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Abstract
We present an experimental study of complex noise-like pulse dynamics in a passively mode-locked figure-eight fiber laser, by performing simultaneous temporal and spectral mapping of the waveform sequences. The simultaneous measurements allow us to relate temporal and spectral events. We found in particular that the evolution of energy and of temporal features such as the number and width of the wave packets is correlated to spectral variations, namely of the central wavelength and bandwidth of the instantaneous spectrum. The simultaneous temporal and spectral measurements also allowed a substantial improvement in the precision of the latter, which was performed using the dispersive Fourier transform method. In particular, this enhanced precision allowed measuring the subtle spectral differences between the two laser outputs and tracking their evolution over the cycles, providing crucial information that allowed to determine the physical phenomena involved in the observed dynamics.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It has been observed that passively mode-locked fiber lasers can operate in a wide variety of regimes, such as conservative solitons [1,2], similaritons [3–5] and dissipative solitons [6,7], etc. The regimes mentioned above correspond to stationary or close to stationary regimes; however, operating a fiber laser far from a steady state, we can also find the so-called noise-like pulses (NLPs) [8–33]. NLPs are long (~ns) bunches of optical radiation with a fine inner structure (sub-ps) that present a complex and chaotic behavior, with constantly and randomly varying time duration and amplitude. The statistical study of this type of dynamics has come to reveal extreme fluctuations, which under some conditions result in rare events like optical rogue waves [34–38]. The first studies on NLPs were published in the 90s [8]. Due to the extremely complex nature and variability of NLPs, their characterization is very challenging, and until now there is no solid understanding of their formation mechanisms and their behavior, even though several theoretical and experimental works have been carried out for this purpose [8–10,13,16]. The complexity and the chaotic variations of the fine inner structure (at sub-ps scale) of NLPs justify their classification among the non-stationary (incomplete, partial mode locking) regimes. In many cases however, the NLP envelope (as observed on a scope) is relatively stable, producing a regular pulse train at the laser output. In some cases, however, the envelope suffers drastic changes at sub-ns scale, triggering complex dynamics that includes the fragmentation of the bunch into multiple packets, their fusion, their emergence from background radiation, drift, fading etc [22,24,29]. In general, only limited information about these regimes can be obtained by conventional measurements (average optical spectrum using an optical spectrum analyzer, autocorrelation trace, conventional single-shot scope captures, etc.). These limitations encourage the development of novel techniques to allow a better analysis and characterization of these non-stationary regimes [16,22,26,31,35,38]. Firstly, the dynamical evolution of the temporal waveform can be tracked by the time-domain mapping technique, which consists in stacking together a large number of periodic single-shot measurements from a fast oscilloscope [26,29]. On the other hand, one powerful technique to achieve real-time measurements of fast dynamical events in the spectral domain is the dispersive Fourier transformation (DFT). DFT allows real-time spectroscopy overcoming the sweeping speed limitations of traditional optical spectrometers, mapping an optical spectrum onto a time-stretched temporal waveform using the group-velocity dispersion (GVD) taking place when a pulse travels inside a dispersive medium [39,40], enabling to realize single-shot spectral measurements with an ultrafast real-time oscilloscope. Many works have been reported that analyze non-stationary regimes in fiber lasers in the time domain [11,20,22,28,29], whereas others carry out a real-time spectroscopy analysis by applying the DFT [16,31,35,40–42]. A recent work presents the spectral evolution of a femtosecond pulse train through ~900000 consecutive cycles by using DFT [43]. This type of time-stretching technique has also been implemented to study and resolve the evolution of sub-ps soliton molecules [44]. DFT has also been implemented to realize a statistical analysis of the spectral fluctuations of NLPs, in search for rare high-intensity events called spectral rogue waves [37], as well as Raman rogue waves [35]. It should be noted however that, although the study of such complex dynamics has been the focus of a substantial amount of research work already, it seems that no study has been carried out yet based on the simultaneous measurement of the temporal and spectral evolutions. Such simultaneous measurements are interesting, as they would allow relating events in both temporal and spectral domains, offering much more valuable information than separate measurements and allowing a much deeper understanding of the underlying physics. For example in [37], spectral rogue waves were identified by applying the DFT technique; however their effects on the temporal waveform evolution or the connections that these spectral events bear with their temporal counterparts are not clear, as discussed in that paper. On the other hand, some experimental studies have reported very complex temporal dynamics of NLPs, revealing in particular that they tend to divide into sub-ps packets taking roughly discrete energy levels, or to exhibit quasi-stationary behaviors [29,45,46]. In order to unveil the mechanisms of such puzzling dynamics, it would be very convenient to perform simultaneous temporal and spectral measurements in real time.
