1. Introduction

Dissipative solitons, which are localized wavepackets formed by a delicate balance of gain/loss, dispersion and nonlinearity/diffraction, have been widely demonstrated in many nonlinear dissipative physical systems, including optical fiber lasers [1–5]. Typically, different types of dissipative soliton states can be formed with distinct pulse-shaping mechanisms in mode-locked fiber lasers depending on the cavity dispersion. For example, pulses generated in an all-anomalous dispersion cavity can form dissipative solitons mainly due to the interplay between anomalous dispersion and nonlinearity [3]. Such pulses are sometimes called ‘conventional solitons’, but here they will be referred to as dissipative solitons with an anomalous-dispersion pulse-shaping mechanism (DSA) to avoid confusion with conservative solitons generated from Hamiltonian systems. The same terminology can also be extended to solitonic pulses generated in a dispersion-managed cavity with a net-anomalous dispersion where pulse breathing occurs but the overall pulse breathing ratio is relatively small [6]. In the net-normal/all-normal dispersion regimes of mode-locked lasers, up-chirped dissipative solitons with characteristic steep spectral edges can be generated and their pulse shaping typically involves more significant energy dissipation effects [4]. Here, they will be referred to as dissipative solitons with a normal-dispersion pulse-shaping mechanism (DSN). While mode-locking has been explored in general in great detail, recently combinations of different phenomena and cavity parameters have allowed the discovery of novel dissipative soliton solutions. Our group previously demonstrated the generation of DSA and DSN in one single laser cavity with a fixed net-anomalous dispersion [5]. Phase-matching-induced close to chirp-free solitons were generated in the same normal-dispersion fiber laser where DSN were obtained [7].

Aside from steady-state mode-locked states with high coherence, which are characterized by the same profile evolution every roundtrip, partially mode-locked states with reduced coherence can exist for a specific range of laser cavity parameters. A wide range of partially mode-locked states have been explored during the past decades since the first demonstration of soliton fiber lasers, including non-stationary states in the form of noise-like pulses [8–12], soliton explosions [13–16], and breathing solitons with chaotic behavior [17–20]. However, the parameter space for these laser pulses during the transitionary evolution of mode-locked states remains fairly unexplored. Nonetheless, it offers an interesting research field which could lead to improved understanding of the underlying physics behind soliton instability (and stability) and better control of laser operations. These partially mode-locked states with reduced coherence share common characteristics including localized temporal structures and a certain degree of stochasticity. Thus, the concept of incoherent dissipative solitons was proposed to encompass all these chaotic-pulse dynamics which can be considered as chaotic attractors of laser cavities [21–23]. Here, this terminology will be used for the transitionary states generated from the laser cavity that feature reduced coherence compared to steady-state mode-locking solutions. Although all these incoherent pulses are of non-stationary nature, they can be distinguished by their unique pulse dynamics and different degrees of pulse coherence. These incoherent dissipative solitons can also be characterized by pulse shaping mechanisms typically observed in fiber laser cavities with different dispersion regime [22]. So far they have not been demonstrated in a single fiber laser.

Various methods have been proposed to initiate different mode-locking states by adjusting laser parameters, including pump level [16,24,25], spectral filtering [26,27], saturable absorption [28], polarization [5,12,29], etc. For cavities featuring nonlinear polarization evolution (NPE), an intracavity polarization controller (PC) can significantly affect the pulse dynamics. For example, by changing the PC settings, three types of localized pulse structures including steady-state dissipative solitons, noise-like pulses and an intermediate state which bears a partial coherence property were obtained in an all-normal dispersion fiber laser mode-locked with NPE [12]. Chaotic events could be induced in pulsating soliton states by fine-tuning the pump power and the intracavity PC of a NPE cavity [20]. In a laser cavity that experiences NPE phenomena, the adjustment of the PCs can change several laser parameters, including the linear cavity phase delay, spectral filtering, net cavity loss, saturable absorption and reverse saturable absorption. As a result, although the effect of the polarization state on mode-locking has been widely acknowledged, the underlying mechanism of the PC-induced mode-locking dynamics can vary between different cavities or even different states and has not been fully understood.

