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The vector nature of multi-soliton dynamic patterns was investigated in a passively mode-locked figure-eight fiber laser based on the nonlinear amplifying loop mirror (NALM). By properly adjusting the cavity parameters such as the pump power level and intra-cavity polarization controllers (PCs), in addition to the fundamental vector soliton, various vector multi-soliton regimes were observed, such as the random static distribution of vector multiple solitons, vector soliton cluster, vector soliton flow, and the state of vector multiple solitons occupying the whole cavity. Both the polarization-locked vector solitons (PLVSs) and the polarization-rotating vector solitons (PRVSs) were observed for fundamental soliton and each type of multi-soliton patterns. The obtained results further reveal the fundamental physics of multi-soliton patterns and demonstrate that the figure-eight fiber lasers are indeed a good platform for investigating the vector nature of different soliton types.
© 2014 Optical Society of America
1. Introduction
In recent decades, temporal solitons have been extensively investigated since they were first observed in optical fibers by Mollenauer et al. in 1980 [1]. As we know, passively mode-locked fiber lasers have been regarded as excellent tools to investigate both the generation and the dynamics of soliton pulses. Generally, only single pulse propagates in the cavity of passively mode-locked fiber lasers. However, supposing that the pump power is high enough, more than one pulse will emerge due to the quantization of soliton energy; namely, multi-soliton operation is achieved in fiber lasers [2–4]. Because of the interactions among solitons, dispersive waves and continuous waves (CW), it was found that the multi-soliton operation presented more physical features than those of the single soliton in the fiber lasers [5,6]. Therefore, it is always interesting to investigate multi-soliton evolutions and interactions in fiber lasers via skillfully selecting the cavity parameters.
To date, various modes of multi-soliton have been observed under different laser cavity conditions. When multiple solitons are formed in the cavity, they can be tightly spaced, i.e., soliton cluster [7,8], or dispersedly and randomly spaced, e.g., random static distribution of soliton [9]. Apart from the disorder interval between multiple solitons in the time domain, multiple solitons could also be regularly spaced in the laser cavity, for instance, the harmonic mode-locking [10–12]. The above-mentioned multi-soliton operations are static. However, it has been recently demonstrated that the soliton rains/flow as a dynamic state is another possible behavior of multi-soliton [13–15]. The demonstrations of versatile multi-soliton types are beneficial for complementing the understanding of soliton family. However, the previously reported multi-soliton operation generally focused on the scalar characteristics.
Actually, solitons propagating in single mode fiber (SMF) can exhibit complicated polarization dynamics as the result of the SMF supporting the two orthogonal polarization modes [16,17]. Except that the polarization-maintaining fibers are utilized in the fiber laser, the laser cavity always displays small amounts of random birefringence. Consequently, it is also significative to consider the vector nature of solitons generated in the fiber laser. Based on the laser cavities without the polarization discrimination devices, different vector soliton dynamics were observed in passively mode-locked fiber lasers, such as polarization-locked vector solitons (PLVSs) [18–21], polarization-rotating vector solitons (PRVSs) [22–25], and group velocity locked vector solitons (GVLVSs) [26–29]. As mentioned above, the generation of multi-soliton patterns is an important physical phenomenon in fiber lasers. Thus, investigating vector nature of multi-soliton patterns is beneficial for further revealing the intrinsically physical features of multi-soliton dynamic patterns. For example, as a type of the multi-soliton patterns, soliton flow in fiber laser is a dynamic process. From the viewpoint of fundamental physics, it would be interesting to know whether the polarization-locked and the polarization-rotating vector soliton flow could be observed during its dynamic flowing or not. Thus, for further comprehending the fundamental physics of multi-soliton operations, there would be a strong motivation to investigate the vector nature of multi-soliton patterns.
