1. Introduction

As one of the most fascinating nonlinear interactions, soliton collisions have been extensively investigated in the fields of Bose-Einstein condensates (BECs) [1,2], cell biology [3], fluid mechanics [4], nonlinear fiber optics [5–10], etc. In an integrable Hamiltonian system governed by the nonlinear Schrödinger equation, a soliton maintains its original envelope stability after the collision [1,11]. However, the loss of integrability results in the annihilation or fusion of soliton pairs [12]. Another report on the interaction between atomic BECs and light fields in an optical ring cavity demonstrated that the collision outcomes depend on the approaching velocity of the solitons, where medium velocity collisions induce a severe scattering and partial annihilation of the BEC atoms (the typical behavior of dissipative solitons) [13]. That is, soliton collisions exist in a wide range of non-integrable and non-conservative systems. In a dissipative system, the collision dynamics are more complicated because of another set of nonlinear gain and loss balance. For example, non-chemotactic mutants of the cellular slime mould exhibiting soliton-like structures pass through one another in multicellular movements [3].

Moreover, due to the typical dissipative properties of dual-wavelength mode-locked fiber lasers (MLFLs), numerous efforts have been devoted to the study of their internal complex nonlinear dynamics [14–18]. However, limited by the compromise of scanning speed and accuracy, conventional measurement methods can only exhibit time-averaged results, which are insufficient to reveal the complex physical mechanisms in the oscillator. With the rapid development of real-time spectroscopy, the perfect match between time-stretch Fourier transform (TS-DFT) and fiber lasers provides an optimal route to explore the transient processes of soliton dynamics. Inspired by the pioneering observation of internal SM motions, endeavors have been undertaken to demystify the nonlinear interactions between solitons, such as the buildup of solitons [19–21], stationary [22–24] and vibrating SMs [25–27], soliton explosions [28–32], and rogue waves [33–35].

In MLFLs, collisions originate from the differences between the phase and group velocities of the solitons. Generally, collisions are triggered randomly and instantaneously respond to changes in the cavity parameters. The peak-power clamping effect suppresses the excessive accumulation of energy, resulting in the generation of multi-pulses or self-organized structures of the SMs [36]. Besides, a soliton pair and a single soliton appear simultaneously for certain cavity parameters and must collide at some stage due to their group velocity difference [7]. The different group velocities of the colliding pulses may be caused by stimulated Raman scattering that is shifting the carrier frequencies differently [6]. Under the influence of gain dynamics, the fast gain response also enables different organized soliton structures to exhibit different drift velocities [9]. Another experimental result identifies that the increase of the intracavity energy drives the stable double pulses to generate the relative velocity difference, and then the collisions trigger the extreme event of soliton explosion [37].

However, the collisions mentioned above and observed in single-wavelength MLFLs appeared randomly. In recent years, due to the application of dual-comb sources in such fields as optical fiber sensing, absolute distance measurement, and molecular spectroscopy [38–40], dual-wavelength MLFLs have gradually aroused researcher interests. From the fundamental scientific research perspective, the inherent periodic collisions are a perfect entry point for exploration of the complex nonlinear interactions in dual-wavelength MLFLs. With the support of the TS-DFT technique, various nonlinear phenomena during the collision have been revealed. In the buildup process of dual-wavelength mode-locking, soliton collisions enable noise to evolve into a mode-locked state at the new wavelength [41]. In an anomalous-dispersion dual-color fiber laser, the collisions result in dispersive wave shedding associated with Kelly sidebands and reorganization of multi-pulse SMs [42,43]. In addition, periodic collisions also cause periodic soliton explosions and the generation of dissipative rogue waves [44]. Despite the fact that the buildup process and collision-induced exotic dynamics of dual-wavelength MLFLs have been investigated, there is still an issue worth considering: how bifurcation behaviors ubiquitous in dissipative systems are influenced by collisions? It is well known that the bifurcation transition is reversible, such as the transitions between fixed-point attractors and limit-cycle attractors [11]. However, the bifurcation reversible transitions during the collision have not yet been discovered. Exploring the influence of collisions on Hopf-type bifurcations is of great significance for investigating the complex nonlinear dynamics in dual-wavelength MLFLs.

