High-amplitude “rogue” waves have attracted significant interest during the past few years. Originally a niche area of oceanography [1], the subject exploded into a multidisciplinary field of research triggered by the pioneering experiments of Solli et al. [2]. This study highlighted that long-tailed probability distributions characteristic to oceanic rogue waves could also be observed in the context of ultrafast fiber supercontinuum generation. The initial experiments stimulated a number of studies, and “optical rogue waves” were rapidly identified across the field of photonics [3–10]. While the majority of studies have focused on conservative systems, characteristic intensity fluctuations have also been reported in fiber amplifiers [11–13] as well as in continuous wave Raman lasers [14,15]. More recently, studies have ventured toward the limit of highly dissipative systems, and extreme events have been studied in both Kerr lens mode-locked Ti:Sapphire lasers [16] as well as in ultrafast fiber lasers [17–20]. In this context, fiber lasers are of special interest because their 1D character renders the platform ideal for the study of complex nonlinear dynamics [21].

Noise and fluctuations play a critical role in the performance of mode-locked fiber lasers, and so their study is rooted into a deeper history. An operation regime that has attracted particular attention is one where the laser output comprises a train of temporally localized bursts of “noise-like pulses” [22]. First observed in 1997 at the output of an Er-doped device, the operation regime has since been shown to be ubiquitous in the domain of passively mode-locked fiber lasers [23,24]. Recently, a new type of noise-like operation has been identified, where the destabilization of uniform mode-locking is correlated with the simultaneous appearance of a frequency down-shifted Raman pulse [25,26]. Other studies have found that Raman pulses do not necessarily lead to noise-like operation—they can coexist with a stable dissipative soliton owing to the formation of a bound Raman-soliton complex [27,28]. These contrasting regimes clearly highlight that further studies are required to better understand the underlying dynamics and lasing characteristics. In particular, the multitude of systems establishing Raman scattering as the culprit of rogue-wave-like fluctuations [6,7,11,13–15] renders Raman-triggered noise-like pulses a potentially fertile nonlinear setting for studying the emergence of dissipative rogue waves. Yet, so far this mode of operation has been characterized only using ensemble-averaged techniques, and no statistical treatment has been presented.

In this Letter, we identify a new kind of dissipative rogue wave phenomenon in a partially mode-locked fiber laser. Specifically, we perform single-shot spectral measurements in order to investigate the roundtrip-to-roundtrip fluctuations of a Raman-triggered noise-like Yb-doped fiber laser, and observe sporadic emission of strong Raman pulses. Our experiments use the real time dispersive Fourier transformation technique [29–32], which we have recently exploited to distinguish shot-to-shot spectral fluctuations in a noise-like Er-doped oscillator [33]. Here, we show that, for low intracavity gain, the shot-to-shot energy in the Raman band displays a long-tailed probability distribution, with the most energetic events fulfilling criteria proposed for rogue waves. For higher gain levels Raman emission displays a transition into a saturated regime with Gaussian statistics. We discuss our observations in light of the nonlinear transmission characteristics of the cavity mode-locker.

The laser used in our experiments is schematically illustrated in Fig. 1(a). It is an all-normal dispersion, all-polarization maintaining passively mode-locked Yb-doped fiber laser, similar to the one used in [26,34]. The cavity is 200 m long, yielding a fundamental repetition rate of 950 kHz, and it incorporates a nonlinear amplifying loop mirror (NALM) to promote ultrashort pulse formation [35]. The total cavity length is dominated by the main laser loop, which contains a 180 m long section of single-mode fiber. The cavity hosts two gain segments driven by identical laser diodes, one in the main loop and one in the NALM. The laser output is extracted immediately after the NALM using an 80% output coupler. The narrow bandpass filter (1.7 nm bandwidth) centered at 1028 nm compensates for the large normal dispersion accumulated during a single roundtrip. Fixing the NALM pump power at 250 mW while varying the pump power in the main loop allows for stable or unstable mode-locking, and for the output characteristics to be controlled. Fig. 1(b) shows the output spectrum measured using an optical spectrum analyzer (OSA) when the pump power in the main loop is set to 170 mW. For this pump level the laser emits a stable train of pulses, spectrally centered at 1028 nm. When the pump power is increased to 190 mW the output undergoes a significant change, as shown in Fig. 1(c). Indeed, it is clear that a secondary spectral peak appears at 1075 nm [26]. This peak originates from intracavity Raman conversion, as evidenced by the 12.8 THz frequency difference between the 1028 and 1075 nm center wavelengths. Further increasing the main loop pump power results in the growth of the Raman component. An example spectrum is shown in Fig. 1(d), where we have increased the pump power to 230 mW.

