1. Introduction

Solitons, localized structures ubiquitous in dissipative systems, arise from the composite balance among dispersion, nonlinearity, gain and loss [1]. Passively mode-locked fiber lasers constitute ideal playgrounds for exploring various dissipative solitons and revealing rich nonlinear dynamics. However, early experiments were limited due to the lack of ultrafast characterization approaches to track the soliton evolution and interactions [2]. Fortunately, in recent years, the dramatic advances in real-time measurements, including dispersive Fourier transform (DFT) and time-lens techniques, make the full-field characterization of transient dissipative soliton dynamics available [3–6]. For DFT, the spectral information is mapped in time domain by using group velocity dispersion and detected via a real-time oscilloscope. This relatively simple but powerful technique has been widespread in resolving complex nonlinear phenomena in ultrafast lasers, such as soliton buildup dynamics [7,8], soliton rain [9], soliton explosions [10,11], soliton molecules [12–16], soliton pulsations [17–20] and dissipative soliton resonance [21]. In a word, DFT technique is of great significance to reveal the transient nonlinear dynamics and optimize the laser systems.

Among complex nonlinear dynamics, multi-pulse operation has long been a hotspot of theoretical and experimental research [21–24]. Generally, multiple pulses coexisting in a laser cavity always interact with each other, leading to a variety of unusual self-organized structures, such as harmonic mode locking [25], soliton bunch [26], soliton molecules [27], soliton rain [9], etc. Soliton molecules, exhibiting striking analogies with matter molecules, are under intense research focus. They are bound states of several interacting solitons that are separated typically by several or dozens pulse widths. The early investigations of soliton molecules were focused on theoretical analyses and experimental observations based on averaged spectral and autocorrelation measurements. The internal dynamics escaped direct experimental detection. A major turn came with the development of DFT technique, enabling real-time visualization of the internal motion. In 2017, different categories of internal pulsations of soliton molecules, including vibration-like and phase drifting dynamics, were revealed successively [12,27]. Later, the entire buildup process of soliton molecules and rich nonlinear processes involved in were detected [15,16]. Soon after, Wang et al. demonstrated the observation of optical soliton molecular complexes and highlighted the important differences between the intra-molecular and inter-molecular bonds [28]. In a word, the powerful DFT technique has enabled fruitful and diverse experimental studies on soliton molecules, including in normal-dispersion regime [29] or 2 μm wavelength range [14].

However, all the works mentioned above revealed only a partial of the molecular dynamics: the solitons making up the molecule were assumed as rigid bodies with identical profiles; the main dynamics were characterized by the evolutions of temporal separation and relative phase between adjacent solitons. The pulsating or chaotic behaviors of the soliton itself were ignored. As matter of fact, due to the dissipative feature of laser cavity, solitons generally behave like deformable entities. Recall that the pulses in partial parameter space vary in adjacent roundtrip (RT) but repeat periodically after multiple RTs, exhibiting complex and interesting pulsating behaviors [30]. To date, based on the complex Ginzburg-Landau equation (CGLE), a variety of pulsating solutions have been found theoretically [31,32], a few have been detected experimentally via DFT technique, such as pure soliton pulsations [17], periodic soliton explosions [20], period doubling [20,33], etc. Moreover, soliton molecules may deform asymmetrically in the course of propagation because of the inequalities in energy [34]. Existing experimental reports are far from covering all the intriguing molecule or pulsating dynamics in nonlinear optical systems. More complex molecule dynamical experiments, especially where the soliton constituents are treated as deformable entities, are still desiderata.

In this work, by means of DFT technique, we experimentally investigate the internal dynamics of pulsating soliton molecules generated in an L-band normal-dispersion mode-locked fiber laser. Moreover, we demonstrate for the first time that the dissociation dynamics of pulsating soliton molecule could be induced through soliton interactions mediated by energy exchange. These results uncover the diversity of soliton molecule and the universality of pulsating behaviors in a mode locked fiber laser, promoting the understanding about complex dynamics in nonlinear optical systems.

