Open Access
Add to BibSonomy
Abstract
We carry out systematic and dedicated experimental investigations on the shot-to-shot coherence and wave-packet formation in quasi-mode-locked (QML) regimes, including noise-like-pulse, symbiotic, and multi-soliton regimes in an anomalous-dispersion fiber ring cavity. To analyze the regimes in real-time, we take advantage of the spatio-temporal shot-to-shot measurement technique. We show that the individual regimes exhibit significantly different coherence characteristics, depending not only on the amount of nonlinear phase shift accumulated per roundtrip but also on the degree of soliton interaction, the latter of which crucially governs the bunching (i.e., the wave-packet formation) or anti-bunching mechanisms in the corresponding QML regimes. In fact, solitons with higher intensities tend to undergo higher nonlinear phase shift and stronger soliton interactions. Subsequently, the intensified soliton interactions among the individual solitons in the multi-soliton-regime cavity trigger them to form a bunched soliton-group, i.e., a wave packet, thereby resulting in QML pulses in the noise-like pulse or symbiotic regime. This complicated nonlinear process, in turn, causes a severe degradation in the shot-to-shot coherence of the resultant QML pulses. In addition, the shot-to-shot coherence trends observed in the experiment are in good agreement with our previous numerical predictions, so that the strong correlation between the shot-to-shot coherence of QML pulses and the corresponding nonlinear phase shift accumulated per roundtrip is confirmed by this experimental observation.
© 2017 Optical Society of America
1. Introduction
Whilst passively mode-locked fiber lasers (PMLFLs) have extensively been studied as a compact, flexible, and efficient platform for generating coherent and stable ultrashort pulses, such as soliton, Gaussian, dispersion-managed solitons, and dissipative soliton pulses [1–14], they are, in fact, a complex nonlinear dissipative system of a multitude of cavity parameters, including dispersion, nonlinearity, spectral selectivity, amplification, and saturable absorption [15], thereby tending to exhibit a chaotic or stochastic nature of photon dynamics if forced to operate in an extraordinary condition [16–38]. Quasi-mode-locked (QML) regimes that distinctively generate partially coherent or incoherent multiple optical pulses are among such stochastic regimes of PMLFLs [39]. In general, QML regimes emerge when a laser cavity is excessively pumped or amplitude-modulated, regardless of its configuration or dispersion regime, which can be classified into incoherent/partially-coherent noise-like-pulse (NLP), symbiotic, and coherent/partially-coherent multiple-soliton (MS) regimes from the perspective of their coherence and packet-forming natures [38–40]. In recent years a number of investigations on QML regimes have been carried out, aiming to clarify their formation mechanisms and coherence characteristics by means of controlling various cavity parameters, including the degree of nonlinear polarization rotation (NPR) [16–28], the level of pumping [27,38], and the amount of cavity dispersion [16,20,23,38]: Horowitz et al., for the first time, reported the formation of the NLP and described it in terms of the cavity nonlinear transmission and the birefringence of the propagation medium [16]. Tang et al. explained the NLP formation process as a combined effect of the soliton collapse and positive feedback, reporting the MS formation mechanism resulting from the peak-power-limiting effect of the given laser cavity [20,21]. Yet, the detailed chaotic or stochastic nature of QML regimes could not be fully unveiled by means of conventional measurement techniques, such as simple spectral- or temporal-domain measurement techniques [16–24]. Thus, novel techniques, such as the dispersive Fourier transform and spatio-temporal measurement methods have recently been implemented, providing real-time shot-to-shot information of pulse dynamics in QML regimes [34–37]. In fact, with the aid of the real-time measurement techniques, Runge et al. analyzed the coherence property of a QML regime, reporting the lack of coherence in an NLP regime as a result of substantial roundtrip-to-roundtrip spectral fluctuation [36]. Gao et al. attributed the QML formation process to the vector parametric frequency conversion and coherence loss by the intra-cavity polarization rotation [37]. Some of the authors have shown experimental evidences of the existence of partial coherence in QML pulses, while investigating the characteristic origin of the NLP formation in a fiber ring cavity [38,40].
Notwithstanding, QML regimes have remained obscured in many aspects, including their shot-to-shot coherence properties and wave-packet formation mechanisms, in particular. More recently, the authors have conducted a phenomenological numerical study on the shot-to-shot coherence characteristics of QML regimes in terms of the nonlinear phase shift (NPS) accumulated per roundtrip [39]. In addition to the numerical analysis, the authors have also presented a preliminary experimental observation that the wave-packet formation in QML regimes could also be initiated by various dynamic interactions among the MS pulses generated in the cavity [40].