 

Fig. 1. (a) Schematic illustration of the laser cavity. SMF: single-mode fiber. (b)–(d) The optical spectrum recorded with an OSA when the pump power in the main loop is set to  170, 190, and 230 mW. The NALM pump power is 250 mW.

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The modes of operation shown in Figs. 1(c) and 1(d) fulfill the criteria used to identify noise-like pulses [22,26]: broad and smooth spectrum and an autocorrelation trace comprising a sharp peak atop a broad pedestal (a well known coherence artifact [36]). Both features are suggestive of large shot-to-shot fluctuations. To gain more insights, we measured the roundtrip-to-roundtrip spectra at the laser output for both operation regimes using dispersive Fourier transformation [29–33]. In our experiment we stretch the laser’s output using 10 km of single-mode fiber (group-velocity dispersion ${\beta }_2=18.9{\mathrm{ps}}^2{\mathrm{km}}^{-1}$) and record 4000 consecutive spectra in the time domain using a 12 GHz photodetector and a 12.5 GHz real time oscilloscope [33]. A control measurement (not shown) was performed on the stable laser regime shown in Fig. 1(b) to verify the measurement technique to be free of error.

Results of our real time measurements are shown in Fig. 2. We first discuss the results obtained for the lower pump value of 190 mW shown in Fig. 2(a). The left-hand panel shows the pulse spectrum over 350 consecutive roundtrips as a 2D density map. The single-shot results confirm that both the main pulse at 1028 nm, as well as the Raman component at 1075 nm, exhibit strong shot-to-shot spectral fluctuations. More surprisingly, we can see that a strong Raman component appears only intermittently, with the vast majority of roundtrips displaying negligible Raman conversion. The panels on the right-hand side of Fig. 2(a) highlight the large roundtrip-to-roundtrip contrast in more detail. Here, we plot individual spectral realizations corresponding (from top to bottom) to events displaying the minimum, the mean, and the maximum energies in the Raman band. It is apparent that no Raman component can be observed in the minimum case, while the maximum case shows almost 6% of the overall pulse energy residing in the Raman band. When the cavity pump power is increased to 230 mW the probability of Raman emission increases dramatically. Indeed, strong emission can be observed in almost all the 350 spectra shown Fig. 2(b). Although the spectral intensity of the most extreme Raman realizations is comparable to the main pulse at 1028 nm, this pumping level still supports events in which no Raman conversion can be observed.

 

Fig. 2. Single-shot spectral measurement results when the main loop pump power is set to (a) 190 mW and (b) 230 mW. The 2D linear scale density maps display optical spectra over 350 consecutive roundtrips. The side panels highlight Raman fluctuations in more detail by showing spectra with the smallest (top), mean (middle), and largest (bottom) energy in the Raman band.

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In order to investigate the statistical properties of the Raman events in more detail, we extract the roundtrip-to-roundtrip Raman energy by integrating the measured single-shot spectra over the wavelength range 1050–1090 nm. This yields statistical samples comprised of 4000 individual realizations. The resulting energy histograms are shown in Figs. 3(a) and 3(b) for both the low and the high pump power, respectively. For low pumping, the energy histogram [Fig. 3(a)] is highly skewed and displays a long tail. In contrast, for increased pump power, the Raman energy histogram exhibits a clear Gaussian distribution as seen in Fig. 3(b).

 

Fig. 3. (a) and (b) show Raman energy histograms when the main loop pump power is set to (a) 190 mW and (b) 230 mW. The inset in (a) shows a zoom on the tail. (c) and (d) show Raman energy correlations corresponding to the histogram shown in (a). The scatter plot in (c) highlights the lack of correlation between the Raman energy and the total pulse energy. Similarly, the scatter plot in (d) shows that Raman energies are not correlated across consecutive rountrips.

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The long-tailed distribution in Fig. 3(a) indicates that, for low pump power, the emission of an energetic Raman pulse is rare, yet more frequent than expected for Gaussian statistics. These statistics do not trivially reflect the intrinsic energy fluctuations of the noise-like pulse train as shown in Fig. 3(c), where we illustrate the dependence of Raman energy on the total roundtrip pulse energy (obtained by integrating over the full single-shot spectrum). No apparent correlations are observed and, in fact, we can see that the most energetic Raman events occur for roundtrips whose total energy is comparatively close to the mean. Finally, the scatter plot in Fig. 3(d) highlights that the roundtrip-to-roundtrip Raman energies are not correlated with one another; energy at the nth roundtrip does not depend on the energy at the (n-1)th roundtrip. This near total absence of system memory can be understood by recalling that the cavity implements a narrow bandpass filter centered at 1028 nm, such that the Raman component is fully filtered at the end of each roundtrip.

