1. Introduction

Thulium (Tm) based compact laser sources operating in the 2 µm regime with femtosecond pulse durations have enabled a wide range of applications from tissue ablation [1] and metrology [2] to spectroscopy [3]. Most applications rely on stable mode-locked lasers (MLLs) to provide high temporal precision. When excessive gain is available in a soliton mode-locked laser cavity, multiple pulses can be formed as a result of energy quantization [4]. Depending on the laser parameters, harmonic mode-locking [5], soliton bunching [6], asynchronous pulsing [7] and other different forms of stable multi-pulsing states can be generated, extending the versatility of a single laser source. Yet, the complex dynamics of such highly nonlinear systems can result in unstable events as well, which can hinder the normal operation of a MLL. The transition dynamics from N to N+1 pulses have been studied using the nonlinear Schrödinger equation (NLSE), and it was predicted that the laser undergoes a Hopf bifurcation, so that pulses in the cavity can exhibit oscillatory and chaotic behaviors before entering another stable operation regime [8]. The NLSE can be extended to the cubic-quintic Ginzburg-Landau equation (CQGLE), which can include higher order nonlinear terms such as nonlinear gain and losses, allowing for a wider range of solutions for a given nonlinear optical system [9]. With these models, various phenomena associated with localized solutions have been numerically predicted, including pulsating solitons, soliton explosions, period-doubling, creeping solitons and chaotic solitons [10]. Moreover, route to chaos via a period-doubling bifurcation and the co-existence of multiple solutions for a given set of parameters were presented. For soliton explosions that were first experimentally observed in a Ti:sapphire laser, it was found that oscillatory behavior preceded spontaneous explosions [11]. These soliton explosions represent fascinating phenomena of a nonlinear optical system, as pulses experience drastic collapse before being restored back to their original shape. Thanks to the time-stretched dispersive Fourier transform (DFT) [12], a novel shot-to-shot spatial-spectral measurement technique allowing the recording of transient events in a MLL, the rich nonlinear dynamics of transient phenomena, such as internal dynamics of soliton molecule [13], rogue waves emerging from breathers [14], transition dynamics between multi-pulsing states [15] and various forms of soliton explosion dynamics [16–21] have been unveiled with enhanced details [22]. In particular, extensive real-time experiments of soliton explosions have been reported in laser configurations with different dispersion maps. Regardless of the net-cavity dispersion of the laser, most soliton explosions occur in between stable pulsing regimes, during a transition e.g. from single pulsing to noise like pulses [16,18], from single to double pulsing [17,19,23] or between two mode-locking states in a stretched-pulse laser [24].

In the above mentioned cases, explosions occurred spontaneously due to over driving excessive nonlinear effects or instability. At the same time soliton collision induced explosions [19–21] have also been reported, governed by the interaction dynamics between propagating solitons. Colliding solitons, one of the most intriguing phenomena, can fuse to one entity [25,26], or undergo energy exchange and redistribution [27–30], depending on the initial conditions of the two solitons. Collisions have also been observed in laser systems with multiple solitons propagating at different group velocities in a single laser cavity [20,21,31–33]. To the best of our knowledge, most research focused on fiber ring laser systems that relied nonlinear polarization rotation or carbon nanotube saturable absorbers for mode-locking. We aim to advance our insight into soliton dynamics in linear fiber lasers that utilize semiconductor saturable absorbers. The gain dynamics and recovery processes can potentially vary whether pulses pass unidirectionally through the gain medium once (as e.g. in a ring laser) or from both directions (as in a linear laser). The saturable absorber forms an important parameter since it has been shown in modeling with the CQGLE that nonlinear terms associated with the saturable absorber can impact the dynamics such as period-doubling bifurcations [10]. While colliding soliton studies have been conducted mostly for ytterbium and erbium based fiber laser systems, given the different gain dynamics and transient gain responses of Tm-doped fibers, it is important to investigate these rich dynamics in the 2 µm wavelength regime with real-time analysis techniques. Studying the impact of such collisions can enrich our knowledge of various pulse formation evolution. Further, they can be critical to understand how single free-running sources generating two pulses with slightly different repetition rates can be used for the emerging field of dual-comb spectroscopy [34,35].