Recently, we have suggested that the figure-eight fiber laser could be utilized to investigate the vector characterizes of various soliton types [30]. It is worth noting that there were no polarization-loss dependent components (i.e. polarization sensitive isolator) in the laser cavity. Moreover, comparing to the mode-locking fiber lasers based on semiconductor saturable absorber mirror (SESAM) [31,32], carbon nanotube [33,34], graphene [35–37], or topological insulator [12,38–41], the intra-cavity polarization controller (PC) offers one more degree of freedom to adjust the laser operation regime in the figure-eight fiber laser based on a nonlinear amplifier loop mirror (NALM) or nonlinear optical loop mirror (NOLM). Hence, the figure-eight fiber laser could be treated as an excellent platform to investigate vector features of different soliton types. In this work, we demonstrate experimentally the fundamental vector soliton and various vector multi-soliton patterns (i.e., random static distribution of vector multiple solitons, vector soliton cluster, vector soliton flow and the state of vector multiple solitons occupying the whole cavity) in a passively mode-locked fiber figure-eight laser based on a NALM. In our experiment, via tuning the pump power and properly adjusting the intra-cavity polarization controllers (PCs), diverse types of vector multiple solitons were observed. Both the PLVSs and the PRVSs were observed for single soliton and each type of multi-soliton patterns by using polarization resolved measurement. The obtained results further reveal the fundamental physics of multi-soliton dynamics in the passively mode-locked fiber lasers.
2. Experimental setup
The experimental setup is shown in Fig. 1. It is a figure-eight fiber laser based on a NALM that is coupled to a unidirectional ring (UR) cavity by a 50/50 fiber coupler. The NALM contains 1.8-m-long erbium-doped fiber (EDF) with a dispersion parameter of D = −15 ps/nm/km, 30-m-long SMF, a PC, and a wavelength division multiplexer (WDM). The net cavity dispersion is about −1.125 ps2. The UR cavity consists of a PC, a polarization insensitive isolator (PI-ISO), and a 10/90 fused optical coupler (OC). The total length of the laser is ~58 m, corresponding to the fundamental repetition rate of ~3.64MHz. The EDF is pumped by a 980-nm laser diode with the highest power of 350 mW through the WDM. The output pulse train is monitored with a photodetector (EOT ET-3000A-FC, 2GHz) and visualized with an oscilloscope (LeCroy WaveRunner 620Zi, 2 GHz). The spectral properties are analyzed with an optical spectrum analyzer (OSA, Anritsu MS9710C). In order to observe the vector nature of multi-soliton patterns, a PC and a fiber-based polarization beam splitter (PBS) are used to be externally connected with the OC.
3. Experimental results
3.1 Fundamental vector soliton
Experimentally, when the pump power was increased to the mode-locked threshold of 45 mW, the mode-locking of the laser was obtained via carefully adjusting the intra-cavity PCs. Initially, the multiple mode-locked pulses could be easily formed in the fiber laser. Moreover, the number of the mode-locked pulses was able to be controlled by carefully tuning the pump power. In particular, the single soliton operation could be achieved at the 11 mW pump power by virtue of the pump hysteresis phenomenon [42]. As mentioned above, no polarizer was utilized in the cavity, and all fibers used had weak birefringence, vector soliton generation was therefore naturally obtained in the figure-eight fiber laser. By using polarization resolved measurement, both the fundamental PLVS and PRVS are able to be identified.
Figure 2 shows the typical fundamental PLVS operation in the fiber laser. Figure 2(a) shows the temporal waveforms measured without and with passing through a PBS, respectively. As can be seen here, the fundamental repetition rate is 3.64 MHz and the two soliton polarization components have uniform pulse trains. Figure 2(b) presents the corresponding spectra, in which the central wavelength is 1566.25 nm and the 3-dB bandwidth of the spectrum is about 2.1 nm. The spectra exhibit Kelly sidebands, confirming that the mode-locked pulse is solitary wave of a fiber laser with anomalous dispersion regime [43]. Then the paddles of the PC before the PBS input port were tuning arbitrarily. All the solitons passing after the PBS simultaneously have the same pulse height and the same central wavelength, which is the typical characteristic of PLVS [44]. The autocorrelation trace of total pulse is shown in Fig. 2(c). If the sech2 pulse profile is assumed, the duration of the autocorrelation trace is ~1.4 ps. Thus, the time-bandwidth product (TBP) is ~0.359.
Fig. 2 Vector nature of fundamental PLVS. (a) Oscilloscope traces; (b) Polarization-resolved spectra; (c) Autocorrelation trace of total pulse.
