for the first time, rogue waves generation in a mode-locked fiber laser that worked in multiple-soliton state in which hundreds of solitons occupied the whole laser cavity. Using real-time spatio-temporal intensity dynamics measurements, it is unveiled that nonlinear soliton collision accounts for the formation of rogue waves in this laser state. The nature of interactions between solitons are also discussed. Our observation may suggest similar formation mechanisms of rogue waves in other systems. © 2016 Optical Society of America OCIS codes: (140.3510) Lasers, fiber; (140.4050) Mode-locked lasers; (060.5530) Pulse propagation and temporal solitons. References and links 1. N. Akhmediev and E. Pelinovsky, “Editorial–Introductory remarks on “Discussion & Debate: Rogue waves– towards a unifying concept?”,” Eur. Phys. J. Spec. Top. 185(1), 1–4 (2010). 2. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007). 3. N. 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Vol. 24, No. 19 | 19 Sep 2016 | OPTICS EXPRESS 24256 #267420 http://dx.doi.org/10.1364/OE.24.021256 Journal © 2016 Received 31 May 2016; revised 16 Aug 2016; accepted 29 Aug 2016; published 6 Sep 2016 18. F. Amrani, A. Haboucha, M. Salhi, H. Leblond, A. Komarov, and F. Sanchez, “Dissipative solitons compounds in a fiber laser. Analogy with the states of the matter,” Appl. Phys. B 99(1–2), 107–114 (2010). 19. F. Amrani, M. Salhi, P. Grelu, H. Leblond, and F. Sanchez, “Universal soliton pattern formations in passively mode-locked fiber lasers,” Opt. Lett. 36(9), 1545–1547 (2011). 20. F. T. Arecchi, G. Giacomelli, A. Lapucci, and R. Meucci, “Two-dimensional representation of a delayed dynamical system,” Phys. Rev. A 45(7), 4225–4228 (1992). 21. M. Marconi, J. Javaloyes, S. Barland, S. Balle, and M. Giudici, “Vectorial dissipative solitons in vertical-cavity surface-emitting lasers with delays,” Nat. Photonics 9(7), 450–455 (2015). 22. F. Gustave, L. Columbo, G. Tissoni, M. Brambilla, F. 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Introduction
Rogue waves (RWs) have drawn widespread interest in physics systems ranging from hydrodynamics to superfluidity [1].They have extremely large amplitude and seem unpredictable.In optics, RWs were observed for the first time in supercontinuum generation in optical fibers [2].This subject has been extensively studied in different optics systems since then [3], as these convenient optical experiments relate to giant waves in oceans and many other scientific fields [4].Mode-locked fiber lasers which have abundant physical effects including nonlinearity, dispersion, gain and loss, are excellent platforms for nonlinear sciences investigation such as RWs.Indeed, recently, RWs were observed in mode-locked fiber lasers under different mode-locking states [5][6][7][8][9][10][11][12].As well known, there are several mode-locking states in fiber lasers.Soliton bunch is a state in which many solitons bound into a group [13][14][15].This state provides the possibility to simplify the burst-mode amplifier systems [16,17].Depending on the laser conditions, the solitons in the bunch can stay static, or interact with each other resulting in chaotic soliton bunch.Chaotic soliton bunch state can induce RWs resulting from interaction of solitons [5,8], but it is a challenge to identify experimentally how solitons interact mutually to generate RWs.
Besides soliton bunch, there is a mode-locking state in which multiple solitons spread across the whole cavity; for example, a total number of 380 solitons per roundtrip were observed in high-power double-cladding erbium-doped fiber lasers, and this state was dubbed as "soliton gas" [18,19].Obviously, there are interactions between solitons in this modelocking state.Whether RWs exist in such state is still an open question.
Spatio-temporal intensity measurements have been widely used in various systems to trace the features otherwise hidden in the one-dimensional intensity measurement [20], including discovering dynamics of vector dissipative solitons in vertical-cavity surfaceemitting lasers [21], uncovering the existence of dissipative phase solitons and their interactions with turbulent state in semiconductor lasers [22], unveiling the interactions between acoustic waves and temporal cavity solitons [23,24], and manipulating solitons [25][26][27].In the context of fiber lasers, such technique helped to observe the laminar-turbulent transition in a fiber laser [28], reveal non-trivial periodicity and long-scale correlations of radiation in partially mode-locked fiber lasers [29], observe soliton explosion in a passively mode-locked fiber laser [30], as well as investigate distinct regimes in quasi-CW Raman fiberlasers [31,32].
In this work, we report for the first time that RWs can be generated in another modelocking state of a fiber laser, namely the multiple-soliton state in which a total number of ~400 solitons occupied the whole laser cavity.By using spatio-temporal intensity measurements, the interaction processes between solitons can be captured and it is found that nonlinear soliton collision accounts for RWs generation in this mode-locking state.The formation and destruction processes of RWs are revealed.Instead of using high-power double-cladding fiber lasers, a long cavity enables us to access the multiple-soliton state with solitons spreading the whole cavity in a traditional single-cladding erbium-doped fiber laser, under low pump power.The laser setup is shown in Fig. 1(a).A 1.5-m long segment of EDF with nominal absorption coefficient of 52 dB/m at 976 nm was used as the gain medium, the dispersion of which is normal (−50 ps/nm/km).This fiber was pumped through a 980/1550 wavelength-division multiplexer (WDM) by a 976-nm laser diode.Nonlinearity of the laser was enhanced by introducing a 12-m SMF after EDF, which made the aforementioned multi-soliton state accessible under low pump power.An in-fiber polarization-dependent isolator (PDI) sandwiched with two polarization controllers (PCs), converted nonlinear polarization rotation to amplitude modulation, initiating and stabilizing mode-locked operation, and it also ensured single direction oscillation.A 10:90 fiber coupler was employed to tap 10% of laser power out of the cavity.The net dispersion of the laser cavity is + 12 ps/nm/km.

