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Abstract
Some complex dissipative dynamics associated with the noise-like pulse (NLP) regime of a passively mode-locked erbium-doped fiber laser are studied numerically. By means of a convenient 3D mapping of the spatio-temporal pulse evolution, for properly chosen dispersion parameters, several puzzling dissipative dynamics of NLPs are identified, including the expelling of sub-packets that move away from the main bunch, the sudden extinction of isolated sub-pulses, the collision between different internal fragments travelling at different speeds, the rising of sub-pulses, the formation of complex trajectories by substructures that first move away and then return to the main bunch, and so on. In addition, the emergence of optical rogue waves (ORWs) within NLPs is also demonstrated numerically; to help understand these behaviors evidenced in the time domain, spectral analyzes were also performed that show, among other things, that the spectrum of a NLP is notoriously distorted when it hosts an ORW phenomenon. These numerical results are consistent with previously published experimental results.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Noise-like pulses (NLPs) are extravagant optical pulses produced by passively mode-locked fiber lasers (PML-FLs), which have attracted the attention of many researchers in recent years. This interest is mainly motivated by the fact that NLPs can reach high energies and still maintain a quasi-stationary behavior. In particular, in fundamental mode-locking operation, this behavior is compatible with applications such as supercontinuum generation (SCG) [1–5], low-coherence spectral interferometry [6], sensing [7], nonlinear frequency conversion [8] and imaging [9,10], among others. On the other hand, NLPs can give rise to very complex puzzling dynamics whose understanding is challenging and therefore encourages many research efforts in non-stationary operation. Here, these pulses display very interesting and rich dissipative dynamics, some of which were studied experimentally in [11–13]. Among possible scenarios, one can witness the splitting of the main bunch, the drifting of sub-packets following complex trajectories, the emergence of sub-pulses from the background radiation, as well as their progressive decay. In addition, it has also been reported that NLPs are a fertile background for the emergence of extreme events also known as optical rogue waves (ORWs) [14–17]. All these manifestations illustrate the fact that NLPs are not simple conventional pulses but complex wave packets, with a total duration of the order of ns experimentally, displaying a chaotic internal evolution at the fs-ps inner scale. The NLP emission (quasi- and non-stationary operations) is traditionally characterized by a wide and smooth average spectrum (>10 nm 3-dB bandwidth) and by a double-scaled autocorrelation trace, where a fine coherence spike rides a broad pedestal [1,18,19]. Interestingly, the recent development of novel characterization techniques, in particular relying on single-shot measurements, revealed a much deeper complexity than previously conceived [12,13,15,20–22]. In spite of these important experimental advances and a handful of numerical works [23–26], to date the formation and dynamics of NLPs are still a broadly debated subject of study.
Although NLPs and solitons are radically different types of objects (associated with chaotic and stable mode-locking regimes, respectively), when the former is discussed one cannot leave aside the latter and their mutual interaction. For instance, in [20], grey and dark solitons were experimentally observed in the envelope of a NLP formed in a ring laser cavity; in [27], the authors observed in a ring laser multi-soliton bunches in an intermediate regime between continuous wave and NLPs. On the other hand, dual soliton-NLP regimes are presented in [28–30]: such regimes can be found in a figure-eight laser (F8L) [28], or in a ring cavity [29,30]. Besides, a comprehensive study where a wide variety of regimes involving NLPs and solitons were identified, was performed in [31]. In [11,12], fragments of NLPs with quasi-discrete amplitudes were observed, alluding the presence of solitons within their envelope. Furthermore, through numerical simulations, a hybrid regime between NLP and solitons was discussed in [26] in a ring laser cavity, suggesting that the solitons may originate from inner sub-pulses that manage to escape from a NLP. In [23], bifurcation limits of the co-existence of soliton-like and NLP numerical solutions are established by the proposed of different models that represents a nonlinear polarization evolution mode-locked ring laser.