In this paper, we present detailed experimental evidences to expand the frontier of comprehension further to our previous numerical result [39] on the shot-to-shot coherence and wave-packet formation in QML regimes in a fiber ring cavity of anomalous dispersion mode-locked by NPR. With fine adjustment of the cavity saturation power, we make the laser cavity operate in three distinct constitutional regimes of NLP, symbiotic, and MS regimes. We investigate the individual regimes through experimentally analyzing their optical spectra, autocorrelation traces, and the shot-to-shot spatio-temporal evolutions for every roundtrip up to 800-roundtrip times. Based on the experimental results, we discuss the transitional processes among different QML regimes and explain the role of the peak powers of the solitonic pulses in their pulse-to-pulse (i.e., MS) interactions, which subsequently determines their coherence characteristics and bunching dynamics. In order to elucidate the consequences by MS interactions, we conduct a very dedicated experiment, closely analyzing intermediate states across two adjacent QML regimes in relation with their modulus of the complex first-order degree of coherence (MDOC) characteristics, in which MS pulses are about to cluster or break up right before forming a wave packet or losing it, respectively. We verify that the shot-to-shot coherence trends observed in the experiment are in good agreement with our previous numerical result [39]. We highlight that our experimental results will justify how critically the formation of QML pulses of the three distinct constitutional regimes depends on the degree of interactions among the MS pulses within the cavity, which is, in fact, closely correlated with the peak-power levels of the MS pulses and the NPS accumulated within the cavity.
2. Experiment setup
Figure 1 depicts the schematic of the fiber laser utilized in our investigation. It was a passively mode-locked fiber ring laser based on NPR. The active medium was an erbium-doped fiber (EDF) with an absorption coefficient of 6 dB/m at 1530 nm. It was core-pumped at 980 nm with a single-mode diode laser through a wavelength division multiplexer (WDM). An additional WDM placed after the EDF removed unabsorbed pump power. The combination of a fiberized polarizing beam splitter (PBS) and two in-line polarization controllers (PCs: λ/4-λ/2-λ/4) spliced before and after the PBS, offered the functionality of fast-response saturable absorption.
The two 1/99 tap couplers were placed before and after the PBS in the cavity to monitor pulse evolution inside the cavity. The polarization insensitive isolator on the left side was incorporated to maintain unidirectional operation of the cavity. The total length of the cavity was ~120 m and had a net anomalous dispersion of β2 ~−2.95 ps2. At low pump power (< ~65 mW) the cavity operated in the single-soliton regime, generating a single soliton per cavity roundtrip having pulse width of ~3 ps and bandwidth of ~4 nm, which varied slightly, depending on the specific conditions of the PCs. In fact, the single-soliton regime tended to be obtained by first increasing the pump power up to ~65 mW and carefully decreasing it down to ~40 mW owing to the pump hysteresis effect [21]. In contrast, at high pump power (> ~151 mW) the cavity operated in QML regimes, including NLP, symbiotic, and MS regimes. It is worth noting that there existed various intermediate or transitional states that did not clearly fall into the categories of the aforementioned regimes for very specific conditions of the PCs even without changing the pump power level. We are not going to discuss all of them in this paper, but believe that some of them may be worth to report elsewhere with a more thorough examination and further discussion.
Figure 2 depicts the fiberized Michelson interferometer utilized to measure the shot-to-shot coherence of the QML pulses generated from the laser cavity shown in Fig. 1. The 50/50 tap coupler equally split the input signal (i.e., the output QML pulses from the laser cavity) into two arms: A half of the input signal into one end of the coupler was reflected by an optical loop mirror through a dispersion-shifted fiber (DSF). The use of the DSF was mainly to avoid unnecessary pulse broadening due to dispersion of the standard single-mode fiber (SMF) at ~1550 nm. The other half of the input signal into the other end of the coupler was reflected by a high-reflecting broadband dichroic mirror through a free-space variable delay line. It is worth noting that the length of the DSF was roughly determined to provide a single-roundtrip path difference between the optical pulses returning through both ends of the coupler, and that the free-space delay line was fine-adjusted to maximize the overlap between the two returning optical pulses once they had been combined together through the third end of the coupler connected to the optical spectrum analyzer (OSA). In fact, the combined optical pulses with a single-roundtrip path difference produced an interference fringe pattern, the fringe visibility (FV) of which corresponded to the MDOC of the output signal of the laser cavity.