Experimental principle and setups
Here, the laser's spatio-temporal intensity dynamics, I(t, T), was characterized, instead of its one-dimensional intensity dynamics, I(t); that is the evolution of the instantaneous intensity pattern, I(t), over many-cavity roundtrips, T, was recorded.An 80-GSa/s real-time oscilloscope (Agilent DSOX93204A) together with a 50-GHz fast photodetector (Finisar XPDV2320R) were used to record spatio-temporal intensity dynamics.The temporal resolution of the measurement is ~30 ps limited by the bandwidth of the oscilloscope (33 GHz), and all the following measurements were based on this temporal resolution.It is to note that probability distribution functions (PDFs) shown in the followings were built following the way in [2].A time trace around 0.3 ms was used for all the PDFs in the followings, which gives 1.3 × 10 6 events in the soliton gas states; then the pulse intensities were used to build the histograms.The laser can work in different mode-locking states.In particular, stable soliton bunch can be observed at a pump power of 22 mW.The optical spectrum of this stable soliton bunch is shown in Fig. 1(b) (blue); Kelly sidebands on the spectrum are characteristics of solitons [33].The corresponding temporal traces on the oscilloscope are shown in Fig. 2(a).As seen in the figure, the soliton bunch is circulated in the laser at the fundamental cavity repetition period of 96.6 ns (10.35 MHz) corresponding to the total cavity length of 19.8 m.By magnifying the soliton bunch, it can be seen that there are six solitons in the bunch as shown in Fig. 2(b).The soliton bunch duration is 1.6 ns, which only occupies 1.66% of the cavity space.Figure 3 shows the corresponding soliton bunch intensity evolution over 8000 consecutive roundtrips.The evolution traces of the solitons show straight lines on the figure, indicating there are no interactions between solitons.The PDF is shown in Fig. 4(a).The histogram displays a quasirectangular shape in log scale, which does not show rare events.The intensity is normalized to the average intensity of the solitons.The significant wave height (SWH) defined as the mean amplitude of the highest third of the waves, is 1.09.Thus, RWs are not present in this stationary soliton bunch state, in comparison with the chaotic pulse bunch state in which pulses within the bunch move relatively and interact resulting in RWs generation [5].It is worthy to emphasize that the following experiments were carried out by increasing the pump power solely without changing in other laser parameters.

~200 solitons occupied the whole cavity exhibiting weak interactions
Starting from the aforementioned stable soliton bunch state, increasing the pump power created more solitons in the cavity.When the pump power was increased to 303 mW, the total number of solitons reached 186 and they occupied the whole cavity as seen in Fig. 5(a).This multiple-soliton state has been observed in high-power double-cladding fiber lasers [18,19].Figure 5(b) shows the magnified portion of Fig. 5(a).The corresponding optical spectrum is shown in Fig. 1(b) (red) the shape of which is nearly the same as the one of the stable soliton bunch state.
Figure 6 shows the corresponding multiple-soliton intensity evolution over 8000 consecutive roundtrips.The time span is 3 ns the same as that in Fig. 5(b) to see the evolution clearly (the remaining solitons within the roundtrip shows the similar evolution).It can be seen that there are weak interactions between solitons, indicated by the slightly bending evolution traces of the solitons, in contrast to Fig. 3 where the traces are straight.For example, the soliton around 0.9 ns, moves weakly with respect to the neighboring solitons.In addition, there are also solitons staying static, as a result of long temporal separations from other solitons.For instance, the evolution trace of the soliton around 2 ns, shows a straight line.The PDF of this mode-locking state is shown in Fig. 4(b).The histogram displays a quasi-rectangular shape close to that in Fig. 4(a).Moreover, the SWH is also 1.09.Again, there are no rogue waves in this state, as the interactions between solitons are very weak.

