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Abstract
A novel optical soliton dynamics phenomenon, called “invisible” pulsation, has gradually attracted extensive interest in recent years, which can only be identified effectively with the help of real-time spectroscopy technique, i.e., dispersive Fourier transformation (DFT). In this paper, based on a new bidirectional passively mode-locked fiber laser (MLFL), the “invisible” pulsation dynamics of soliton molecules (SMs) is systematically studied. It is indicated that the spectral center intensity, pulse peak power and relative phase of SMs are periodically changed during the “invisible” pulsation, while the temporal separation inside the SMs is constant. The degree of spectral distortion is positively correlated with the pulse peak power, which verifies that self-phase modulation (SPM) is the inducement of spectral distortion. Finally, the universality of the SMs “invisible” pulsation is further experimentally verified. We believe our work is not only conducive to the development of compact and reliable bidirectional ultrafast light sources, but also of great significance to enrich the study of nonlinear dynamics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The concept of soliton, which stems from fluid mechanics [1], is firstly introduced to the field of optical fibers by H. Akira et al. [2] in 1973. Optical solitons have broad application prospect in optical communication due to their inherent stability [3,4]. Meanwhile, solitons are ubiquitous localized structures in dissipative systems, which is controlled by the composite balance between dispersion, nonlinearity, gain, and loss [5]. Worth to note that, the passively MLFL, with complex nonlinear effects, is an ideal platform for exploring various soliton dynamics. In recent years, the rise of DFT, a powerful real-time spectroscopy technique, has enabled scientists to reveal a series of complex and interesting soliton transient dynamics in dissipative systems, such as soliton explosion [6,7], soliton buildup dynamics [8,9], soliton rain [10] and soliton pulsations [11,12]. In addition, the double-Hopf-type breathers were observed in real time in an all-normal dispersion all-polarization maintaining ytterbium-doped fiber laser by DFT technology [13]. And the self-optimization of the breather regime in a MLFL is realized through a four-parameter nonlinear polarization evolution with the help of DFT technology [14]. In a word, the DFT technology has played a huge role in the study of soliton nonlinear dynamics, which is not only helpful to understand the physical mechanism of dissipative systems but also has important significance for developing compact, efficient and reliable laser sources.
Recently, a novel type of nonlinear dynamical phenomenon, namely “invisible” soliton pulsation, which occurs without obvious energy fluctuation, has attracted increasing attention. Unfortunately, because of the “invisible” pulse energy fluctuation, this phenomenon cannot be directly monitored by traditional measurement instruments such as oscilloscope and radio frequency (RF) spectrum analyzer. Benefiting from the shot-to-shot pulse spectrum acquired by DFT technique, it is found that when this phenomenon occurs, there are periodic and drastic changes in the stretched spectral profile. J. Peng et al. [15] and M. Liu et al. [16] reported the triple-state dissipative soliton induced by self-parametric amplification (SPA) and the double periodic “invisible” soliton pulsation, respectively. Subsequently, similar phenomena are successively reported under different conditions [17–21]. Although the names of the phenomena in the above reports are different, their essence is that when the energy is constant, the pulse spectrum changes drastically and periodically. Since the oscilloscope cannot directly identify the occurrence of the above phenomenon due to the constant pulse energy, this phenomenon is referred to as “invisible” soliton pulsation as defined in Ref. [16]. Considering that the occurrence of “invisible” pulsation will have a serious impact on the stability of the MLFL, it is of great significance to study its evolution in-depth to optimize ultrafast laser performance and reveal the underlying physical mechanism of the complex soliton dynamics. Note that, up to now the experimental study of the “invisible” pulsation phenomenon is limited to single-soliton state. In fact, when the pump energy in ultrafast fiber laser is increased, multi-soliton patterns will be formed, and the interactions between the multiple solitons enable them to self-organize into bound states, namely SMs [5,22,23]. In addition to displaying many fascinating properties in the fundamental soliton physics [24–26], SMs are also attractive for applications in optical communication [27] and all-optical storage [28].
