A pulse–pulse interaction that leads to rogue wave (RW) generation in lasers was previously attributed either to soliton–soliton or soliton–dispersive-wave interaction. The beating between polarization modes in the absence of a saturable absorber causes similar effects. Accounting for these polarization modes in a laser resonator is the purpose of the distributed vector model of laser resonators. Furthermore, high pump power, high amplitude, and short pulse duration are not necessary conditions to observe pulse attraction, repulsion, and collisions and the resonance exchange of energy between among them. The regimes of interest can be tuned just by changing the birefringence in the cavity with the pump power slightly higher than the laser threshold. This allows the observation of a wide range of RW patterns in the same experiment, as well as to classify them. The dynamics of the interaction between pulses leads us to the conclusion that all of these effects occur due to nonlinearity induced by the inverse population in the active fiber as well as an intrinsic nonlinearity in the passive part of the cavity. Most of the mechanisms of pulse–pulse interaction were found to be mutually exclusive. This means that all the observed RW patterns, namely, the “lonely,” “twins,” “three sisters,” and “cross,” are probably different cases of the same process.
© 2016 Optical Society of America
The rogue wave (RW) as a concept was initially introduced in oceanography to describe rare events commonly called either “freak” or “rogue” waves. These waves have an amplitude that is much larger than the average, resulting in a destructive impact on nature and society . Investigation of mechanisms that cause RWs has practical importance due to the huge damage they cause in real-world scenarios. Meanwhile, an intrinsic scarcity of these events as well as the evident technical difficulties in performing full-scale experiments have been the main obstacles to understanding and predicting RWs. Mode-locked lasers are perfect candidates to be test-bed systems for investigating the conditions for the emergence of RWs and their mitigation . The use of high repetition rate pulsed lasers gives us the opportunity to collect a huge amount of data in a relatively short time under laboratory-controlled conditions. In the context of mode-locked fiber lasers, optical RWs have mostly been observed in the regimes of chaotic bunches of noise-like pulses  or the regime of soliton rain . Previously it has been found that RWs can be generated in mode-locked lasers both as a result of the interaction of dissipative solitons through the overlapping of their tails and by dispersive pulses [5,6]. The result of these interactions is a coupling enhancement, which leads to chaotic bunching with periods shorter than the round-trip time. An extensive study of the mechanisms of formation of optical rogue waves (ORWs) has been done both experimentally and theoretically in fiber lasers with nonlinearly driven cavities , Raman fiber amplifiers and lasers [8,9], and fiber lasers via modulation of the pump .
The absence of the mode locker in the laser cavity opens the way to a very rich polarization dynamics based on competition between polarization modes. The set of equations for this kind of oscillator needs to include the equations for both polarizations, treating the electric and magnetic fields as full 3D vectors (vector model). The solution of this problem, similar to the problem of propagation of a pulse in polarization-dependent media, leads to a set of polarization instabilities near the lasing threshold. Fortunately, for a standard optical fiber with artificially induced birefringence, in a linear approach, the characteristic polynomial of the differential operator associated with the system of equations can be factorized in two terms, one of which depends only on birefringence and another one only on pump power. This happens due to the intrinsic symmetry of the associated operator and allows the possibility to realize and handle an extremely large range of regimes in the laser oscillator (vector oscillating regimes). Some of these regimes are stable and others are chaotic.
Unlike in previous publications, in this paper, we report observations of RW generation at pump powers close to the lasing threshold in strongly birefringent media (vector RWs). These processes can be controlled by changing both the birefringence in a cavity and the polarization of the pump by means of polarization controllers. The change of the pump power in this scenario does not lead to new RW patterns but changes the likelihood of RWs occurring.
We have observed two classes of RW patterns. The first one comprises patterns with duration less than the characteristic time of the decay of the autocorrelation function (AF) (fast optical RW or FORW). Meanwhile, the second includes events with a duration significantly longer than this (slow optical RW or SORW) [11,12]. In the first case, the patterns are affected by periodic amplification in each round trip, which can be considered a kind of intrinsic “memory.” During the second scenario, the pattern is averaged along the resonator. This issue affects the RW statistics and, as we have found, the first scenario excludes the second one. In the third section, we have classified and illustrated patterns of fast optical rogue waves (FORW).
2. EXPERIMENTAL SETUP
The schematic of the laser is illustrated in Fig. 1. Unlike a mode-locking scheme based on nonlinear polarization rotation, we have used just one polarization controller (CPoC) inside the laser cavity. Another controller (PPoC) was used to control the state of polarization (SOP) of a pump wave. The cavity includes 1 m of -doped fiber (Liekki Er80-8/125) and 614 m of single-mode fiber (SMF-28) () (see Supplement 1).