We also measured its autocorrelation trace using an autocorrelator (FR-103XL, Femtochrome), which gave us the information on the spike-to-pedestal ratio (SPR) and the average pulse-width when it formed a wave packet. An autocorrelation trace of a typical NLP normally consisted of a sub-ps coherent spike on top of a broad shoulder from a hundreds of picosecond (ps) to nanosecond (ns) pulse-width [38]. This characteristic trace represented a group of sub-ps pulses irregularly distributed and bunched in an even wider temporal window. In contrast, the trace of the signal optical pulse in an MS regime only exhibited a single peak without forming a broad shoulder. Besides recording the spectral interference pattern and autocorrelation trace, we also carried out spatio-temporal measurements on the output QML pulses for up to 800 consecutive roundtrip times, using a photodetector having a 45-GHz bandwidth and a high-speed oscilloscope having an ~80-ps-time-scale resolution (DSO91204A, Agilent: 12-GHz bandwidth and 40-GS/s sampling rate).
3. Experiment results
3.1. QML regimes: NLP, symbiotic, and MS regimes
We pumped the EDF in the cavity at ~220 mW to make it operate in QML regimes. A specific QML regime could be picked up by adjusting the PCs. Most of them resembled the regimes already observed in previous studies [16–38], whereas some others appeared to be exclusively different from any previously reported regimes in very particular conditions. Notwithstanding, relying on the MDOC and SPR measures, we could identify and classify them into the most dominant, three constitutional regimes, i.e., the NLP, symbiotic, and MS regimes [39,40]. In fact, the specific regimes discussed in our previous numerical study [39] could readily be reproduced within the operation windows of the three constitutional regimes. They exhibited clear distinctions in terms of the spectrum, interference fringe pattern, and autocorrelation trace for the output signal, as shown in Fig. 3. It should be noted that the maximal scan range of the autocorrelator used in our experiment was set to ~200 ps.
Fig. 3 Spectra, interference fringe patterns, and autocorrelation traces of typical QML pulses: (a) the NLP regime, (b) the symbiotic regime, and (c) the MS regime.
In addition, we conducted the real-time spatio-temporal measurement, tracing the shot-to-shot pulse evolutions up to 800-roundtrip times, as shown in Fig. 4. (These graphics can also be viewed in a movie version available in a separate file: Visualization 1). It is worth noting that all the intensities are normalized relative to the maximum value obtained from the NLP regime, which will hereafter hold unless stated otherwise.
Fig. 4 Spatio-temporal shot-to-shot measurement results: (a) the NLP regime, (b) the symbiotic regime, and (c) the MS regime. Note that the color bars represent the matched normalized intensities.
NLPs are normally characterized by their broad spectrums, wave-packet containing sub-pulses, and an autocorrelation trace of a coherent spike on top of a broad shoulder, which may vary slightly depending on the precise cavity conditions [16–20]. From the viewpoint of specific experimental observation on our laser system, we specifically defined the NLP regime as the case in which the output signal had a broad spectrum of > 10-nm bandwidth, a moderate pulse width of < 1 ns, and an autocorrelation trace with SPR of < 0.293. In the NLP regime, the output signal was comprised of a group of optical pulses tightly confined within a wave packet, which had substantially higher intensities in comparison with those operating in the other two regimes. Satellite pulses outside the main wave packet were hardly present as clearly shown in Fig. 4(a). In addition, it is worth noting that this regime resulted in the lowest MDOC, i.e., the shot-to-shot coherence measure, as reported in our previous numerical study [39].
By slowly tweaking the PCs further, we could make the cavity operate in the symbiotic regime [39,40], observing that the bandwidth of the output spectrum was reduced to as low as ~5 nm. The tightly bunched optical pulses within a wave packet (previously in the case of the NLP regime) started to be released a bit, thereby holding a loosely confined wave packet approximately 4 to 5 times as broad as that of the NLP regime. In the autocorrelation trace, we could observe that the relative height of the coherent spike on top of the pedestal became taller than that of the NLP regime. The distinct, characteristic feature of the symbiotic regime was the fact that a main wave packet and satellite solitons could coexist in the cavity. They could interact or propagate independently, depending on the precise cavity conditions. For example, satellite solitons could remain separate from the main wave-packet, being stationary or running away from it; in some cases they could rejoin the main wave packet after moving around and interacting with others; etc. Thus, in Fig. 5, we illustrate in more detail various characteristic features of the symbiotic regime obtained from the cavity with fine adjustments of the PCs. (These graphics can also be viewed in a movie version available in a separate file: Visualization 2).
Fig. 5 Spatio-temporal shot-to-shot measurement results for various states of the symbiotic regime: (a) the tear-drop state, (b) the inclined rain state, (c) the straight rain state, (d) the radiating state, (e) the tornado state, and (f) the hurricane state. Note that the color bars represent the matched normalized intensities.