On the other hand, as an emerging ultrafast laser source, bidirectional ultrafast fiber lasers without direction-dependent devices are capable of simultaneously generating counter-propagating ultrashort pulses, which have great application potential in dual-comb spectroscopy [29] and gyroscopic effect detection [30]. Hitherto, due to its unique physical characteristics different from unidirectional MLFLs, nonlinear dynamics in bidirectional ultrafast fiber lasers have been extensively studied and reported, which will not only promote the optimization of dual-comb light sources and ultrafast laser ring gyroscopes, but also deepen and enrich the study of nonlinear dynamics process [31–35]. It is a remarkable fact that, because of the gain competition that can easily result in quasi-unidirectional operation, traditional bidirectional ultrafast fiber lasers might only output counter-propagating ultrashort pulses with low-energy, which is disadvantageous for practical applications [36,37].
In this paper, a novel bidirectional passive MLFL without gain competition is designed, where the internal dynamics of SMs during “invisible” pulsation has been studied in detail for the first time. Firstly, stable SM pulses are simultaneously obtained in the two operation directions of the built laser, verifying the effectiveness of the laser. Then, “invisible” pulsation phenomena, respectively with pulsating periods of 7 roundtrips (RTs) and 2 RTs, of SMs are realized simultaneously in clockwise (CW) and counterclockwise (CCW) propagating directions. Moreover, not only the variation of the spectrum and pulse duration are studied, but also the evolutions of the temporal separation and relative phase inside the SMs are studied. Finally, in order to confirm the universality, the “invisible” pulsating SMs with a pulsating period of 5 RTs are obtained simultaneously in both operation directions by adjusting the polarization controllers (PCs) in cavities. This work enriches the research on “invisible” pulsation and is of great significance for promoting the development of ultrafast fiber lasers.
2. Experimental setup
So as to avoid the adverse effect of the gain competition on the bidirectional mode-locking operation, a multiplexed fiber ring laser consisting of a bidirectional main cavity and two counter-propagating branches inserted with heavily Er3+-doped fibers (EDFs) (LIEKKI, Er110-4/125) is constructed, as shown in Fig. 1. Here, the EDFs in CW and CCW propagating directions are backward-pumped by two laser diodes (LDs) respectively via a 980/1550 nm wavelength division multiplexer (WDM). Two 3-port circulators (CIRs) are used to connect the two branches to the main cavity, and ensure the unidirectional operation of each branch. In this work, the mode-locking operation of bidirectional laser is realized by the nonlinear polarization rotation (NPR) technique consisting of a fiber-based polarizer and three PCs. In the constructed bidirectional ultrafast fiber laser, by adjusting PC1/PC2 and PC3, the mode-locking state in the CW/CCW direction can be independently controlled. Finally, a 2 × 2 optical coupler (OC1) with a 20% output ratio is used to extract counter-propagating ultrashort pulses. The lengths of the two EDFs, with a group velocity dispersion (GVD) of 12 ps2/km at 1550 nm, used in this experiment are both 0.5 m, and the rest of the cavity is composed of standard single-mode fibers (SMFs) with GVD of −22 ps2/km at 1550 nm. The total cavity lengths in CW and CCW directions are ∼3.60 m and ∼3.58 m, corresponding to net dispersions of −0.0622 ps2 and −0.0618 ps2 at 1550 nm, respectively.
Fig. 1. Schematic diagram of the experimental setup for observing counter-propagating soliton dynamics in a bidirectional ultrafast fiber laser.