In this work we report a simultaneous temporal and spectral experimental analysis of NLPs in a mode-locked figure-eight fiber laser. We implemented the DFT using 15 km of single-mode fiber (SMF-28), to measure spectral fluctuations in real-time, and simultaneously we captured the temporal traces in order to match the temporal and spectral dynamics. Temporal and spectral events can then be related.
2. Experimental setup
The experimental arrangement used to generate the NLPs is an erbium-doped figure-eight fiber laser (F8L) shown in Fig. 1. The F8L consists of a nonlinear optical loop mirror (NOLM) inserted in a ring laser cavity. The laser has an approximate length of ~218 m, corresponding to a round-trip time of 1.1 μs.
The ring cavity is formed by two sections of the same erbium-doped fiber, EDF1 and EDF2 (3-m and 2-m long respectively) with a 30-dB/m absorption at 1530 nm, each one pumped at 980 nm. The use of two amplifiers favors the formation of pulses with higher energy and more complex structure. Besides, independent pump power adjustment yields the enhanced flexibility that is useful to obtain a type of emission that fulfils the specific requirements of this work (in particular, a relatively complex dynamics but confined within a compact waveform, see below). The cavity also includes an optical isolator (ISO) to ensure unidirectional laser operation; 100 m of dispersion compensating fiber (DCF, D = −3 ps/nm/km); a polarizer (POL); a polarization controller (PC), and 2 couplers (90/10) to provide 2 outputs. The polarization controller (PC) is made of a half-wave retarder (HWR1) and a quarter-wave retarder (QWR1). The PC is used to control the power transmission through the POL, and the HWR2 to control the inclination angle of the linear polarization at the NOLM input, and in this way to control the switching power of the NOLM [47,48].
The NOLM used in our arrangement consists in a power-symmetric and polarization-imbalanced scheme, in such a way that the switching mechanism is based on the control of polarization through the loop. The NOLM consists of a 50/50 coupler, a quarter-wave retarder (QWR2) to break the polarization symmetry, and 100 m of low-birefringence SMF-28 fiber (D = 17 ps/nm/km), twisted at a rate of 5 turns per meter, in order to eliminate the loop residual birefringence and ensure that light ellipticity is maintained during propagation [49]. Indeed, twist causes a precession of the fiber principal axes, which tends to cancel the random linear birefringence; twist also introduces circular birefringence, which rotates the polarization ellipse in a predictable manner, but does not alter ellipticity.
In our study the NLPs are measured mainly at output 2, which is located at the NOLM output. At output 2 we use a coupler (50/50) in order to split the signal in two, one of the signals is detected directly to analyze the NLPs temporal evolution on the scope channel 1, whereas the other one first passes through 15 km of standard single-mode fiber (SMF-28) to perform the dispersive Fourier transformation (DFT), before being detected and measured on the scope channel 2 in order to obtain information on the NLPs spectral evolution. Both signals were detected using two identical high-speed photodetectors (25-GHz) and a fast real-time oscilloscope (16-GHz). A single event on channel 2 triggered the simultaneous measurement on both channels. We worked with single-shot captures of different lengths, reaching up to 1.25 ms (1129 laser periods). The data were then segmented into successive periods and processed to build the spectral and temporal maps.
With an appropriate position of the wave retarders and an appropriate pump power, it is possible to obtain self-starting mode locking. However, we could observe that the temperature is a factor that also influences the self-starting mode locking, and the more interesting dynamics were obtained close to 298 K. In all cases, mode locking yielded the generation of a NLP, whose duration was variable depending on pump power and retarders settings. To apply the DFT it is important to work with short enough temporal pulses, in such a way that the temporally stretched waveform leaving the dispersive medium has a duration at least one order of magnitude larger than the input pulse [37]. We worked with pulses with the minimum possible temporal duration (~0.80 ns), where fluctuations could be observed in the pulse shape or in the overall behavior. The regime in which the laser was operating is characterized by the frequent fragmentation of the main bunch, giving rise to the emergence of sub-packets, which subsequently may merge, fade away etc.
