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Abstract
We have experimentally and theoretically investigated optical rogue waves (ORWs) in a net negative dispersion Tm-doped fiber laser with a long cavity, adopting nonlinear polarization evolution as a mode-locker as well as a spectral filter. We obtained a state with numerous pulses bunched in a burst accompanied by perturbation within the burst, in which the spectrum was partially perturbed. After statistical analysis, we found that ORWs have existed in this bunching state. By adjusting the intra-cavity polarization controllers, the perturbed pulse bunching turned into a chaotic pulse bunching state, which gave rise to giant pulses with ultra-high amplitudes, and the giant pulses were a precursor of a broad-spectrum noise-like pulse. The probability of occurrence of ORWs was increased in the chaotic state, which is caused by multi-pulse instability induced by the spectral filtering effect. Simulation results confirm the experimental results and demonstrate that the spectral filter bandwidth (SFB) is directly related to the probability of the emergence of ORWs. When increasing the SFB across the range of multi-pulse instability at a fixed pump power, the frequency with which ORWs appear increases.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
8 September 2021: Typographical corrections were made to the body text, the caption of Fig. 7, and Refs. 5, 7, 9, 15, 26, 36, and 37.
1. Introduction
Rogue waves (RWs) were firstly observed on the ocean and have been the cause of many sunken ships and lost human lives. However, due to their unpredictable nature, it is difficult to investigate RW phenomena in their natural environment thoroughly. On the other hand, an external pump and a gain medium in a laser are analogous to the wind over ocean water. More importantly, the Nonlinear Schrödinger equation can describe pulse propagation in optical fibers, the same equation used by oceanographers to describe water waves. Consequently, many researchers have focused efforts on optical rogue waves (ORWs) in high-nonlinearity optical fibers. In 2007, Solli et al. first reported the observation of ORWs with long-tailed histograms in measurements of intensity fluctuations at long wavelengths in the fiber supercontinuum. They demonstrated that infrequently arise from initially smooth pulses owing to power transfer seeded by a slight noise perturbation [1]. Since then, the physical mechanism behind the ORW has been hotly debated, with modulation instability [2,3], breathers [4], higher-order dispersion [5], and nonlinear spectral shaping [6] as striking points of connection in highly nonlinear fibers [7], and other fiber-based optical systems [8].
A theoretical prediction regarding ORW generation in a passively mode-locked fiber laser (MLFL) with anomalous cavity dispersion was presented by Soto-Crespo et al. in 2011 [9]. Shortly thereafter, Zaviyalov et al. numerically investigated the ORWs existing in a normal dispersion MLFL [10]. In recently, Lauterio-Cruz et al. numerically studied complex dynamics and extreme events in an erbium figure-eight laser. Through a temporal-spectral correspondence, they evidenced a notorious distortion that the spectrum undergoes around an ORW, which provided a novel method of identifying ORWs through spectral deformation [11]. In all of the above theoretical studies, the researchers chose unique simulation parameters in which the laser could operate in a nonstationary chaotic state, i.e., the intensity had random fluctuations. Experimentally, ORWs have been observed in passively MLFLs with various pulse dynamics, such as soliton explosion [12–15], noise-like pulses (NLPs) [16–19], and chaotic state multi-soliton [20–26]. Among modes of operation, ORWs are particularly abundant in the presence of multi-soliton states.
The first experimental study of ORW formation in a chaotic multiple-pulse regime in an MLFL was reported by Grelu’s group in 2012, where the researchers pointed out that the ORWs resulted from the ceaseless relative motion and nonlinear interaction of pulses within a temporally localized multi-soliton phase [20]. Subsequently, Peng et al. further confirmed that nonlinear soliton collisions could give rise to the formation of ORWs in the multi-soliton state [22]. In 2015, Luo et al. reported the ORW generation in a fiber laser with a microfiber-based topological insulator serving as a saturable absorber (SA). They pointed out that the high nonlinearity of the SA could result in a chaotic multi-pulse wave packet with strong long-range nonlinear interactions, which induced the formation of ORWs [23]. In 2018, Klein et al. showed that non-instantaneous relaxation of the SA and the polarization mode dispersion of the cavity could generate ORWs in chaotic multiple-pulse MLFL. They further verified that the dynamics of the SA were directly related to the formation of the ultrafast ORWs [24]. That same year, Jeong et al. reported that the combination of the probabilistically started soliton interactions and the accompanying dispersive waves might generate ORWs [25].
