We demonstrate the coexisting dynamics of loosely bound solitons and noise-like pulses (NLPs) in a passively mode-locked fiber laser with net-normal dispersion. The total pulse number of the single soliton bunch under the NLP operation regime almost increases linearly with increasing pump power, whereas the average pulse spacing decreases accordingly. Furthermore, pulse-to-pulse separation between adjacent soliton pulses in one soliton bunch keeps in the range of hundreds of picoseconds, which decreases from left to right with the change of time. Besides, the real-time observation has been performed by utilizing the time-stretch method, showing that each one of the loosely bound solitons on the NLP operation is actually composed of chaotic wave packets with random intensities. These findings obtained will facilitate the in-depth understanding of nonlinear pulse behaviors in ultrafast optics.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
During the past several years, passively mode-locked fiber lasers (PMLFLs) have made important progress and have been gradually developed as the mature ultrashort pulse sources, which can be widely used in various fields ranging from optical communication, optical frequency comb, micromachining, medicine imaging to material processing [1,2]. Furthermore, PMLFLs can also constitute an excellent platform for studying a wide range of intriguing soliton dynamics [3–5], attracting a considerable attention in the fields of fundamental physics and nonlinear optics. So far, there has been a variety of pulse regimes demonstrated in PMLFLs, such as the conventional solitons (CSs) , self-similar solitons  and dissipative solitons (DSs) . When the fiber laser system is subjected to the interplay of gain, loss, the nonlinearity and dispersion, DSs can be usually generated [9–11]. Interestingly, DSs can exist in the normal dispersion or anomalous dispersion regime . Commonly, the DSs possess much higher pulse energy than CSs, attributed to accumulated large chirp, which can make them easily amplified and compressed .
Apart from the single soliton regime, multiple-soliton bound state can also be formed by properly increasing the cavity energy supply in the DS fiber laser. Actually, multiple-soliton operation is a common regime in PMLFLs . Bound solitons, also known as soliton molecules, have been theoretically studied and experimentally demonstrated in PMLFLs with the normal dispersion regime [14,15]. Generally, based on the pulse-to-pulse separation, bound solitons can be classified to two types, namely the tightly and loosely bound solitons . In the tightly bound solitons, small pulse separations less than several pulse durations are formed under the situation with strong soliton interaction . In contrast, loosely bound solitons possess large pulse interval, resulting from the weak soliton interaction . At present, the real-time dynamics of loosely bound solitons in the anomalous dispersion fiber laser has been studied by using the time-stretch technique . Although the loosely bound solitons were demonstrated in the net-normal-dispersion fiber laser , their real-time dynamics, which may have more interesting and complicated behaviors, has not been fully explored.
Under certain conditions, PMLFLs can also operate in a so-called the noise-like pulse (NLP) regime . Since the first report in the 90s , NLPs have aroused great interest and extensive research because of their distinctive characteristics of broad spectrum, high pulse energy and low spectral coherence, making themselves useful in fields of supercontinuum generation  and low coherence interferometry . In the frequency domain, NLPs generally exhibit a broad and smooth spectrum and a relatively stable pulse packet in time domain. Actually, NLPs are the chaotic wave-packets consisting of loads of short pulses with random peak power and pulse widths. In addition, the corresponding autocorrelation trace exhibits a typical high peak in the femtosecond time range riding on a wide pedestal in the picosecond time scale. Although plenty of experimental and theoretical researches on the NLPs have been carried out [11,21,25–28], their dynamical behaviors and the underlying physical mechanisms still require further studies. Recently, the simultaneous generation of harmonic solitons and NLPs has been presented in the anomalous dispersion fiber laser . Moreover, the coexistence regime of solitons and NLPs has also been demonstrated . And the coexisting observation of harmonic mode-locking and NLPs has experimentally reported in a mode-locked Tm-doped fiber laser . Thus, it is necessary to explore the complicated dynamics of the coexisting pattern between NLPs and other soliton regime, allowing a further understanding of NLPs in PMLFLs.
In this work, the simultaneous generation and real-time dynamics of loosely bound solitons and NLPs in the PMLFL with the net normal dispersion are first presented to the best of our knowledge. By increasing the pump power, the total pulse number in single bound bunch shows a nearly linearity increment while the average temporal interval in one soliton bunch decreases almost linearly. The adjacent pulse spacing in a bunch gradually decreases with the increment of time. The spectral dynamics of the coexistence regime based on the time-stretch method represents that the each of the loosely bound solitons is a series of noise-burst pulses with stochastically varying pulse power.