3. Technique for simultaneous temporal and spectral characterization
In our study we first implemented the temporal mapping technique, organizing the data, and then processing and analyzing them. After acquiring a long sequence of consecutive periods by a single-shot capture, the data was segmented into successive cycles, and the traces were superposed to form a surface that shows us the temporal evolution of the NLP through the cycles. A similar procedure on channel 2 allowed forming a surface showing the spectral evolution of the NLP. Figure 2 shows an example of temporal and DFT signals obtained simultaneously from a 1.25-ms single-shot measurement, corresponding to 1129 consecutive laser periods.
Fig. 2 Simultaneously measured sequences of 1129 consecutive cycles. (a) NLP temporal evolution; (b) DFT signal evolution.
Although the sequences of Fig. 2 were measured simultaneously, a delay arises from the difference between their optical paths, which is mainly due to the 15 km dispersive fiber. This distance corresponds to ~68 cycles of the 218-m laser cavity. Hence the DFT sequence is delayed by this amount with respect to the temporal sequence. The delay between both signals can be obtained more precisely by comparing the evolution of the NLP energy for both sequences. NLP energy is obtained at each cycle by integration of the NLP temporal profile. Figure 3 shows that a perfect match between the two curves is found for a delay of 67 cycles, in good agreement with the estimation. Therefore, after discarding the last 67 cycles of the temporal sequence and the first 67 of the DFT sequence, we conserve in each case only the data of the 1062 cycles that are synchronous.
Fig. 3 Energy anlysis to relate the temporal an DFT-signals. (a) Energy evolution through the cycles, for both temporal and DFT sequences. (b) Matched curves.
Figure 2 shows that the trajectories of the waveforms across the cycles are significantly skewed. The accumulated linear drift of the temporal waveform by ~3 ns towards shorter times (Fig. 2(a)) is due to some mismatch between the actual laser period and the value that was used to segment the data. This mismatch is small however: less than 3 ps, which is 6 orders of magnitude smaller than the value of the period, and also much smaller than the scope sampling period that was used (40 ps).
By virtue of the DFT principle, this drift obviously affects Fig. 2(b) as well, and should not be interpreted as a wavelength shift. Such a linear drift can be easily detected and corrected; however, it should be remembered that in passively mode-locked lasers, there is no reference fixing the lasing period (or fundamental frequency), which is allowed to drift e.g. due to environmental changes, such as temperature, in an unpredictable and uncontrolled way, during the measurement. Besides, in the case of complex waveforms subject to constant evolution such as NLPs, the central position of the waveform is substantially altered across the cycles (see, e.g., the vanishing of sub-packets observed in Fig. 2(b). All these effects that affect the position of the temporal waveform are directly mirrored on the DFT evolution, and should not be mistaken for spectral drifts, so that a correction should be performed. This is possible if both temporal and DFT sequences are measured simultaneously, as illustrated in the following.
From the data of Fig. 2, we determine the evolution of the central position of both temporal and DFT waveforms along the cycles. The central position of a waveform is approximated by the first central moment (tc), and can be calculated at each cycle from the intensity profile I(t) with the following expression, which is adapted for the complex and asymmetric temporal and spectral profiles of NLPs:
The latter formalism is adapted for complex and asymmetric temporal and DFT profiles of NLPs. The curves showing the evolution of the first central moment of both Figs. 2(a) and 2(b) are shown in Fig. 4(a), where the initial values at cycle 1 were set to zero. The linear 3-ns shift due to period mismatch is clearly visible; however, some fluctuations of the central position of the temporal waveform still remain when the curve is corrected for this drift (Fig. 4(b), blue curve), which in turn affect the position of the DFT spectrum (red curve). The central position of the temporal profile is thus subtracted from the central position of the spectral waveform, yielding the corrected evolution of the spectral central position shown in Fig. 4(c). This correction is then applied to the spectral map. This correction ensures that the fluctuations observed in the central position of the waveforms of the DFT signal, actually correspond to displacements of the optical spectrum, independently of the position of the temporal waveform. It has to be stressed that, through this procedure, a precise choice of the period used for segmenting the data is irrelevant, as any mismatch will affect both sets of data by the same amount, and will thus automatically vanish in the subtraction (even if this mismatch varies during the measurement). In other words, Fig. 4(c) can be determined directly, by subtraction, from the data in Fig. 4(a). It has to be noted finally that a small slope in the evolution of the central position of the spectrum is visible in Fig. 4(c), which could not be distinguished from the drift of Fig. 4(a) without this correction procedure. Therefore, for precise DFT measurements, it is important to measure simultaneously the evolution of the temporal waveform, in particular when the period is subject to random changes and/or the waveform suffers important alterations across the cycles, as it is typically the case for NLPs and other related non-stationary regimes.