In MLFLs, besides the effects of dispersion, nonlinearity, and saturable absorption, other cavity parameters such as nonlinear gain and spectral filtering can also affect the dissipative soliton pulse dynamics [27,28]. Specifically, recent theoretical studies have shown that chaotic states resulted from the intra-cavity spectral filtering resulting in multi-pulse instability, which induced the formation of ORWs [29]. Pulse instability induced by spectral filtering has also been experimentally demonstrated as a novel mechanism for ORW emission [30,31]. However, to calculate the extreme events in their studies, they always used the dispersive Fourier transform (DFT) technique to map the intensity evolution of single-shot spectra, i.e., they carried on their study in the frequency domain added to the complexity of the experiment. Since the multi-pulse instability shows rich pulse dynamics, it is desirable to explore the formation mechanism of ORWs in the time domain.
In the present work, adopting the nonlinear polarization evolution (NPE) as a mode-locker and a spectral filtering mechanism in a Tm-doped fiber laser with a long cavity length, the interaction through dispersive waves agglomerates dissipative solitons into a bunching state. Under the perturbation of the pulse inside the pulse bunching, the ORWs were generated. By properly adjusting the polarization controllers (PCs) in the cavity, a nonstationary pulse bunching state with colliding chaotic pulses induced ORWs with a higher probability of occurrence. Furthermore, the probability of the formation of ORWs was found to increase with increasing pulse instability. We also studied the ORW formation mechanism for the experiment conditions numerically, and the simulation results indicated that the dynamics of the spectral filter were directly related to the emergence of extreme waves in our fiber laser.
2. Experimental setup
The experimental setup of the fiber ring laser is shown in Fig. 1. It utilized a 4.2-m-long thulium-doped fiber (TDF, SM-TSF-9/125) with a group velocity dispersion (GVD) of -73 ps2 /km at 1900 nm as gain fiber, a 1570-nm fiber laser as pump source, a 1570/1930nm wavelength-division multiplexer (WDM), and a section of 50-m-long single-mode fiber (SMF, SMF28e) to increase the nonlinear phase shift accumulation of the cavity. The self-starting mode-locking operation resulted from NPE implemented with two PCs and a polarization-dependent isolator (PD-ISO) [32]. An output coupler was placed after the NPE device, and a 6-m-long dispersion compensation fiber (DCF, UHNA7), which had a splice loss of 0.7 dB with SMF28e, was used in front of the TDF to control the GVD. The total length of the whole cavity was 65.9 m, and the net cavity GVD was about -3.77 ps2. An optical spectrum analyzer (OSA, Yokogawa AQ6375B) with a minimum resolution of 0.05 nm, a 33-GHz real-time oscilloscope (OSC, DSAZ504A) with a 12.5 GHz photodiode detector (PD, 818-BB-51F) were used to monitor the laser outputs simultaneously.
3. Results and discussion
The laser threshold pumping power for mode-locking was 450 mW. When the pump power exceeded this threshold, self-starting mode-locking could be easily achieved by adjusting the PCs, and the average output power was 2.67 mW. The spectrum of a single pulse exhibits traditional soliton characteristics, as shown in Fig. 2(a). The Kelly sidebands appeared clearly on both sides of the central wavelength of 1879 nm, a product of soliton interfering with the dispersive wave [33]. Figure 2(c) shows the pulse trains with a cavity period of 318.9 ns, corresponding to the total cavity length of 65.9 m. To test the stability of individual pulses, we recorded ∼2000 traces of single pulses covering a round trip in each trace, as shown in Fig. 2(b). The bright and straight-line represented the pulse remained steady.
Fig. 2. Characteristics of the single pulse. Spectrum (a); temporal diagram showing the pulse stability (b); pulse train (c).