2. Experimental setup
Figure 1 illustrates the experimental set-up of the net-normal-dispersion mode-locked fiber laser. A piece of 10 m Er-doped fiber (EDF) is the gain fiber pumped by a 980 nm laser source through a 980/1550 wavelength-division multiplexer (WDM). A polarization-dependent isolator (PD-ISO) sandwiched between two polarization controllers (PCs, PC1 and PC2) forms the nonlinear polarization rotation (NPR) structure, which is employed to achieve the mode locking. The single direction operation is enforced through the PD-ISO. The laser output is taken by a 10:90 optical coupler (OC). Except for EDF, the other fiber in the experiment is standard single-mode fiber with a length of 6 m. The corresponding total cavity length is 16 m. The dispersion parameters of EDF and SMF at 1550 nm are −12.16 ps/nm/km and 18 ps/nm/km, respectively. Thus, the net cavity dispersion is calculated to be ∼ 0.017 ps2. An optical spectrum analyzer (Yokogawa, AQ 6375B), an oscilloscope (Tektronix, MDO 4104C) with a 1-GHz bandwidth, a radio-frequency (RF) analyzer (Keysight, N9322C) and a commercial autocorrelator (APE-150 Pulsecheck) are utilized to monitor the laser output. The internal temporal characteristics of output pulses are evaluated by a high-speed photodiode (New Focus, Model 1444) plugged into a 12.5 GHz real-time oscilloscope (Tektronix DPO71254C).
3. Experimental results and discussion
In the experiment, continuous wave (CW) emission starts to appear at the low pump threshold of 6 mW. Upon increasing the pump power, mode-locked pulses are observed by finely adjusting the PCs. Figure 2 illustrates the results of stable output pulses at the pump power of 18 mW. As depicted in Fig. 2(a), the optical spectrum is characterized by a quasi-rectangular shape with steep edges, due to the net-normal-dispersion laser cavity , which is a representative characteristic of DSs. The corresponding central wavelength and 3 dB spectral width are 1566 nm and 15 nm, respectively. Figure 2(b) shows the temporal trains with a pulse-to-pulse separation of 80.1 ns, corresponding to the fundamental repetition rate of 12.49 MHz. As shown in Fig. 2(c), the RF spectrum illustrates the high signal-to-noise ratio (SNR) of 55.4 dB at a peak of 12.49 MHz, indicating the good stability in the laser operation. Besides, the real-time shot-to-shot spectra based on the dispersive Fourier-transform (DFT) method are recorded to further investigate the fine details of these pulses . Herein, the output pulse is stretched by using the dispersion compensating fibers (DCFs) with a total dispersion of −331 ps/nm at 1550 nm. Therefore, the DFT spectral resolution is about 0.24 nm. Figure 2(d) demonstrates the evolution of shot-to-shot spectra at consecutive roundtrips. Clearly, it shows that the profiles and intensities of these spectra are both nearly indistinguishable from each other, confirming that the apparent smooth spectra measured through the commercial optical spectrum analyzer (see Fig. 2(a)).
Subsequently, when the pump power is increased to 50 mW, the DSs can switch to the NLP regime with properly changing the intra-cavity polarization state. The corresponding measured results at the pump power of 209 mW are illustrated in Fig. 3. Differ from the DSs, there are no steep edges on the NLP spectrum, as depicted in Fig. 3(a). Furthermore, the optical spectrum, at the wavebands over 1545 nm to 1620 nm, is broad and smooth. The 3 dB spectral width is 34.33 nm and the central wavelength is 1580.38 nm. Figure 3(b) presents the temporal trains of NLPs recorded by using the low-speed oscilloscope (1 GHz). The pulse interval between adjacent pulses is 80.1 ns. Figure 3(c) presents the corresponding autocorrelation traces, which are featured by a high spike located on a broad base with the picosecond time scale. The close-up of the spike is depicted in Fig. 3(d), which has a small width in the femtosecond time range. These typical characteristics are similar to the observations in the previous reports [21,28]. In order to investigate the internal details of these output pulses, their temporal traces based on the high-speed oscilloscope (12.5 GHz) are depicted in the Fig. 3(e). The enlarged portion shown in Fig. 3(f) indicates that the output pulses are multiple-pulse bunches, including three pulses. The first pulse-to-pulse separation is ∼ 713 ps and the second one is ∼ 508 ps.