Usually, ORWs are defined as those unpredictable and ephemeral events whose intensities exceed 2.2 times the significant wave-height (SWH; average amplitude of the highest third of all events) [32]; for the observation of these extreme events, NLPs constitute an ideal benchmark. The highly nonlinear chaotic dynamics of NLPs can lead to the generation of ORWs, as observed experimentally in a ring laser [15]. Again, in a ring laser cavity, we reported recently the observation of a soliton creation process that occurs during extreme-intensity episodes affecting a NLP, as well as the inverse process, through which a NLP occasionally emerges from an extreme event that affects a packet of solitons [16]. In a similar laser architecture, although in the normal-dispersion regime, some authors observed experimentally that the formation of NLPs is related to the generation of a high-intensity structure formed of 2 or 3 consecutive intense pulses [33]. In addition, numerically, using a ring laser model, it was shown that in the chaotic dynamics of a NLP, transient pulses of extreme intensity appear through nonlinear collisions [34]. Finally, it was proposed in a numerical study that the mechanism that governs the formation of extreme events is related to the temporal behavior of the saturable absorber and the polarization mode dispersion of the cavity [35].
In this numerical work, diverse dissipative phenomena are evidenced within NLPs produced by means of a F8L model. Using 3D mapping a spatial-temporal evolution is presented, exposing dynamics such as the expelling of sub-packets from the main bunch and their subsequent drifting or extinction, the collision of sub-packets at different speeds as well as the emergence of ORWs. Besides, using the formalism of the statistical central moments, a substructure that describes a complex trajectory with respect to the main bunch is examined in detail. Finally, the temporal-spectral correspondence during an ORW within a NLP is analyzed, revealing that this intense event is associated with an abrupt alteration of the spectrum.
2. Numerical setup and model
To produce NLPs numerically, a F8L model with temporal field components in circular base was used. The PML-F8L with a total length of 29 m, consists of a ring cavity plus a nonlinear optical loop mirror (NOLM), as depicted in Fig. 1. In the ring section, a 4-m-long erbium-doped fiber (EDF, $D ={-} 17\textrm{ ps/nm/km)}$ was implemented; this active fiber has uniform gain over its entire length. The gain saturates on pulse energy ${E_p}$ $({g_0}$ is the small-signal gain and ${E_{sat}}$ the saturation energy) as
A value $\gamma = 1.5\textrm{ }{\textrm{W}^{ - 1}}\textrm{k}{\textrm{m}^{ - 1}}$ of the nonlinear Kerr coefficient is taken for all sections of fiber. For input polarization orthogonal to the QWR, the NOLM switching power reaches its minimal value (calculated in the continuous wave approximation) of ${P_\pi } = {{6\pi } \mathord{\left/ {\vphantom {{6\pi } {(\gamma {L_N})}}} \right.} {(\gamma {L_N})}} \approx 1250\textrm{ W}$ (where ${L_N}$ is the length of the NOLM loop). Light propagation is calculated by integrating the coupled extended nonlinear Schrödinger equations [39], which write as
3. Results and discussions
An important number of simulations were carried out by means of our numerical laser model, obtaining several temporal sequences of NLPs. One single waveform of one sequence is shown in Fig. 2(a); in contrast to experimental measurements, through numerical simulations it is easy to appreciate the complex internal structure that these pulses possess in the time domain, as well as the corresponding noisy single-pulse optical spectrum (Fig. 2(b)); it is worth mentioning that it is possible to get similar single-shot spectra experimentally though methods such as the dispersive Fourier-transform [21,43]. On the other hand, compared with these single-shot spectra, those obtained experimentally using an optical spectrum analyzer are much smoother because the individual fluctuations of millions of pulses are averaged in the measurement [1,19,44,45]; for comparison, in Fig. 2(b), the single-shot numerical spectral profile (red line) and the average spectrum of 200 numerical spectra (black line; 19.2 nm 3 dB bandwidth) are depicted. Finally, Fig. 2(c) shows the autocorrelation trace, displaying a typical characteristic of NLPs. For the whole study, only the circular left polarization state $({C^ - })$ was taken into account (a similar behavior is observed for the right polarization state).
Fig. 2. (a) Single numerical NLP waveform. (b) Individual spectrum (red line); averaged spectrum (black line) of the 200 spectral profiles of Sequence 1, studied below. (c) Autocorrelation trace of the single NLP waveform.
