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Abstract
Bidirectional ultrafast fiber lasers capable of generating counter-propagating (CP) coherent solitons are promising to be served as a dual-comb light source for the applications in spectroscopy and gyroscope. In the absence of efficient numerical model, the understanding of the operation of bidirectional fiber lasers is very limited. In this paper, we experimentally explore the pulsating dynamics of CP solitons in a bidirectional mode locked fiber laser and present a set of rich complex dynamics of CP solitons, revealing for the first time, to the best of our knowledge, the periodic breathing and acyclic pulsating dynamics of CP solitons. With a bi-directional pumping configuration, the impacts of gain distribution along the fiber on the dynamics of CP solitons have been investigated and discussed. These results provide further evidence of the universality of breathing dynamics of solitons. More importantly, the abundant dynamical behavior of CP solitons demonstrated in this paper, collaborating with a handful of previous reports on the buildup dynamics of CP solitons in bidirectional fiber lasers, underline further the independent evolving dynamics of CP solitons. These findings contribute to the understanding of how bidirectional lasers work and, consequently, will accelerate the development of bidirectional lasers in the applications such as gyroscope.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Bidirectional ultrafast lasers, usually referred to laser resonators without isolators inside that allow the light dual-directional propagating in the cavity, generate counter-propagating (CP) coherent solitons in the laser cavity. By virtue of the capability to generate dual optical comb simultaneously as well as its compact structures, bidirectional ultrafast lasers have been paid much attention in recent years, owing to the potential of dual-combs in a diversity of applications such as precision metrology [1] and dual-comb spectroscopy [2–4]. Basically, the formation of solitons in ultrafast lasers is a result of the balances among dispersion and nonlinearity as well as dissipation effects [5]. Contrary to the one-way propagating of solitons in traditional unidirectional lasers, CP solitons propagate in an opposite direction in the cavity and hence collide with each other once every roundtrip. In contrast to conservative system in which soliton can reserve its energy and recover their shape after the collision [6], in dissipative system, the interaction among solitons can lead to rich and more complex dynamics, such as the dispersive wave radiation [7], soliton pulsating [8], the formations of soliton molecule [9,10] and soliton molecular complexes [11], as well as the generation of incoherent pulse structure [12] and rogue wave [13–15]. Considering the interactions between CP solitons in bidirectional ultrafast lasers pose difficulties in building effective numerical model to simulate the soliton behaviors in such lasers, so as to obstruct the performances that can be achieved in bidirectional lasers compared with unidirectional lasers.
The needs of ultrafast lasers with superior performances stimulate people to understand the working principle of lasers and search for novel operating regimes, reciprocally, the discovery of new regimes in ultrafast lasers does lead to the boost of the laser performances [16,17]. For instance, the pulse energy from ultrafast lasers operating in similariton [18] or dissipative soliton regimes [19,20] is typically two or three orders higher than traditional solitons. Understanding the operation of bidirectional ultrafast laser will improve its performance and push the application of this type of lasers forward. However, in the absence of efficient numerical model, the understanding of the operation of bidirectional fiber lasers is very limited. In bidirectional lasers, CW and CCW solitons propagate in an opposite direction and collide with each other once every roundtrip in the cavity. Understanding the nature of solitons collision is nontrivial but not easy, as the resultant dynamics of soliton collisions are strongly dependent on the initial pulse features, such as the relative intensity, polarization as well as their pulse durations and group velocity [21,22], which poses the difficulty in building efficient model for simulation of CP solitons.
The recent developed real-time measurement technology, termed as dispersion Fourier transformation (DFT) technology [23,24], allows to single shot capture the spectrum in the time domain at a repetition rate of dozens of megahertz and to visualize the evolution on the oscilloscope in real time. This measurement has been successfully used to study the ultrafast transient dynamics in nonlinear optics field [25,26]. With the implementation of DFT, the buildup dynamics of CP solitons reported in Ref. [27] show that the CP solitons in a normal dispersion bidirectional laser exhibit similar evolutional dynamic, as they experience the same gain and loss modulation when propagating. Paradoxically, a very recent study in Ref. [22] clearly demonstrate the different scenario of the buildup dynamics of CP solitons in an anomalous dispersion cavity, revealing the independent evolution of CP solitons in bidirectional lasers, suggesting that the soliton collision might play an indispensable role in the formation and stable operation of CP solitons. Do the CP solitons evolve truly independently? How much does one soliton affect its counter propagating counterpart during propagation? These questions are still tricky problems.