Fig. 4 Evolution of the central position of the waveforms associated to the DFT-signal and temporal-signal, over 1062 cycles; (a) determined from Fig. 2, taking as reference the central position of the waveform in the first cycle (straight line shows a linear fit to the temporal trace); (b) fluctuations around the linear fit; (c) difference between DFT and temporal curves in (b) (straight line again shows a linear fit).
Next, the scale of the DFT spectra (correspondence between ns and nm) is determined. For this purpose, the data from the spectral map (after correction through the procedure described above) is used to calculate the average of all the waveforms which was compared with the average spectrum measured with an optical spectrum analyzer (OSA). Figure 5(a) shows the average waveform obtained from the DFT signal and the individual waveforms of a couple of cycles. From this figure we can observe that the spectrum varies in amplitude and width through the cycles, showing a spiky aspect and random behavior, which is characteristic of NLPs. With the OSA we obtained a spectrum with a maximum centered at 1561 nm with 13-nm 3-dB bandwidth. The factor used to fit the average waveform from DFT to the spectrum yields the relation between the time and spectral domains: 1 ns = 3.33 nm (Fig. 5(b)). Both curves show an excellent agreement down to ~10 dB below the maximum. For comparison, Fig. 5(c) shows the result of fitting the OSA spectrum to the spectrum obtained by averaging directly the data of Fig. 2(b), i.e., without applying the correction. Although Fig. 5(c) shows that a good adjustment between the curves is still achieved in this case, the drift of the DFT waveform (Fig. 2(b)) distorts the average spectrum, which in turn translates into an underestimated scaling factor: 1 ns = 2.94 nm. This value represents an error of 11.67% if the correction is not carried out. Such a large error could be strongly reduced in this case without this correction, by performing a more precise choice of the period, which would eliminate the linear drift observed in Fig. 2(b). However, as already mentioned, this drift may include a genuine wavelength shift, in which case this procedure could lead to significant errors. This again stresses the importance of the correction procedure described in this section for precise spectral measurements through the DFT technique. As a comparison, Fig. 5(d) presents the results of applying the same measurement to solitons produced by a ~20 m fiber ring laser. The single-shot spectrum presents in this case a sinusoidal modulation, which indicates the presence of two (or more) solitons in the packet; however it does not present the noisy aspect of Fig. 5(a). On the other hand, the average DFT spectrum in Fig. 5(d) reproduces quite well the spectrum measured with the OSA, including the Kelly sidebands.
Fig. 5 Adjustment between the average-temporal waveform obtained from the DFT and the spectrum measured with an OSA. (a) a few individual waveforms and the average waveform obtained from single-shot measurements, applying the scale factor to present the data in nm;(b) fit after applying the correction to the DFT traces (1 ns = 3.33 nm); (c) fit before applying the correction (1 ns = 2.94 nm); (d) Example of a soliton spectrum obtained with the DFT technique.
After processing the data of Fig. 2 as described above, the temporal and spectral sequences are shown in Fig. 6. In Fig. 6(b), each profile was also normalized with respect to its energy.
Fig. 6 Processed sequences; (a) temporal evolution (the linear drift has been subtracted). (b) corrected and normalized spectral evolution.
Finally, to estimate the temporal and spectral widths of the waveforms, considering the complex shape of the NLPs, we calculated the RMS width with the following expression:
where tc is calculated using Eq. (1), and I(t) represents the NLP profile (in the case of DFT profiles, the spectral width in nm is readily determined from this value and the scaling factor established above).To ensure that the spectral width fluctuations observed in Fig. 6(b), obtained with DFT actually correspond to spectral variations, and are not just the reflection of the temporal width fluctuations, we observed and compared both traces (Fig. 7). On this figure, we note that the spectral fluctuations are much larger than the temporal ones and do not necessarily reproduce the same pattern (in particular, the nearly periodic oscillation of the spectral width that is observed over the whole evolution is not present over the first 600 cycles of the temporal curve). Hence we can conclude that the RMS width of the DFT traces is a trustworthy representation of spectral width.
The study described in the next section is based on data that was previously processed as described in this section.