Real-time measurements like time-stretched dispersive Fourier transform (DFT) can offer insights into the transient dynamics of incoherent dissipative solitons, which are challenging to reveal with conventional averaged measurement techniques. However, obtaining sufficient temporal stretching at 2 µm is challenging due to high propagation losses in commercially available fibers. So far, most of the DFT systems have been designed for Yb/Er-doped fiber lasers [16,23,30,31] and only a few studies of laser dynamics in Tm-doped fiber lasers based on DFT [32–34]. However, Tm lasers can provide unique opportunities to explore novel pulse solutions as the gain bandwidth, transient gain dynamics, and fiber dispersions/nonlinearities are different from those of Yb/Er- lasers. Thus, the dynamics of incoherent dissipative solitons at the wavelength of ∼2 µm remain largely unexplored. In this paper, we analyze the pulse dynamics of partially mode-locked states with different types of instabilities that can occur in between two steady-state dissipative soliton solutions (DSA and DSN) in a mode-locked Tm-doped fiber laser cavity with a fixed net-anomalous-dispersion. By only rotating the intracavity PC, the cavity parameters can be tuned between two stable mode-locking states and diverse incoherent dissipative solitons in the partial mode-locking regime are induced (while keeping the pump power constant). With a custom-built dispersive Fourier transform setup, the shot-by-shot dynamics of the 2 µm pulses are captured in real-time. The spectral correlation of the DFT results is analyzed to gain insights into the degree of coherence and the underlying physics associated these incoherent states. The coherence degradation and recovery process during the transition between two steady mode-locked states of a single fiber laser is analyzed, which, to the best of our knowledge, has not been reported before.

2. Experimental setup

In this paper, the real-time dynamics in a Tm-doped ring laser cavity are investigated with the experimental setup shown in Fig. 1. The total cavity length is 15 m (corresponding to a cavity roundtrip time of 73.1 ns), and is composed of 1.2 m of gain fiber (TH512, Coractive) and 5  m of normal dispersion fiber (NDF) (UHNA4, Nufern), while the remaining passive fibers are SMF-28e + . The net cavity dispersion is calculated to be -0.176 ps² based on estimated fiber dispersions values [35], thus falling into a net-anomalous dispersion regime. Although the cavity is dispersion-managed, the dispersion map is imbalanced (towards anomalous dispersion), which limits the pulse breathing ratio in the cavity and thus enables solitonic operation of the laser. Mode-locking is initiated by a saturable Bragg reflector (SBR). The SBR is butt-coupled to one arm of the fiber circulator, which also ensures unidirectional light propagation in the cavity. Due to the slight polarization-dependent loss (PDL) of the circulator, both a birefringent spectral filter and an artificial saturable absorber (NPE) are formed in the cavity, which can assist mode-locking. An intracavity PC facilitates to optimize the laser operation and to alter the mode-locking dynamics. The laser output is obtained from the 75% port of an optical fiber coupler. Overall, the oscillator can generate ultrashort pulses with different profiles from vector DSA to DSN [5].

 

Fig. 1. Schematic diagram of the Tm fiber laser cavity and the custom-built DFT setup for real-time characterization of the 2 µm pulse dynamics. TDF: thulium-doped gain fiber. NDF: normal dispersion fiber. WDM: wavelength division multiplexer. CIR: circulator. SBR: saturable Bragg reflector. C: coupler. ISO: isolator. PBS: polarization beam splitter. AMP: thulium-doped fiber amplifier. L: lens. PPLN: periodically-poled lithium niobate. OSC: oscilloscope.

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To analyze the ultrafast laser dynamics around the wavelength of 2 µm in real time, a DFT setup combined with second-harmonic generation (SHG-DFT) was designed [32]. Due to the vector soliton nature of the oscillator output, the pulses are first decomposed into two orthogonal polarizations which pass through a fiber delay line before recombining into a single fiber with polarizations aligned by an external PC. After propagation in a fiber amplifier, the laser output is then frequency-doubled before being stretched to nanosecond duration in a dispersive fiber spool. A periodically-poled lithium niobate (PPLN) crystal is used to implement the wavelength conversion from ∼2 µm to ∼1 µm via the SHG process. After propagation in a fiber spool with 5.4 km of SMF, the spectrum of the pulse is mapped into the time domain and the stretched pulses are then coupled to a 22 GHz InGaAs fast photodetector and the converted electronic signal is captured by a 20-GHz digital oscilloscope. The spectral resolution of the DFT setup is ∼0.29 nm for a wavelength of 1910 nm with a spectral SNR of >20 dB. Aside from slight spectral narrowing due to the power-squared relationship of the SHG, no significant spectral distortion is observed during both the amplification and SHG stage.