So far, only a few studies of laser dynamics in lasers at 2 µm and beyond have been conducted utilizing the DFT real-time measurement technique [36,37]. While standard DFT measurements rely on long fiber spools to accumulate the respective dispersion to stretch the pulses accordingly, this is more challenging around wavelengths of 2 µm due to the high propagation losses encountered in most commercially available passive fibers. In this paper, an alternative method, which first up-converts pulses at 2 µm via second harmonic generation (SHG), then stretches the pulses temporally in standard single-mode fiber (SMF) [37,38], is employed to study two different types of soliton explosion dynamics. We experimentally study the real-time dynamics of chaotic and periodic soliton explosion in a Tm-doped MLL. A chaotic soliton state exists in the transition region between a single pulsing and double pulsing state and irregular soliton explosions due to perturbation induced fluctuations are investigated. In addition to soliton explosions, period-doubling bifurcation and multi-period pulsation are observed during this chaotic evolution, demonstrating that multiple solutions can co-exist for the same set of parameters as predicted in [10]. Dual-wavelength vector solitons with orthogonal polarization states can be generated at a higher pump power in the presented laser system, and the pulses collide periodically because of a slight difference in repetition frequency. These soliton explosions, which are induced by collisions, are followed by a soliton recovery process. To the best of our knowledge, explosive events in a chaotic regime between a single pulsing and doubling-pulsing regime, as well as collision induced explosions in a dual-wavelength state in a Tm-doped fiber laser are investigated for the first time. These studies can provide additional insights into the Tm laser dynamics and stable and unstable fiber laser operating regimes.

2. Experiment setup

The real-time dynamics of Tm ultrafast fiber lasers are experimentally studied in the setup illustrated in Fig. 1. A linear Tm-doped mode-locked fiber laser (see also [39]) is studied, as shown in Fig. 1(a), operating based on soliton mode-locking with a semiconductor saturable absorber (SBR). The SBR used here has a modulation depth of 12%, a saturation fluence of 65 µJ/cm² and a relaxation time of 10 ps at a wavelength of 2000 nm. The total length of the fiber cavity is 132 cm, which consists of 112 cm of gain fiber (TH512) and 20 cm of single-mode fiber SMF-28e+, thus operating in the net anomalous dispersion regime. This yields a fundamental repetition rate of 77.5 MHz, which corresponds to a round-trip pulse spacing of 12.9 ns. A polarization controller (PC) is placed on the gain fiber, and polarization rotation locked vector soliton (PRLVS) states with different polarization evolution frequencies can be obtained with different PC settings [39]. The blue solid curve in Fig. 1(b) shows a typical optical spectrum of a fundamentally mode-locked state with an average output power of 2.6 mW (for a coupled pump power of 197 mW), featuring a full-width at half maximum spectral bandwidth (BW) of 6.4 nm, which corresponds to a transform-limited pulse duration of 625 fs. The pulse train then passes through a linear polarizer (LP) so that a polarization resolved analysis of the output pulse train directly after the laser cavity can be conducted. The pulse train is amplified in a single-stage Tm-doped fiber amplifier, during which the optical spectrum undergoes slight spectral narrowing, see red dashed curve in Fig. 1(b). Nonetheless, the pulse spectrum maintains its major features and single pulse temporal shape, as verified by the autocorrelation measurement of the amplified pulse train shown in the inset of Fig. 1(b). For the real-time analysis of the pulse dynamics around the wavelength of 2 µm, we developed a modified DFT measurement setup where the pulses are frequency-doubled first before they are stretched and sampled with a fast oscilloscope (20 GHz). To perform SHG up-conversion from the 2 µm light to 1 µm light, a periodic-poled lithium niobate (PPLN) crystal is used. The length of the PPLN is chosen to be 1 mm, so that the acceptance input spectral BW can be as large as ∼40 nm. In addition, a quarter waveplate (QWP) and a half waveplate (HWP) are used to alter the polarization of the amplified light so that the polarization is maximized for SHG at the PPLN input. The optical spectrum of the up-converted pulse is shown in Fig. 1(c) as the purple solid curve, with a central wavelength at 977 nm. The signal contrast (of the center of the pulse to the hump on the right in the amplified 2 µm spectrum) in Fig. 1(b) is 15 dB, while for the 1 µm signal it is measured to be 30.4 dB, indicating an effective conversion according to a power-squared relationship between the signal and pump light. Moreover, the spectral sidebands are successfully up-converted and visible in the optical spectrum, although the intensity is lower due to the quasi-coherent nature of them. The frequency-doubled light is spilt by the polarization beam splitter, whose splitting ratio can be controlled by the HWP. The pulses are directly coupled to a 12.5 GHz InGaAs photodetector (PD1) for temporal measurements and into a 5.4 km -long SMF fiber spool for performing DFT measurements before they are converted to an electronic signal with a 22 GHz InGaAs photodetector (PD2). Although only ∼ 1 mW of power is coupled to the fiber spool, the optical spectrum recorded with an optical spectrum analyzer (OSA, Anritsu), shown as dash-dotted dark green curve in Fig. 1(c), illustrates that the stretched pulse is slightly spectrally broadened on account of self-phase modulation (SPM).