Stacking waveforms consecutively, significant fluctuations over the round-trips can be observed, but at the same time their global features are conserved, as shown in Fig. 3. In this figure it can be appreciated that a sub-packet (pointed out by a red arrow) moves away from the main bunch (thick green arrow) toward longer times; on the other hand, an intense peak protrudes in cycle 33; both dynamics are analyzed in the following. However, in this figure, the small number of temporal waveforms limits the analysis of the bunch evolution.
Fig. 3. Temporal NLP waveforms of 10 consecutive cycles. The red arrow indicates the trajectory of a sub-packet that moves away toward longer times from the main bunch, whose trajectory is highlighted with a thick green arrow.
When the same waveforms are arranged consecutively in larger numbers and a 3D mapping is used, the overall evolution of the NLP can be appreciated more clearly. Figure 4 presents the first sequence (Sequence 1) of a globally stable main bunch, displaying a rich dynamics; the temporal waveforms illustrated in Fig. 3 constitute a section of this whole evolution. In the following, a few selected puzzling dynamics are first analyzed with more details, and finally the manifestations of ORWs are studied. The 3D view of Sequence 1 is shown in Fig. 4(a); it presents the evolution from cycle 11 (where the NLP is already fully formed) to cycle 210. In this figure the temporal walk-off of several sub-packets can be appreciated. It can also be seen that some sub-pulses to the right of the main bunch extinguish without evident collisions; for example, that first one expelled to longer times, which extinguishes at cycle 20 (at 53.2 ps), as well as some others between cycles 160 and 190 (∼60 ps to ∼75 ps). The small number of cycles in the sequences allows to observe some fine scale dynamics in detail.
Fig. 4. Spatio-temporal evolution of Sequence 1, formed by 200 consecutive cycles of a NLP (cycles 11–210). (a) 3D mapping of the spatio-temporal pulse evolution, displaying some puzzling dynamics. (b) Top view: trajectories of sub-packets sp1, sp2 and sp3 discussed in the text are highlighted by yellow, red, and pink arrows, respectively; an interesting collision between sp1 and sp2 is marked with a black circle; a dashed gray box indicates a substructure that makes a notorious C-shaped trajectory.
The top-view of Sequence 1 is presented in Fig. 4(b). A clear and interesting collision between two sub-packets at cycle 87 is marked with a black circle. In the collision, the two sub-packets involved travel at different speeds: “sp1” (“sub-packet 1”, detached from the main bunch) and “sp2” (which appears around cycle 65). After the collision, another sub-packet (“sp3”) comes into existence a couple of cycles later (near cycle 90), following a curved trajectory that eventually gets straight and parallel to those of previously expelled sub-packets (right side of Fig. 4(b)). Sub-packets sp1, sp2 and sp3 are pointed out with yellow, red and pink arrows in the figure, respectively; a close-up view of the sp1-sp2 collision is shown in Fig. 5(a), using the same mark colors. It seems that sp3 results from sp2 absorbing almost all the energy of sp1 at the collision and subsequently deflecting its original trajectory progressively, whereas the remaining of sp1 forms a short decaying tail that is soon extinguished.
Fig. 5. (a) Close-up view of the temporal evolution near the sp1-sp2 collision at cycle 87 (at 71.6 ps). (b) Central wavelengths and (c) spectral width of sp1, sp2 and sp3.