In this work, in addition to the stable state pulse, the laser can also operate in the pulsation regime. As some references have reported, pulse stability is associate with gain saturation and the effects of spectral filtering [9,28]. In particular, when the pump power was increased, the stability and the number of pulses were drastically affected. When the pumping power was increased to 550 mW, spectra at four different moments was recorded (Note: these four spectra were time-averaged spectra recorded by OSA, not real-time spectra, though they were captured in different moments.), as shown in Fig. 3(a). Overall, they demonstrated a typical net negative dispersion spectrum similar to in Fig. 2(a). The Kelly sidebands were clear and distinct. However, we found a slight difference between the four spectra, particularly in the range from 1874.5 to 1878.5 nm shown in the dotted box of Fig. 3(a). This wavelength range is expanded in Fig. 3(b), which can observe apparent fluctuations. As for the time domain, the pulses aggregated and formed a bunching state, as illustrated in Fig. 3(c). We can see that in each pulse bundle, there existed a high peak riding on a broad temporal envelope, which was constantly fluctuating (Note: Fig. 3(c) was only the selected pulse bunching with spikes in several successive round trips to visually show the high amplitude that the pulse bunching may burst. In order not to cause the reader's wrong understanding, we added Visualization 1). The inset of Fig. 3(d) shows an expanded bunching envelope for observing the clarity of the inner structure of the bunching envelope. The contour of the pulse pedestal looked like a sturdy square envelope rather than a chaotic state, as Ref. [23] showed. However, since every square envelope collected a number of pulses, and the interaction between the pulses was difficult to perceive, we were prompted to consider whether or not ORWs could be generated in this case. We, therefore, recorded ∼20000 events using the 33 GHz real-time OSC (The actual application bandwidth was 12.5 GHz, limited by the PD) and added the maximum values of each event into a histogram as shown in Fig. 3(d). This was completed with continuous sweeping as in Ref. [20]. Since the pulse bundle has an envelope width as large as 25 ns (Inset of Fig. 3(d)), while our photodetector has a bandwidth of 12.5 GHz, corresponding to a rise time of about 40 ps, with which it would be possible to identify those ORW-events around or greater than such temporal resolution. Figure 3(d) shows the statistical distribution of peak amplitudes. We should note that the wave amplitude distribution is non-Gaussian in that it has a long tail corresponding to the occurrence of extreme events. The significant wave height (SWH: average amplitude of the highest third of all events) [20] was calculated to be 9.05 mV. By calculating the sum of the y-axis data in the three red boxes in Fig. 3(d), one can work out the number of ORW events, defined as those with amplitudes more prominent than 2*SWH, was 6. Thus, the ORWs showed a probability of appearance of 0.3‰. This number was scarce compared to the 20000 recorded events. In addition, the laser average output power, in this case, was 7.23 mW.
Fig. 3. Characteristics of the pulses bunching. Time-averaged spectra for four pulses, shown with different colors, with the same pump power and polarization state (a); expanded view of the spectra under wavelengths ranging from 1874 to 1878.5 nm (b); bunching pulse train with high amplitude peaks riding on the pulse bunching envelopes (c); histogram of the peak amplitudes (log-scale) (Inset: expansion of a pulse bundle), SWH: significant wave height, the right red line shows where the amplitude 2*SWH is located, and the red boxes show the value and occurrence number of ORWs (d); See the text for further detail.
It is known that the NPE regime provides a filtering effect in the frequency domain and that the spectral filter bandwidth (SFB) could be changed by adjusting the PCs in the cavity [34,35]. When the pump power was set to 550 mW and the PCs were carefully adjusted, bunching pulses with a central wavelength of 1902 nm were easily observed (See Fig. 4(a)). In this case, the average output power was about 2.84 mW, and the spectrum of Fig. 4(a) is essentially the same type as Fig. 2(a) except for stronger filtering in frequency, which can be appreciated in the modulations on the spectrum. The bunching pulse remained in a relatively stable state (See Fig. 4(b)).
Fig. 4. The spectrum of bunching pulse with Kelly sidebands (a); temporal evolution of bunching pulse concerning sweeping times (b).