 

Fig. 1. (a) Tm-doped linear soliton fiber laser and SHG-DFT setup for real-time pulse dynamics analysis. A PPLN is used for frequency doubling so that pulses can be temporally stretched and captured with a fast oscilloscope. (b) Optical spectrum of the oscillator (solid blue) and amplified pulse (dashed red). Inset: intensity autocorrelation trace of the amplified pulse. (c) Optical spectrum of the frequency doubled pulse before (solid purple) and after (dash-dotted dark green) DFT recorded with an OSA.

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3. Chaotic state

For higher pump powers (increasing the pump power to ∼210 mW), the laser leaves its stable mode-locking state and enters a regime where multiple operation states, including single pulsing, double pulsing and chaotic states can exist depending on the PC settings. The PC can modify the polarization dependent gain and loss dynamics in the linear fiber laser cavity. The optical spectrum of this transient state is shown in Fig. 2(a), which is narrower than that of the stable mode-locking state. The spectral BW is around 4.4 nm. The recorded optical spectrum is not as smooth as it is in stable mode-locking state, indicating that the optical spectrum fluctuates from roundtrip to roundtrip, making this state particularly interesting to capture its real-time dynamics. The pulse-to-pulse fluctuations are reflected in the radio-frequency (RF) spectrum shown in Fig. 2(b). The RF spectrum spanning from 30 MHz to 80 MHz shows the fundamental repetition rate peak at 77.5 MHz and a half-harmonics peak at 38.7 MHz, with a pronounced pedestal on both peaks. The half-harmonic peaks feature a modulation with a periodicity of two cavity roundtrips. The pedestal on both peaks indicates additional rapidly varying intensity modulations, which underlie the nature of a chaotic state. The slight difference in the intensity of the harmonic peaks in the RF spectrum with a 500-MHz-span, shown as the subplot in Fig. 2(b), suggests pulsing instabilities. A 5 ms–long oscilloscope trace and the DFT trace of the chaotic state are recorded simultaneously with a high-speed oscilloscope. In Fig. 2(c), the 1 µm oscilloscope trace with a time span of ∼200 µs or ∼15,000 cavity roundtrips in the chaotic state shows a semi-periodic periodicity around 2,500 cavity roundtrips for pulse constructs (sectioned by dashed lines) despite the chaotic nature. Within these semi-periodic pulse constructs, some evolutions result in a higher spike, as shown in the first and fourth pulse construct in Fig. 2(c). Although the other constructs do not feature this spike, they all have in common that the pulses experience a short period in a quasi-stable regime, though the duration of it varies (cf. to the plateau duration in Fig. 2(c)). The occurrence of either type of pulse construct evolution is stochastic and not necessarily repetitive after a certain number of roundtrips or pulse constructs. In the following, a detailed analysis of the evolution for those two types of pulse constructs is conducted, specifically, for the third and fourth pulse construct evolution.