As shown in Fig. 1, OC2 and OC3 with coupling ratios of 30:70 are used to send the CW and CCW pulses into two branches respectively. Considering the loss of DFT technique containing dispersion compensation fiber (DCF) (YOFC, DM1010-D, −131.34 ps/(nm⋅km) @1545 nm), 70% of the output is used for monitoring of real-time spectral evolution. When the dispersed pulse signal is captured by a photodetector (PD) (DiscoverySemi, DSC10H) with a bandwidth of 45 GHz and sent to a high-speed oscilloscope (Tektronix, DPO75902SX) with a bandwidth of 33 GHz, the spectral evolution can be visually observed. The lengths of DCF1 and DCF2 in the measurement system are 1050 m and 1500 m, providing spectral resolutions of ∼0.297 nm and ∼0.274 nm, respectively. In another branch, optical spectrum analyzer (OSA) (Yokogawa, AQ6370D), oscilloscope (LeCroy, WaveRunner 620Zi), and RF spectrum analyzer (Rigol, RSA3045) are used for time average measurement of the pulses.
3. Results and discussion
3.1 Stationary SMs
When the powers of LD1 and LD2 are respectively set to 126 mW and 132 mW, the stationary counter-propagating SMs can be obtained by properly adjusting the PCs. The fundamental characteristics of pulses in CW and CCW directions are shown in Figs. 2(a)-(d) and (e)-(h), respectively. The shot-to-shot spectra of CW SMs, illustrated in Fig. 2(a), are measured via DFT technology, and the stability is well confirmed by the nearly identical spectral profiles. As can be seen from the illustration on the left of Fig. 2(a), as expected, the pulse energy calculated by spectral integration is always at a fixed level with the cavity RTs, where the small fluctuations are mainly introduced by measurement noise [16]. In Fig. 2(b), the two average spectra obtained from the OSA and the 100 RTs of shot-to-shot spectra, respectively, are in good agreement, confirming the effectiveness of our DFT setup. According to the average spectrum, it can be found that the interference fringes are equidistant and the interval is 0.734 nm, which can be considered as two-beam interference, that is, the obtained SMs are bi-soliton molecules. And within the time window of 10 µs, no obvious intensity modulation can be observed in the pulse sequence shown in Fig. 2(c). Moreover, it can be clearly seen from the RF spectrum in Fig. 2(d) that the repetition frequency of the pulses in CW direction is 56.5 MHz, corresponding to the CW cavity length, and the signal-to-noise ratio (SNR) at the fundamental frequency is about 58 dB. The RF spectrum measurement spanning 500 MHz is further performed and the results are shown in the inset of Fig. 2(d). The signal peak spacing of 56.5 MHz coincides with the fundamental repetition frequency. Similar to the test results of SMs in CW direction, it is indicated that the stationary SMs are also obtained in CCW direction. It is noteworthy that the interval of interference fringes in the average spectrum of SMs obtained in CCW direction is 0.646 nm. Moreover, the repetition frequency of pulses in CCW direction is 56.9 MHz, corresponding to the length of CCW cavity.
Fig. 2. Output characteristics of the stationary SMs pulses. (a) Shot-to-shot spectra (inset: pulse energy calculated by spectral integration), (b) average spectra respectively obtained from OSA and 100 RTs of shot-to-shot spectra, (c) oscilloscope trace, and (d) RF spectrum with 1 Hz resolution within 2 kHz at fundamental frequency (inset: the wideband RF spectrum with 3 kHz resolution) of CW pulses. (e) Shot-to-shot spectra (inset: pulse energy calculated by spectral integration), (f) average spectra respectively obtained from OSA and 100 RTs of shot-to-shot spectra, (g) oscilloscope trace, and (h) RF spectrum with 1 Hz resolution within 2 kHz at fundamental frequency (inset: the wideband RF spectrum with 3 kHz resolution) of CCW pulses.
It is shown by all of the above results that the stationary counter-propagating SMs are obtained simultaneously in the constructed bidirectional ultrafast fiber lasers. And the noise levels of the CW and CCW pulses are estimated to be 1.05% and 1.63%, respectively.