By further slightly rotating the orientation of intra-cavity PCs, the fundamental PRVS was achieved, as shown in Fig. 3. The oscilloscope trace of the single PRVS operation in the cavity is shown in Fig. 3(a). Being different from the state of fundamental PLVS, the pulse intensities of the two polarization components varied periodically with the twice cavity round-trip times, meaning that the polarization of the soliton is rotating [23]. Figure 3(b) shows the corresponding optical spectra of the fundamental PRVS. The central wavelength is 1565.1 nm and the 3-dB bandwidth of the mode-locked spectrum is about 2.2 nm. Comparing with the optical spectra of the fundamental PLVS, there is an extra set of spectral sidebands on the polarization resolved spectra. A more detailed inspection is presented in the inset of Fig. 3(b). Before an external PBS, the set of optical sidebands is similar to the ordinary Kelly sidebands. When measured after the PBS, the intensities of the two optical sidebands alternated between two values in the two polarization components. Distinctly, it is indicated that this kind of spectral sidebands is a special type of Kelly sidebands, which are formed due to the constructive interference between the dispersive waves. In addition, it is also believed that the locations of the additionally spectral sidebands of PRVS are related to the rotation period of PRVS in the laser cavity [25,44]. Figure 3(c) displays the autocorrelation trace of total pulse. If the fit of hyperbolic secant pulse shape is assumed, the pulse duration is 1.3 ps. Hence, the TBP is ~0.35.
Fig. 3 Vector nature of fundamental PRVS. (a) Oscilloscope traces; (b) Polarization-resolved spectra; (c) Autocorrelation trace of total pulse.
3.2 Random static distribution of vector multiple solitons
For gradually increasing the pump power and adjusting the intra-cavity PCs, various multi-soliton patterns could appear in the cavity. One of the most common states is that several stable vector solitons randomly scattered over the cavity under a pump power of ~49 mW. Similarly, both the PLVSs and PRVSs were also observed for the state of random static distribution of multiple solitons.
Figure 4 exhibits the state of random static distribution of multiple PLVSs. Here, all the PLVSs in the fiber laser have uniform polarization and their characteristics would keep consistent when they propagated in the cavity. Thus, as can be seen in Fig. 4(a), all the solitons have the same pulse height after every round trip when passing through the PBS. Figure 4(b) shows the corresponding optical spectra. Here, in addition to the Kelly sidebands, there is another set of optical sidebands (indicated by black arrows) in the polarization resolved spectra. These special sidebands present peak-dip alteration between the two polarization axes, whose appearance was resulted from the four-wave-mixing coupling (also called coherent energy exchange) between the two polarization components of the PLVSs [45]. It should be noted that the locations of peak-dip spectral sidebands are dependent on the cavity birefringence. Therefore, the location of the additionally spectral sidebands of PLVS could be dynamically changed by properly adjusting the PCs, while the Kelly sidebands still stayed on the original positions.
Fig. 4 The state of random static distribution of multiple PLVSs. (a) Oscilloscope traces; (b) Polarization-resolved spectra.
By rotating the paddles of intra-cavity PCs, we could further obtain the multiple PRVSs of the state of random static distribution, as shown in Fig. 5. Figure 5(a) exhibits that 6 PRVSs emerge randomly and dispersedly in the cavity. As presented in Fig. 5(a), without passing through the PBS, the intensities of all the pulses in the oscilloscope trace are equal. However, after passing through the PBS, the intensities of pulses subsequently altered between two values, suggesting that the soliton polarization is rotating in the cavity. It is to note that the soliton polarization rotation was locked to twice the cavity round-trip time, suggesting that both the intensity and the polarization state of the vector soliton returned to its original value every two cavity round trip time. It should be also noted that the soliton polarization rotation could be locked to different times of the cavity round trip by adjusting the linear cavity birefringence [46]. Therefore, by further carefully adjusting the cavity parameters, different periods of soliton polarization rotation might be obtained. Figure 5(b) shows the corresponding spectra of the PRVSs. Here, there are also two extra sets of spectral sidebands (indicated by black arrows) caused by the periodic polarization rotation of the vector solitons.
Fig. 5 The state of random static distribution of multiple PRVSs. (a) Oscilloscope traces; (b) Polarization-resolved spectra.
3.3 Vector soliton cluster
By further increasing the pumping power to ~155mW, the number of solitons increased and the solitons tended to be dynamic. The solitons relatively moved and finally came together. Then, by changing the PC settings slightly, the solitons would form a soliton cluster in which the solitons bunched together tightly, as shown in Fig. 6 and Fig. 7. When the soliton cluster was achieved, the soliton bunched unit operated at the fundamental repetition rate and they spanned over 100 nanoseconds. Due to the limitation of measurement instruments, all the pulses in soliton bunch could not be resolved accurately. However, different types of vector soliton cluster still could be recognized based on the measured oscilloscope traces and the corresponding spectra.