 

Fig. 2. (a) Optical spectrum of a chaotic-transient state. (b) Zoom-in RF spectrum shows both fundamental and half-harmonic peaks with pedestal. Subplot, RF spectrum in 500 MHz span. (c) Oscilloscope trace of consecutive semi-periodic chaotic pulse construct transient events at 1 µm. Dotted lines indicate pulse constructs analyzed below.

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Figure 3 shows the transient dynamics of the third pulse construct in Fig. 2(c) in greater detail, featuring the temporal trace for 2,500 roundtrips in Fig. 3(a). We chose a starting point for the evolution of the pulse construct so that the pulses are in a relatively stable pulsing regime at the beginning before decaying abruptly, and finally evolving towards another quasi-stable pulsing regime again with enhanced oscillations. In the zoom-in of the transient pulsing regime, shown in Fig. 3(b) around the 6,900^th roundtrip, pulses at even cavity-roundtrips are significantly suppressed compared to the intensity of the pulses at odd cavity-roundtrips. The observed period-doubling pulsing matches with the half-harmonic peaks in RF spectrum. The DFT trace is sectioned based on the cavity roundtrip time and stacked back-to-back to form a 2-D spatial-spectral dynamics contour plot, cf. Figure 3(c). The pulse starts decaying from the transient pulsing regime around the ∼7,100^th roundtrip, and weak spectral interference fringes are observed close to the ∼7,400^th roundtrip. The autocorrelation function (ACF) across the decaying region from the 7,100^th roundtrip to 7,900^th roundtrip, see Fig. 3(d) is calculated from the Fourier transform of the DFT trace according to the Wiener–Khinchin theorem. It shows that there is energy dissipating from the main pulse in the form of dispersive waves (DW) starting around the ∼7,300^th roundtrip, as indicated by the arrows in Fig. 3(d). The DWs can be fairly hard to distinguish from the background signal in this case due to the lower conversion rate of the DW in the SHG process. The spatial drift of the pulse in Fig. 3(c) indicates a shift of the center wavelength until the ∼8600^th roundtrip. During the following restabilization process, the pulse is restored to the original state due to gain competition, as revealed by the observed peaks in the frequency spectrum. Further insights are obtained from the energy evolution plot shown in Fig. 3(e), plotted by integrating each roundtrip and separating the energy values by odd (green) and even (purple) roundtrips: The energy of the pulses from even and odd roundtrips is constant until the ∼7,200^th roundtrip, as is expected for a period-doubling pulsing regime. After the pulses at odd roundtrips decay, the pulse energy envelope at every other roundtrip is oscillating out of phase, while the energy buildup and subsequent oscillation envelope alternates between odd and even roundtrips. At around the ∼8,100^th roundtrip, the odd roundtrip pulses decay while the even roundtrip pulses complete the pulse restoration process with damped oscillations before entering a transient pulsing regime again. In summary, this type of evolution can be separated into different stages: Due to the higher pulse energy (beyond the stable soliton mode-locking regime), instabilities lead to radiative emission in form of DWs and subsequent pulse decay [40]. The set of pulses (odd and even roundtrips) undergo a pulse reshaping process before stabilizing themselves. The laser returns back to the transient pulsing state before another evolution cycle for the pulse constructs starts with the decay of one pulse at the projected polarization. Similar general trends as discussed here are observed for all the different evolutions although the periodicity and amplitude change slightly.