Here, we reveal various nonlinear phenomena during the collision in a dual-wavelength dispersion-managed mode-locked fiber laser based on nonlinear polarization rotation. The whole and clear collision process has been observed by DFT-based real-time spectral measurements and an extra-cavity delay line. Firstly, we observe the details of collision between single solitons. Subsequently, the smooth transitions of dual-color SMs are observed with a higher pump power and appropriate polarization controller (PC) orientations. Remarkably, regardless of the stationary SM or the vibrating SM, the smooth transition is reflected by the fact that the collisions do not change the mode-locked state. With the same PC orientations and pump power, bifurcation reversible transitions during the collision are recorded at another moment. Numerical results reproduce the bifurcation transition process of tightly spaced SMs from the stationary phase to the sliding phase during the collision. To the best of our knowledge, it is the first in-deep study of the collision between dual-color optical SMs. Our results contribute to the understanding of the complex nonlinear interactions between solitons in dual-wavelength MLFLs.

2. Experimental setup

The experimental setup to analyze the intra-cavity exotic dynamics includes a dual-wavelength dispersion-managed MLFL and a real-time DFT-based spectral measurement system, as shown in Fig. 1(a). Similar to our previous work [45], a dual-wavelength dispersion-managed MLFL mode-locked by the nonlinear polarization rotation mechanism is used for dual-color soliton generation. A 51-cm segment of erbium-doped fiber (EDF, LIEKKI Er110-4/125) is backward-pumped by a 976 nm laser diode through a 980/1550 nm wavelength division multiplexer (WDM). The pigtail fiber of the WDM is an 85-cm OFS980 fiber. The group-velocity dispersion (GVD) of the EDF and the OFS980 fiber are +0.012 ps²/m and +0.005 ps²/m, respectively. The sandwich structure consisting of two PCs and a polarization-dependent isolator (PD-ISO) is employed as an artificial saturable absorber (SA). A 14-cm polarization maintaining fiber (PMF) provides the birefringence required for dual wavelength mode-locking, corresponding to a spectral filtering bandwidth of ∼40 nm. In order to compensate for the intra-cavity dispersion to be near-zero, a 2.1-m long dispersion compensation fiber (DCF, Thorlabs DCF38) with a GVD of +0.048 ps²/m is inserted into the ring cavity. The pigtails of the optical components are all single-mode fiber (SMF) with a GVD of -0.022 ps²/m. The overall fiber length of the ring cavity is 9.83 m corresponding to a repetition rate of 20.882672 MHz (47.89 ns roundtrip time). The net dispersion of −0.029 ps² indicates that the ring laser operates in the near-zero anomalous dispersion regime. The pulses are output through a 10:90 output coupler (OC) from the ring cavity. Part of the output signals are sent directly to an optical spectrum analyzer (OSA, Yokogawa AQ6370B) to monitor the current operation. The rest is used for shot-to-shot spectral evolution measurements.

 

Fig. 1. (a) Schematic of the dual-wavelength dispersion-managed MLFL and the real-time DFT-based measurement system; (b) real-time spectral evolution of the dual-wavelength soliton singlet mode-locking during 3000 roundtrips, and the collision happens during the 911th roundtrip; (c) close-up of the collision process in (b); (d) energies of the short- (blue) and long-wavelength (red) mode-locked pulses and their total energy (black).

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The asynchronous pulse sequences at different center wavelengths emitted from the laser oscillator are separated by a bandpass filter (T:1530-1562 nm, R:1260-1520 nm & 1570-1675 nm). In Fig. 1(a), the pulses at short- and long-wavelengths are marked in blue and red, respectively. A 4-m long SMF delay line is inserted into the branch of the red pulses to avoid the aliasing of the intra- and extra-cavity collisions. Then, the two spectra are combined again in a 3-dB fiber coupler, and pass together through a DCF module with a total dispersion of -342 ps/nm in the C band. The spectra of the pulse are mapped by chromatic dispersion to a temporal waveform whose intensity envelope reflects the spectral information. The stretched pulses are detected by a 20-GS/s-sampling-rate real-time oscilloscope (OSC, LeCroy WaveRunner 640Zi, 4 GHz bandwidth) equipped with a fast photodetector (PD, ET-5000F, 12 GHz bandwidth), which renders the spectral resolution to 0.26 nm.