Fig. 6 Vector nature of the PLVSC. (a) Oscilloscope traces; (b) Corresponding oscilloscope traces with small span in detail; (c) Autocorrelation trace of total pulse; (d) Polarization-resolved spectra.
Fig. 7 Vector nature of the PRVSC. (a) Oscilloscope traces; (b) Corresponding oscilloscope traces with small span in detail; (c) Autocorrelation trace of total pulse; (d) Polarization-resolved spectra.
Figure 6 illustrates, for example, the polarization-locked vector soliton cluster (PLVSC). Figures 6(a) and 6(b) show the pulse-trains of PLVSC. It can be seen from Fig. 6(a) that all the widths of the soliton clusters are uniform both in the total output and the polarization-resolved outputs. Figure 6(b) illustrates oscilloscope traces of single soliton packet with small span in detail. As can be seen here, the pulse-trains are uniform among the total pulses and the two orthogonal polarization components. Figure 6(c) shows the autocorrelation trace of pulse before the PBS. The pulse duration is about 1.52 ps. Regarding to the spectra of PLVSC, there also coexist the Kelly sidebands and the peak-dip optical sidebands on the spectra, as shown in Fig. 6(d). It is worth noting that there is no modulation on the mode-locked spectrum, indicating that the PLVSC is tightly composed of numerous solitons with random separations and phases. Therefore, it is suggested that the PLVSC is different from the condensed soliton phase which consists of tens or hundreds of phase-locked solitons in a soliton packet [47,48].
By further adjusting the PCs, the polarization rotation vector soliton cluster (PRVSC) was achieved. Figure 7(a) shows the pulse-trains of PRVSC. Figure 7(b) further shows the intensities of solitons vary between the two orthogonal polarization components in detail. Moreover, we also measured the autocorrelation trace of the PRVSC, as presented in Fig. 7(c). The pulse duration is about 1.63 ps. Figure 7(d) shows the corresponding spectra of PRVSC. As shown in Fig. 7(d), more sets of the spectral sidebands (indicated by black arrows) due to periodic polarization rotation of vector solitons appear on the spectra. Based on the characteristics of the pulse-trains, autocorrelation trace and spectra of the PRVSC, it can be concluded that both the PRVSC and the PLVSC are distinct from the condensed soliton phase mentioned above.
3.4 Vector soliton flow
After the achievement of vector soliton cluster, the pump power was further increased to ~204 mW while the other cavity parameters were fixed. In this case, another interesting dynamic multi-soliton pattern could be obtained; namely, the vector soliton flow was achieved. For better clarity, we have also presented the dynamics of vector soliton flow observed in our fiber laser in Media 1 and Fig. 8. As presented in Media 1, the flowing solitons arise from one soliton cluster, and drift at different speeds to another soliton cluster. In fact, the soliton flow is a reverse phenomenon of the soliton rain, in which the pulses generated from the noisy and falling to the condensed phase or soliton cluster [13,14]. Experimentally, by further increasing the pump strength within certain range, it was easier to acquire more solitons arising from the soliton cluster and the higher speeds of these motion pulses could be observed. Again, the polarization-locked vector soliton flow (PLVSF) and the polarization-rotating vector soliton flow (PRVSF) will be investigated in the following.
Figure 9 presents the case of the PLVSF. According to Fig. 9(a), all the pulses in the soliton cluster and the corresponding flow are uniform among the total output and the two polarization resolved outputs. It was demonstrated that the polarization locked features of the fiber laser operating in the soliton flow region are the same as those of conventional vector solitons. Figure 9(b) shows the spectral components of the PLVSF. The extra spectral sidebands induced by the four-wave-mixing effect were still observed, which were similar to those of PLVSs obtained in our experiment. Then, the PRVSF was observed by tuning the PCs properly, as shown in Fig. 10. As can be seen in Fig. 10(a), the intensities of all the flowing solitons at the two polarization axes varied in twice cycle of fiber laser cavity. However, the polarization rotating characteristics of the soliton flow for the part of cluster could not be clearly distinguished in Fig. 10(a). Figure 10(b) shows the spectra of PRVSF as well as the polarization resolved components. Obviously, the extra sidebands induced by the periodic polarization rotation of vector solitons were also observed.