To find out, we explored the dynamics of CP solitons in a bidirectional Er-doped ultrafast fiber laser and found that the CP solitons exhibit rich and colorful dynamical behaviors. With implementing the DFT measurement, we showed in this paper complex pulsating dynamics of CP solitons including periodic breathing and acyclic pulsating dynamics. The impact of the pumping configuration on the solitons dynamics has also been discussed. Our results shed new lights on the operation of bidirectional ultrafast lasers, which will not only be helpful for building higher performance bidirectional lasers, but will also motivate people to study the nonlinear dynamics in bidirectional ultrafast lasers.
2. Experimental setup
The schematic of the system we use to investigate the CP soliton dynamics is shown in Fig. 1. It is an Er-doped fiber (EDF) ring laser with bidirectional pumping configuration. No optical isolator exits in the cavity, consequently, the pulse can bidirectionally propagate inside the cavity once it is generated. A piece of single-wall carbon nanotube (SWNT) inserted in the cavity acts as a saturable absorber [28], which helps the soliton formation and stabilizes the pulse operation. A polarization controller placed in the cavity is to optimize the light polarization states before/after entering the SWNT, which is beneficial for initiating the mode locking operation. A 2 × 2 optical coupler extracts 50% of the intracavity power out for the measurements. The cavity consists of 0.7 m EDF with a group velocity dispersion (GVD) of 0.0769 ps2/m at 1550 nm and 4.9 m single mode fiber (SMF) with GVD of -0.0228 ps2/m at 1550 nm, resulting in a total cavity length of 5.6 m, corresponding to a pulse repetition rate of 35.8 MHz. The net cavity dispersion is -0.058 ps2, therefore, the laser is an anomalous dispersion cavity but it is dispersion managed. Consequently, no Kelly sidebands are observed in the later experiments.
Fig. 1. Schematic of the experimental setup for observing the CP soliton dynamics in a bidirectional Er-doped fiber laser mode locked by a SWNT-SA. WDM: wavelength division multiplexer; EDF: Er-doped fiber; SWNT: single-walled carbon nanotubes; PC: polarization controller; OC: optical coupler; ISO: isolator; CW: clockwise; CCW: counter-clockwise; DCF: dispersion compensation fiber; PD: photodiode.
The real-time measurement is implemented via sending the output clockwise (CW) and counterclockwise (CCW) pulses into 8.8 km dispersion compensation fiber (DCF). The low intensity pulses are linear stretched in DCF due to the different GVD at different frequencies, and hence the input spectrum will be mapped into the temporal intensity profile in the time domain provided the far-field diffraction condition is satisfied. A 15 GHz photodiode is used to convert the light signal to electrical signal and an 8 GHz 50 GSa/s real-time digital oscilloscope is applied to visualize the pulse and spectrum evolution in real-time. Considering the DCF has a GVD of 0.056 ps2/m, the electronical-determined spectral resolution of the DFT measurement is hence 0.1 nm. The validity and correctness of the DFT measurement is verified by comprising the average of the DFT spectra over roundtrips with the averaged spectrum measured with traditional spectrum analyzer (Yokogawa spectrum analyzer with optical resolution of 0.02 nm).
3. Experimental results
3.1 Periodic breathing CP solitons
A breather is a localized periodic solution that can be achieved by solving the nonlinear Schrödinger equations (NLSEs) [29,30]. In nonlinear optics, breathers typically refer to a solitonic wave whose amplitude and/or spectral/temporal width vary in time [31]. Given that the light propagating in optical fibers is governed by NLSEs that possess breather solutions, it is not surprising to find breathers in bidirectional ultrafast fiber lasers.