4. Experimental results
The first set of data that is analyzed in this section is the one that was used in the previous section. Figures 8(a) and 8(b) reproduce the temporal and spectral evolutions of Fig. 6. Figure 8(a) shows a complex waveform evolution, including the emergence and decay of sub-packets, and a quasi-periodic dynamics marked by significant changes in the waveform extension and the alternating fragmentation and reconstitution of the main packet, which is particularly visible during the last 500 cycles of the sequence. A quasi-periodic evolution is also perceptible in Fig. 8(b). We recall that we normalized this data at each cycle with respect to its energy, so that, contrary to Fig. 2(b), the observed fluctuations actually reflect spectral shape variations, independently of energy fluctuations. Figure 8(c) shows the evolution of the complete packet energy. Here we see that the energy also fluctuates quasi-periodically, with a periodicity of 120-140 cycles. We can relate these energy fluctuations with some features of the temporal waveform evolution, comparing Figs. 8(a) and 8(c). It turns out that, during cycles where energy is high, there is a single, broad main bunch, sometimes accompanied by a few sub-pulses. In contrast, when energy decreases, the main bunch tends to compress temporally, or to fragment into sub-pulses. Figure 8(d), which shows the evolution of the NLP duration, confirms that this parameter follows the quasi-periodic evolution of the energy, in particular beyond cycle 600. In contrast, between cycles 100 and 600, the temporal width does not follow this pattern of quasi-periodic oscillations. This can be understood by observing in Fig. 8(a), that around cycle 100, two sub-packets emerge from the main bunch and substantially diverge towards shorter times, until they eventually vanish. Due to the substantial temporal separation between the different NLP components during cycles 100-600, the RMS width does not reflect the cumulated width of the different sub-packets over this range (note the abrupt decrease in the RMS duration taking place when each of the two above-mentioned sub-packets disappears).
Fig. 8 Temporal and spectral NLP measurements in a quasi-stationary dynamics over 1062 cycles. (a) temporal evolution; (b) spectral evolution; (c) NLP energy; (d) temporal RMS width; (e) spectral central position; (f) spectral RMS width.
The DFT also provides interesting information that can be related with the temporal behavior. Figure 8(e) shows the evolution of the spectral central position, which is affected by the same quasi-periodic evolution. Specifically, we observe that the spectrum presents a redshift (e.g. red-highlighted cycles) when energy is high and a blue shift (e.g. blue-highlighted cycles), when energy is low. Finally, Fig. 8(f) shows that the variations of the spectral width mimic the evolution of NLP energy as well (higher energy meaning a broader spectrum, and conversely). In summary, we observe that variations in the NLP energy are connected not only to changes in the NLP temporal behavior, but affect its properties in the spectral domain as well.
As the laser is operating in a nonstationary regime, some variations in the laser behavior can be expected, so that a large number of single-shot measurements were performed in order to capture different types of dynamics and analyze them. Figure 9 shows the temporal and spectral information obtained with a 0.628-ms single-shot measurement, corresponding to 564 cycles (497 after discarding the 67 non-synchronous cycles in each sequence). Compared with the data of Fig. 8, a reduction in the measurement duration allowed a twofold improvement in terms of sampling rate (here performed with a 20-ps period). Figure 9(a) again shows a quasi-periodic temporal evolution of the NLP profile, which oscillates between two different states, one in which only one bunch is present, and one where 2 or 3 smaller packets coexist. These oscillations are also visible in the spectral evolution of Fig. 9(b). Figure 9(c) also shows a quasi-periodic evolution of the NLP energy, which in this case presents a ramp-like evolution, with phases of nearly linear gradual increase alternating with abrupt decays. A comparison between Figs. 9(a) and 9(c) indicates that multiple packets gradually broaden and eventually merge into one single bunch as energy increases, whereas this bunch disintegrates back into multiple sub-packets as energy drops. Hence the highest energy values correspond to one single bunch, and the lowest energy values to multiple packets. This periodic evolution is also visible in the evolution of the RMS width of the NLP shown in Fig. 9(d).
Fig. 9 Related temporal and spectral NLP measurements (497 cycles). (a) Temporal evolution; (b) specrtral evolution; (c) normalized energy; (d) temporal RMS width; (e) central position of the spectrum; (f) spectral width.
Like in the previous case (Fig. 8), comparing Figs. 9(a) and 9(b) allows us to see that the time-domain dynamics described previously is accompanied by spectral changes; in particular, the spectral central position (Fig. 9(e)) and the spectral width (Fig. 9(f)) present an evolution that tends to mimic the variations of energy. Hence, the broadest spectra with the largest redshifts are reached mainly when high energy levels concentrate in one broad single bunch.