 

Fig. 3. Transient dynamics of the 3^rd pulse construct in Fig. 2(c). (a) Oscilloscope trace over 2,500 roundtrips. (b) Zoom-in into the transient mode-locking region. (c) 2-D contour plot of the spatial-spectral dynamics. (d) Autocorrelation function from the 7,100^th roundtrip to 7,900^th roundtrip. (e) Energy evolution of odd (green) and even (purple) roundtrip pulses.

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Another type of representative evolution of the pulse construct is the following event with a spike in Fig. 2(c). Figure 4(a) presents the 2-D spatial-spectral evolution from the 9,100^th roundtrip to 11,600^th roundtrip. Similar to the third pulse construct described above, the evolution starts with a transient pulsing stage before decaying and getting restored after intensity modulations and a center wavelength shift. However, unlike the smooth decay process shown in the previous case, there is a sharp increase in the peak intensity, leading to a broadened spectrum, spectral interference fringes and an intensity modulation. In the region from the 9350^th roundtrip to 9550^th roundtrip, shown in Fig. 4(b), the pulse gradually increases its intensity. It explodes before the 9472^th roundtrip, noticeable by the presence of ripples on the wide spectrum, characteristic for a soliton explosion [10]. The ACF from the 9,300^th roundtrip to the 9,600^th roundtrip in Fig. 4(c) shows a collision event when one pulse gets close to the main pulse so that they collide at the 9472^th roundtrip. Then there is a pulse walk-off from the main pulse after the collision until it disappears eventually. This process can be interpreted as follows: Pulse splitting caused by SPM-induced modulation instability gives rise to the generation of a secondary weak pulse [41]. Since the two pulses partially overlap spectrally, attractive forces pull them together temporally and catalyze a soliton collision [27]. The explosion enhances the soliton interaction and induces a strong temporal phase perturbation, which subsequently leads to a pulse destruction in the temporal domain. Thus, an abrupt separation of the two pulses associated with the shedding of a DW is observed [42]. The traditional oscilloscope trace, which is integrated over the photodetector response time which is at least one to two orders of magnitude longer than the actual pulse duration of ultrafast pulses, provides an averaged approximate intensity profile of the pulses. At the same time, the time-stretched DFT trace records each pulse and its corresponding spectral intensity profile, which provides a different aspect into the laser dynamics. As illustrated in the upper plot of Fig. 4(e), the peak intensity during the explosion at the 9,472^th roundtrip is slightly higher than that after the explosion at the 9,506^th roundtrip in the single-shot oscilloscope recording (before DFT). The green and pink colors correspond to the roundtrip indicated by the two dashed lines in Fig. 4(d). The stretched spectrum at the 9,472^th roundtrip recorded after DFT is shown in the lower plot of Fig. 4(e), featuring a chaotic localized structure, which gets spreads out by virtue of strong soliton-soliton interactions. It has a lower peak intensity in the spectral domain when comparing to that at 9,506^th roundtrip. The spectrum at the 9,506^th roundtrip features spectral fringes which result from the strong constructive interference of the two pulses, indicating a transient soliton molecule after the collision. This side-by-side comparison clearly emphasizes the importance of high resolution real-time measurements to analyze pulsation in instability regions and to understand the associated phenomena, which cannot be revealed in detail in conventional averaged OSA and temporal oscilloscope measurements. The energy evolution of the fourth pulse construct, cf. Figure 4(f), continues directly in the transient pulsing regime at the end of Fig. 3(e). An abrupt energy jump is observed at the 9,472^th roundtrip, indicated by an arrow in Fig. 4(f), when the explosion characterized by a chaotic spectrum occurs, followed by a decreased energy oscillatory behavior. This can be potentially attributed to dynamic instabilities and relaxation oscillations [11]. The pulses then evolve further until they re-enter the transient pulsing regime.

