## Abstract

Mode-locked lasers are prone to exhibit a wide range of dynamics, many of them analyzed within the concept of dissipative solitons. However, ultrafast laser dynamics goes beyond the denomination of mode locking, for instance, with the possible emission of incoherent pulse bursts. Therefore, we propose to expand the notion of dissipative solitons to incoherent localized formations, which we illustrate experimentally by reporting original vector dynamics of incoherent dissipative solitons generated in an ultrafast fiber laser. The laser uses a semiconductor saturable absorber mirror and is operated in the anomalous dispersion regime. We provide a real-time spectral characterization of the vector dynamics, highlighting several salient features of the self-organization. Both polarization locking and polarization switching dynamics of the incoherent soliton are observed, generalizing previous observations made in the coherent case. We also demonstrate that transient spectro-temporal ordering takes place among the incoherent pulse, analogous to the coalescence of well-separated droplets in liquids. In the vicinity of polarization switching dynamics, we show that intermittent behaviors can be associated with the generation of vector rogue waves bearing common features with the soliton explosion effect.

© 2017 Optical Society of America

## 1. INTRODUCTION

The robustness of the optical soliton concept is remarkable. Conceived originally in the frame of a coherent formation resulting from the balance between nonlinearity and dispersion, the concept has evolved in a multifarious way, responding to the diversity of the propagation media of practical interest. In particular, the solitary waves obtained in presence of nonlinear gain and loss were instrumental to shape the concept of dissipative solitons, which has become an important research avenue of nonlinear photonics during the past decade [1,2]. The stability and robustness of dissipative solitons stem from the existence of dynamical attractors in dissipative nonlinear systems. However, beyond fixed point attractors, dissipative systems host a wide range of nonlinear dynamics, including pulsations and chaos. Ultrafast lasers constitute a particularly convenient platform for investigating the versatility of dissipative soliton dynamics [3], and they lead to the paradoxical discovery of chaotic yet localized temporal structures in the moving reference frame, such as soliton explosions [4,5], noise-like pulse (NLP) emission [6], soliton liquids [7–9], and pulse-bunching dissipative rogue waves (DRWs) [10]. Soliton explosions, which were first predicted numerically [11], are defined by intermittent periods of pulse instability followed by recovery, the pulse returning to quasi-stationary coherent mode-locked dynamics before the next instability develops. They have been observed in Ti:sapphire and fiber laser experiments [4,5]. NLP emission, soliton liquids, and DRWs are characterized by an internal field structure that behaves chaotically at all times, in contrast to the intermittency of soliton explosions. NLP emission appears as the most chaotic, though localized, state [6]. Compared with the latter, a soliton liquid possesses an internal structure made of interacting dissipative solitons, which move chaotically. However, depending on the finite resolution of ultrafast experimental measurements, such distinction can be quite faint. The soliton liquid is also a precursor of pulse-bunching DRWs [10,12], and one of the three components of the soliton rain dynamics [7,8]. Interestingly, the soliton explosions in fiber lasers were, until now, observed exclusively in the normal dispersion regime, where they can also lead to DRW observation [13,14]. Actually, a given sign of the chromatic dispersion in dissipative systems can be considered to facilitate the observation of a specific dynamics, but is not a prerequisite to its existence in general, in contrast to the situation for conservative systems [3]. For instance, DRWs and NLP dynamics are observed in both anomalous and normal dispersion laser architectures [6,10,12,13,15–17].

Therefore, NLP emission, soliton liquid, and pulse-bunching rogue waves can be encompassed in the denomination of *incoherent dissipative solitons*. Such terminology presents an interesting parallel with the notion of incoherent solitons of conservative systems. The latter involves the self-trapping of incoherent light in either the spatial or temporal domain, using a nonlinear medium characterized by a slow response [18,19] or nonlocality [20]. Whereas several physical mechanisms can lead to the formation of incoherent dissipative solitons, dissipation itself will always take a leading role, enabling the pulse localization in both the temporal and spectral domains, in contrast, for instance, to the spectrally accelerated temporal incoherent solitons [19].

According to the cavity architecture, the polarization degree of freedom can be frozen or, conversely, can manifest in a dramatic way. The latter can take place within ultrafast fiber lasers that use standard single-mode fibers (SMFs) and optical components with polarization-independent losses. A SMF supports two orthogonal polarization modes due to the presence of small amounts of random birefringence induced by manufacturing imperfections and externally applied random strain or bending. The use of a semiconductor saturable absorber mirror (SESAM) allows a variety of ultrafast polarization dynamics to manifest, which can be tuned by an intracavity polarization controller (PC) that adjusts the overall cavity birefringence. This way, Cundiff and coworkers studied the transition from polarization-rotating (PRVS) to polarization-locked vector dissipative soliton (PLVS) [21,22]. Subsequent investigations reported locked, switching, and rotating states of polarization for various dynamical regimes, which include single-pulse, harmonic mode locking, soliton molecules, and other multi-pulse operations [23–29].