3.2 “Invisible” pulsations of counter-propagating SMs with different periods
The generation of the “Invisible” soliton pulsation is dominated by the change of the chirp sign of the pulses in the cavity [15]. The pulse amplified by the positive dispersion EDF has a slight positive chirp, and then it is compressed after passing through the negative dispersion SMF. Excessive peak power produces strong SPM in the SMF, resulting in the double-peaked spectrum. After the chirp sign of the pulse in the cavity is reversed, due to the SPA effect, the two peaks of the pulse spectrum approach each other and eventually merge to form a single peak in the EDF, that is, the spectral center is amplified by obtaining power from the spectral wing. It is precisely because the sub-pulse in the soliton molecule undergoes the above-mentioned process that the spectrum of the soliton molecule changes periodically.
With the aid of DFT technology for real-time detection of spectral information, when the power of LD1 and LD2 is set to 148.3 mW and 150.8 mW, respectively, by fine-tuning the PCs, the “invisible” pulsation phenomena of counter-propagating SMs are realized simultaneously. The fundamental characteristics of CW and CCW pulsating SMs are recorded in Figs. 3(a)-(d) and (e)-(h), respectively. The spectrum evolution of CW and CCW pulsating SMs are respectively depicted in Figs. 3(a) and (e). It is clear that the spectra of CW and CCW SMs change drastically and periodically, and their periods are 7 RTs and 2 RTs, respectively. However, as shown in the insets of Figs. 3(a) and (e), the energies of both CW and CCW SMs are kept at a fixed level during the “invisible” pulsations, and only small irregular fluctuations caused by measurement noise is found. The average spectra respectively obtained by OSA and 50 RTs of shot-to-shot spectra in Figs. 3(b) and (f) are both very consistent, verifying the accuracy of the DFT measurement. In addition, a series of additional sidebands, beyond the Kelly sidebands, are found in the average spectra, which will be discussed below. Furthermore, the sequences and RF spectra of counter-propagating pulses are directly recorded without passing through any dispersive element. It is illustrated in Figs. 3(c) and (d) that the CW pulse sequence is still uniform without obvious intensity modulation, while the SNR at the fundamental frequency of 56.5 MHz is 58 dB without any sideband. The measurement results of CCW pulses shown in Figs. 3(g) and (h) are completely similar to those of CW, except that the SNR of CCW pulses at the fundamental frequency of 56.9 MHz is 61 dB. Again, it is strongly proved that there is no fluctuation of pulse energy during the pulsations. Altogether, the “invisible” pulsations of counter-propagating SMs are strongly verified by the above experimental results. It is noteworthy that the “invisible” pulsation phenomenon is significantly different from the higher-order soliton and multi-mode soliton phenomena. During the “invisible” pulsation, the periodic broadening of the spectrum is symmetrical, and the pulse envelope is only slightly periodic broadening in the time domain without splitting.
Fig. 3. The fundamental characteristics of “invisible” pulsation of counter-propagating SMs. (a) Shot-to-shot spectra (inset: pulse energy calculated by spectral integration), (b) average spectra respectively obtained from OSA and 50 RTs of shot-to-shot spectra, (c) oscilloscope trace, and (d) RF spectrum with 1 Hz resolution within 2 kHz at fundamental frequency (inset: the wideband RF spectrum with 3 kHz resolution) of CW pulses. (e) Shot-to-shot spectra (inset: pulse energy calculated by spectral integration), (f) average spectra respectively obtained from OSA and 50 RTs of shot-to-shot spectra, (g) oscilloscope trace, and (h) RF spectrum with 1 Hz resolution within 2 kHz at fundamental frequency (inset: the wideband RF spectrum with 3 kHz resolution) of CCW pulses.