Figure 2 shows the experimental results of the periodic breathing dynamics of CP solitons obtained in our bidirectional lasers. The total pump power is 150 mW (Forward pump power: 75 mW, backward pump power: 75 mW.). From the real time recording of the single shot spectra evolution of CW and CCW solitons over 6000 roundtrips (See in Fig. 2(a)), we clearly observe the periodic oscillation in the soliton spectral width. The weak interference-like pattern on the DFT background is the noise background. Figures 2(d) and (e) are the single-shot spectrum and the averaged DFT spectrum over 6000 consecutive roundtrips of CW and CCW pulses, reflecting that the ratio between the noise background and the maximum value of the spectral intensity is less than 10%. Therefore, the pulsating dynamics of CP solitons are scarcely influenced by the noise background. The normalized pulse energy is calculated by integrating the DFT spectrum intensity over frequency. Figure 2(b) shows the pulse energy evolution over 2000 roundtrips, evidencing that the pulse energy of CP solitons does oscillate periodically in time with a period of ∼490 roundtrips. Figure 2(c) presents the evolution of the normalized 3dB spectral width. The thorns existing on the curves come from the low signal-to-noise ratio of the DFT data due to the low pulse power before the DFT measurement, but the spectral width for both CW and CCW solitons is periodically oscillating. Two main features of the dynamics are reflected in this scenario: First, it is the first time to observe the CW/CCW breathers simultaneously in bidirectional ultrafast lasers. Second, the breathing dynamics of CP solitons are synchronized in time, with the same period. But we keep in mind that achieving pulsating dynamics of CP solitons with different pulsating periods will be helpful to get insight into the dynamics of CP solitons, as the different oscillating period for CW/CCW solitons might reflect much more information about mutual interactions between CP solitons.
Fig. 2. Periodical breathing dynamics of CP solitons. In this case, the forward pump power is 75 mW, and the backward pump power is 75 mW. (a): DFT recorded single-shot spectra evolution over 6000 roundtrips. (b): the evolution of the pulse energy and (c) the spectral width over 2000 roundtrips. Ep represents the pulse energy that calculated through integrating the DFT spectrum over a certain range. δλ represent the 3-dB spectral width. (d) and (e): the single-shot spectrum and the averaged DFT spectrum over 6000 consecutive roundtrips of CW and CCW pulses.
3.2 Acyclic pulsating dynamics of CP solitons
Turning off the backward pump source enables us to observe an intriguing pulsating dynamics of CP solitons. In this case, the forward pump power is 75 mW, while the backward pump power sets to 0 mW. The DFT recorded spectrum over 5000 roundtrips shown in Fig. 3(a) exhibits pulsating dynamics for both CP solitons without following discernible cycle. We call the dynamics in this scenario as acyclic pulsating dynamics. The noticeable feature of the dynamics is: the oscillating dynamics of the CW solitons is unsynchronized with CCW solitons in time. The evolution of the pulse energy and spectral width of CP solitons shown in Figs. 3(b) and (c) provides a clear evidence of the unsynchronized pulsating dynamics. Compared to the periodic breathing dynamics of CP solitons shown in Fig. 2, the acyclic unsynchronized pulsating dynamics of CP solitons reflect the truly independent evolving dynamics of CW/CCW solitons. Figures 3(d) and (e) show the single-shot spectrum of CW and CCW solitons with the smallest width (navy line) and with the largest width (blue line) during the evolution, clearly demonstrating the acyclic pulsating dynamics of CP solitons without explosion. Considering the different pumping configurations for periodic breathing dynamics and acyclic pulsating dynamics, we speculate that the gain distribution along the fibers affects the CP soliton features, indicating that the performance of CP solitons can be separately controlled and optimized through optimizing the pump configuration and pump power. Based on the dynamics shown in Fig. 3, slightly rotating the polarization controllers allow us to observe a periodic breathing dynamic for CW solitons, while the CCW solitons is in a stable mode locking state (Shown in Fig. 4), further confirming the independent dynamics of CP solitons.
Fig. 3. Acyclic pulsating dynamics of CP solitons. The forward pump power is 75 mW, while the backward pump power sets to 0 mW in this case. (a): DFT recorded single-shot spectra evolution over 6000 roundtrips. (b): the evolution of the pulse energy and (c) the spectral width over 6000 roundtrips. Ep represents the pulse energy that calculated through integrating the DFT spectrum over a certain range. δλ represent the 3-dB spectral width. (d) and (e): the single-shot spectrum at RT 522 (RT 1390) and RT 1446 (RT 2705) as well as the averaged DFT spectrum over 5000 consecutive roundtrips of CW and CCW pulses.
Fig. 4. Dynamics of breathing CW solitons and stable CCW solitons. (a): DFT recorded single-shot spectra evolution over 5000 roundtrips. (b) and (c) single-shot spectrum at specific roundtrips to show the breathing dynamics.