A system of reference in which the speed of propagation of a circularly polarized pulse (average speed of light) was taken as zero has been chosen as most appropriate for analysis.
3. RESULTS AND DISCUSSION
A. Previous Considerations
FORW patterns have been classified into two groups: patterns that occur with a relatively high likelihood (“lonely,” “twins,” and “three sisters”) and the rare patterns. In the group of rare patterns, we have considered two mechanisms of pulse–pulse interaction: resonance interaction (the “accelerated” pattern) and collision (the “cross” pattern).
Resonance interaction occurred when the difference of relative speeds between two or more pulses was small enough () to allow interaction between them through polarization “hole burning” in the orientation distribution of the population inversion ; meanwhile, the collision occurred at the highest relative speeds. Both patterns belonging to this group were observed only for the regime similar to soliton rain . The lonely and three sisters patterns have been grouped together because they have similar probability distribution functions (PDFs) (see Fig. 2) and similar lifetimes. The twins pattern probably belongs to the other group. This pattern has a different PDF; also, it has been observed only close to the edges of mode-locked pulses and has significantly longer lifetime than the lonely and three sisters patterns.
Through the interaction between pulses, a period of “uncertainty” with duration of less than 10 round trips (the typical duration was around three or four round trips) was observed. During this period, the interacting pulses look “smeared-out.” Probably, the duration of this uncertainty period marks a frontier between the cross and lonely/three sisters patterns. The lonely and three sisters patterns are representing the case when the pulses remain together longer than two uncertainty periods. Moreover, the three sisters pattern itself is perhaps just a result of short-lived coupling among three lonely patterns. The twins patterns were observed mostly when the relative speed between pulses was very close to zero; in this situation, two pulses propagating with the same speed probably realized a kind of coupling, which made the pattern very stable (note that there are no records about the twins RW pattern in water). The cross pattern is the exhibition of another extreme. When the relative speed is very high, the pulses have no time to interact and just collide.
Twins patterns typically are found near the main pulse (see pulse patterns in Supplement 1). This would explain the extremely long lifetime of these patterns. Close to the main pulse, the system has enough energy to amplify and maintain the pattern during thousands of round trips (the memory effect). Meanwhile, between pulses, the inverse population was depleted and the pattern could not be maintained for a long period.
In the quasi-soliton-rain regime, the energy stored in the active fiber was probably spread out among many polarization states and was not enough to maintain multiple FORW patterns, which dissipate energy either because of coupling between pulses of the pattern or interaction with precursor and follower. Hence, the multiple patterns become unstable, degenerating to short-living lonely patterns.
The PDFs (normalized histograms) of the amplitudes of pulses for the regimes in which we have observed the patterns are illustrated in Fig. 2. The PDFs were calculated by subtracting the mean value from the dataset and dividing the data by the standard deviation. After that, the PDFs were plotted on a logarithmic scale. The use of PDFs instead of non-normalized histograms in our opinion makes comparison among them easier. All the PDFs have a characteristic L shape. The PDFs for the three sisters and lonely patterns have similar shapes, giving us reason to classify these patterns in the same class of events. The PDFs for the cross and accelerated patterns also have similar shapes because these patterns appear together; meanwhile, the twins patterns belong to a statistically different class of events.
The PDFs for cross/accelerated and twins patterns look quite similar; a significant difference between them is observed only for the L-shaped tail of the distribution. Comparison of the data from the optical spectra analyzer (OSA) (Fig. 3) shows that these regimes, in fact, are different and probably should be considered as two different groups of successes.
1. Lonely Pattern
The pattern of lonely FORWs, illustrated in Fig. 4, was detected when the pump polarization controller (PPoC) has been set at and the polarization controller in the cavity (CPoC) at 20° (the angles were measured from the vertical position of the knob). This pattern was observed for almost all positions of the polarization controllers where RW generation has been observed. The pattern illustrated in Fig. 4 corresponds to polarization with refractive index ; it was propagating faster than the average speed of light.
A detailed description of the regime and method of estimation of the pattern speed is given in Supplement 1. The pattern itself is shown in Fig. 4(a). The axes labeled and represent a new system of reference that moves 1624.8 m/s faster than the average speed of light. This system of reference does not match accurately with the system of reference related to the event because the oscilloscope has a finite time of sampling. For this reason, the event looks tilted. The result of filtering is shown in Fig. 4(b), as well as a 3D view of the front [Fig. 4(c)] and the back [Fig. 4(d)] of the pattern. The pattern oscillated during round trips or (the red head and shoulders figure); the next top [the yellow top in Figs. 4(c) and 4(d)] appeared later, this time not surpassing the RW threshold. The precursor showed similar behavior; however, the follower has just vanished without oscillating. After that, the significant wave height (SWH) pulse became unstable over 83 round trips () and, finally, after a short period of uncertainty, became split into two different pulses [see encircled area in Fig. 4(a)]. Both the follower and the precursor were affected during the uncertainty [see the areas of the Fig. 4(a) labeled 1 and 2].