2. Experimental setup

As a test-bed system, we build a typical normal-dispersion mode-locked fiber laser in L-band, as shown in Fig. 1. A segment of 9.5 m long normal-dispersion erbium-doped fiber [EDF, Fibercore, I-25(980/125)] is used as gain medium and dispersion manage component, which is forward pumped by a 976 nm laser diode (LD) through a fused wavelength division multiplexer (WDM). Mode locking is realized by nonlinear polarization rotation technology with a polarization dependent isolator (PD-ISO) sandwiched by two polarization controllers (PCs). The laser output is detected through the 79% port of an optical coupler (OC). The dispersions of devices pigtails (single-mode fiber, ∼9 m in total) and EDF are −23 and 40 ps²/km at 1550 nm, respectively, resulting in a net cavity dispersion of 0.19 ps². The total cavity length ∼18.5 m entails a RT time of ∼89.2 ns.

 

Fig. 1. Schematic of the L-band normal-dispersion mode-locked fiber laser and its detection system.

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As shown in Fig. 1, the laser output is split into three branches by two OCs, for synchronously monitoring of average spectrum, temporal evolutions and real-time spectra. An optical spectrum analyzer (OSA, Yokogawa, AQ6370D) and a high-speed real-time oscilloscope (33-GHz bandwidth, 100 Gsamples/s, Tektronix DPO75902SX) together with two photodetectors (PD1, 45-GHz bandwidth, DiscoverySemi, DSC10H; PD2, 50-GHz bandwidth, Finisar, XPDV2320R) are employed. The DFT is implemented by temporally stretching the solitons in a 1.5 km long dispersion compensating fiber (DCF, YOFC DM1010-D, −131.34 ps/(nm·km) @ 1545 nm), providing ∼0.152 nm real-time spectral resolution.

3. Results and discussion

By increasing the pump power and adjusting the PCs appropriately, both stable and pulsating soliton states can be easily achieved, as detailed in our previous works [20,26]. The subject of this paper is pulsating soliton molecules, where the soliton constituents can periodically change their intensity profiles and spectral bandwidths, and recover the original states after multiple RTs rather than one. It's worth mentioning that we have ruled out the possibility of polarization oscillation by synchronously monitoring of two orthogonal polarization components of pulses. In fact, if mode locking is realized by nonlinear polarization rotation technology, the polarization dependent device will fix the soliton polarization state, so the fiber laser generally forms scalar solitons [35]. For an appropriate setting of PCs, soliton bunch state with more than 5 pulsating solitons is obtained at a high pump power above 400 mW. Subsequently, we annihilate pulses one by one by decreasing the pump power, pulse self-assembly takes place during this process. Meanwhile, the pulse evolution is monitored by real-time spectral measurement to capture pulsating soliton molecules in time. Noteworthily, molecules in our case are always accompanied by redundant unbound solitons, and rarely observed under 300 mW. The recorded time series are segmented with respect to the RT time and then the transient dynamics of solitons can be depicted by the single RT time and the RT number. Note that it is inaccurate to give unitive wavelength coordinate on the X axis of the spectral evolutions due to the spectral overlaps between neighboring solitons. Therefore, we label bandwidth information in the real-time spectra for easier visualization of the spectral characteristics. In addition, the energy evolution provides an effective way to understand the transient dynamics of nonlinear systems and is employed here to reveal the pulsating and dissociation dynamics of soliton molecules.

Figures 2(a)–2(g) summarize the performances of pulsating soliton molecules at the pump power of 362.2 mW. The real-time spectra of 2000 consecutive RTs are mapped in Fig. 2(a), showing bandwidth pulsating property with a period of ∼200 RTs (∼89.2 ns per cavity RT). Correspondingly, the energy and temporal evolutions are presented in Figs. 2(b) and 2(c). For pulsating solitons, the changes in group velocity induced by the variety of spectral composition generally lead to periodical temporal shifts [20], which are invisible in this case due to the insufficient detection resolution. Combining Figs. 2(a)–2(c), there are 5 synchronous pulsating solitons self-assembling into two soliton molecules, where the trailing molecule is accompanied by an unbound soliton. We define the leading molecule as Molecule 1 and the trailing one as Molecule 2. Figures 2(d) and 2(e) show autocorrelation traces obtained by performing the fast Fourier transformations of the real-time spectra of Molecules 1 and 2, respectively. The distance between the central peak and the satellite peaks in the autocorrelation trace refers to the temporal separation inside a molecule. As we can see, the separations inside both the two molecules jitter over time, slightly, chaotically and asynchronously. For better displaying the spectral fringes arising from interference between bound pulsating solitons, we also present the spectra close-ups in Figs. 2(f) and 2(g), respectively for Molecules 1 and 2. The irregular fringes indicate that the temporal separations and phase differences between the molecule constituents irregularly evolve with RT time.