4. Discussion
In order to get insight into the independent dynamics of CP solitons, study on the impact of the gain distribution along the fibers on CP soliton dynamics has been conducted. Figure 5(a) represents the DFT spectral evolution of CP solitons under a total pump power of 125 mW where the forward pump power is 75 mW and the backward pump power is 50 mW. The CCW soliton is breathing with a period of 490 roundtrips, whereas the CW soliton exhibits hybrid dynamics which include the soliton collapsing mixed with pulsating dynamics [32]. Remarkably, reversing the pumping configuration can switch the dynamics between CP solitons. Under a forward pump power of 50 mW and the backward pump power of 75 mW, the CW soliton switches to breathing dynamics, while the CCW soliton changes from the breathing dynamics into periodic soliton collapsing dynamics (See in Fig. 5(b)). The zoom-in 3D plots of the spectrum evolution shown in Figs. 5(c) and (d) confirm the distinct soliton dynamics of the CP solitons under the unsymmetrical pumping conditions [33], confirming that the gain distribution along the fiber affects the CP soliton dynamics.
Fig. 5. Separate and distinct pulsating dynamics for CP solitons achieved under different pumping conditions. (a) spectral evolution of CP solitons under a forward pump power of 75 mW and the backward pump power of 50 mW; (b) spectral evolution of CP solitons under a forward pump power of 50 mW and the backward pump power of 75 mW; (c) and (d): zoom-in spectral evolution over 1200 roundtrips to show the different dynamics for CP solitons.
We remind that the pulsating dynamics of solitons in different platforms such as ultrafast fiber lasers [8,34,35], continuous wave driven fiber resonators [36] as well as microresonators [37,38] have been extensively studied and reported. Reporting on the discovery of pulsating solitons in bidirectional ultrafast lasers hence is not the point, while the innovation of this article is reflected in the following aspects:
First and foremost, complex pulsating dynamics of CP solitons, especially the coexisting of distinct dynamics of CP solitons, confirm the independent dynamical evolution of CP solitons. Under different pumping configurations (symmetric or unsymmetrical pumping condition) and/or in different polarization states, CP solitons exhibit completely different evolving dynamics, indicating that the pumping configuration and pump power as well as polarization in bi-pumped bidirectional mode locked lasers can be used as a degree of freedom to control and optimize the performance of CP solitons, making such a laser structure an attractive dual-comb coherent pulse source for a variety of applications.
Second, the results suggest that a numerical model including the dispersion and nonlinearity effects as well as the gain evolution along the gain fiber could be used to simulate most of the major features of CP solitons in bidirectional lasers, although the soliton interaction between CP solitons during the counter propagation in the cavity has been proved as a factor for the formation and the stable operation of CP solitons [21]. A simplified model hence will make up the current lack in an efficient numerical model to simulate and guide the design of bidirectional mode locked lasers with high performances.
Finally, the periodic breathing and acyclic pulsating dynamics of CP solitons sustain the universality of breathers. As a unique solution of NLSEs, the breathers are linked to rogue waves that are usually defined as the suddenly appearance of unexpected wave with extremely high amplitude on surface wave. Previous literatures have shown that the rogue waves can be induced by soliton interactions [31]. Considering the soliton collision might play a role in the formation of CP solitons, the bidirectional ultrafast lasers hence could provide a good platform to study the dynamics of rogue waves caused by soliton collisions.
5. Summary
In summary, we report, for the first time, the breathing dynamics and acyclic pulsating dynamics of CP solitons in a dual-pumped bidirectional ultrafast fiber laser. Different soliton dynamics for CP solitons under symmetric and asymmetric pumping conditions have been studied. These results validate the independent evolving dynamics of CP solitons and the universality of breathers in nonlinear optics. Also, the findings of distinct pulsating dynamics of CP solitons existing simultaneously in the bidirectional lasers demonstrate the potential of improving the performances of bidirectional fiber lasers through controlling the pumping conditions. These results deepen our understanding of the operation of bidirectional ultrafast lasers, and open a way to build high performance bidirectional soliton lasers for various applications in metrology and dual-comb spectroscopy.
Funding
National Natural Science Foundation of China (91950105); Nanjing University of Posts and Telecommunications (NY218114, NY219050); Jiangsu Shuangchuang Outstanding Doctor Talents Support Program (CZ1060619002); Natural Science Foundation of Jiangsu Province (BK20180742).
Disclosures
The authors declare no conflicts of interests.
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