When keeping the PCs unchanged, the amount of multiple pulses in the NLP operation increases with the pump power raised. The three-dimensional (3D) contour diagrams of more than 3000 consecutive roundtrips at the pump powers of 300 mW and 495 mW are illustrated in Figs. 4(a) and 4(b). Clearly, the multiple-pulse bunch in single cavity roundtrip consists of four pulses with unequal pulse-to-pulse separation, as presented in Fig. 4(a). The temporal spacing between adjacent pulses gradually decreases from 751 ps to 382 ps. For the pump power of 495 mW, there are six pulses in the multiple-pulse bunch, as depicted in Fig. 4(b). The pulse-to-pulse interval gradually decreases from 612 ps to 270 ps. These change trends of pulse spacing are different from previous results in the reports [19,34]. In their observations, the pulse-to-pulse separation changes randomly. However, in this experiment, the pulse-to-pulse separation is declining, suggesting that the long-distance interactions among these multiple pulses grow stronger and stronger. Actually, the slow recovery and depletion processes of the gain in the fiber cavity can play a vital role in the formation of the varying pulse-to-pulse separations . Moreover, the number of multiple pulses in a bunch for different pump powers is evaluated, as depicted in Fig. 5(a). The total pulse number of a bunch almost increases linearly from 1 to 6 with increasing the pump power. Furthermore, the average pulse-to-pulse interval in a bunch is summarized. As can be illustrated in Fig. 5(a), the pulse interval decreases nearly linearly with the increment of pump power, which differs from these changing trends in previous observation [19,34]. That is, the pulse-to-pulse interactions in a bunch are gradually stronger as improving the pump power. Apparently, these temporal separations are more than 270 ps (see Figs. 3(f), 4(a) and 4(b)), which are bigger than the temporal widths in the Figs. 3(c) and 3(d) and beyond the scope of the commercial autocorrelator. Thus, these multiple pulses in the NLP operation can be recognized as the special loosely bound solitons [18,36]. Besides, we also investigated the dependence of total pulse number of soliton bunch on the pump power. Figure 5(b) illustrates the corresponding results. The red steps present the descending process while the black ones show the ascending procedure. At the pump power of 495 mW, the pulse amount in the bunch is 6. However, the pulse amount decreases to 5 with reducing the pump power to 367.8 mW. Obviously, it has a hysteresis characteristic, which is consistent with the similar observation for the multiple soliton regime in the ultrafast fiber lasers [19,37]. The trend of the width of the noise-like pulse bunch as a function of pump power is also studied, as illustrated in Fig. 5(c). Clearly, the temporal width of the single bunch gradually increases with the increment of pump power, suggesting that the total pulses in a bunch are growing as improving the pump power.
Next, the optical spectrum evolution at different pump powers is experimentally studied. And Fig. 6(a) shows the recorded results. With the incease in the pump power, the spectral shape nearly keeps unchanged while the optical spectrum gradually gets broader. Besides, there are no obvious interference fringes appearing in these smooth spectra, resulting from large pulse-to-pulse separations and incoherence of the pulses in the bunch. Moreover, the average output power of multiple pulses in the NLP operation with different pump powers is also evaluated, and the corresponding results are illustrated in Fig. 6(b). The increasing trend is almost linear and the slope efficiency is about 9.96%. Under the maximum pump power of 495 mW, the output power is approximately 49 mW. Actually, the pulse energy of NLPs cannot be precisely calculated because the NLPs are actually made up of many small ultrashort pulses with stochastically varying pulse number, peak intensity and pulse width [21,38]. But the total pulse energy of the NLPs bunch can be estimated by using the reported method [11,39]. Therefore, the total pulse energy of the multiple pulses in the NLPs operation at 495 mW is estimated to ∼3.92 nJ, which is larger than the pulse energy of CSs in the anomalous-dispersion regime.