3. Experimental results and discussions

3.1 Collision between dual-color single solitons

The inherent repetition rate difference between the dual-wavelength asynchronous pulses drives inevitable periodic collisions. Figure 1(b) exhibits the real-time spectral evolution obtained by the DFT technique during 3000 roundtrips at a pump power of 110 mW. The stable evolution is hardly affected by the collision, as can be seen from Fig. 1(d) where the energies before and after the collision are at similar levels. The collision and reconstruction process lasts for about 60 roundtrips, accounting for only 0.13% of a collision period (corresponding to a repetition rate difference of 443 Hz). Thus, the process is not enough to influence the short-term stability of dual-wavelength mode-locking and its application in some precision measurement fields [46–48]. Surprisingly, contrary to previous experimental evidences for dual-wavelength mode-locked collisions in the anomalous dispersion regime [42], the overlap of the pulse tails at the collision point causes their spectra to repulse each other instead of attracting, as shown in Fig. 1(c). The white dotted line marks the location of the collision point. The energy fluctuations during the collision process in Fig. 1(d) correspond to the first spectral repulsing, then attracting and finally recovering from the asymmetric spectral structure. The total energy suddenly increases and then gradually dissipates after the collision. In a dissipative system, soliton interaction manifests as attraction or repulsion between adjacent pulses. Long-range interaction is mediated by the acoustic response dominated by the electrostrictive effects in optical fibers [49]. Short-range interaction is caused by the direct overlap of pulse tails. Attractive and repulsive forces are provided by cross-phase modulation (XPM) and cross-amplitude modulation (XAM), respectively [50,51]. In Ref. [42], it was found that XPM triggers dispersive wave shedding during the collision process. However, for sideband-free dispersion-managed solitons, XAM may act before XPM at the collision point, causing the spectral repulsion. Another appealing difference is the soliton interaction length during the collision. Dispersion management makes the interaction length larger than one roundtrip, as will be illustrated later. Periodic collisions are the basis of our investigation into complex nonlinear dynamics in the dual-wavelength MLFL.

3.2 Collision between dual-color SMs

The energy boost in the oscillator facilitates the formation of multi-soliton patterns. In the near-zero dispersion regime, the interaction between solitons leading to the formation of bound states originates from the direct overlap of stretched pulses [50]. The shot-to-shot real-time interferograms in Fig. 2(a) depict the smooth transitions of dual-color stationary SMs at a pump power of 248 mW. A close-up of the internal motion of the SMs during the collision process is shown in Fig. 2(b), in which their fast destabilization and reconstruction are clearly observed. For convenience, we define the short- and long-wavelength SMs as blue and red SMs, respectively. Taking the blue SM with sparse fringes as an example, we can see that the separation and intensity of the leading and tailing pulses undergo extreme to slight oscillations, and recover to the initial stable point after a collision period. Generally, relative phase and temporal separation are considered as the degrees of freedom and intuitive representations of internal dynamics, dominating the intra-molecular temporal and spatial motions. The interaction planes, with the temporal separation taken as the radius and the relative phase as the angle, are used to reveal how dynamic attractors evolute during the collision in the phase space [Fig. 2(d) and 2(e)]. The blue SM fluctuates in the region near the fixed point before and after the collision, while the red SM moves away from the previous fixed point after the collision. The temporal separations and relative phases all fluctuate around the fixed point before and after the collision, which mainly originate from the quantum diffusion of the pulses caused by quantum-limited noises and from the intensity difference between the bound pulses within SMs [52–56]. The collision breaks the relative phase-locking of the bound solitons, appearing as scattered green dots in the interaction planes. The relative phase differences during the collision of the blue and red SMs are ∼4π and ∼3π, respectively, which is evident in Fig. 2(c).

 

Fig. 2. Smooth transitions of dual-color temporal SMs during the collision (for convenience, SMs at short- and long-wavelengths are defined as blue and red SMs, respectively). (a)-(e) Collision of dual-color stationary SMs: (a) experimental real-time interferograms; (b) close-up of the collision and reconstruction process during 120 roundtrips in (a); (c) relative phase fluctuations of blue and red SMs; (d) and (e) interaction planes of blue and red SMs, respectively. (f)-(i) Collision between vibrating blue SM and stationary red SM: (f) experimental real-time interferograms, and the inset shows a close-up; (g) relative phase of blue SM and the insets show the close-ups before (blue line) and after the collision (red line); (h) 3D interaction space of blue SM; (i) interaction plane of the blue SM between 1750 and 2000 roundtrips.