3. Mode-locking transitions

With 550 mW of pump power coupled into the cavity, stable mode-locking operation can be initiated and two types of steady-state pulses are generated in the net-anomalous-dispersion laser cavity – DSA and DSN, at different angles of the intracavity PC. The DSA pulses operate in a multi-pulsing regime, with each pulse featuring a hyperbolic-secant spectral shape and a slight chirp. For the DSN, only a single pulse is circulating in the cavity per roundtrip and the DSN pulses are highly up-chirped with distinct steep spectral edges. The detailed characterization of the DSA and DSN in both temporal and spectral domains is included in the Supplement 1. With the real-time DFT setup, the spectral profiles of both states are analyzed with a frame rate equivalent to the cavity repetition rate and both of them show a high pulse-to-pulse coherence as expected for stable mode-locking, see Supplement 1 for more details. While DSA mode-locking can be initiated for lower pump powers as well, a focus will be on these described states where two different types of dissipative solitons can be generated without changing the pump power.

By solely rotating the intracavity PC angle, different pulse solutions and transitions between DSA and DSN states can be generated for the same pump power condition. During the transition process, the mode-locking is not completely lost and diverse partially coherent states can be generated in a controllable and reversible manner.

Figure 2 shows the evolution of the pulsed laser output with respect to different angles of the intracavity PC. Due to a polarization-dependent loss and fiber birefringence, it is well known that a periodic Lyot birefringent filter can be formed in the cavity, imposing a polarization-dependent filtering effect on intracavity pulses [36,37]. As shown in Fig. 2(a), by monitoring the cavity output power during the rotation of PC, the periodic pattern in the output power indicates the existence of such a Lyot filter. The experimental results are consistent with the theoretical analysis of a Lyot filter modulation depth based on a Jones matrix calculation [37], as depicted in Fig. 2(a) (for details see Supplement 1). The modulation depth of the simulated Lyot filter follows an absolute sinusoidal function with a periodicity of 90° while the output power evolution is quasi-periodic with a similar period. Small discrepancies can exist between the two curves since the output power is also affected by other laser dynamics. For example, the sharp drop of the output power around 85° can be attributed to the complete loss of mode-locking at that PC angle since a continuous-wave lasing state experiences a higher net loss due to an unsaturated SBR.

 

Fig. 2. Pulse evolution and the different states in the laser oscillator when solely the intracavity PC angle is modified. (a) The modulation depth (MD) of a periodic Lyot filter simulated with a Jones matrix calculation (green, top) and the average laser output power with respect to the intracavity PC angle (black, bottom). The color coding is representative for different states. IDS-A: incoherent dissipative solitons (Type-A). IDS-N: incoherent dissipative solitons (Type-N). EXPL: dissipative soliton explosions. (b) Spectral evolution during adjustment of the intracavity PC angle.

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Here, although several other mode-locking dynamics were captured in the full range of the PC angle, we focus on the evolution of mode-locking between the two distinctive types of steady-state pulses (DSA and DSN) in order to gain insights into the transitionary pulsing regimes and their corresponding pulse shaping mechanisms. Three types of incoherent dissipative solitons with characteristic pulsing dynamics are discovered between the DSA and DSN states in the PC angle range of 0°∼76.2°: incoherent dissipative solitons (Type-A, IDS-A), incoherent dissipative solitons (Type-N, IDS-N) and dissipative soliton explosions (EXPL). The terminology of IDS-A and IDS-N is also related to their pulse shaping mechanisms typically observed in distinctive dispersion regimes (net-anomalous/net-normal). In Fig. 2, the three regimes are color-coded within their corresponding PC angle ranges. The corresponding spectral evolution from a DSA to DSN state is presented in Fig. 2(b), revealing spectral broadening/narrowing during the mode-locking transition. In the following sections, a detailed analysis of the different intermediate states is provided.