To deepen the analysis of the collision, we examined the spectral evolution of the individual sub-packets. For this purpose, at each cycle, each individual sub-packet (sp1 or sp2 before the collision, sp3 after the collision) was extracted from the temporal data (this was done simply by setting to zero the remaining data in the numerical mesh, conserving only the values of the corresponding sub-packet). The waveform was then multiplied by a Gaussian smoothing window centered on the corresponding sub-packet, and its spectrum was calculated by applying the fast Fourier transform (FFT). Firstly, we focused on the evolution of the central wavelength and bandwidth of each spectrum across the cycles. For this purpose, the formalism of the statistical central moments adapted for characterizing arbitrary shaped pulses [25,46] was used. By applying the first statistical central moment concept, the central position of the spectral profiles is defined as
Additionally, we analyze in more detail the spectra of the individual sub-pulses. In the following, we focus on two particular cycles: cycles 78 (before the collision) and 120 (after the collision). The results can be seen in Fig. 6. Figures 6(a) and (b) show the spectra of sp1 and sp2 before the collision. The spectra are similar, but are centered at different wavelengths, presenting maxima at 1554.1 nm and 1550.9 nm, respectively. The shift of $\Delta \lambda = 3.2$ nm between the two spectra can be clearly observed in the close-up of Fig. 6(c). This displacement is consistent with the value calculated via the total cavity dispersion $({D_T} = 0.357$ps/nm; anomalous) [47] and a sp1-sp2 temporal separation reducing at a rate of 1.1 ps per cycle in Sequence 1: these values yield $\Delta \lambda = 3.1\textrm{ nm}$, which closely matches the value found directly from the calculated spectra. These detailed calculations thus confirm the conclusions drawn from the analysis of Fig. 5: the spectral shift allows understanding that sp1 and sp2 travel at different speeds and eventually collide. Slight wavelength displacements between different temporal components can thus explain that sub-pulses move at diverse velocities in the NLP internal dynamics, which in turn promotes their frequent interaction. Finally, Fig. 6(d) presents the instantaneous spectrum of sp3 at cycle 120 (after the collision); unlike the spectra of Figs. 6(a) and (b) before the collision, it displays lateral peaks, which could be interpreted as Kelly sidebands, being equidistant from the central peak by 3 nm (inset).
Fig. 6. At cycle 78, before the collision: instantaneous spectra of (a) sp1 and (b) sp2; (c) close-up of the main peaks of sp1 and sp2, showing a spectral shift. At cycle 120, after the collision: (d) instantaneous spectrum of sp3, where Kelly-sidebands-like peaks are displayed (inset).
As a complement, Fig. 7 presents the temporal profile, unwrapped phase and instantaneous frequency representations of the complete NLP at cycle 111. Again, a roughly parabolic temporal phase (Fig. 7(b)) and an overall positive slope of the instantaneous frequency (Fig. 7(c)) are observed. The existence of linearly chirped NLPs was revealed in a previous work; in [24], a numerical study was carried out with a similar laser model, focusing on the intracavity dynamics within each cycle under a strong dispersion mapping; in that work, it can be seen that a NLP stretches and compresses alternately in the cavity, with a chirp that switches sign due to the alternating dispersion sign of each section.
Fig. 7. (a) Temporal profile of the complete NLP at cycle 111. (b) Unwrapped phase. (c) Instantaneous frequency.
Another interesting dissipative manifestation studied is the substructure which generates the C-shaped protruding trajectory over round-trips 65–100, which can be seen on the left-hand side of Sequence 1 (pointed out by a dashed gray box in Fig. 4(b)). This substructure, expelled from the main bunch near cycle 63, moves away to shorter times until it stays still, then slows down until its temporal walk-off is reverted and eventually returns towards longer times to the main bunch, reaching it back near cycle 100. More examples of this kind of trajectories can be noticed on the left-hand side of the whole Sequence 1. The complete NLP between cycles 65 and 100 (temporal and spectral evolution) is presented in Figs. 8(a) and (b). Figures 8(c) and (d) depict the evolution of the substructure over the same cycles. It can be observed that no significant spectral displacement was detected in Fig. 8(b); nonetheless, isolating the temporal data of the C-shaped structure and performing the corresponding spectral calculation, a notorious spectral shift can be seen (Fig. 8(d)). A comparison made between the complete NLP and the substructure through the first statistical central moment is shown in Fig. 8(e). As can be observed, the central wavelength of the former remains roughly constant at ${\lambda _c}^{NLP} \approx 1552\textrm{ nm;}$ as this wavelength is slightly above 1550 nm (chosen as the central wavelength in the simulations), the NLP temporal waveform is slightly deflected to the right in Fig. 8(a). In contrast, the substructure undergoes a significant redshift from 1546 nm to ${\lambda _c}^{sub} \approx 1550\textrm{ nm,}$ reaching the central wavelength of the simulation near cycle 83 (Fig. 8(e)), where the deflection of the substructure occurs (Fig. 8(c)). Subsequently, the wavelength of the substructure remains close to (although slightly below) the NLP central wavelength. In this case, the temporal evolution is only partly consistent with the spectral evolution.