Since the counter-propagating SMs have independent periodic characteristics, for the sake of clarity, the periodic evolutions of CW and CCW pulses are recorded respectively in Fig. 4 and Fig. 5. In order to capture more details, the shot-to-shot spectra within one pulsating period are shown in Fig. 4(a). It can be clearly found that the spectra of the SMs between adjacent cavity RTs oscillates violently at the central wavelength (see Visualization 1). Among them, the decrease of the spectral intensity is caused by the SPM in the SMF, while the increase is derived from the SPA in the EDF with normal-dispersion [15,16]. In addition, two distinct parametric sidebands (PSs) are observed at ∼1559.5 and ∼1573.9 nm (as indicated by the black arrows) in the spectra, which are generated by the four-wave mixing-based parametric process induced by periodic changes in pulse peak power [38,39]. In this experiment, the generation of PSs is a facilitator for the internal flow of pulse energy during the “invisible” pulsation process. In order to qualitatively analyze the time profiles of SM pulses, the corresponding field autocorrelation traces are obtained by Fourier transform of the shot-to-shot spectra in Fig. 4(a), as shown in Fig. 4(b). Moreover, the fluctuation of pulse durations over one pulsating period is further described in the right inset. It can be concluded from the experimental results that the temporal separation inside the SMs remain unchanged, while the pulse duration is vibrating, and the pulse with more obvious spectral distortion corresponds to a shorter pulse duration. When the pulse energy is constant, the pulse peak power is inversely proportional to the pulse duration. Therefore, it is verified again that the distortion of the spectrum is caused by SPM introduced from the excessive peak power. In order to reveal the evolution characteristics more deeply and precisely, the evolution of the temporal separation τ and the relative phase φ of pulse pairs inside the SMs, with the cavity RTs, are further calculated based on the shot-to-shot spectra, as shown in Fig. 4(c). The results show that although the temporal separation is constant during the pulsation process, the relative phase presents a sharp oscillation. The phase oscillation means that the bond between the pulse pair is weak [40], and the relative intensity between the pulse pair is undergoing an oscillation process with the same period as the phase oscillation [41]. It is noteworthy that within the period of pulse pairs state evolution, there may be several evolution periods of the coherent interference spectrum i. e. SM spectrum. Therefore, the oscillation period of the relative phase is exactly an integral multiple of the SM spectral evolution period. As can be seen from Fig. 4(c), the oscillation period of relative phase between pulse pairs is 28 RTs, which is 4 times the evolution period of the SMs spectral intensity, which indicates that all parameters of the SM pulse are self-consistent after 28 RTs, rather than evolving to chaotic behavior with the disturbed periodicity [42]. Furthermore, the evolution of vector (τ, φ) is expressed in 3D interaction space in the cylindrical coordinate system, with cavity RTs as the Z-axis, to visualize the internal dynamics, as shown in Fig. 4(d). The invariance of the temporal separation of pulse pairs is demonstrated in the projection plot depicted in the inset. It is indicated that in the “invisible” pulsation process, although the peak power of the SM pulse has no effect on the separation between pulse pairs, its relative phase is periodically oscillatory, and the oscillation period is an integer multiple of the spectral evolution period.
Fig. 4. The real-time evolution of CW pulsating SMs obtained by DFT technology. (a) Consecutive seven shot-to-shot spectra and (b) field autocorrelation traces (inset: the pulse duration changes with RTs) within one period. (c) Temporal separation and relative phase evolution versus RTs. (d) Trajectories of multiple internal motions (temporal separations τ and relative phases φ) of SMs in the interaction space (inset shows the trajectories of SMs on the phase plane).
Fig. 5. The real-time evolution of CCW pulsating SMs obtained by DFT technology. (a) Consecutive two shot-to-shot spectra and (b) field autocorrelation traces (inset: the pulse duration changes with RTs) within one period. (c) Temporal separation and relative phase evolution versus RTs. (d) Trajectories of multiple internal motions (temporal separations τ and relative phases φ) of SMs in the interaction space (inset shows the trajectories of SMs on the phase plane).