In general, the symbiotic regime had a wide operational range in between the NLP and MS regimes, so that it tended to exhibit various characteristic features in terms of wave-packet-to-soliton interactions. We classified them into six sub-groups in terms of phenotypic behaviors of the regime, which included the tear-drop, inclined rain, straight rain, radiating, tornado, and hurricane states (see Fig. 5). It is worth noting that some of them seemed to resemble what had been reported [25], but others did not. We noticed that the cavity conditions for triggering the symbiotic regime was quite close to that for the NLP regime. As a result, the overall shape of the main wave packet was quite similar to that of the NLP regime. It is, however, worth noting that the intensity of the main wave packet in the presence of satellite solitons, i.e., in the symbiotic regime, became significantly lower than that in the NLP regime [see the relative intensity levels in Figs. 4(b) and 5 in comparison with that in Fig. 4(a)].
By tweaking the PCs even further, we observed the main wave packet eventually collapse into MS pulses [21], so that the symbiotic regime turned into the MS regime. The collapse of the wave packet can be explained as a result of the decrease in binding forces among solitons that will be discussed in detail in Section 3.2. During this process, the shot-to-shot coherence gradually increased as the cavity condition turned away from the NLP regime. In the MS regime, well-separated solitons were the only optical pulses that could exist in the cavity, which were, however, capable of forming neither bunched pulses nor a wave packet. These solitons travelled at the same group velocity. The spectrum of the output signal in the MS regime was nearly identical to that of a conventional single-soliton pulse with Kelly sidebands. In the MS regime, the autocorrelation trace was comprised of a single coherent spike without having any broad pedestal, indicating the complete temporal separation between adjacent solitons in the span of at least 200 ps (i.e., the maximal scan range of the autocorrelator used in our experiment). The shot-to-shot coherence reached close to unity in this regime
On the whole, we could clearly notice that the MDOC tended to increase as we altered the cavity condition from the NLP regime to the symbiotic regime, and subsequently to the MS regime (see Fig. 3). In particular, the NLP regime exhibited the lowest MDOC, ranging from 0.0204 to 0.125. We also noticed that the evolution of the SPR with respect to the PC adjustment showed the very similar trend as the MDOC did. In fact, the NLP regime had the lowest SPR, decreasing down to 0.293 [see Fig. 3(a)]. The MDOC of the symbiotic regime ranged from 0.1 to 0.562. The corresponding SPR could increase significantly up to 0.412 [see Fig. 3(b)]. The measured MDOC and SPR values indicate that QML pulses in the symbiotic regime exhibited significantly increased partial coherence and decreased pulse complexity in comparison with those in the NLP regime. The MDOC kept on increasing in the symbiotic regime as the cavity condition turned further away from the NLP regime. However, once the MDOC had reached beyond 0.869, the main wave packet eventually collapsed, and the operating regime of the cavity was switched to the MS regime. In the MS regime, the MDOC was greater than 0.869, and the SPR was as close as unity. That is, QML pulses operating in the MS regime exhibited nearly complete coherence as well as clear separation between consecutive pulses. The general trends of these experimental observations were in accord with our previous numerical work [39]. Table 1 summarizes the operation parameters of the QML regimes.
Table 1. Operation parameters of the QML regimes.
3.2. Shot-to-shot coherence and wave-packet formation in QML regimes
In this section we further discuss the shot-to-shot coherence properties of QML pulses, paying more attention to their correlation with the formation of a wave packet within the cavity. In the experiment we kept the cavity length constant and precisely varied the saturation power via tweaking the PCs. Subsequently, we were able to monitor in real-time the subtle changes in the optical pulses built up in the cavity across the transitional moment between two different QML regimes, fully taking advantage of the spatio-temporal measurement technique. Phenomenologically, it was very important to resolve out such transitional behaviors in conjunction with measuring the corresponding MDOC and autocorrelation trace at the same time, because this allowed us to identify the cause and effect of soliton interactions in relation with the formation of a specific type of QML pulses.
While taking spatio-temporal measurement on QML pulses triggered under various cavity conditions, we observed two types of soliton interactions take place dominantly, which included a “local” type of soliton interaction via radiative dispersive waves and the “direct” soliton interaction [41–44]. It is worth noting that a “global” type of soliton interaction induced by unstable continuous-wave (CW) components [44] was not apparent, because the transitional QML pulses were primarily initiated from the MS regime, so that there were initially no noticeable CW components built up in the cavity. In addition, we could control the overall strengths of solitons generated in the cavity as well as their separation intervals by means of varying the saturation power of the cavity via tweaking the PCs. In fact, the higher saturation power led to the higher loss being imposed onto the optical pulses in the cavity, thereby resulting in the lower intra-cavity soliton intensity, and vice versa.