Fig. 8. Analysis over cycles 65–100. (a) Temporal profile of the complete NLP evolution, and (b) its spectral profile. (c) Temporal profile of the substructure evolution, and (d) its spectral profile. Comparison between their respective (e) central wavelengths and (f) spectral widths.
Before cycle 85, the substructure is significantly blue-shifted, which explains that it moves towards shorter times, away from the NLP. This drift then slows down as the wavelength of the substructure approaches 1550 nm. The substructure then stops drifting; this happens at cycle 83 (its spectrum actually reaches 1550 nm at cycle 85). Beyond this point, the trajectory of the substructure is reverted as it starts moving quickly to the right (back towards the NLP); this observation is not consistent with the spectral position of the substructure, whose central wavelength is only slightly higher than 1550 nm, and decreases down to 1550 nm between cycles 90 and 100. This result points out towards the conclusion that the intervention of additional mechanisms, in particular dissipative effects, must intervene in the onset of this kind of dynamics. This means that the substructure initially moves faster than the main bunch, drifting towards lesser times (to the left), then it stops drifting (cycle 83) and finally starts moving slower (to the right), until it is reached by the main bunch. Possible dissipative mechanisms that could be involved in such behavior include the interaction between pulses by means of dispersive waves, or a temporal gradient of nonlinear losses in the cavity. The latter effect is similar to that occurring in experimental systems, when a pulse passes through the amplifier. In those cases, the pulse experiences a slightly smaller gain in the trailing edge than in the leading edge, due to saturation by the same pulse, which generates a slight temporal displacement towards negative times [48,49]. Such gain-induced gradient cannot exist in the present case because fast gain saturation is not accounted for in our model; however, it can be produced by nonlinear losses, in particular because ultrashort pulses that counterpropagate in the NOLM undergo slightly different evolutions, and their interference is not uniform across the profile when they recombine. The existence of such gradient and its variability over the cycles could contribute to alter the trajectories of substructures. In addition, the spectral width of both pulses is presented in Fig. 8(f): the spectral width of the former (SWNLP) remains roughly constant around 9 nm, whereas the spectrum of the substructure (SWsub) evolves over the cycles, having initially a lower value than the complete NLP but getting eventually a slightly higher SW. Experimentally, we reported dynamics very similar to the one presented by this substructure, in [12,13]; this work confirms that those experimental results are consistent with numerical simulation.
The formation of ORWs in the internal structure of the NLP during its evolution is another very interesting aspect of this regime that received attention. Figure 9 presents a comparison between some waveforms that present high-intensity events, corresponding to single spikes whose amplitude is several times higher than average, and some that do not. Three ORW manifestations are depicted in Figs. 9(a), (c), and (e); for comparison, three waveforms that do not contain extreme events are shown in Figs. 9(b), (d), and (f). The waveforms are presented according to their sequence order. Note that the first waveform (Fig. 9(a)) corresponds to the last waveform of Fig. 3.
Fig. 9. Selected single waveforms of Sequence 1 where extreme events are evidenced: (a) cycle 33, (c) cycle 74, and (e) cycle 132; and waveforms where such events are absent: (b) cycle 38, (d) cycle 96, (f) cycle 188.