 

Fig. 4. Transient dynamics of the 4^th pulse construct in Fig. 2(c). (a) 2-D contour plot of the spatial-spectral dynamics. (b) 200-roundtrip 3-D plot highlighting the intensity spike region. (c) Autocorrelation function from the 9,300^th roundtrip to the 9,600^th roundtrip. (d) Comparing continuous oscilloscope traces measured before DFT on top (blue) and after DFT below (black). (e) Pulses at the 9472^th (green solid) and 9506^th (pink dotted) roundtrip before (top) and after (bottom) DFT showing clear interference fringes in the latter. (f) Energy evolution of pulses at odd (green) and even (purple) roundtrips.

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In this section, a chaotic state which contains two types of evolutions for the pulse constructs were studied in the transition region between a single-pulsing and a doubling-pulsing state. For the presented laser parameters, the laser does not converge to a particular solution, rather, there exist chaotic attraction basins so that the pulse experiences various phenomena in the complex solution space: The evolution always starts and ends at the same point but takes different pathways, exhibiting various routes to chaos through period-doubling [10,43]. This chaotic state belongs to a region where multiple solutions coexist, offering various localized structures including period-doubling pulsations and soliton explosions. Since the discussed chaotic dynamics are polarization resolved after the polarization-sensitive SHG process, it cannot be determined directly whether the observed period-doubling are dominated by polarization rotation or inherent laser dynamics. However, it is clear from the energy evolution of the even and odd pulse roundtrips that a period-doubling bifurcation exists throughout the whole process. An irregular evolution of polarization states in general was previously observed in a chaotic state [44]. The occurrence of two types of evolution for the pulse constructs is attributed to the perturbation induced gain fluctuations, so that either the pulse decays by shedding DWs or by generating a new pulse which can lead to soliton collision induced explosions, depending on the effective gain-loss dynamics [4]. Similar explosive evolutions have also been observed in the transition between multi-pulsing regimes from double-pulsing to three-pulse states in the presented laser and they undergo even more complex dynamics.

4. Dual-wavelength vector soliton collision

When the pump power is increased (to 216 mW), the laser starts operating in a double-pulsing regime. Resolving the output pulse train with an LP, different polarization rotation periods are observed depending on the PC settings, indicating that the laser operates in a PRLVS state. When the pump power is increased even further (to 248 mW), by carefully tuning the PC, a flatter-top soliton-like optical spectrum, shown as dark grey curve in Fig. 5(a), is obtained. The spectra of two orthogonal polarizations, resolved for a LP angle of 0° and 90°, are presented in the same plot as red and blue curves. The two pulses corresponding to orthogonal polarizations are centered at different wavelengths, at 1964.8 nm and 1970.1 nm, respectively, and feature a distinct sets of sidebands. The optical spectrum with the LP set to an angle of 45° (orange curve in Fig. 5(a)) matches in shape with the grey curve. The corresponding 2 µm oscilloscope trace for a LP angle of 45° in Fig. 5(b) features spikes of a value 1.2 beyond the normalized peak intensity that repeat every ∼27,910 cavity roundtrips (frequency of ∼2.8 kHz). This corresponds to the repetition rate difference between the two pulse trains, indicating that periodic collision events occur, similar to [20]. To gain real-time insight into the dynamics during the collision, we performed SHG-DFT measurements. The inset of Fig. 5(a) shows the up-converted pulses at 1 µm for an LP angle of 0°, 45° and 90°, with a center wavelength at 982.4 nm for 0° and at 985.1 nm for 90°, respectively, while the spectrum at 45° almost overlaps with the other two. Over 4,800 cavity roundtrips of spatial-spectral dynamics before, during and after the collision, resolved by DFT, are shown in Fig. 5(c). The two pulses propagate asynchronously with slightly different group velocities, as derived from the different slopes in the plot, collide and recover to their original state after ∼2,500 cavity roundtrips. The soliton collision process starts around the 4,000^th roundtrip in Fig. 5(d), and an interference pattern is observed at the ∼4,240^th roundtrip. In the ACF, presented as Fig. 5(e) from the 4,150^th to 4,350^th roundtrip, the symmetric peaks gradually move towards the main lobe at the center with a speed of ∼455 fs/RT, which corresponds to the frequency difference of the two pulse trains. Afterwards, they deviate away from the main lobe right after the collision, suggesting the generation of a weak pulse [21]. At the moment of collision, a signature collapse of the wide spectrum of a soliton explosion is observed around the ∼4,240^th roundtrip, see Fig. 5(c). The explosion enhances the interaction between the two solitons, potentially leading to soliton trapping [45]. The two solitons gradually experience a shift in their center wavelength through cross phase modulation (XPM). They attempt to merge to a singlet, similar to the merging behavior of two solitons in [26]. After the collision, during a recovering period a significant breathing behavior is noticeable starting around the ∼5,000^th roundtrip in Fig. 5(d), and that can most likely be attributed to polarization rotation resulting from perturbations after the collision. In Fig. 5(f), an energy spike is observed during the collision for the pulse energy evolution. Since the recorded dynamics are polarization resolved with an LP at an angle of 45°, which can represent a partial superposition of two eigenmodes of the system, a change in the polarization state of the two underlying solitons during collision can give rise to such an abrupt energy change [28]. After the spike, a damped energy oscillation is observed, indicating the nonlinear system is approaching an equilibrium state after the collision. During the first modulation lobe (around 200 roundtrips long), shown as inset of Fig. 5(f) from the 4270^th to the 4470^th roundtrip, two additional modulations with shorter periods can be observed. Thus, another modulation with approximate periodicity of ∼20 roundtrips on top of a period-doubled modulation form the contributions to the first modulation lobe, as highlighted in the inset. This multi-period modulation continues throughout the soliton recovery process and can be attributed to polarization rotation effects and energy exchange between different pulse polarization components [29,30]. Beyond the ∼7,000^th roundtrip, the two solitons are restored to steady state and propagate with their original different group velocities before the collision.