The splitting of a RW in two SWH pulses was observed to be the most likely scenario for the destruction of the FORW. The pulse power has been integrated before the splitting (the area below the pulse is the energy carried by the pulse), and the result of the integration has been compared with the result of the integration after the splitting. The comparison has indicated that the overall energy was conserved to within an error of . The ratio between the overall energy and the pulse speed also remained approximately unchanged, that is to say, the splitting of this pattern was an elastic process. This pattern was the most frequently observed. We have estimated the likelihood of the pattern as the ratio between the number of observations of this pattern and the whole number of observations of RW patterns for all orientations of the polarization controllers and have obtained a value of 0.6.
2. Twins Pattern
The pattern of two RWs propagating one just after the other (the twins pattern) has been observed for only a few positions of the polarization controllers and has a significantly smaller likelihood than the lonely pattern. The pattern illustrated in Fig. 5 was detected close to the back edge of the main pulse when the PPoC and CPoC were set to and 40°, respectively. This pattern was similar to the two-pulse soliton molecule , although the pattern in the illustrated case has significantly longer lifetime. Also, the pulses that form the twins pattern were significantly longer than the pulses illustrated in .
This pattern has shown oscillations in a breather-like style with a period of (). This frequency of oscillation matches the frequency of satellites observed in the RF spectra (see Fig. S2 in Supplement 1). At the same time, the pattern of pulses was propagating faster than the average speed of light, and the refractive index for the pattern was calculated to be . The lifetime of the whole pattern was approximately 750 round trips (). After that, the continuous pattern vanished and later surpassed the RW threshold only in oscillation maxima, forming a kind of chained lonely pattern. At the same time, the first pulse of the pattern became split in two. The remains of it interacted with the precursor and the follower, forming two practically equal SWH pulses separated with a small pulse in between them. This mechanism of the destruction of a long-lived RW pattern when the pulse loses part of its energy after being split in two was observed frequently, being one of the most common causes of destruction of accelerated and twins patterns.
We have observed that this kind of pattern appearing either close to the main pulse or close to a high subharmonic pulse (in the case of the multipulsing regime). The extremely long lifetime of the twins pattern can probably be explained by a periodic amplification of the pattern in each round trip; that is to say, the pattern cannot appear between pulses because the energy of the system is depleted and is not enough to amplify a pulse above the RW threshold. The likelihood of this pattern was .
3. Three Sisters Pattern
The pattern illustrated in Fig. 6 has three FORW propagating in a bunch . The direction of propagation is indicated by the arrow in Figs. 6(a) and 6(b) labeled “Time axis.” If the observer moves in the direction against this arrow, then they will observe three consecutive RWs. The pattern was visually similar to the twins pattern, although it was not related to the main mode-locked pulse or a periodic subpulse. Unlike the twins pattern, this pattern showed a structure quite similar to Akhmediev’s breather structure . The foregoing RW was created after two SWH pulses had been merged [see Fig. 6(a)]; in this moment, the precursor of the third RW rose until it surpassed the RW threshold.
Attraction and repulsion between pulses were observed immediately before merging and after splitting, respectively [17,18] [see labels “A” (attraction) and “R” (repulsion) in Fig. 6(a)]. We have observed that the attraction process has a delay threshold. When two pulses were getting closer to each other than to this threshold, the pulses were mutually attracted.
The lifetime of the pattern was . After the first wave was destroyed, being split in two SWH, this pattern did not appear again. During the next oscillation, only one of the remaining SWH pulses has surpassed for a moment the RW threshold. The energy has been conserved during the lifetime of this pattern. The speed of propagation of the pattern was less than the average speed of light. The period of oscillations of the pattern was similar to the period observed in the case of the twins pattern. The likelihood of this pattern was low (). What is more, it has appeared for only a few positions of the polarization controllers.