On the other hand, the evolution of CCW SMs is depicted in Fig. 5. As shown in Fig. 5(a), similar to the CW SMs mentioned above, when the “invisible” pulsation occurs, the spectral intensity of CCW SMs is also sharply oscillatory at the central wavelength with a period of 2 RTs (see Visualization 2). And the generation of PSs, resulting from the four-wave mixing-based parametric process, at ∼1545.1 nm and ∼1595.8 nm (as indicated by the black arrows) is observed, respectively. The field autocorrelation traces obtained from shot-to-shot spectra within one pulsating period is shown in Fig. 5(b), from which it can be seen that the temporal separation inside the SM is also a constant value, and the peak power reflected by the pulse duration is fluctuating correspondingly with the spectrum. It is demonstrated that the distortion of the spectra of SMs in the CCW direction is caused by the same reason as that of the SMs in the CW direction, which is jointly caused by SPM and SPA. The evolution of the temporal separation and the relative phase of SMs for 20 consecutive RTs is depicted in Fig. 5(c). While the temporal separation is kept constant, the relative phase is found to fluctuate periodically with the change of RTs. Here the period of relative phase fluctuation is 2 RTs, which is one time the period of the spectral oscillation. It is shown in the 3D interaction space in Fig. 5(d) that the relative phase is always oscillating along a circle with a fixed radius of τ, which is due to the fact that the temporal separation is always unchanged during the “invisible” pulsation.
It is indicated by the above experimental results that the “invisible” pulsation phenomenon of SMs, with the same internal mechanism, are simultaneously realized in both propagating directions. Moreover, the independence of the two directions of the bidirectional passively MLFL in this work is also demonstrated through the different pulsating periods. In addition, the noise levels in the CW and CCW directions are estimated to be 31.53% and 35.62%, respectively.
3.3 “Invisible” pulsations of counter-propagating SMs with same periods
For the verification of universality, by adjusting the birefringence parameters in the cavity, the “invisible” pulsation phenomena of SMs with pulsating periods of 5 RTs are further realized simultaneously in the CW and CCW directions, and their output characteristics are respectively shown in Figs. 6(a)-(d) and Figs. 6(e)-(h). It is clearly observed from Figs. 6(a) and (e) that while the pulse energies of both CW SMs and CCW SMs are kept at a fixed level, their spectra fluctuate significantly with periods of 5 RTs, which is a typical feature of the “invisible” pulsation. Moreover, the effectiveness of the DFT measurement used in the experiment is verified by the two well-matched average spectra shown in Fig. 6(b) and Fig. 6(f), respectively. As another typical feature of the “invisible” pulsation, PSs caused by the periodic variation of pulse peak power, similar to the experimental results discussed in Section 3.2, can still be clearly displayed on the average spectra. And the invariance of pulse energies of counter-propagating SMs are strongly confirmed by the stability of the pulse sequences respectively shown in Figs. 6(c) and (g). The RF spectra without any sidebands are respectively shown in Figs. 6(c) and (g). It is noteworthy that the repetition frequencies of the counter-propagating pulses do not change, and the SNRs at the fundamental frequencies of CW SMs and CCW SMs are ∼53 dB and ∼61 dB, respectively, indicating the high stability of the two pulse trains.
Fig. 6. The fundamental characteristics of “invisible” pulsation of counter-propagating SMs. (a) Shot-to-shot spectra (inset: pulse energy calculated by spectral integration), (b) average spectra respectively obtained from OSA and 50 RTs of shot-to-shot spectra, (c) oscilloscope trace, and (d) RF spectrum with 1 Hz resolution within 2 kHz at fundamental frequency (inset: the wideband RF spectrum with 3 kHz resolution) of CW pulses. (e) Shot-to-shot spectra (inset: pulse energy calculated by spectral integration), (f) average spectra respectively obtained from OSA and 50 RTs of shot-to-shot spectra, (g) oscilloscope trace, and (h) RF spectrum with 1 Hz resolution within 2 kHz at fundamental frequency (inset: the wideband RF spectrum with 3 kHz resolution) of CCW pulses.