 

Fig. 2. Characteristics of pulsating soliton molecules. (a) Spatio-spectral dynamics. (b) Energy evolutions of Molecules 1 (black curve) and 2 (red curve), respectively. (c) Spatio-temporal dynamics. (d) and (e) Autocorrelation traces calculated from the real-time spectra of Molecules 1 and 2, respectively. (f) and (g) Close-ups of the data from the A and B regions of Fig. 2(a), respectively.

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As shown in Fig. 3(a), the average spectrum measured by the OSA presents arcuate edges, which is typical for pulsating solitons [20]. Note that no interference pattern can be observed on the average spectrum due to the chaotic evolutions of both temporal separations and phase differences inside the two molecules. For better characterizing the pulsating soliton molecules, Figs. 3(b) and 3(c) illustrate the real-time spectra corresponding to the extrema of the oscillations of Molecules 1 and 2, respectively. The spectral fringes corresponding to the interference inside molecules are clearly visible. The spectral modulation period of both molecules is close to 0.304 nm, well consistent with the temporal separations in autocorrelation traces (shown in Figs. 2(d) and 2(e)).

 

Fig. 3. (a) Optical spectrum directly recorded by the OSA. (b) and (c) Real-time spectra of Molecules 1 and 2, respectively; corresponding to the cross sections at the RT 151 (black curve) and 256 (red curve) of Fig. 2(a).

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In addition, it is procurable to induce pulsating bound triplet states by identical pump power adjustment manner. At 365.8 mW, we observe triplets with dynamically evolving temporal separations and relative phases, as shown in Figs. 4(a)–4(c). Figure 4(a) shows the real-time spectral evolution. For better clarity, the enlarged spectral evolution is shown in Fig. 4(b). Two temporal profiles corresponding to the extrema of the oscillations are shown in Fig. 4(c), demonstrating that three solitons bind as a molecule accompanied by two unbound solitons.

 

Fig. 4. Characteristics of pulsating triple-soliton molecule. (a) Spatio-spectral dynamics. (b) Close-ups of the box in Fig. 4(a). (c) Temporal profiles at the RT 131 (black curve) and 208 (red curve).

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Generally, the energy is equally distributed among the molecule constituents [36]. However, soliton molecule can deform asymmetrically in the course of propagation, which may induce molecule dissociation. Here, we present the first experimental observation of the dissociation dynamics of pulsating soliton molecule. Figure 5(a) shows the spectral evolution of this fascinating transient dynamics. Corresponding cavity energy evolution is presented in Fig. 5(b). Figure 5(c) provides the autocorrelation trace for tracing the separation within the molecule. Moreover, the enlargements of the spectral evolution are shown in Figs. 5(d) and 5(e) to illustrate more spectral details of this dynamic event. As can be seen, multiple pulses coexist in the laser cavity with the closest two initially exhibiting bound state akin to molecule. At the outset, the pulsating soliton molecule experiences typical spectral bandwidth breathing with a period of ∼160 RTs, spectral interferogram is clearly visible. Meanwhile, the corresponding energy exhibited in Fig. 5(b) undergoes periodically oscillations. Then, the total energy oscillates damply and rests at a relatively stable value, near the middle value of the oscillation, indicating that no pulse arises or annihilates. Further, as shown in Figs. 5(d) and 5(e), the molecule constituents evolve asynchronously (RTs from ∼820 to 1630), demonstrating a marked energy exchange among solitons. The spectral fringe patterns exhibit declining modulated period, and gradually fade out due to the insufficient detection resolution, which implies the solitons within the molecule repel each other. The repulsive interaction can be further confirmed by the autocorrelation trace in Fig. 5(c). Note that the pulses number n within a molecule corresponds to 2n–1 peaks in the autocorrelation trace, the real-time spectral resolution of ∼0.152 nm corresponds to a unilateral range of ∼56 ps in the autocorrelation trace. The red arrow in Fig. 5(c) denotes partial dissociation process: the two molecule constituents are initially separated by ∼23 ps, and then repel each other (increasing in separation), finally, the separation is out of detected range as the three peaks is transformed to a single peak. After an energy redistribution process, solitons restore synchronous pulsating pattern as displayed in Fig. 5(a).