4. Characterization of incoherent dissipative solitons

4.1 Incoherent dissipative solitons (Type-A)

Starting from the multi-pulsing DSA state, by rotating the PC slightly, the mode-locking is perturbed and a transition occurs from steady-state pulses with high coherence to non-stationary pulses. For a PC angle range of 0° to 36°, a group of dissipative solitons with reduced coherence and similar dynamics are obtained and categorized as incoherent dissipative solitons (Type-A).

In Fig. 3, the dynamics of three representative states are presented in this regime of incoherent dissipative solitons (Type-A) (for a PC angle of 7° for IDS-A1, 10° for IDS-A2 and 28.5° for IDS-A3, respectively). These incoherent states with localized temporal structures are characterized by substantial pulse-to-pulse fluctuations. The spatio-spectral dynamics of the three IDS-A states from the 1^st roundtrip to the 6000^th roundtrip are shown in Figs. 3(a)-(c). For a more straightforward comparison between the different states, a relative time scale is applied by aligning the peaks of the averaged DFT signal to time zero. Due to the time-frequency relation and Fourier transform mapping with the DFT setup, the temporal envelope shown in the plots is representative of the corresponding optical spectrum with the leading/trailing edge of the pulse representing longer/shorter wavelengths. Although the spectral evolution is stochastic and noisy in all three instances, different degrees of randomness are experienced. Depending on the PC angle, various levels of spectral fringe visibility are observed, specifically in the single-shot spectra of these states at the 1000^th roundtrip in Fig. 3(g). It is noticeable that the spectral fringe visibility gradually degrades when the mode-locking deviates further from the multi-pulsing DSA state at 0°. As the spectral interference fringe pattern is directly correlated with its temporal structure, the fringe visibility can reflect the degree of partial ordering in the temporal domain, i.e., the coherence level of the pulses. In general, the spectrum of a stable temporal soliton molecule has spectral fringes with a high modulation depth. In contrast, spectral fringes disappear for a completely stochastic pulse bunch like noise-like pulses, due to the randomly distributed temporal sub-structure [11]. The degradation of spectral visibility from IDS-A1 to IDS-A3 suggests that these pulses gradually lose their coherence and become more stochastic with increasing PC rotation angle.

 

Fig. 3. Characterization of incoherent dissipative solitons (Type-A). (a)-(c) Real-time spatio-spectral dynamics and (d)-(f) corresponding spectral intensity correlation maps (plotted with respect to the peak wavelength ${\lambda _0}$) for three representative states in this regime: IDS-A1, IDS-A2 and IDS-A3. (g) Single-shot spectrum for each of the three states at the 1000^th roundtrip, offset in intensity. (h) Average field autocorrelation function over 6000 roundtrips. Inset: zoom-in of the ACF pedestal.

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The field autocorrelation functions (ACFs) of these states are calculated from the Fourier transform of the DFT traces according to the Wiener–Khinchin theorem and the ACF for each state averaged over 6000 roundtrips is plotted in Fig. 3(h). As reflected by the zoom-in plot of the ACFs, the maximum and shape of the pedestal varies between different states. The position of the side peaks in the ACFs can reflect the temporal separation between sub-pulses within the pulse bunch unit independent of the pulse chirp, offering insights into the temporal profile of these states. The ACF of IDS-A1 with an obvious single-peak structure in the pedestal shows a relatively strong transient partial ordering, resembling the coalescence of well separated droplets in liquids [23]. For IDS-A2 and IDS-A3, the pedestal is more spread out, indicating that the distribution of those sub-pulses becomes more stochastic with a weaker transient ordering. In other words, the pulse-to-pulse coherence degrades, which agrees well with our observation in the spectral domain. By measuring the autocorrelation (AC) signal of these states with a commercial intensity autocorrelator, a similar trend is observed.