Finally, to demonstrate that the observed high-intensity events qualify formally as ORWs, a statistical analysis over Sequence 1 was performed. Figure 10 shows a noticeable L-shaped distribution in a semi-log scale histogram, where the presence of ORWs is verified (events over 2.2 times the SWH [32]). To build this histogram, the values of all peaks over 17 W were taken into account instead of only the highest peak of each waveform, in order to increase the amount of data, considering the relatively small total number of waveforms. Although typically, a very small value is taken as the threshold (in order to exclude the quasi-CW background), this consideration decreases the average value and would overestimate the number of ORWs. Hence, a relatively high value (17 W: the average power of the sequence), was chosen as threshold to establish a stronger condition for the observation of extreme events. Statistically, 11 manifestations are a statistically significant value taking as reference the size of Sequence 1. In addition, it is worth mentioning that very short sequences were intentionally presented, in order to exhibit all these phenomena over few cycles; if simulating larger sequences, the same behavior was observed, where ORWs still appear with similar probability of manifestations. Our simulation of ORW waveforms and the shape of the histogram (Fig. 10) matches well with a recent experimental observation in [50], where the internal details of NLPs were resolved in single-shot for the first time.
Fig. 10. Peak power distribution of Sequence 1, exhibiting that ORWs are more frequent than expected for a conventional distribution (solid orange line: exponential decay for comparison).
One important observation from these data is that the emergence of an ORW event is associated with significant spectral alterations. Figure 11 presents corresponding temporal and spectral profiles of the NLP to evidence the changes that the spectrum undergoes during an extreme event. The analysis was carried out around two different ORWs (Figs. 9(a) and (c)), over cycles 31–35 and 72–76, respectively. As can be seen in Figs. 11(a) and (b), spectra of cycles 31, 32, 35, and 72, 73, 76 are quite consistent with the typical spectra of NLPs [19,44]. However, when an ORW event occurs (cycles 33 and 74, highlighted by box and green arrows in Fig. 11), the shape of the spectrum changes notoriously; in particular, the spurious fluctuations reduce on the sides of the maximum, where lateral lobes start to emerge. It is also noticeable that the spectral manifestation of these extreme events reaches its maximum only one cycle later (cycles 34 and 75, where well developed side lobes are visible in each case around 1520 nm and 1580 nm), when the intense spike is no longer present. This makes sense considering that, once the intense peak is formed, enhanced nonlinear spectral broadening takes place after further propagation in the cavity; on the other hand, the newly created frequencies survive for some time in the cavity after the temporal spike has vanished. However, another cycle later, the sidelobes have completely disappeared and the spectrum has recovered its typical spurious NLP appearance.
Fig. 11. Correspondence between the temporal profile and the instantaneous spectrum, showing the change that the spectrum undergoes around two ORWs: (a) at cycle 33, and (b) cycle 74.
Further insight is provided by Fig. 12, where the extreme event of cycle 33 was isolated from the NLP waveform in the time domain; the spectra of the intense spike and of the complementary NLP waveform are then calculated separately in order to analyze their contributions to the total spectrum. As expected, the intense spur has a smooth, broad spectrum marked by the emergence of side lobes originating from the Kerr nonlinearity (Fig. 12(b)). In contrast, the complementary waveform displays a comparatively narrow spectrum carrying a nearly uniform spurious modulation (Fig. 12(c)), which makes it look like the spectrum of NLP waveforms containing no extreme event. Because the intense spike is spectrally broader, its influence can become dominant on the sides of the NLP spectrum, where in some cases the spurious modulation virtually disappears (see right side of spectrum in Fig. 12(a)), even though this spike does not contain a major part of the total NLP energy.
Fig. 12. Temporal-spectral analysis over an extreme event: (a) original waveform (cycle 33), (b) isolated intense peak of the same waveform, and (c) the waveform without the intense peak.