We initially set the cavity conditions such that the operation regime was in an intermediate state in between the MS and symbiotic regimes, in order to take precise snapshots on the transitional behaviors of QML pulses across them. For example, two typical intermediate states are depicted in Fig. 6. (These graphics can also be viewed in a movie version available in a separate file: Visualization 3). The graphics on the upper half represent a state that was set closer to the MS regime [see Figs. 6(a)-6(c)], whereas those on the lower half represent a state that was set closer to the symbiotic regime [see Figs. 6(d)-6(f)]. Both intermediate states were accessible from the stabilized MS regime, in which the separation intervals among solitons were normally maintained more than 1 ns [see Fig. 4(c)]. By lowering the saturation power of the cavity, the overall intra-cavity soliton intensities tended to grow, and subsequently, the separation intervals among solitons became reduced.
Fig. 6 Solitons in intermediate states: An intermediate state closer to the MS regime (a) in a 400-ns span, (b) 20-ns span, and (c) 3-ns span; an intermediate state closer to the symbiotic regime (d) in a 400-ns span, (e) 20-ns span, and (f) 3-ns span. Note that the color bars represent the matched normalized intensities.
In the former case shown in Figs. 6(a)-6(c), the intra-cavity solitons started to line up closely to one another with the increase of their overall intensities, and their separation intervals were significantly reduced to less than ~250 ps, so that the individual solitons seemed to have non-negligible influences on the adjacent solitons. It is worth noting that in this case the corresponding soliton interactions incorporated mainly a local type of soliton interaction [44], in which the increase in the soliton intensities resulted in elongating the effective range of the radiative dispersive waves generated from the individual solitons, thereby giving rise to long-range resonant soliton interactions. However, the characteristic features of the MS regime was still more or less maintained, such that its optical spectrum remained within ~3-nm bandwidth, including Kelly sidebands, and the MDOC remained above at least 0.8. Figures 6(b) and 6(c) show the intermediate state in smaller spans of 20 and 3 ns, respectively. Owing to the effect of the local type of soliton interaction, the trail of solitons evolved with roundtrip times, lining up in a rather wiggly manner in the spatio-temporal domain as compared with that of the exact MS regime [compare Fig. 6(c) with Fig. 4(c)]. In general, solitons in the MS regime did not exhibit such behaviors, mainly because they had even wider soliton separation than 1 ns and even lower soliton peak intensity than 0.026, so that individual solitons and the corresponding dispersive waves could hardly interact within the cavity length, even between the nearest neighboring solitons.
By tweaking the PC even further, we could make the cavity operate with relatively higher peak intensity and with more soliton interactions [see Figs. 6(d)-6(f)]. This resulted in even higher degree of local-type soliton interactions and eventually making adjacent solitons interact directly. Figures 6(e) and 6(f) show that both local-type and direct soliton interactions were taking place simultaneously in smaller spans of 20 and 3 ns, respectively. In result, the solitons tended to evolve with roundtrip times in a tree-branch or lightning-strike manner in the spatio-temporal domain, maintaining an average group velocity but with inter-solitonic collisions, attractions, and cross-overs, depending on their inter-solitonic phase differences [41,44]. Their intensities were noticeably higher than those of the solitons in the MS regime as well as in the former intermediate state. It is important to note that in this intermediate state, the optical spectrum looked more like that of the symbiotic regime, having bandwidth of ~5 nm, and the MDOC was reduced down to ~0.55, which was significantly lower than that of the former intermediate state. Actually, the further intensified soliton interactions increased soliton complexity, which in turn led to the optical spectrum losing the Kelly sidebands as shown in Fig. 3(b). We also observed that even stronger soliton interactions could occasionally trigger rogue-wave-like behaviors during some inter-solitonic collisions [45], which are, however, outside the scope of our current discussion.
On the whole, both intermediate states exhibited significantly different features in their optical spectra and MDOC values, depending on the degree of soliton interactions in the cavity. In fact, the stronger soliton interactions led to the higher shot-to-shot fluctuations, thereby causing the lower MDOC values. The cavity condition in which such intermediate states could hold was relatively very tight that they tended to be short-lived, readily settling down into the symbiotic or MS regime if they were not exactly fit to survive. This was due to the fact that the degrees of soliton interactions grow “exponentially” with respect to increase in the intra-cavity soliton intensity as well as to decrease in the soliton separation interval [41–44]. For example, when we further increased the intra-cavity soliton intensity from the intermediate states discussed above, the cavity regime abruptly made transition into the symbiotic regime close to the states shown in Figs. 5(e) and 5(f) where the formations of the solitons were already exhibiting a wave-packet-like feature, such that they were closely spaced, vividly interacting within a ~10-ns time window and having high relative intra-cavity intensities of well above 0.1. The whole wave-packet formation process can also explain the collapse of the wave packet that tended to happen when the intra-cavity soliton intensity was reduced beyond a certain critical value, with which the cavity could manage to bind individual solitons within a packet. Just like the wave-packet formation process, the transition from a wave packet to free running solitons was also an abrupt process.