With further adjustment of the PCs, the Kelly sidebands disappeared, and the envelope of the spectrum took on a triangular form, as shown in Fig. 5(a), which offered three spectra obtained at different times. The 3-dB bandwidth was about 0.7 nm, excluding NLP because the latter has a broad spectrum [36,37]. Compared with Fig. 3(a), only part of the spectrum fluctuated in Fig. 3(a); however, in the spectra in Fig. 5(a), peaks and dips of varying magnitudes occurred at random wavelengths. These observations suggest that a soliton explosion may have occurred considering that DFT has previously detected the explosion spectrum by Luo’s group [38]. In this case, the average output power was about 1.63 mW, lower than that of the stable bunching state in Fig. 4.
Fig. 5. Characteristics of the ORW and the NLP. Captured spectra at different moments with slight changes (a) and dramatic changes (b) in the spectrum profiles; temporal evolution of the chaotic pulse bunching, where the ORW appears with a low probability (See Visualization 2) (c) and a higher probability (See Visualization 3) (d); corresponding peak power evolution (e) and (f); ORW examples at different trace numbers (g) and (h); histogram of maximum amplitudes for each event (Inset: the autocorrelation trace of NLP) (i); the spectrum of NLP with a 3-dB bandwidth of 19 nm (Inset: pulse train at the fundamental period) (j).
This eruption of extreme wave dynamics, which lies behind the instability of the spectrum, is displayed in the time domain and can be seen using the 12.5 GHz bandwidth PD connected with the 33-GHz OSC. Figure 5(c) (See Visualization 2) shows the temporal evolution of the pulses in the chaotic bunching state, where some hot spots (see the dotted box) appear out of nowhere and then disappearing without a trace. Figure 5(e) shows the intensity evolution corresponding to Fig. 5(c). We can see that there exist three regions with large intensities, which submerge the low-intensity bunching pulses. We show three traces (Trace number: R=360, 910, and 1600) in Fig. 5(g). The upper trace of Fig. 5(g) exhibits a low-intensity pulse bundle with a high peak rising out from the chaotic multi-pulses (R=1600). The middle and bottom are giant pulses seen in trace numbers 910 and 360. Compared to the upper trace, in the lower portion of Fig. 5(g), the low-intensity bunching pulse seems to be swallowed by the giant pulses since these giant pulses manifest as a high peak about 15 times the size of the bunching pulse.
A higher probability of giant pulses occurring was also detected when the PCs were finely tuned to modulate the SFB and drive the pulses to a more unstable state, as shown in Figs. 5(d) and (f) (See Visualization 3). Figure 5(h) shows selected typical pulse envelopes for different traces. They show similar features to those in Fig. 5(g), where the giant pulses seem to swallow as much energy as possible. The highest intensity from the upper part of Fig. 5(h) was higher than at the bottom of Fig. 5(g), though the rare events were randomly generated. The spectra in this state tended to show more spikes and dips, as shown in Fig. 5(b). We can consider that the spectral filtering effect caused multi-pulse instability, which also induced mutual soliton collisions and resulted in dispersion wave formation. Under the interaction of dispersion waves and solitons, chaotic pulses formed and led to a more vigorous collision, which led in turn to the appearance of extreme events. Pulse dynamics and spectral dynamics, therefore, complement each other. Thus, the time-domain intensity fluctuation during the soliton collision also led to the fluctuations in the spectrum domain. To confirm ORW formation (see Visualization 4), we measured the histogram of the peak amplitudes as shown in Fig. 5(i). The L-shape statistics added evidence supporting the formation of ORWs. The 2*SWH was 36.6 mV, and a considerable portion of events was higher than 2*SWH. Instability in pulses and spectra is always characteristic of a transition state between steady states [39]. In our experiment, this ORW state was a precursor of NLP. We can see from Visualization 4 that, though the spectrum is trying to expand to a broad spectrum (NLP spectrum), it is limited to the triangular spectrum. Interestingly, by finely adjusting the PCs, NLP formed with 3-dB bandwidth of 19-nm in the spectrum (See Fig. 5(j)), and the pulse train shows the fundamental round trip periodicity in the time domain (Inset of Fig. 5(j)). The corresponding autocorrelation trace is double-scaled with a spike riding on a broad pedestal, as shown in the inset of Fig. 5(i), which is the typical character of the NLP [40,41].