Figure 10 shows another interesting example of NLPs dynamics. In this case, as in the previous one, we performed a 0.628-ms single‐shot measurement (497 related cycles). Figures 10(a) and 10(b) show an evolution that in this case does not clearly reveal a quasi-periodic behavior. From cycle 1 we can observe 4 sub-packets, one of which disappears near cycle 270 whereas the remaining three merge to form a single bunch around cycle 355. This bunch is then maintained until the end of the measurement. However, near cycle 415 an interesting, although ephemeral process takes place: the bunch tends to fragment again into 4 sub-packets, although they never separate completely and quickly merge again into a single envelope, as highlighted in the inset in Fig. 10(a).
Fig. 10 Related temporal and spectral NLP measurements (497 cycles). (a) Temporal evolution; (b) spectral evolution; (c) normalized energy; (d) temporal width; (e) central position of the spectrum; (f) spectral width.
Although quasi-periodic oscillations can still be discerned in the energy evolution of Fig. 10(c), they are significantly smaller than in the previous cases. Besides, a similar evolution is observed only in the optical bandwidth (Fig. 10(f)); in contrast, the evolution of the temporal width (Fig. 10(d)) and of the spectral central position (Fig. 10(e)) overall display quite different patterns. It is also noticeable that, in this case, there is no clear-cut connection between the number of bunches in Fig. 10(a) and the energy level in Fig. 10(c). Still, the ephemeral appearance of 4 sub-packets near cycle 415 is associated with a sharp dip in the energy evolution. This event is also characterized by a short-lived though significant blue shift of the spectrum (Fig. 10(e)), as well as spectral narrowing (Fig. 10(f)).
Another noticeable event appearing in Fig. 10(a) is the rapid vanishing of a packet near cycle 270, which is associated with a step-like decrease in the energy and pulse width. This event also coincides with a sharp decrease of the spectral width, which then quickly returns to its original value (Fig. 10(f)). Finally, by comparing Figs. 10(a) and 10(e), it appears that, quite independently of the energy, the merging of multiple packets into a single bunch is accompanied by a red shift of the spectrum (around cycle 355), and their splitting by a blue shift.
In order to refine the analysis of the dynamics observed in Fig. 10, we analyze the evolution of each sub-packet individually. Figure 11(a) reproduces the time-domain evolution. Multiple packets can be clearly individualized up to cycle ~350. In this analysis we did not consider the main packet fragmentation near cycle 415, since it is incomplete and short-lived. The evolution of the energy of each sub-packet can be observed in Fig. 11(b), which shows that the sub-packets possess roughly the same energy, which corresponds to ~25% of the bunch energy. This stability over hundreds of cycles contrasts with the rapid decay of one of the sub-packets around cycle 270, which takes as few as ~10 cycles to vanish completely, as seen in Fig. 11(b).
Fig. 11 Temporal parameters evolution for each sub-packet from Fig. 10. (a) Temporal sequence; (b) Normalized energy; (c) temporal width; (d) central position of the waveforms; (e) phase space diagram.
The temporal extension of each sub-packet is also found to be closely related to their energy. Figure 11(c) shows the evolution of the RMS width values of each sub-packet, which mimics the evolution of energy. Similarly, values close to 25% of the RMS width of the main packet are observed for each sub-packet. The evolution of the central position of each sub-packet was also tracked (Fig. 11(d)). The figure indicates that the sub-packets describe complex trajectories whose details are all different, although they all globally drift towards shorter times. Such features may be dictated by slight differences in their individual spectral evolution, as well as dissipative effects and the mutual interaction between sub-packets.
Figure 11(e) shows the phase-space representation where the peak intensity versus energy for each packet is plotted at each cycle. This figure shows two main zones, one corresponding to the sub-packets and the other one to the single bunch. In this figure, we can observe that the energy of the main packet varies over a broad range whereas its envelope intensity remains almost constant. In contrast, all the sub-packets concentrate in the same region centered on relatively fixed values of intensity and energy. This region also presents a thin connection to the origin, which describes some tendency of the sub-packets to vanish.
To investigate more in-depth, the possible causes of the spectral alterations when the pulse energy varies, we simultaneously applied the DFT to both laser outputs. Both signals were combined with a 50/50 coupler. After having the signal of both outputs combined, we connect one output of the coupler to another 50/50 coupler to split the signal in two. One of them is captured directly to register the temporal evolution of both outputs, and the other one is introduced to the 15-km dispersive fiber to apply the DFT, as we can see in Fig. 12.