We emphasize that both intermediate states, which we managed to spot and analyze by means of the real-time shot-to-shot measurement as discussed above, provide a way to finding the missing links between the formation of a wave packet of quasi-randomly-packed solitons and the intra-cavity soliton interactions. This aspect has been remaining unclear to date due to the fact that such regime transitions take place too abruptly and instantaneously to detect via conventional measurement methods. Whilst we were resolution-limited in analyzing the more detailed inter-soliton dynamics within the wave packet, our experimental observations somehow manifest that the growth of the intra-cavity soliton complexity, i.e., the degradation of the MDOC after the formation of a wave packet out of multiple intra-cavity solitons was, in fact, triggered by strong soliton interactions, which were further parameterized by the intra-cavity soliton intensity and inter-soliton separation. Subsequently, it has become obvious that further increasing the intra-cavity soliton intensity from the symbiotic regime could readily lead to the regime transition into the NLP regime, which has the tightest soliton confinement in the wave packet and the highest intra-cavity soliton intensity among the three constitutional regimes. Therefore, one can expect that the corresponding MDOC would further degrade certainly via the strong NPS induced every roundtrip and the chaotic nature of the inter-soliton dynamics inside the wave packet. It is fair to say that the NLP or symbiotic regime in an anomalous-dispersion fiber ring cavity is an extreme operation state achievable when the soliton interaction is sufficiently intense.
In our previous numerical study [39], we showed that the MDOC is strongly correlated with the NPS induced to the intra-cavity solitons per roundtrip. In fact, an absolute experimental measurement of the NPS induced per cavity roundtrip would make a direct comparison with the numerical result previously obtained [39]; however, our spatio-temporal detection was resolution-limited to ~80 ps, so that it was not fast enough to measure and trace out the exact peak powers of the individual intra-cavity pulses that tended to be as short as sub- or a few ps (see the autocorrelation traces shown in Fig. 3). Thus, we would instead like to take the total intra-cavity intensity in order to make a relative comparison between our experimental and numerical results, because the total intra-cavity intensity should obviously be proportional to the NPS accumulated per roundtrip. It is worth noting that the total intra-cavity intensity can readily be estimated by simply integrating the spatio-temporal detection data without justifying the exact peak powers of the individual intra-cavity pulses. Figure 7 shows the experimentally measured MDOC values for the 34 individual QML states classified into the three constitutional QML regimes, relying on the criteria given in Section 3.1, with respect to the relative total intra-cavity soliton intensity that is normalized to the maximum value obtained in the case of the incoherent NLP regime.
Fig. 7 Experimental data on the MDOC with respect to the relative total intra-cavity intensity; red solid line: Lorentzian fitting curve. All the data points are classified into the corresponding QML regimes, relying on the criteria given in Section 3.1.
Qualitatively speaking, the general trend of the MDOC distribution is in good agreement with that obtained in the numerical study [39], which was also well fitted with a Lorentzian fitting curve. (See Ref [39]: Roughly, the normalized total-intra-cavity intensity parameter multiplied by a factor of ~10 would lead to a good match with the corresponding NPS per roundtrip.) Thus, we stress that the strong correlation between the MDOC and the NPS per roundtrip, i.e., the total intra-cavity soliton intensity, predicted by the numerical study has now been confirmed by the experimental observation.
4. Conclusion
We have experimentally characterized the three constitutional QML regimes of a PMLFL of an anomalous-dispersion cavity, including the NLP, symbiotic, and MS regimes, analyzing and detailing out their shot-to-shot coherence properties, wave-packet-forming/bunching processes, and correlations with soliton interactions. We verified that solitons with higher intra-cavity intensities not only experienced greater NPSs but also triggering stronger soliton interactions. Carefully controlling the intra-cavity intensities and thereby the degree of soliton interactions by adjusting the saturation power of the cavity, we could analyze the inter-soliton dynamics within narrow windows of transitional states across different QML regimes in real-time and showed that increase in the degree of soliton interactions could result in drastic degradation of the shot-to-shot coherence of the corresponding QML pulses generated in the cavity. In an anomalous-dispersion cavity mainly governed by soliton dynamics, the NLP and symbiotic regimes could be regarded as mutant forms of the MS regime that are fit to survive in the cavity even when individual solitons are heavily interacting one another, thereby settling in a quasi-stable, soliton-clustered wave packet and incorporating chaotic or stochastic consequences in their spectral, temporal, and coherence properties. To the best of our knowledge, there have been no systematic and dedicated experimental investigations on the wave-packet formation mechanisms in the QML regimes and the corresponding consequences in their coherence properties, which have now been clarified by our discussion presented above. Our experimental investigations also verified and confirmed our previous numerical work, claiming good agreement with it. We hope our close experimental observations on the intra-cavity soliton interactions and their detailed dynamics and consequences, would fill the gaps in the existing knowledge on the QML regimes of PMLFLs, as well as putting a stepping stone to the further expansion of them.