It is therefore of great interest to investigate how vector dynamics will unfold with incoherent dissipative solitons. This is precisely the goal of this paper, where we explore further the self-organizing ability of dissipative optical solitons, providing the first experimental demonstration of vector dynamics of incoherent solitons. The dynamical regime is spectrally resolved in real time for both polarization components, using a dispersive Fourier-transform (DFT) spectro-temporal imaging technique [30]. Several original features are highlighted. Either polarization locking or polarization switching of the incoherent soliton are obtained. We show that transient spectro-temporal ordering takes place among the incoherent pulse, analogous to the coalescence of well-separated droplets in liquids. Finally, we demonstrate that, in the vicinity of polarization switching dynamics, intermittent behaviors can take place, such as the generation of vector rogue waves observed in the spectral domain, which also bear common features with the soliton explosion effect.

## 2. EXPERIMENTAL SETUP

Figure 1 illustrates the experimental setup, based on an erbium-doped fiber laser emitting around 1.53 μm. Short pulse operation is obtained using a SESAM with a relaxation time of 10 ps that is inserted in the cavity through a three-port polarization-insensitive circulator. The amplifier section is composed of a 0.55-m-long erbium-doped silica fiber (EDF, $110\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{dB}/\mathrm{m}$ absorption at 1530 nm, 4 μm core diameter, $\mathrm{NA}=0.2$), which is backward-pumped by a 980 nm laser diode (LD). The cavity is completed with a standard SMF (SMF-28), whose total length is 7.35 m. The group velocity dispersion at 1.55 μm is $+13.5\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{ps}}^{2}\xb7{\mathrm{km}}^{-1}$ for the EDF and $-22.9\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{ps}}^{2}\xb7{\mathrm{km}}^{-1}$ for the SMF, providing a net anomalous path-averaged dispersion of ${\beta}_{2}=-20.4\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{ps}}^{2}\xb7{\mathrm{km}}^{-1}$. The total cavity length, 7.90 m, entails a fundamental repetition rate of 26.28 MHz. The polarization-insensitive optical isolator ensures the unidirectional light propagation, while the intracavity PC allows to target different vector multi-soliton operation regimes through a precise adjustment of the net cavity birefringence.

Experimental characterization is performed through the output beam extracted from the cavity by the 50/50 fiber coupler. The average measurements are made with an optical spectrum analyzer (OSA) and a multi-shot second-order optical autocorrelator. In order to observe vector dynamics, an extra-cavity PC and a fiber-based polarization beam splitter (PBS) are placed in series along one output line. We have then adapted the dispersive Fourier-transform spectro-temporal imaging technique (DFT) detailed in Ref. [30], in order to access to the polarization-resolved shot-to-shot spectral dynamics. The DFT is implemented by temporally stretching the laser output in a 6354-m-long dispersion compensating fiber (DCF) that provides a total accumulated dispersion of $874\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{ps}}^{2}$. The equivalent far-field pulse stretching in the linear regime maps the spectrum of each optical pulse into a nanosecond temporal waveform, which can be conveniently analyzed by fast electronics. Our adaptation of the technique sends both propagation components of the laser output field into counterpropagating directions of the same DCF, facilitating temporal and spectral calibrations of the two stretched waveforms in tandem, which we record via two high-speed photodiodes (12-GHz and 45-GHz) connected to a 6-GHz, 40-GSa/s real-time oscilloscope. Our DFT configuration leads to an electronic-based spectral resolution of 0.03 THz (0.24 nm), which is suitable for analyzing ultrafast dynamics of $\sim 5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ wide spectra generated from our laser cavity. Self-starting short pulse operation is routinely obtained for a pump power ${P}_{p}$ in the range of 220–380 mW.