The delay between the pulses of the two outputs was sufficient to allow simultaneous measurement of the two DFT signals on the scope without overlap. Using this scheme, we were able to capture simultaneously the temporal and DFT signals of both outputs, allowing to carry out for each of them the correction described in the previous section.
As we can see in Fig. 1, output 2 is inserted immediately after the NOLM output, and output 1 after the first amplifier. Figure 13 shows the results obtained from this simultaneous analysis. Similarly to the previous cases, the spectral width at both outputs mimics the NLP energy evolution, as we can see in Figs. 13(a) and 13(c). Figure 13(b) also shows that the spectral central position is red-shifted when energy increases (red-highlighted-regions in Fig. 13(b)) and blue-shifted when energy decreases (blue-highlighted-regions in Fig. 13(b)). Although both curves of Fig. 13(b) are almost superimposed over the whole sequence, a small difference exists, as shown in Fig. 13(d).
Fig. 13 Simultaneous analysis of outputs 1 and 2. (a) Normalized energies; (b) spectral central positions; (c) spectral widths; (d) difference between the central positions of the spectra at ouputs 2 and 1.
Figure 13(c) shows that the spectrum is wider at output 2 (NOLM output) than at output 1 (EDF1 output). This can be explained by the nonlinear effects (probably mainly the Kerr effect) taking place in the DCF and NOLM sections, and possibly in EDF2. The nonlinear spectral broadening is believed to be stronger in the DCF section, which has low dispersion value and larger nonlinear coefficient than the standard single-mode fiber, and is inserted in the section of the cavity where the pulse energy is the highest (after EDF2). The NOLM section can also contribute to spectral broadening, although to a lesser extent, as it is made of standard fiber and because after the 50/50 coupler the power circulating in the loop in each direction is only half the input power. Besides, the wavelength dependence of twist-induced circular birefringence causes spectral filtering through the NOLM [50]. This dependence is small however at the scale of the pulse spectrum, and only causes bandpass filtering (yielding some reduction of the spectral bandwidth) if the NOLM is biased at maximal low-power transmission, and not for saturable absorber action (small, non-zero low-power transmission). On the other hand, the RMS bandwidth decreases after passing through the EDF1 amplifier section, due to the bandpass filtering effect caused by the limited bandwidth of the doped fiber. We believe that nonlinear broadening does not take place in EDF1, where signal power is smaller than in EDF2 (EDF1 output power roughly corresponds to EDF2 input power) and which only provides a small amount of amplification (for the regimes described in this work, EDF1 pump power was set to only ~30% that of EDF2).
An interesting measurement is that of the difference between the spectral center position at outputs 2 and 1 (Fig. 13(d)), which oscillates between positive and negative values; these oscillations appear again related to the energy fluctuations of Fig. 13(a) but are in this case in phase opposition with them. Such an evolution can be explained in terms of gain dynamics. First of all, it is important to note from Fig. 1 that laser outputs 1 and 2 are located at the output and input of amplifier EDF1, respectively. When the pulse energy increases over successive cycles, the gain saturates, which causes its maximum to progressively shift towards longer wavelengths [51]. As a consequence, one can expect that the pulse spectrum will progressively shift towards longer wavelengths over the cycles, too. This means that, during those cycles where energy increases, the central wavelength at output 1 (EDF1 output) will be slightly longer that at output 2 (EDF1 input), so that the difference (output 2 – output 1) will be negative. If now the pulse energy decreases over the cycles, the gain progressively recovers from saturation, and its maximum shifts towards shorter wavelengths. In this case, the difference between central wavelengths at outputs 2 and 1 will be positive. Finally, it is interesting to note that these wavelength shifts are small: they only amount to ~1-2 nm over several tens of cycles (Fig. 13(b)), which means that, over each pass through the EDF1, they are still smaller; in spite of this, the technique still has the sufficient precision to measure these tiny spectral variations.
5. Discussion
Due to the chaotic evolution of the bunches at fine inner scale, the NLP regime is always non-stationary by nature. However, from the relatively stable bunch that produces a periodic train of pulses at the laser output, to the complex collective dynamics involving a large number of packets distributed across the whole cavity [29,45], this regime encompasses a broad variety of behaviors. The use of the DFT technique restricts the study to regimes where all components of radiation are confined into a temporal extension of the order of ~1 ns (as the signal duration must be kept much smaller than the extension reached by the waveform at the dispersive fiber output). In spite of this, the present study is not limited to relatively stable, single-NLP regimes, but covers scenarios where multiple packets are present, merge, split, or drift away, vanishing after a short drift (similar dynamics where sub-packets drift away from the main bunch, although over much longer distances, were previously reported in [29]).