Funding
Ministry of Trade, Industry and Energy (Project no. 10060150); National Research Foundation of Korea (2017R1D1A1B03036201); Brain Korea 21 Plus Program.
References and links
1. L. Nelson, D. Jones, K. Tamura, H. Haus, and E. Ippen, “Ultrashort-pulse fiber ring lasers,” Appl. Phys. B 65, 277–294 (1997).
2. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18(13), 1080–1082 (1993). [PubMed]
3. F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004). [PubMed]
4. A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14(21), 10095–10100 (2006). [PubMed]
5. N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124–3128 (2008).
6. W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances in laser models with parameter management,” J. Opt. Soc. Am. B 25, 1972–1977 (2008).
7. L. Shah and M. E. Fermann, “High power femtosecond fiber chirped pulse amplification system for high speed micromachining,” J. Laser Micro Nanoeng. 1(3), 176–180 (2006).
8. H. Lim, Y. Jiang, Y. Wang, Y.-C. Huang, Z. Chen, and F. W. Wise, “Ultrahigh-resolution optical coherence tomography with a fiber laser source at 1 μm,” Opt. Lett. 30(10), 1171–1173 (2005). [PubMed]
9. T. R. Schibli, K. Minoshima, F.-L. Hong, H. Inaba, A. Onae, H. Matsumoto, I. Hartl, and M. E. Fermann, “Frequency metrology with a turnkey all-fiber system,” Opt. Lett. 29(21), 2467–2469 (2004). [PubMed]
10. J. R. Unruh, E. S. Price, R. G. Molla, L. Stehno-Bittel, C. K. Johnson, and R. Hui, “Two-photon microscopy with wavelength switchable fiber laser excitation,” Opt. Express 14(21), 9825–9831 (2006). [PubMed]
11. J. Clowes, “Next generation light sources for biomedical applications,” Opt. Photon. 3(1), 36–38 (2008).
12. N. Nishizawa and J. Takayanagi, “Octave spanning high-quality supercontinuum generation in all-fiber system,” J. Opt. Soc. Am. B 24(8), 1786–1792 (2007).
13. Y. Kwon, L. A. V. -Zuniga, K. Park, S. Lee, H. Chang, and Y. Jeong, “Combinatorial study of supercontinuum generation dynamics in photonic crystal fibers pumped by ultrafast fiber lasers,” IEEE J. Quantum Electron. 52(6), 6400311 (2016).
14. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6, 84–92 (2012).
15. M. E. Fermann and I. Hartl, “Ultrafast fiber lasers,” Nat. Photonics 7, 868–874 (2013).
16. M. Horowitz, Y. Barad, and Y. Silberberg, “Noiselike pulses with a broadband spectrum generated from an erbium-doped fiber laser,” Opt. Lett. 22(11), 799–801 (1997). [PubMed]
17. A. I. Chernykh and S. K. Turitsyn, “Soliton and collapse regimes of pulse generation in passively mode-locking laser systems,” Opt. Lett. 20(4), 398–400 (1995). [PubMed]
18. D. E. Pelinovsky, V. V. Afanasjev, and Y. S. Kivshar, “Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(2), 1940–1953 (1996). [PubMed]
19. J. U. Kang, “Broadband quasi-stationary pulses in mode-locked fiber ring laser,” Opt. Commun. 182(4–6), 433–436 (2000).
20. D. Tang, L. Zhao, and B. Zhao, “Soliton collapse and bunched noise-like pulse generation in a passively mode-locked fiber ring laser,” Opt. Express 13(7), 2289–2294 (2005). [PubMed]
21. D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005).
22. A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71(5), 053809 (2005).
23. L. M. Zhao, D. Y. Tang, J. Wu, X. Q. Fu, and S. C. Wen, “Noise-like pulse in a gain-guided soliton fiber laser,” Opt. Express 15(5), 2145–2150 (2007). [PubMed]
24. L. M. Zhao, D. Y. Tang, T. H. Cheng, H. Y. Tam, and C. Lu, “120 nm bandwidth noise-like pulse generation in an erbium-doped fiber laser,” Opt. Commun. 281(1), 157–161 (2008).
25. S. Chouli and P. Grelu, “Soliton rains in a fiber laser: An experimental study,” Phys. Rev. A 81(6), 063829 (2010).
26. O. Pottiez, R. G. -Coutiño, B. I. -Escamilla, E. A. Kuzin, and J. C. H. -García, “Adjustable noiselike pulses from a figure-eight fiber laser,” Appl. Opt. 50(25), E24–E31 (2011).