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Beyond the phase-locked bound states, the collision relating to the vibrating SMs with evolving phase and pulse separation is also experimentally recorded, as shown in Fig. 2(f)–2(i). The close-up in Fig. 2(f) exhibits an entire sliding process including two separation jump points (a, d) and two phase turning points (b, c). To further resolve such vibrating SMs with a compound sliding phase, the relative phase evolution and interaction plane are plotted in Fig. 2(g) and 2(i), respectively. The blue and red lines in Fig. 2(g) are close-ups of the relative phase evolutions before and after the collision, respectively. The consistency of the undulation proves that the sliding period of the blue SM is not perturbed by the collision. There is a two-torus formed by the black dotted line in the interaction plane where steps of the evolution are marked with numbers, as shown in Fig. 2(i). The relative phase jumps at point a to the limit cycle with a larger separation (in contrast to the situation at point d) and U-turns at point b and c on the same limit cycle. The 3D interaction space of the blue SM in Fig. 2(h) more intuitively exhibits the stepping phase evolution and flipping motions. By combining the relative phase evolution and the interaction plane [Fig. 2(g) and 2(i)], the direction of the phase sliding is tracked from the evolution trajectories of successive interferograms [see the inset of Fig. 2(f)]: the spectral fringes shift towards longer wavelengths, corresponding to a counterclockwise rotation of the patterns in the interaction plane, indicating that the intensity of the tailing pulse is stronger than that of the leading pulse. This slight difference in intensity can be explained by their different phase velocities in a dispersive medium induced by the intensity-dependent Kerr effect, and results in a different phase shift of the carrier envelope after each round trip [22]. This simple relative phase evolution changes dramatically at point a. The temporal separations of the bound solitons from 2.43 ps to 2.61 ps causes a stepping speed change of the phase evolution, as shown by the slope between points a and b of the blue line in Fig. 2(g). Subsequently, the intensities of the bound pulses alternately dominate, corresponding to changes in the shifting direction of the fringes. Finally, the separation is restored to 2.43 ps at point d.

In the frame of nonlinear dynamics, the local instability of the fixed-point attractor leads to the appearance of a limit cycle attractor via a Hopf-type bifurcation. Considering the SMs in fiber lasers, the excitation of vibrating SMs is generally achieved by the changes in the pump power or polarization states. Instead, we observe the collision-induced Hopf-type bifurcation, in which energy exchange and complex nonlinear interaction have been witnessed [37]. Clear and well-resolved Hopf-type bifurcation reversible transitions are shown in Fig. 3. The bifurcation transition during the collision consists of four stages: fixed phase and separation (stage I), collision-induced extreme changes in phase and separation (stage II), separation-dominated phase sliding (stage III), and phase-dominated sliding (stage IV). The inverse process of bifurcation can also be divided into four stages; however, phase-dominated sliding (stage I) and fixed phase and separation (stage IV) are opposite to those in the bifurcation transition process. The spectral evolutions and the corresponding shot-to-shot Fourier transform based first-order autocorrelation traces are divided according to the above four stages [Fig. 3(a) and 3(b), 3(e) and 3(f)]. Since the change of pulse separation and phase of the red SM during the collision are consistent with those in Fig. 2, only the blue SM is discussed here. In stage II of collision-induced extreme changes in phase and separation, the collision process of the two-color SMs is more complicated due to the existence of inter- and intra-molecular interactions. The XPM induced by the overlap between the molecules leads to the internal motions of the SMs. The difference in the separation of the bound pulses in the SM with stationary and sliding phases originates from the phase space topology of different cavity parameters. In order to reflect the intra-molecular relative phase evolutions clearly and intuitively before and after the collision, discrete interaction points during the collision are hidden [Fig. 3(d) and 3(h)]. Different evolutions of stage II and stage III are manifested as a fast inward orbital transfer and a stepped outward orbital transfer in the interaction plane, respectively. The phase-dominated sliding of the inverse bifurcation process is divided into two stages: double limit-cycle phase sliding which is similar to that in Fig. 2(g) and 2(i) and phase sliding on a fixed limit cycle. In addition, the sign reversal of the slope in the stages of phase-dominated sliding illustrates a brief intensity reversal of the bound pulses, as shown in Fig. 3(c) and 3(g).

 

Fig. 3. Collision-induced bifurcation reversible transitions of dual-color temporal SMs. Since the change of pulse separation and phase of the red SM during the collision are consistent with that in Fig. 2, only the blue SM is represented. (a)-(d) Transition from stationary SM to vibrating SM during the collision, and (e)-(h) transition from vibrating SM to stationary SM during the collision; (a) and (e) experimental real-time interferograms. (b) and (f) 2D contour plot of shot-to-shot first-order autocorrelation traces; (c) and (g) relative phase curves retrieved from autocorrelation traces; (d) and (h) interaction planes before and after the collision.