## 3. FIBER LASER DYNAMICS

#### A. Incoherent Dissipative Solitons with Transient Order

Operating the laser, the first notable trait is the recurrent observation of chaotic yet temporally localized dynamics, akin to NLP emission, which are, therefore, not precluded by the SESAM with 10-ps relaxation time, at least not in the anomalous dispersion regime and pump power range investigated here. A typical characterization of this regime is illustrated in Fig. 2. The overall lack of coherence is clearly visible from the variability of consecutive single-shot spectra displayed in Fig. 2(b), whereas the mean duration of the incoherent pulse, $\sim 20\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{ps}$, is on the order of the SESAM relaxation time, and the average autocorrelation trace in Fig. 2(c) reveals ultrafast internal dynamics characterized by much shorter intensity variations, below the picosecond. In contrast to common observation from NLP dynamics, the central coherence peak of the autocorrelation trace is here flanked by cross-correlation peaks of large magnitude spaced by 2.65 ps. These additional peaks are reminiscent of an internal structure made of multiple pulses with a defined separation between them—a feature best exemplified in dissipative soliton molecules, for instance [3,31,32]. However, the absence of modulation in the average OSA spectrum shown in Fig. 2(d) (blue curve) suggests the lack of a stable phase relationship among the internal pulse constituents. To remove this apparent contradiction, additional insight is provided by the 2D contour plot of Fig. 2(a), which displays 1000 consecutive single-shot first-order autocorrelation traces that are obtained by Fourier transform of the corresponding shot-to-shot spectra of Fig. 2(b). Clearly, Fig. 2(a) demonstrates that some partial and transient ordering takes place inside the incoherent pulse and vanishes, within time scales of 10–400 cavity roundtrips. In Fig. 2(c), the mean of these 1000 first-order correlation traces is displayed (red curve), showing indeed a good temporal match of the average first-order correlation side peaks with those obtained from the multi-shot autocorrelation recording. Therefore, the incoherent dissipative soliton observed here features transient spectro-temporal ordering, analogous to the coalescence of well-separated droplets in liquids. Yet, we note that this regime is different from the regime of *soliton liquid* that has been emphasized in prior studies [7–9], in which the internal structure made of chaotically interacting dissipative solitons could, in principle, be followed over arbitrary long times.

#### B. Polarization Locking and Switching

We now investigate the polarization dynamics of this incoherent dissipative soliton regime. By the slight adjustment of the intracavity PC, we can obtain two different vector dynamics, which are illustrated in Fig. 3, after the polarization-resolved DFT spectro-temporal measurements. Figure 3(a) shows the existence of polarization-locked vector incoherent solitons. Indeed, after a suitable orientation of the polarization projection basis is obtained by adjusting the extra-cavity PC, the output laser field can be projected onto a single polarization axis, called here the y-polarization. This projection does not change for successive cavity roundtrips—in this figure, as many as 8000 cavity roundtrip spectra are displayed.

In contrast, Fig. 3(b) demonstrates the existence of polarization switching vector solitons, with temporally alternating x- and y-polarizations over a long quasi-period of $\sim 2000$ roundtrips, which corresponds to $\sim 76\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu s}$. By additional fine-tuning of the intracavity PC, we can further control the value of the duty cycle ratio, which is for instance $\sim 0.5$ in Fig. 3(b). Though the origin of the long-period polarization switching dynamics is not fully understood, we note that in the case of *coherent* vector dissipative solitons, slow precessions of the state of polarization were observed [24,25]. In Ref. [25], the authors explained that the slow vector dynamics originated from a complex interplay between linear and circular birefringence of the fiber laser cavity, along with a light-induced anisotropy caused by polarization hole burning. To a large extent, this explanation rules out the influence of the saturable absorber dynamic response to focus on the cross-polarization gain saturation effect. The latter physical effect has also been identified as a key generating effect of antiphase vector dynamics, which has been predicted and observed in a fiber laser configuration without any explicit saturable absorber [33]. Still, in the present situation depicted by Fig. 3(b), the observation of polarization switching in the case of *incoherent* vector solitons is remarkable.

#### C. Explosive Dynamics

By exploring the vicinity of the transition between the above two polarization dynamics through careful adjustment of the intracavity PC, we observe a peculiar instability, which bears common features with soliton explosion dynamics [4,5,13,14]. Usually, soliton explosion can be understood as a critical regime between stable soliton pulses and NLPs, where a solitary pulse intermittently suffers an explosive instability and then recovers to its original stable state. Interestingly, soliton explosions may produce DRWs, as was recently demonstrated in the scalar limit of normally dispersive fiber lasers [13,14].

The combined vector and spectral real-time analysis is presented in Fig. 4. Around the explosive event, four sets of single-shot spectra are displayed, see Fig. 4(b). Whereas the laser operates in the regime of polarization switching dynamics, we can see that the x-polarization spectrum collapses at roundtrip number $\sim 3390$, entailing an important corresponding spectral development on the y-polarization, red shifted by nearly 20 nm from the central wavelength at 1.53 μm. This association of the spectral collapse with a red shifted development was also emphasized in the case of scalar soliton explosions [5]. In our vector case, the abrupt spectral instability is followed by the recovery of the initial incoherent soliton on the x-polarization. We also investigate the time-domain dynamics of this explosive phenomenon: Fig. 4(c) presents the evolution of the temporal intensity envelope that is directly measured at the cavity output (i.e., without propagation in the DCF), as a function of roundtrip numbers. We can clearly identify a sudden temporal shift of about 1 ns, which is also typical for soliton explosion events [5,34]. The same value of the temporal jump is taken into account in the spectral DFT measurements. Thus, the main common feature between the present observation and soliton explosion dynamics [4] resides in the manifestation of an abrupt spectral expansion followed by recovery. There is nevertheless an important difference with respect to prior investigations: here, the central dynamics from which the explosion develops and vanishes is that of an *incoherent* dissipative soliton. The other important specificity of the present dynamics is represented by its vector expression and associated with polarization switching.