The evolution of SMs in the CW direction and CCW direction within one pulsating period are summarized in Fig. 7 and Fig. 8, respectively. The sharp fluctuation of spectral intensity at the central wavelength can be observed from the shot-to-shot spectra shown in Fig. 7(a), accompanied by the generation of PSs at ∼1557.0 and ∼1575.2 nm, respectively (see Visualization 3). Meanwhile, the field autocorrelation traces, shown in Fig. 7(b) obtained by the Fourier transform of real-time spectrum, clearly shows that the pulse duration has the same evolutionary trend as spectral intensity at the central wavelength. And this is visualized in the right inset of Fig. 7(b). At this point, the temporal separation inside the SMs is constant as expected, as shown in Fig. 7(c). In contrast, the relative phase exhibits an oscillation with a period of 10 RTs, which is twice the period of the spectral evolution. In the 3D interaction space shown in Fig. 7(d), the vector (τ, φ) always oscillates periodically with the cavity RTs along a fixed circle with a radius of τ, which can also prove that the relative phase oscillates periodically while the temporal separation remains constant during the “invisible” pulsation of SMs.
Fig. 7. The real-time evolution of CW pulsating SMs obtained by DFT technology. (a) Consecutive five shot-to-shot spectra and (b) field autocorrelation traces (inset: the pulse duration changes with RTs) within one period. (c) Temporal separation and relative phase evolution versus RTs. (d) Trajectories of multiple internal motions (temporal separations τ and relative phases φ) of SMs in the interaction space (inset shows the trajectories of SMs on the phase plane).
Fig. 8. The real-time evolution of CCW pulsating SMs obtained by DFT technology. (a) Consecutive five shot-to-shot spectra and (b) field autocorrelation traces (inset: the pulse duration changes with RTs) within one period. (c) Temporal separation and relative phase evolution versus RTs. (d) Trajectories of multiple internal motions (temporal separations τ and relative phases φ) of SMs in the interaction space (inset shows the trajectories of SMs on the phase plane).
In the shot-to-shot spectra of CCW SMs within one pulsating period, as shown in Fig. 8(a), the considerably peak-dip alternation caused by both SPM and SPA is found (see Visualization 4). And obvious PSs are also observed at ∼1546.2 nm and ∼1594.8 nm, respectively. It is indicated by the experimental results shown in Figs. 8(b) and (c) that although the temporal separations inside the SMs are constant, the pulse duration of the SMs is varying, corresponding to the shot-to-shot spectra. This indicates the noticeably fluctuating peak powers of SM pulses. Meanwhile, the relative phase is also periodically oscillatory during the “invisible” pulsation with a period of 10 RTs which is twice the period of the spectral oscillation. The relative phase oscillation can be clearly seen in the 3D interaction space, as shown in Fig. 5(d), where the trajectory of the vector (τ, φ) evolves on a fixed orbit with temporal separation of τ. Finally, the noise levels in the two directions are estimated in this state, which are 40.35% and 18.64% in the CW and CCW directions, respectively. In short, the occurrence of “invisible” pulsation will bring significant noise to the MLFL.
In addition, the “invisible” pulsation phenomena with other pulsation periods can also be observed in experiments, and their evolution characteristics are very similar to the above results.
4. Conclusion
In summary, the “invisible” pulsation phenomena of SM are realized simultaneously in two operation directions of a bidirectional passively MLFL. Similar to the “invisible” pulsation of single-soliton state, the obvious periodic distortion of SM spectrum with constant energy can also be detected during “invisible” pulsation. Moreover, the generation of PSs attributable to four-wave mixing can also be observed. Furthermore, through the field autocorrelation traces of SMs, it can be found that the pulse duration of SMs is periodically changed, indicating the periodic variance of pulse peak power during the pulsation. On the other hand, it is shown by the internal evolution analysis of SM that the temporal separation between pulse pairs is constant, while the relative phase is periodically and violently oscillating. And the oscillation period is an integral multiple of the spectral evolution period. Additionally, both the independence of the two counter-propagating directions and the universality of the “invisible” pulsation are verified in this work. In summary, our work will provide new insights into the complex nonlinear dynamics of soliton, and is of great significance to optimize the stability of bidirectional laser sources.
Funding
National Natural Science Foundation of China (62205159, 61835006, 12274238); Fundamental Research Funds for the Central Universities, Nankai University (63221007).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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