In order to evidence a different type of dynamics, another spatio-temporal evolution (Sequence 2) is presented in Fig. 13(a). Overall the dynamics is similar to the case of Sequence 1. However, in this second sequence of 100 cycles (cycles 11–110), a sub-packet expelled towards longer times, labelled “sp4” (green arrow in Figs. 13(b) and 13(c)), extinguishes around cycle 95 (∼138 ps) in a peculiar manner. Indeed, the extinction process occurs in this case in steps, as highlighted with two circles: before sp4 decays, its intensity increases (dashed yellow circle in Figs. 13(b) and (c), at cycle 74); then, it maintains itself almost constant over 21 cycles until it finally vanishes abruptly from one cycle to the next (solid red circle). Experimentally, similar fast decays of sub-packets moving away from the main bunch were observed, although the vanishing extended over several tens of cycles, as can be found in [12,13]. Even the short-lived intensity increase of sub-packets previous to their sudden vanishing was observed experimentally [12]. The overall evolution is highlighted in the close-up view of Fig. 13(c), inset. A spectral analysis over this sub-packet shows some peaks that could be interpreted as Kelly sidebands, as shown in Fig. 13(d), in particular the peaks in the upper inset. On the other hand, the sub-packet labelled “sp5” (white arrow in Fig. 13(b)) was analyzed to see the temporal walk-off suffered by the sub-pulses. It was observed a temporal shift to the right of 1.461 ps per cycle (Fig. 13(a)); this is consistent with the blue shift of $\Delta \lambda = 4.1$ nm with respect to the central wavelength (1550 nm) that was observed in the spectral profile (centered at ∼1546 nm).
Fig. 13. (a) Spatio-temporal evolution of Sequence 2. (b) Top view; sp4 extinguishes at cycle 95 (solid red circle) but before increases its intensity (dashed yellow circle). (c) Close-up view of this vanishing; inset: step-like manner decaying of sp4. (d) Spectrum of sp4 at cycle 65; upper inset: close-up of the central peak; lower inset: temporal profile.
Several sequences (as the one depicted in Fig. 14: Sequence 3) where no evident pulse splitting, notorious collisions, drifting sub-packets nor ORWs were achieve by introducing a large normal dispersion fiber into the cavity (D = –150 ps/nm/km), by changing to normal the total dispersion of the cavity (DT = –0.175 ps/nm), in the same laser scheme (Fig. 1). In this completely different regime, where the dispersion and nonlinear effects interact completely unlike, the conservative soliton effect no longer exists, and the sub-pulses are wider and chirped; hence, it is not surprising that the same type of dynamics as in anomalous dispersion are not observed here. Nonetheless, it is important to stress that there are experimental works where ORWs appear in normal dispersion under certain conditions [21,51]. Figure 14 is useful in this study to clearly see that, although 1550 nm is the central wavelength in the simulations, the NLP temporal waveform is slightly deflected to the right (at 1552 nm).
Fig. 14. Sequence 3, where no pulse splitting, evident collisions, drifting sub-packets nor extreme events are manifested: (a) spatio-temporal evolution, (b) top view.
It is worth mentioning that all the results presented in this study used a temporal (0.07814 ps) and spectral (0.0125 nm) scale partition smaller than the size of the finest details of the temporal and spectral shapes of the pulse. Changing the level of resolution, the details of the NLP change; however, the same phenomena are qualitatively reproduced.
4. Conclusion
Through the numerical model of an erbium F8L, several sequences of NLPs fashioned by dozen of consecutive round-trips each one were performed. Selecting a couple of sequences, some chosen rich dynamics were studied, for instance. Such is the case of two sub-packets with different central wavelengths that collide each other and become extinct, thereby confirming there are inner fragments traveling at different temporal walk-off rates into a NLP; the temporal study was complemented with spectral analysis and by the formalism of the first and second statistical central moments. Otherwise, the extinction of a sub-pulse decaying in a step-like manner without colliding was observed. On the other hand, besides confirming the manifestation of extreme events within NLPs, a notorious distortion that the spectrum undergoes around an ORW was evidenced through a temporal-spectral correspondence; in addition, it was seen that more than one cycle is necessary to “fully recover” the typical spectral profile of a NLP after an extreme event. The present results are favorable to confirm our previously published experimental studies. In addition, this work helps to better understand the dynamics of NLPs at a fine inner scale, which in turn will allow to design laser sources to generate NLPs with specific characteristics for applications, such as high energy or broad bandwidth. On the other hand, this study suggests a novel experimental method to detect temporal ORWs in NLPs through the widening and distortion of the single-shot spectrum
Funding
Consejo Nacional de Ciencia y Tecnología (CONACYT) (FC2016-1961, CB 253925).
Disclosures
The authors declare no conflicts of interest.
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