The quasi-periodic instability that is reported in this work is related to the peculiar “Q-switched-like” NLP operation that was described in several reports [45,46,52–54], although the magnitude of the variation is smaller in this case than in most of these references. We believe that this kind of instability at frequencies in the kHz range is related to the spiking and relaxation oscillations phenomena that arise in many types of lasers [55]. In passively mode-locked lasers, damping of these energy oscillations can be suppressed by the action of the saturable absorber (whose role is played by the NOLM in this setup); besides, in the NLP regime, the strong variability of the waveform causes significant changes in the nonlinear transmission at each cycle, which constantly disturb the laser operation and prevent damping of the instability, which thus tends to become permanent. Quite interestingly, these oscillations are associated with a quasi-periodic evolution in both temporal and spectral domains. In the time domain, a decrease in energy appears to be related to a shrinking of a large main bunch or its splitting into multiple bunches, whereas an energy increase induces a broadening of the packets that tend to merge together into one single bunch. These results are to some degree contrary to those of [45], where an increase in the NLP energy is associated to its temporal compression. On the other hand, the appearance of multiple packets with roughly the same duration and energy has already been observed experimentally with similar setups [22,24,29,56], and was also recently reproduced numerically, and explained for a cavity with strong dispersion map (which is the case of the cavity under study in this work) [57]. In that model however, an energy increase results in the increase in the number of bunches, each of them conserving roughly the same energy and duration: they do not broaden temporally, nor do they tend to merge together, as it happens in the present case. This indicates that, in spite of the ongoing progress, we are still far from reaching a complete understanding of such phenomena. Finally, it can be useful to note that the present evolution in the number of sub-packets, which decreases with increasing intracavity energy, is opposite to the case of solitons, whose number increases with energy, due to soliton energy quantization [58].
In the spectral domain, oscillations of the pulse energy are mirrored by the evolution of both the spectral bandwidth and central wavelength of the NLPs. By looking simultaneously at the evolutions of bandwidth and temporal waveform, one notes that in general a high-energy, broad NLP has a broader spectrum than the smaller multiple packets. Because spectral broadening is related to nonlinearities (principally the Kerr effect), this means that small packets overall have lower intensities (and not only smaller energies) than a large bunch. This could indicate that small packets are less robust than large bunches and are more susceptible to disappear. Indeed, considering the fluctuations that NLPs are constantly submitted to, low-intensity small packets may at some point become unable to produce a sufficient amount of nonlinear phase shift to ensure proper transmission through the NOLM, therefore suffering high losses, which would lead to their quick vanishing, as it was observed experimentally in some cases. On the other hand, the quasi-periodic evolution of the NLP central wavelength can be interpreted as a consequence of gain dynamics, in particular by comparing the spectra measured before and after a section of amplifier, at both laser outputs. Although the measured spectral shifts are tiny (they correspond to spectral changes taking place over a portion of cycle only), they clearly point out towards the gain dynamics interpretation. For example, the alternating sign of the central wavelength difference rules out an interpretation in terms of Raman self-frequency shift in the DCF (as it would yield a spectrum at output 2 that is systematically red-shifted with respect to output 1). Finally, it should be stressed that, with wavelength excursions not larger than ± 1 nm, the magnitude of this spectral dynamics is moderate in comparison with others where wavelength shifts of several tens of nm were reported [45].
6. Conclusions
In this work we analyzed some NLP dynamics in a figure-eight fiber laser, applying simultaneously temporal and spectral mapping to the measured sequences. The analysis of the measurements allows to analyze subtle details and to observe the highly variable nature of NLPs regimes. The simultaneous temporal and spectral analysis allows relating spectral and temporal parameters, which are both intimately connected to energy fluctuations. In particular, contrary to previously reported results, we noticed that larger values of energy tend to associate with the formation of a large single bunch, whereas smaller energies are marked by the coexistence of multiple small packets. Contrasting spectral and temporal evolutions also allows enhanced measurement precision and gives clues about the physical mechanisms involved. In particular, the simultaneous analysis of the two laser outputs allows extracting precise information that point to the involvement of gain dynamics in the spectral evolution. Finally, this work demonstrated that the simultaneous spectral and temporal analysis can provide a deeper insight into complex NLP regimes and in general into non-stationary regimes of passively mode-locked fiber lasers.
Funding
Consejo Nacional de Ciencia y Tecnología (CONACYT) (project 253925, grant 128282).
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