3.5 The state of vector multiple solitons occupying the whole cavity
When the pump power was adjusted to above 260 mW, vector soliton flow evolved into plentiful vector solitons, which occupied irregularly all the available space along the laser cavity. Depending on the intra-cavity birefringence, the multiple PLVSs and multiple PRVSs occupying the whole cavity could be also observed, as illustrated respectively in Figs. 11 and 12. Figure 11(a) shows the pulse-trains of the state of multiple PLVSs occupying the whole cavity. From Fig. 11(a), it is shown that the pulse-trains of the two soliton polarization components have no polarization rotating characteristics. Here it should be noted that the number of solitons in the cavity could be not counted accurately because of the limitation of the measurement instruments. Nevertheless, there could be hundreds of PLVSs in the whole laser cavity. Figure 11(b) further presents the corresponding optical spectra of the multiple PLVSs, which are similar to the optical spectra of PLVSF. Figure 12(a) shows the oscilloscope traces of the state of multiple PRVSs occupying the whole cavity, in which can be seen that the polarization state of solitons is rotating. In addition, there could be also hundreds of PRVSs in one round-trip time of the fiber laser. The corresponding spectra of the multiple PRVSs are shown in Fig. 12(b), which are also similar to the optical spectra of PRVSF.
Fig. 11 The state of multiple PLVSs occupying the whole cavity. (a) Oscilloscope traces; (b) Polarization-resolved spectra.
Fig. 12 The state of multiple PRVSs occupying the whole cavity. (a) Oscilloscope traces; (b) Polarization-resolved spectra.
4. Discussions
In our experiment, it was found that various regimes of vector multiple solitons were sensitive to the pump power and the orientations of the intra-cavity PCs. It was easily understood that the multiple pulses could be observed constantly due to soliton energy quantization effect [2–4] when the pump power or the accumulated nonlinear effect was increased to a high value in the fiber laser. Once the multiple solitons were generated, they would evolve into various regimes based on the interactions among the solitons, continuous waves and dispersive waves in the cavity. Therefore, through tuning the pump power and adjusting the PCs, different vector multi-soliton dynamic patterns would be formed and converted from one to another in our fiber laser. In addition, it was shown that there were peak-dip optical sidebands on the spectra of every type of multiple PLVSs, indicating that the coherent energy exchange always existed between the two orthogonal polarization components of multiple PLVSs [45]. However, the coherent energy exchange of vector fundamental soliton was not so evident, which was probably due to the low vector spectral intensities. As for the PRVSs, there were always extra sidebands on the spectra of the fundamental PRVS and every type of multiple PRVSs, whose formation was caused by the periodic polarization rotation of the vector solitons [25,44]. In addition, it should be noted that the PLVS and the PRVS were observed only for the fundamental soliton and the four types of vector multiple solitons in this experiment. However, by further selecting the cavity parameters, it is believed that other kinds of vector solitons such as PLV or PRV dissipative soliton resonance (DSR) [49–52], soliton molecules [53,54] and soliton crystal [47,48] may be obtained, which will be investigated in our future work.
5. Conclusion
In summary, we have experimentally investigated the vector nature of single soliton and four multi-soliton patterns in the passively mode-locked figure-eight fiber laser based on a NALM. In the experiment, the fundamental vector soliton and various types of vector multi-soliton operations could be alternately observed via carefully adjusting the intra-cavity PCs and tuning the pump power. Both the PLVSs and the PRVSs could be further identified for fundamental soliton and every type of multi-soliton operation by using the polarization resolved measurement. The obtained results further reveal the fundamental physics of multi-soliton patterns and suggest that the figure-eight fiber lasers mode-locked by NALM (NOLM) indeed could be an excellent platform for studying the vector characteristics of different soliton types.
Acknowledgments
This work was supported in part from the National Natural Science Foundation of China (Grant Nos. 61378036, 61307058, 11304101, 11074078), the PhD Start-up Fund of Natural Science Foundation of Guangdong Province, China (Grant No. S2013040016320), and the Scientific Research Foundation of Graduate School of South China Normal University, China (Grant No. 2013kyjj011). Z.-C. Luo acknowledges the financial support from Zhujiang New-star Plan of Science & Technology in Guangzhou City (Grant No. 2014J2200008).
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