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Remarkably, the collisions of the dual-color SMs above are obtained with the same cavity setups (the fixed PC orientations and pump power) [Fig. 2 and Fig. 3]. We capture the smooth and bifurcation reversible transitions at different moments, revealing that the cavity parameters sometimes pass through the Hopf-type bifurcation point during the collision. Specifically, the phase space of the cavity parameters randomly switches between two or more different topologies through periodic collisions. In other words, collisions can be considered as the intersection of different topologies. A specific observation captures only a projection of the intracavity complex nonlinear dynamics at a certain moment. On the other hand, SMs with molecular-like properties are more sensitive to external interventions, such as changes of the pump power and intracavity polarization state, so that the bonds between their internal constituents break through the critical stabilization and eventually cause the bifurcation [31]. However, the bifurcation evolution around the moment of external intervention is difficult to capture. In contrast, it is not difficult to achieve this for intracavity inherent collisions with obvious characteristics. Single-cavity dual-comb laser systems provide a fantastic platform for investigating bifurcation dynamics.

3.3 Numerical simulation

In order to confirm the universality of collision-induced bifurcation in single-cavity dual-comb laser systems, we numerically simulate the collision process of asynchronous optical pulses in a fiber laser. The simulation model is governed by the coupled Ginzburg-Landau equation [57] including the combination of fiber dispersion, transmission loss, fast saturable absorption, and gain saturation. The EDF gain is described by $g = {g_0}/({1 + {E_p}/{E_{sat}}} )$ where ${g_0}$ is the small-signal gain, ${E_p}$ is the single-pulse energy, and ${E_{sat}}$ is the gain-saturation energy. The SA is modeled by a nonlinear transmittance function [58]:

 

Fig. 4. Collision-induced bifurcation evolution of dual-color SMs in numerical simulation. (a) Spectral evolution of dual-color SMs with different temporal separations during the collision; (b) interaction plane of the blue SM; (c) corresponding temporal evolution during 100 roundtrips; (d) close-up of the collision process in (c) for evolution of asynchronous pulse pairs during successive 25 roundtrips; (e) temporal evolution in a single roundtrip.

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The phase of the blue SM slides linearly after the collision, while the red SM with a large temporal separation still maintains a smooth transition as in the experiment, that is, no bifurcation behavior occurs [Fig. 4(a)], revealing that the experimental results are not accidental. The energy exchange inside the SMs plays the dominant role in the phase-dominated dynamics of tightly spaced bound pulses and is less obvious for loosely spaced bound pulses [25]. As a result, for tightly spaced SMs, the internal energy exchange leads to the dynamic equilibrium in the intensity of pulses, manifesting as the bifurcation behavior [22]. In contrast, the phase-locking of the bound solitons with large separations is easier to recover completely after the collision regarding that the internal energy exchange is weak. The interaction plane in Fig. 4(b) shows that the relative phase starts from a noise-free fixed point with a locked relative phase and converts to a stepping evolution on a limit cycle after the collision. Moreover, the opposing motion of the dual-color pulses in Fig. 4(c) is attributed to the setting in numerical simulation model. The center wavelength in the model is 1560 nm, while the center wavelengths of the dual-wavelength mode-locked pulses are 1540 nm and 1580 nm, respectively. In the simulation, we query the output for each location within the cavity to analyze the details of the collision process, as is evident in Fig. 4(d). Dispersion management enables pulses from different molecules to entangle together within several roundtrips, which is different from conventional solitons that collide completely within a roundtrip [42]. The temporal evolution in a single roundtrip shown in Fig. 4(e) illustrates that the ‘boomerang’ time shift of the pulses in Fig. 4(d) originates from the opposite-signed GVD of different fibers.

4. Conclusion

To summarize, we report abundant exotic nonlinear phenomena in a dual-wavelength dispersion-managed ultrafast fiber laser. Different from traditional solitons with a short interaction length (no more than one roundtrip), dispersion management-dominated ‘boomerang’ time shift maintains the overlap between different colored pulses for several roundtrips, making the collision process more complex and unpredictable. On this basis, experimental evidences of collisions between two-color SMs reveal the particle properties of optical SMs. The evolution direction of the internal degrees of freedom of the SMs during the collision process is random. We demonstrate the existence of collision-induced smooth or reversible transitions of Hopf-type bifurcations without artificially altering the cavity parameters, indicating that the collision process sometimes passes through a bifurcation point. Numerical simulation results corroborate the experimental observation that tightly spaced SMs are more prone to bifurcation. However, further research is needed to verify whether the collision between dual-color SMs is similar to the anti-crossing collision between a soliton pair and a soliton singlet [6]. Although the transient dynamics of collisions in a dual-color soliton fiber laser have been solved by the TS-DFT technique, whether the two solitons or SMs exchange their eigenstates remains experimentally indistinguishable. Moreover, ultracold molecule collisions are an important research direction in condensed-matter physics [60,61]. Therefore, we expect that our results could offer novel insight into inter-molecular interactions ubiquitous in nonlinear science.