 

Fig. 5. (a) Optical spectra without LP (grey), LP at 0°(red), 90° (blue) and 45° (orange) at 2 µm. Inset: corresponding optical spectra for LP at 0°, 45° and 90° after SHG at 1 µm. (b) Single-shot recording of the periodically occurring soliton collisions over 120,000 cavity roundtrips. (c) 3-D plot of the spatial-spectral dynamics of the collision evolution. (d) Close-up 2-D contour map during collision. (e) Field autocorrelation function. (f) Energy evolution during collision period. Inset: Zoom-in of the first modulation lobe from the 4270^th to the 4470^th roundtrip indicating a complex multi-period modulation.

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The collision dynamics of the orthogonally polarized fast (top row) and slow (bottom row) pulse trains are recorded in Fig. 6 and can provide additional insights into the evolution of each individual pulse train. We would like to note that the two data sets are sectioned based on slightly different roundtrip times in Fig. 6 for better visualization purposes, focusing on the behavior of individual pulse train. The dynamics of each pulse train are recorded separately, thus their dynamics are not directly correlated to the same evolution. Although each collision evolution is unique and slightly different, we believe that the collision evolution maintains a similar trend, and the sampled evolution of each pulse train well represents this qualitatively in the following. More than 110,000-roundtrips of spectral-spatial dynamics of the faster pulse centered at shorter wavelength (optical spectrum in red in Fig. 5(a), including four collision events, are shown in Fig. 6(a). The pulse propagates at a constant group velocity in steady state, is slowed down during the collision process, and returns to its original group velocity, showing a stair-like spatial trajectory. Similarly, the slower pulse centered at a longer wavelength (optical spectrum in blue in Fig. 5(a)) accelerates during the collision process, as shown in Fig. 6(d). The shift of the center wavelength of both pulses occurs in opposite directions after the collision process. The amount of wavelength shift is slightly different for the two pulse trains due to the different intensities. The spatial-temporal dynamics obtained before the DFT measurements, presented in Fig. 6(b) and Fig. 6(e), illustrate the change in group velocity of the two pulses due to a shift of their center wavelength. Furthermore, the temporal walk-off of the weaker pulse from the dominant pulse after collision is observed together with its annihilation. The energy evolution of each pulse train, cf. Figure 6(c) and Fig. 6(f), matches well with the above described phenomena. At around the 20,000^th roundtrip, when the collision process starts, the energy transfers abruptly from the slower pulse (shown as a dip in Fig. 6(f)) to the faster pulse (shown as a peak in Fig. 6(c)). After this rapid energy exchange, the energy of each pulse decreases below steady state and slowly recovers around the ∼26,000^th roundtrip. During this recovery process, the two pulses experience opposite energy changes, where the humps and dips are reciprocal in the energy evolution plots, similar to observations in [21]. Such an energy redistribution process enhances the nonlinear birefringence difference experienced by the faster and slower pulse, which balances the linear birefringence associated with the shift in center wavelength. The observed energy “loss” from both pulses can be potentially linked to polarization rotation, and the overall energy recovers to steady state after re-stabilization. The two pulses are completely recovered until a point is reached when the overall birefringence, including the accumulated nonlinear birefringence and linear birefringence get too large as the pulses grow.