#### D. Spectral Rogue Waves

By adjusting the orientation of the PC, we can increase the rate of explosive instabilities, similar to the previous soliton explosion observations [4,5,35]. We perform statistical measurements to further investigate the connection between the explosive instabilities and rogue wave generation. Rogue waves are extreme waves that appear unexpectedly with a huge amplitude, at a rate that exceeds that derived from classical statistical distributions [36]. Dealing with composite or incoherent pulses featuring chaotic internal dynamics, the intensity fluctuations will definitely be higher than the overall energy fluctuations [10,15]. It would then be desirable to access the internal pulse structure in real time, in order to record the statistics of the maximum amplitude, as was done in Ref. [10]. However, the temporal resolution of our electronic measurement cannot resolve the $\sim 20\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{ps}$-long pulse in real time. Therefore, following the approach of [15], we record the maximum spectral intensity, as a voltage amplitude expressed in millivolts, of each single-shot DFT spectrum and add it to a histogram, in order to search the signature of spectral rogue waves. In our case, this is performed for both x- and y-polarizations. Figure 5 illustrates one such set of histograms, built from ${10}^{6}$ successive spectra for x- and y-polarizations, measured in the regime of successive explosive instabilities. As we can see, both histograms exhibit non-Gaussian statistics, showing long-tailed distributions comprising extreme events. The highest recorded voltage amplitude corresponds to 7 times and 3.3 time, that of the significant wave height (SWH) for x-polarization and y-polarization, respectively. We recall that the SWH is calculated as the mean of the highest third of the waves. The large difference in the SWH values corresponding to the two orthogonal polarization components indicates that, in the selected vector dynamics, one polarization is more prone to express the explosive instabilities, which is consistent with the observations related to Fig. 4. The usual statistical criterion for rogue wave generation is that the highest measured amplitude should be larger than $2\times \mathrm{SWH}$ [36]. We note that there is a possible source of controversy while applying this criterion in the optical context. Indeed, the recorded quantity is an optical intensity, not a field amplitude as in hydrodynamics, where the scientific notion of rogue waves originates, following water elevation recordings. As such, it may be more appropriate to consider optical intensity events exceeding four times the SWH as DRW candidates, which is found in our case among the x-polarization statistics. We point out that the statistical distributions obtained in the time-domain also fulfill the $2\times \mathrm{SWH}$ DRW criterion, albeit with smaller maximum amplitudes. It is also worth mentioning that for the incoherent vector dynamics corresponding to Fig. 3, the histograms feature Gaussian statistics for both polarizations: without the explosive instability, we did not find DRW evidence. Therefore, our experiment clearly confirms, through an original vector incoherent dynamics regime, the connection between DRWs and the explosive behaviors in ultrafast fiber laser dynamics.

## 4. CONCLUSION

To summarize, in this paper, we experimentally demonstrated the existence of novel vector dynamics for incoherent short pulses formed in an ultrafast fiber laser. These observations expand the conceptual area of dissipative solitons by showing that self-organized patterns and dynamics can take place for incoherent pulse structures with interacting polarization components. We used real-time, polarization-resolved, DFT spectro-temporal measurements to gain clear insight into the dynamical evolution roundtrip after roundtrip. Our measurements also indicated the existence of transient ordering processes among the incoherent, noise-like, vector pulse, with the coalescence of several succeeding intra-pulse structures with a specific temporal separation between them, followed by their vanishing. We reported the observation of polarization-locked as well as polarization switching vector incoherent solitons, generalizing previous observations made in the case of *coherent* dissipative solitons in ultrafast lasers. Our attempt at generalization also includes the analysis of the following extreme-wave chaotic dynamics: soliton explosions and optical rogue waves. Indeed, the vicinity of the polarization switching dynamics is prone to the manifestation of increased instabilities of explosive type. We have highlighted the similarities and differences with the previous reports of *soliton explosion* dynamics that were performed in the scalar case, and we confirmed the link between the spectral explosive instabilities and rogue wave generation, this time in the more general situation of vector incoherent dynamics. In this context, the experimental demonstration of vector DRWs constitutes an important finding. We believe that our results will stimulate further research of the complex self-organization phenomena enabled in dissipative systems, which now encompass vector and incoherent dynamics, with the challenge of improving numerical modeling and real-time characterization.

## Funding

Centre National de la Recherche Scientifique (CNRS); Indo-French Centre for the Promotion of Advanced Research (IFCPAR) (5104-2); LABEX Action (ANR-11-LABX-0001-01); Agence Nationale de la Recherche (ANR); European Regional Development Fund (ERDF); Région Bourgogne.

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