4. Numerical simulations
In order to verify that the ORW can exist in fiber lasers, we numerically studied the generation and evolution of the ORW. The theoretical modeling for pulse transmission in fiber is based on the Ginzburg-Landau equation [42]:
Here A is the electric field envelope, z is the propagation distance, t is the pulse local time, β2 and γ are the GVD and Kerr nonlinear coefficient, respectively. Ω is related to the gain bandwidth. High-order dispersion and high-order nonlinearity can safely be neglected. For this Tm-doped fiber laser, the gain can be approximated by the equation:
In this expression, ΔT and Psat, which define the modulation depth and saturation power, are 100% and 30 W, respectively. We adopted a ring cavity based on Fig. 1 with the fiber parameters listed in Table 1. The Aeff refers to the effective mode area of fibers. The gain fiber (TSF 9/125) is modeled with a g0 of 25 dB, a gain bandwidth of 120 nm, and an Esat of 0.031 nJ. Based on the experimental results, the NPE regime may be seen to act as a SA and a spectral filter. When the PCs were finely adjusted, which corresponds to adjusting of the SFB, ORWs could form from multiple pulses. Thus, a Gaussian filter was employed owing to the negligible contribution from gain filtering. The function of the spectral filter effect was as follows:
where ω is the instantaneous angular frequency. δω=2πc(δλ/1.665λ2), where c is the speed of light in vacuum, δλ is the SFB, λ is the central operating wavelength. In this simulation, we used a 2-nm SFB to obtain a strong filtering effect.
Table 1. Values of parameters used in the simulation
As an initial signal, a Gaussian pulse was used. Under the above conditions, a stationary single pulse was obtained. Figure 6(a) shows the spectral evolution process of the single pulse. We can find that the initial Gaussian pulse reached stability after propagating for several round trips. The white line in Fig. 6(a) reveals the spectral waveform of the single pulse in a stable state. When the value of Esat was increased to 0.192 nJ, stable multi-pulses were obtained, and Fig. 6(b) shows the spectral evolution process of the multi-pulses. The stable spectrum (white line in Fig. 6(b)) shows stripes, and the stable spectrum of the multi-pulses possesses power dips, which is different from the spectral characteristics of the single pulse. Correspondingly, Figs. 6(c) and 6(d) show the temporal evolution of the single pulse and multi-pulses, respectively. Then, keeping all the other parameters unchanged and increasing the SFB in steps from 2 to 5 nm, the output pulse was found to change from multi-pulses to a state with chaotic pulses. For example, when we set the SFB at 3 nm and injected a single pulse into the cavity, it would break up into multi-pulses.
Fig. 6. Spectral evolution of a single pulse (white line: spectral waveform in round trip 20) (a) and stable multi-pulses (white line: spectral waveform in round trip 30) (b); temporal evolution of a single pulse (c) and stable multi-pulses (d).
They subsequently interacted with each other, aggregated, and diffused during transmission, as shown in Fig. 7(a). In these collisions, a high amplitude pulse might form, which constituted the possibility of giant wave formation. The peak power evolution is given in Fig. 7(b) and shows the evolution of the instability over 1000 round trips. High-amplitude pulses occurred non-periodically. Figures 7(c) and (d) show the evolution of the spectra and pulses for a typical case. In the case illustrated, it may be seen that at round trip number 919, a pulse with ultra-high peak power formed. When the giant pulse formed, the high peak intensity induced self-phase modulation, which resulted in spectral width extension (Bottom of Fig. 7(c)), and the energy quantization effect caused the giant pulse to collapse. This boosted the intensity fluctuations, as did the spectral instability.
Fig. 7. Characteristics of the chaotic multi-pulses. Temporal (a) and peak power evolution (b) vs. round trips; The average spectrum (c) with a triangular shape (upper) and spectral distribution vs. round trips; Several round trips containing the giant pulse in the time domain to show the character of the ORW (d); the corresponding histogram with an SWH of 50.8 W (e).