~400 solitons occupied the whole cavity exhibiting strong interactions
Further boosting the pump power only, the number of solitons continued to increase and highamplitude pulses appeared randomly on the oscilloscope.In the following, it will show that these high-amplitude pulses are RWs.As an example, the PDF of this regime is shown in Fig. 4(c), when the pump power is 730 mW.The histogram in Fig. 4(c) differs drastically from the other two in the figure.The shape is no longer quasi-rectangular and exhibits extended long tails corresponding to extreme events occurring rarely.The SWH is 1.23 in Fig. 4(c).Wave events of amplitude higher than twice the SWH are RWs [1,5].Here the threshold of RWs is 2.46.It can be seen from Fig. 4(c) that there are rare events above this threshold in the long tail.In contrast, Figs.4(a) and 4(b) do not show RWs.
Figure 7(a) displays an example of the temporal trace in the multiple-soliton state discussed above.A total number of 408 solitons occupied the whole cavity, which is more than two times the number of solitons in Fig. 5(a).Moreover, the solitons interacted with each other and thus the corresponding optical spectrum became coarse as seen in Fig. 1(b) (black).Figure 7(b) shows the temporal magnification portion of Fig. 7(a) from 64 to 67 ns, which covers the positon of the highest-amplitude RW.The autocorrelation trace is displayed in Fig. 7(c).There is large pedestal present in the figure, which is typical for this multi-soliton state, resulting from relative movement of solitons during acquisition time of the autocorrelator [18].The soliton width is 1.028 ps.
To reveal how RWs are generated, spatio-temporal intensity dynamics of the above multiple-soliton state are shown in Fig. 8 with solitons evolution over 5000 consecutive roundtrips.The time span in the figure is the same as that in Fig. 7(b) in order to see the evolution clearly.As seen in Fig. 8, there are extensive interactions between solitons.For example, around 64.5 ns, two solitons can collide and separate afterwards.As an example, the dashed square in the figure shows the appearance of RWs.As it can be seen, RW is generated from nonlinear soliton collision; it propagates over 106 consecutive roundtrips indicating its life time of 10.24 µs, then disappears as a result of splitting.

Discussion
Short-and long-range interactions of solitons exist in various systems.Direct soliton-soliton interaction accounts for short-range interaction when solitons are separated by several times their width [34,35].Long-range interactions are mediated by different mechanisms, for instance dispersive waves [36][37][38] and acoustic effects [23,24,39].The weak interactions shown in Fig. 6 in which solitons are separated by several hundred times of their duration (1 ps) indicate that these are long-range interactions.Though acoustic effect inherently exists in fiber, dispersive waves evidenced by strong Kelly sidebands (Fig. 1 (b)) may dominate these long-range interactions; moreover, as shown in Fig. 6 the separations between solitons oscillate with cavity roundtrips, agreeing with the manners of dispersive-wave induced interactions [40].In contrast, Fig. 8 presents both long-and short-range interactions.As it can be seen in the figure, initially, the solitons are spaced by 100 and 50 ps around the positons of 64.5 and 65 ns respectively, which are 100 and 50 times the soliton durations, corresponding to long-range interactions.Then short-range direct soliton-soliton interactions dominate.Theoretical study predicted that periodic attraction and repulsion could happen during shortrange two-soliton interaction [34]; remarkably, the double solitons around 64.5 ns in Fig. 8 shows this behaviour as well as the solitons around 65 ns.

Conclusion
In conclusion, we have observed RWs in another mode-locking state of a soliton fiber laser, namely multiple-soliton state with hundreds of solitons occupying the whole laser cavity.This state can be accessed by increasing the pump power only from a stable soliton bunch state.Benefiting from spatio-temporal intensity measurements, it is shown for the first time in experiment that nonlinear soliton collision is responsible for RWs generation in this state.This finding may also imply the existence of this mechanism in other RWs formation system.The results also show that spatio-temporal measurement technology is promising in charactering the dynamics of mode-locked fiber lasers.

Fig. 2 .
Fig. 2. Stable soliton bunch state at a pump power of 22 mW: (a) temporal trace of the stationary soliton bunch circulating at the fundamental cavity repetition period of 96.6 ns, and (b) temporal magnification of the soliton bunch.

Fig. 3 .
Fig. 3. Spatio-temporal intensity dynamics of the stable soliton bunch over 8000 roundtrips at a pump power of 22 mW.

Fig. 5 .
Fig. 5.The solitons occupied the whole laser cavity when the pump power was increased to 303 mW: (a) temporal trace of the solitons, (b) temporal magnification of the solitons with time span of 3 ns.

Fig. 6 .
Fig.6.Spatio-temporal intensity dynamics of the multiple-soliton state exhibiting weak interactions between solitons at a pump power of 303 mW.

Fig. 7 .
Fig. 7. RWs are generated at a pump power of 730 mW: (a), temporal trace of the multiple solitons; (b), temporal magnification of the solitons with time span of 3 ns, around RW location; (c), the autocorrelation trace.

Fig. 8 .
Fig. 8. Spatio-temporal intensity dynamics of the multiple-soliton state showing strong interactions between solitons at a pump power of 730 mW.The dashed square shows an example of RWs.