In order to further explore the fine details of these multiple pulses in the NLP operation, the DFT method is employed to explore the shot-to-shot spectrum evolution over 3000 cavity roundtrips. Figure 7(a) illustrates the corresponding measured average shot-to-shot spectrum at pump power of 495 mW, which has the smooth and broad characteristics. The overall shape of the average spectrum basically matches with the measure OSA spectrum (see Fig. 6(a)). The shot-to-shot spectral sequence of noise-like pulse bunch is presented in Fig. 7(b). As can be seen from the figure, the spectra show substantial fluctuations in comparison with the DSs (see Fig. 2(d)), which is consistent with the experimental observations in the previous report . Moreover, these shot-to-shot spectra also indicates that the each one of multiple pulses is actually a noise-pulse bunch, including the plenty of noisy pulses with the randomly varying pulse width and peak power. In addition, the noise of the shot-to-shot spectra with consecutive roundtrips is studied. The RF spectra of these multiple pulses are shown in Fig. 8. Clearly, the pulse bunches are at the fundamental repetition rate. And the existence of two small side peaks with low SNR around the fundamental repetition rate indicates that there are random peak modulations in the noise-like pulses [40,41]. Compared with the RF spectrum of stable DSs, the noise-like pulse bunches have a bigger pedestal, which is actually a noisy envelope. Besides, the locations of side peaks in the RF spectra at the pump powers of 495 mW, 399 mW, 209 mW are almost same. For the pump power of 50 mW, there is no side peak observed, indicating that there is less noise in these pulses. As can be seen from Fig. 3(c), the autocorrelation trace of noise-pulse bunches has a high peak on a wide and smooth pedestal. These characteristics suggest that the noise-like pulse bunches have low temporal coherence . Thus, the noise of the shot-to-shot spectra can be regarded as the special noise with the low temporal coherence. Furthermore, the evolution process of the pulse energy at different pump powers are studied. Figure 9 depicts the corresponding measured results. Under the low pump power of 50 mW (black line), the energy shows an unstable evolution. When increasing the pump power, the evolution process has some fluctuations. Under the maximum power of 495 mW (blue line), the energy illustrates a more unstable evolution, which is the one of typical characteristics in the NLP operation .
Recent studies have proved that the total pulse number of one bunch in NLP operation in the dispersion-managed cavity strongly depend on the amplitude of the dispersion map, which may result in the merging events and Raman-induced temporal drifts . Although our dispersion-managed fiber laser system is characterized by a strong dispersion map, there are no merging events and Raman-induced drifts observed in the temporal evolutions. Actually, many nonlinear effects (including modulation instability ) are sensitive to noise characteristics of input pulses, which may lead to the formation of NLPs . Researchers have shown that the modulation instability can play an important role in the mechanisms of coherence loss in the noise-like pulses mode-locked fiber lasers . In our experiment, the fiber cavity includes 10-m EDF and 6-m SMF, thus the dispersion of the fiber cavity is periodically varied. At this condition, the modulation instabilities could occur [46,47]. Then dissipative solitons in the fiber laser will experience periodic intensity modulation caused by the cavity effect, which may finally lead to the generation of noise-like pulses at high pump power. In addition, in this experiment, the mode locking is achieved through the NPR technique. The minimum or maximum value of NPR transmission function can be considered as the critical saturation power (CSP) . If the optical power exceeds that value, the effective feedback transmission would be transformed from one to the other. This conversion effect can play a critical role in the NLP generation in mode-locked fiber lasers with the NPR technique [10,21,48]. Furthermore, adjusting the paddles of PCs also changes the CSR in the NPR fiber lasers . Thus, by appropriately increasing pump power and finely changing the PCs, the pulse evolution of DSs and NLPs can be easily achieved in our fiber laser. In addition, NLPs possess a distinct characteristic i.e., the NLP operation is not only a chaotic multiple-pulse regime in itself, but also consists of loosely bound soliton pulses in the time domain. In other words, the simultaneous generation of loosely bound solitons and NLPs is obtained. In the net-normal-dispersion ultrafast fiber lasers, the rise of pump power may lead to the accumulation of excessive pulse chirps, in which the multiple solitons can be generated . Thus, the loosely bound pulses on the NLP operation exist under the high pump power in our experiment.
In summary, we have proposed the composite regime of loosely bound solitons and NLPs in a dispersion-managed PMLFL with net-normal dispersion. By appropriately increasing pump power and finely adjusting the intracavity polarization state, the stable DSs can be switched to the coexistence state of loosely bound solitons and NLPs. The total pulse number of one loosely bound soliton bunch rises nearly linearly with an increase in pump power and the amount at the maximum power is six. Moreover, the average pulse interval in a bunch decreases almost linearly with increasing pump power. And the temporal separation among adjacent pulses in a single bunch gradually decreases over time. Besides, the real-time spectral characteristic of the coexisting regime is explored through the time-stretch method, exhibiting that each of loosely bound solitons on the NLP operation are localized chaotic pulses. It is believed that our results provide a better understanding of nonlinear soliton dynamics in fiber lasers.
International Cooperation and Exchange Programme (61961136001); Taipei University of Technology-Shenzhen University Joint Research Program (2020007); Science and Technology Planning Project of Shenzhen Municipality (JCYJ20190808143813399); China Postdoctoral Science Foundation (2020M682849); National Natural Science Foundation of China (61705140, 61875132, 61875138, 62005178).
The authors declare no conflicts of interest.
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