 

Fig. 5. Dissociation dynamics within a pulsating soliton molecule. (a) Spatio-spectral dynamics. (b) Energy evolution. (c) Autocorrelation trace. (d) and (e) Close-ups of the data from the A and B regions of Fig. 5(a). (pump power: 355.9 mW).

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We recall that pulses in stationary multi-soliton mode-locked regime generally possess same characteristics due to the so-called “soliton energy quantization” [36]. However, in nonstationary regime, the enhancement or attenuation of any signal will affect all other signals due to the intertwined physical effects including spectral filtering, self-phase modulation, and chromatic dispersion, transient gain response of the gain fiber, etc [34,37], which results in energy exchange. The experimental results above exhibit an interesting phenomenon: the dissociation dynamics of pulsating soliton molecule could be induced through soliton interactions mediated by energy exchange. However, the overlap of the real-time spectra and the deficiency of temporal evolution conceal part information, making it difficult to reveal the individual dynamic evolution of each soliton within molecules.

To further explore the dissociation mechanism of pulsating soliton molecule, we repeat the measurements numerous times by adjusting the PCs and pumping power. Fortunately, we capture an asynchronously evolving pattern without spectral overlap at the pumping power of 311.5 mW, which is complementary with the molecular dynamics above. Figures 6(a) and 6(b) show the spectral evolution and the corresponding energy evolutions of multi-soliton pulsation respectively. The corresponding temporal evolutions are shown in Fig. 6(c). We sort the soliton from 1 to 4 according to their positions in single RT. A marked energy exchange can be observed during the asynchronous evolution, where the energy drop of Soliton 3 is quite complementary with the increases of all the others. For pulsating soliton in normal-dispersion regime, the growth of pulse energy is in sync with the growth of the spectral bandwidth, correspondingly, the decline in energy is accompanied by the spectrum shrink. Further, the changes in group velocity induced by the evolving central wavelength generally lead to temporal shifts [20]. The interaction between the cavity solitons can be studied according to the evolution of their temporal separation [38]. In a word, the energy fluctuations of solitons and their relative positions determine whether the interaction is repulsive or attractive. We speculate that Solitons 1, 2, and 4 all contribute to the temporal shift of soliton 3, since their spectra and energy evolutions are almost identical. As shown in Fig. 6(c), the middle soliton (Soliton 3) gradually approaches the leading ones (Solitons 1 and 2), illustrating attractive interaction. Correspondingly, Soliton 3 can be regarded to be repelled away from the trailing soliton (Soliton 4). Here, the interactions persist for ∼1200 RTs (∼107040 ns), for a total distance of ∼22.2 km, until the solitons restore synchronous pulsating pattern. Over that vast distance, the solitons only shift their relative position by less than 0.1 ns, equivalent to ∼0.7 cm in spatial separation.

 

Fig. 6. Asynchronously evolving dynamics within laser cavity. (a) Spatio-spectral dynamics. (b) Energy evolutions. (c) Spatio-temporal dynamics.

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Accordingly, we speculate that, during the dissociation dynamics of pulsating soliton molecule, the leading soliton within molecule experiences a drastic drop of energy, while the remaining energy in the cavity are shared equally among the trailing one and all other unbound solitons. The solitons within molecule repel each other, consistent well with the dynamics illustrated in Figs. 5(a)–5(c). These experimental results validate that the dissociation dynamics of pulsating soliton molecule is induced through repulsive interactions mediated by energy exchange.

4. Conclusion

As a matter of fact, passively mode-locked fiber lasers are typical nonintegrable systems with a very large number of degrees of freedom, where the existence of soliton molecules is diversiform. One of the most complex dynamics is that the solitons within molecules behave like deformable entities. For one thing, molecule constituents can execute periodic or chaotic evolution synchronously. For another, molecules may exhibit distorted intensity profile, i.e., with inequalities in energy, chirp or spectral width. Herein, a preliminary experimental study of soliton molecules with deformable components is conducted in an L-band normal-dispersion mode-locked fiber laser by means of DFT technique. The concept of soliton molecule is extended to the pulsating regime. Further, we demonstrate for the first time that the dissociation dynamics of pulsating soliton molecule could be induced through the repulsive interaction, which originates from the distorted intensity profile deriving from energy exchange. These results uncover the diversity of molecule dynamics and the universality of pulsating behaviors in mode locked fiber lasers. We expect that this work can stimulate further research both in experimental investigations and theoretical analyses.