 

Fig. 6. (a)-(c) Shorter wavelength pulse dynamics. (d)-(f) Longer wavelength pulse dynamics. Left column: spatial-spectral dynamics of pulse after DFT. Middle column: spatial-temporal dynamics during one collision event before DFT. Right column: energy evolution of pulses.

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In this section, periodically occurring explosions induced by collision of dual-wavelength polarization-multiplexed solitons were analyzed. Overall, although our laser is a linear cavity with a SBR, the collision process reported here is similar to that reported in an erbium [21] and a ytterbium [20] based passive mode-locked fiber ring laser with carbon-nanotube and semiconductor saturable absorbers. Despite the different sign of the net cavity dispersion and the different gain medium with different recovery time constants, two solitons with different propagation velocity collide, followed by energy dissipation in various forms, before recovering to their original pulse shape. Due to the repetition period difference between the two pulse trains, the periodic temporal overlap of the two pulses triggers a soliton collision, which gives rise to a subsequent explosion. While a weaker pulse is generated as an energy dissipation process, the two pulses can co-propagate with very small pulse separations coupled by XPM. Thus, the center wavelengths of the two solitons shift towards each other, leading to a temporal shift of both pulses. In this context it is important to note that while DFT measurements can clearly resolve pulse dynamics that are not accessible by conventional pulse recording methods based on average ensemble values. Due to the time-stretched pulses, pulses that are spaced between 20–60 ps (in this case) could potentially be observed as one unit in DFT measurements. Yet, a different recovering process is observed here for the first time. By studying the polarization projection of the two solitons, a damped multi-period modulation manifests energy exchange between the polarization components due to coherent coupling. Such an energy exchange between the two pulses is also observed when studying the individual pulses. The long recovery period (amounting to thousands of cavity roundtrips compared to shorter recovery times in [20,21] where polarization-maintaining fiber or a bandpass filter were used) can be potentially attributed to the different dynamics associated with various gain media and saturable absorbers.

5. Conclusion

Assisted by SH-DFT, the dynamics of chaotic transition states and periodic soliton explosions are observed in real-time in a mode-locked linear Tm-doped fiber laser. A chaotic state is induced for pump powers beyond the single pulsing regime and below a stable double pulsing state that exhibits a quasi-periodic evolution between transient pulsing states. Complex nonlinear dynamics in this chaotic state feature phenomena including period-doubling bifurcation, pulse splitting, soliton explosion, oscillatory behavior and energy dissipation. Due to perturbation induced gain fluctuations, explosions occur stochastically when instabilities are coupled with a weaker secondary pulse. On the other hand, periodic explosions induced by soliton collisions are investigated in a dual-wavelength polarization multiplexed state. For our Tm-based laser system an extended recovery process accompanied by a multi-period modulation is observed. These results enrich our understanding of explosion dynamics in a nonlinear ultrafast laser system and while the collision dynamics in different gain systems follow similar general patterns, distinct differences were presented and discussed.