As a result, the spectra exhibited respiratory dilation and compression (Bottom of Figs. 7(c)). This process corresponds well to the eruption and disappearance process of ORWs under the chaotic pulse bunching in the experiment, as shown in Fig. 5 and Visualization 4. The average data for the 1000 periods shows a triangular feature, as illustrated in the upper part of Fig. 7(c). A histogram shows the number of events against the maximum peak power of a single round-trip pulse. We can directly count the probability of the occurrence of giant pulses in up to 20000 round-trips from this histogram. The deviation of the distribution from a Gaussian and the long-tailed shape indicated the presence of events that were larger than 2*SWH (2*SWH=101.6 W), as shown in Fig. 7(e). The number of ORW events observed was 71, corresponding to an occurrence probability of about 3.55‰ (The total events number is 20000). We can also find from Fig. 7(e) the peak power of the ORW-events can reach about 150 W, while the peak power in the stable multi-pulse in Fig. 6(d) was calculated to be 25 W. Thus, the peak power of these ORWs was 6 times of the stable multi-pulse.
To study the effect of spectral filtering on ORW formation, the SFB increased to 4 nm and then 5 nm. Figures 8(a), (b), (d), (e) show corresponding temporal evolution and peak power evolution. We found that some pulses drifted away from the main track of the pulse motion and then lost themselves in the background. The histograms shown in Figs. 8(c) and (f) illustrate the peak power distribution for the SFB values of 4 nm and 5 nm. The ORW-event number with peak powers larger than 2*SWH were 77 and 87, corresponding to the occurrence probability of 3.85‰ and 4.35‰, respectively (The total events number is 20000). When comparing Fig. 7(e) with Figs. 8(c) and (f), we can find that as the SFB was increased from 3 to 5 nm, the frequency of occurrence of ORWs was increased. We can understand this as follows, with the increase of SFB, the spectra width of the individual pulse would increase, and the duration of the individual pulse might be decreasing, which would induce the pulse with higher peak power and then split into more pulses under the energy quantization effect. As a result, a more dramatic collision among the pulses might occur, then the probability of the ORWs occurrence was increased. This also demonstrates that the multi-pulse instability, which caused the formation of giant pulses, was ultimately induced by the effects of spectral filtering.
Fig. 8. Temporal evolution of the ORW pulses at SFB of 4 nm (a) and 5 nm (d); the peak power evolution at SFB of 4 nm (b) and 5 nm(e); the corresponding histograms (c) and (f).
5. Conclusion
To conclude, we have experimentally detected ORWs in the time domain in a 65.9 m-long Tm-doped fiber laser with net negative dispersion and demonstrated that the ORWs formation resulted from multi-pulse instability induced by the spectral filtering effect. Using an NPE device as a mode-locker and a spectral filter, and by adjusting the pump power and the PCs, the laser can operate under conditions supporting perturbation pulse and chaotic pulse bunching. When the pump power was increased, through fine-tuning the intra-cavity PCs, the laser can operate from single pulses to a pulse bunching state. Further adjusting the PCs, this multi-pulse state became unstable and perturbated inside the bunching and in part of the spectrum. Through the statistical analysis, we found that ORWs existed in this perturbed bunching state. By adjusting the PCs in the cavity, the perturbed pulse bunching turned into a chaotic pulse bunching state and then gave rise to giant pulses with ultra-high amplitudes, a precursor of the broad-spectrum NLP. In addition, we have also observed that the formation of ORWs in chaotic pulse bunching state was with a higher occurrence frequency than that in the perturbed bunching state, and through adjusting the PCs, the occurrence frequency can even increase. We also studied the evolution of the ORWs numerically. We found that the spectral filtering effect played an essential role in the ORW formation. At a fixed pump power, only with the increase of the SFB, which was within the range of multi-pulse instability, the probability of ORWs formation increases.
Funding
Natural Science Foundation of Hebei Province (F2020205016); National Natural Science Foundation of China (12074098, 61605040).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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