4. Cross Pattern
The pattern illustrated in Fig. 7 is a collision between two pulses [15,19,20]. At first glance, the pattern seems to be similar either to the pattern theoretically illustrated by Antikainen et al. in  for solitons with duration of or to the second-order RW . However, unlike these, the observed pattern has a duration of , and integration of the oscilloscope traces has revealed that the intensity of the peak is just a sum of the intensities of the collided pulses. Moreover, we did not observe out-of-phase patterns, similar to those illustrated in Fig. 1(b) in . All of this probably means that the pattern that we have observed was just a collision of two pulses without interference between them. Moreover, the OSA spectra (see Fig. 3) do not show a double top or ripples. Hence, the most probable cause for the difference in speeds of the interacting pulses is just a different polarization of them. This kind of pattern has not been theoretically discussed from the point of view of the vector polarization model of FORW.
It was observed that, to be able to interact, two pulses should remain close during more than 20 round trips or, in other words, have a relative speed of less than . In the illustrated case, two pulses did not have time for energy exchange due to the very high difference of their speeds (). The collision was very fast (). All of these patterns were observed for a very narrow range of state of polarization. The likelihood of this pattern was extremely low ().
5. Accelerated Pattern
The last pattern (see Fig. 8) corresponds to the situation when the RW changes speed throughout the observation. This pattern of accelerated FORW has been detected along with the cross pattern.
Unlike in , where the acceleration was attributed to the interaction of the soliton, which has a duration of tens of fs and tens of kW peak power with nonlinearities in the passive fiber over the anomalous dispersion regime, in our case the pulse has a very small amplitude () and a duration of nanoseconds. That means the acceleration of the pulse probably was observed due to intrinsic nonlinearities in both the active and passive fibers.
The results of the data fitting are illustrated in Fig. 9. After the fitting, it was noted that the acceleration of the pulse was nonuniform. At the beginning of the observation, the RW propagated far from the oncoming pulse, showing oscillations (see Fig. 10) in a breather-like style. When the pulse labeled 4 interacted with the precursor labeled 2 (relative speed between pulses ) in Figs. 8(c) and 8(d), the RW pattern stopped oscillating (compare Figs. 9–11); after a short transition process, it has acquired an acceleration of , which was lost gradually during approximately 600 round trips (). Through this process, the speed of the pulse (relative to the average light speed) has been changed from to ; the precursor [labeled 3 in Fig. 8(c)] has lost energy and the approaching pulse [labeled 4 in Figs. 8(c) and 8(d)] has gained it. The whole pattern has propagated faster than the average speed of the rest of the pulses.
For a detailed study, we have calculated the mean value for each pixel [see Fig. 11(d)], and we have derived the dataset twice. The results are illustrated in Fig. 11. The horizontal scale in Fig. 11(d) matches the horizontal scale in Fig. 10 and shows the absolute oscilloscope time; the moment when the pattern was detected corresponds to the oscilloscope time of 2.0 ns and the last one to 0.797 ns; in other words, the pattern evolves in a direction against the time axis (the relative speed was less than zero).
In other words, we have observed how the RW was accelerated and then gradually lost acceleration after interacting with a slow SWH pulse. During the interaction between the precursor and the approaching pulse, it got the energy from the precursor. The integration of the oscilloscope traces has revealed that this interaction was elastic. These patterns were observed at two different positions of the polarization controllers. The likelihood of the pattern was extremely low. Also, we have observed similar patterns that have not surpassed the RW threshold.
We have experimentally demonstrated the existence of complex polarization-dependent RW dynamics in a unidirectional ring-cavity fiber laser without a saturable absorber. Unlike previous theoretical studies in which the interaction scenarios were created by the interaction of the dissipative solitons with tens of kW peak power and fs duration, we have observed an interaction mechanism that is more likely based on the interaction between polarization modes. In all illustrated cases, the mechanism of interactions among the RW, the follower, and the precursor cannot be primarily caused by the nonlinearity in a passive fiber, due to the low pulse amplitudes (). That is to say, the interaction between pulses more likely has also occurred in the active part of the cavity, probably through polarization hole burning by means of polarization attraction . In addition, we have observed new patterns: the FORW propagating with variable acceleration and the twins pattern. Although those similar patterns have previously been discussed theoretically, in our experiments we have observed that the amplitudes of the patterns, as well as their duration, were very different from the values that have been considered by theoreticians. The experiment has demonstrated that the concept of a soliton origin of FORWs is more likely universal in the context of coupled nonlinear systems and has supported the thesis about “independence of the dynamics from interactions specific to the optical concept,” which was proposed in .
Leverhulme Trust (RPG-2014-304, CHARISMA); Seventh Framework Programme (FP7) (324391, GRIFFON).
The authors thank the Leverhulme Trust (RPG-2014-304, project CHARISMA) and the Seventh Framework Programme (FP7-PEOPLE-2012-IAPP program, project GRIFFON). We thank professor D. Webb from AIPT for his help with this paper.
See Supplement 1 for supporting content.
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