The most energetic Raman pulses measured with 190 mW pumping fulfill the criteria suggested for rogue waves [20]: (i) they populate the tails of highly-skewed statistical distributions [Fig. 3(a)]; (ii) they are unpredictable and “appear from nowhere and disappear without a trace” [Figs. 3(c) and 3(d)], and (iii) their amplitude exceeds the significant wave height (SWH), defined as the mean amplitude of the one-third largest waves, at least by a factor of two. Indeed, for the statistical distribution shown in Fig. 3(a), the (relative) $\mathrm{SWH}\sim 0.179$, while the energy of the largest Raman event is 1.257 — more than 7 times the SWH.

We believe that the observed Raman statistics originate from stochastic intracavity pulse dynamics combined with the nonlinear transmission of the mode-locker. In particular, the NALM deviates from a “genuine” saturable absorber in that it exhibits a sinusoidal transmission function relative to the signal intensity [35]. The associated periodicity can influence the intracavity energy statistics in a nontrivial fashion, giving rise to long-tailed output distributions. [Note that the laser output is extracted after the NALM as shown in Fig. 1.] To experimentally corroborate our speculation, we performed additional measurements. Specifically, by introducing a 5% tap coupler in the laser cavity immediately prior to the NALM, we are able to inspect the intracavity field before it is influenced by the mode-locker. Fig. 4(a) and 4(b) show OSA-measured spectra before and after the NALM, respectively, when the laser is operating in a regime similar to that shown in Fig. 1(d). Unfortunately, the introduction of the extra coupler reduces the flexibility of the cavity and the regime displayed in Fig. 1(c) is no longer accessible. Our experiments nevertheless give important insights on the origins of Raman fluctuations at the laser output. As shown in Fig. 4(a), we find that the Raman components in fact dominate the spectrum before the NALM, but are heavily attenuated at its output [see Fig. 4(b)]. Single-shot spectral measurements reveal that an intracavity Raman pulse is formed during each roundtrip, with the energy distribution before the NALM following left-skewed statistics as shown in Fig. 4(c). However, upon passing through the NALM, the skewness of the statistic is inverted [see Fig. 4(d)]. While detailed theoretical study is beyond the scope of the present work, we believe this inversion to originate from overdriving of the NALM mode-locker. These measurements highlight the significant role of the mode-locker and, in particular, suggest that sporadic emission of rogue Raman events observed at the laser output may originate from the nonlinear, NALM-mediated selection of recurring intracavity pulses. The fact that nonlinear polarization evolution also exhibits a periodic transmission further hints that similar selection could be central in other regimes of dissipative rogue wave formation [19,20]. Finally, Fig. 4(a) shows that the intracavity field also supports a weak second-order Raman component centered at 1130 nm. Our single-shot measurements establish this component to be intermittent, independently displaying long-tailed probability distributions as shown in Fig. 4(e). These intracavity rogue waves are conceptually distinct from those observed at the output of the laser. Indeed, while the latter are influenced by the nonlinear transmission of the NALM, the former arise solely due to nonlinear optical interactions in the active and passive fiber segments before the NALM. This process resembles the dynamics observed in photonic crystal fibers, where a cascade of Raman processes has been shown to give rise to extreme-value fluctuations [7].

 

Fig. 4. (a) and (b) show OSA-recorded average spectra (a) before and (b) after the NALM when a 5% tap coupler is introduced in the cavity. (c) and (d) show the corresponding single-shot energy histograms for the first Raman order [indicated with blue and green shaded areas in (a) and (b), respectively]; (e) is the energy histogram for the second Raman order observed before the NALM [orange shaded area in (a)]. The inset in (e) shows a zoom on the tail.

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To conclude, we have reported the experimental observation of a new type of dissipative rogue wave phenomenon. We have measured the roundtrip-to-roundtrip spectral fluctuations of a Raman-destabilized, noise-like Yb-fiber laser, and observed the sporadic emission of Raman pulses, both extracavity and intracavity. A statistical analysis has revealed that the most energetic Raman events fulfill the criteria proposed for rogue waves. An interesting future prospect is to explore whether Raman fluctuations manifest themselves in bound complexes consisting of a stable dissipative soliton and an unstable Raman pulse [27,28].