27. J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Dissipative rogue waves: Extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1 Pt 2), 016604 (2011). [PubMed]
28. L. A. V. -Zuniga and Y. Jeong, “Super-broadband noise-like pulse erbium-doped fiber ring laser with a highly nonlinear fiber for Raman gain enhancement,” IEEE Photonics Technol. Lett. 24(17), 1549–1551 (2012).
29. S. Smirnov, S. Kobtsev, S. Kukarin, and A. Ivanenko, “Three key regimes of single pulse generation per round trip of all-normal-dispersion fiber lasers mode-locked with nonlinear polarization rotation,” Opt. Express 20(24), 27447–27453 (2012). [PubMed]
30. H. Santiago-Hernandez, O. Pottiez, M. Duran-Sanchez, R. I. Alvarez-Tamayo, J. P. Lauterio-Cruz, J. C. Hernandez-Garcia, B. Ibarra-Escamilla, and E. A. Kuzin, “Dynamics of noise-like pulsing at sub-ns scale in a passively mode-locked fiber laser,” Opt. Express 23(15), 18840–18849 (2015). [PubMed]
31. Z. Cheng, H. Li, and P. Wang, “Simulation of generation of dissipative soliton, dissipative soliton resonance and noise-like pulse in Yb-doped mode-locked fiber lasers,” Opt. Express 23(5), 5972–5981 (2015). [PubMed]
32. Y.-Q. Huang, Z.-A. Hu, H. Cui, Z.-C. Luo, A.-P. Luo, and W.-C. Xu, “Coexistence of harmonic soliton molecules and rectangular noise-like pulses in a figure-eight fiber laser,” Opt. Lett. 41(17), 4056–4059 (2016). [PubMed]
33. Y. E. B. -Rodriguez, O. Pottiez, E. G. Sánchez, J. L. -Cruz, H. I. -Villalón, J. C. H. -Garcia, M. B. -Jimenez, G. B. Pérez, B. I. -Escamilla, and E. Kuzin, “Dual noise-like pulse and soliton operation of a fiber ring cavity,” J. Opt. 19, 035502 (2017).
34. E. G. -Sanchez, O. Pottiez, Y. B. -Rodriguez, J. P. L. -Cruz, H. E. I. -Villalon, J. C. H. -Garcia, M. B. -Jimenez, and E. A. Kuzin, “Complex dynamics of a fiber laser in nonstationary pulsed operation,” Opt. Express 24(17), 18917–18930 (2016). [PubMed]
35. D. V. Churkin, S. Sugavanam, N. Tarasov, S. Khorev, S. V. Smirnov, S. M. Kobtsev, and S. K. Turitsyn, “Stochasticity, periodicity and localized light structures in partially mode-locked fibre lasers,” Nat. Commun. 6, 7004 (2015). [PubMed]
36. A. F. J. Runge, C. Aguergaray, N. G. R. Broderick, and M. Erkintalo, “Coherence and shot-to-shot spectral fluctuations in noise-like ultrafast fiber lasers,” Opt. Lett. 38(21), 4327–4330 (2013). [PubMed]
37. L. Gao, T. Zhu, S. Wabnitz, M. Liu, and W. Huang, “Coherence loss of partially mode-locked fibre laser,” Sci. Rep. 6, 24995 (2016). [PubMed]
38. Y. Jeong, L. A. V. -Zuniga, S. Lee, and Y. Kwon, “On the formation of noise-like pulses in fiber ring cavity configurations,” Opt. Fiber Technol. 20(6), 575–592 (2014).
39. Y. Kwon, L. A. Vazquez-Zuniga, S. Lee, H. Kim, and Y. Jeong, “Numerical study on multi-pulse dynamics and shot-to-shot coherence property in quasi-mode-locked regimes of a highly-pumped anomalous dispersion fiber ring cavity,” Opt. Express 25(4), 4456–4469 (2017). [PubMed]
40. S. Lee, L. A. V. -Zuniga, Y. Kwon, H. Kim, and Y. Jeong, “Experimental study on the coherence of non-stationary optical pulses generated from a ring laser cavity,” in Proceedings of CLEO 2016, San Jose, USA, 5 Jun.−10 Jun., 2016, paper STh4O.7.
41. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
42. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8(11), 596–598 (1983). [PubMed]
43. F. M. Mitschke and L. F. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12(5), 355–357 (1987). [PubMed]
44. D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016616 (2005). [PubMed]
45. J. Peng, N. Tarasov, S. Sugavanam, and D. Churkin, “Rogue waves generation via nonlinear soliton collision in multiple-soliton state of a mode-locked fiber laser,” Opt. Express 24(19), 21256–21263 (2016). [PubMed]