The spectral intensity correlation maps of these three states, shown in Figs. 3(d)-(f), reveal the intensity fluctuation relation between any two wavelengths in the spectrum. While such an analysis has been more commonly applied in the study of supercontinuum generation [38,39], this is believed to be the first time that spectral intensity correlations are used to analyze the transitionary intracavity pulse dynamics of laser states. Aside from revealing the noise features, it also provides insights into the physics behind the energy transfer between different wavelength components [38]. Without requiring an additional experimental coherence characterization setup like spectral interferometry [9], the coherence feature of these partially mode-locked states can be resolved by the spectral intensity correlation function generated from the ensemble DFT data.

The spectral intensity correlation is calculated from an ensemble of shot-by-shot spectra within 54746 roundtrips captured by the SHG-DFT setup. The spectral intensity correlation function between any two wavelengths ${\lambda _1}$ and ${\lambda _2}$ is given by,

From IDS-A1 to IDS-A3, the incremental increase in stochasticity and the loss of coherence is revealed by the larger close-to-zero correlation region (dark) in the correlation map. For the state of IDS-A3, the spectral coherence is fairly low with most wavelengths being noise-seeded and uncorrelated, thus resembling noise-like pulses [40]. Combining the above analysis, a gradual coherence degradation process from coherent multi-pulsing DSA state to more stochastically distributed incoherent pulses occurs by rotating the intracavity PC.

For laser cavities with artificial saturable absorbers like NPE, either positive or negative feedback can be obtained depending on the linear cavity phase delay, which can be adjusted by polarization controller settings or waveplate orientations. It has been demonstrated that soliton collapse can be induced in NPE cavities within the net-anomalous-dispersion regime to generate noise-like pulses at certain waveplate orientations [41]. The level of soliton collapse can be adjusted with fine-tuning of the linear cavity phase delay in the laser. Numerical simulations show that intracavity pulses can experience a partial loss of phase coherence due to limited soliton collapse/pulse destruction in the NPE cavity at certain intracavity PC angles [42]. In our case, by rotating the polarization controller, we experimentally verified that the strength of the soliton interaction in the cavity can be adjusted and various partially coherent states in the IDS-A regime are obtained with varying degrees of randomness. With a more significant soliton collapse effect, the interaction between the stochastic soliton-like subpulses within the bunched pulse complex increases and quasi noise-like pulses like IDS-A3 can be obtained from the same cavity.

4.2 Incoherent dissipative solitons (Type-N)

For an intracavity PC angle range of 40° to 71°, another group of incoherent dissipative solitons (Type-N) with characteristic dynamics can be generated. The detailed dynamics of three typical states (IDS-N1, IDS-N2, IDS-N3) in this regime are shown in Fig. 4, with corresponding PC angles of 40°, 56° and 71°, respectively. Different from the IDS-A regime, all three states feature strong dissipative dynamics like soliton pulsation and soliton explosions, as illustrated in Figs. 4(a)-(c). Although the average spectra captured by the optical spectrum analyzer feature similar bandwidths for the IDS-A and IDS-N states, as depicted in Fig. 2(b), their shot-by-shot spectra show a significant difference in bandwidth. This emphasizes the importance of real-time measurements to reveal the fundamental pulse dynamics that traditional averaged characterization techniques fail to capture.

 

Fig. 4. Characterization of incoherent dissipative solitons (Type-N). (a)-(c) Spatio-spectral dynamics and (d)-(f) corresponding spectral intensity correlation maps of IDS-N1, IDS-N2 and IDS-N3. (g) 130-roundtrip 3-D plot highlighting the complex breathing and explosion instabilities in the region surrounded by a light blue box in (a) and four single-shot spectra at points indicated by arrows. (h) 420-roundtrip 3-D plot featuring a transient quasi-stable state followed by instabilities in the region surrounded by a light blue box in (c) and four single-shot spectra at points indicated by arrows.

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In Figs. 4(d)-(f), the corresponding spectral intensity maps of IDS-N states reveal the noise features and are indicative of the energy flow in the spectral domain. In general, the regime of IDS-N is characterized by a higher spectral correlation compared to that of the IDS-A states. Due to the pulsation and explosion instabilities, the wavelengths shorter than the peak wavelength become positively correlated with the longer ones. Although these IDS-N states all experience dissipative instabilities, they feature very unique pulse dynamics. Comparing the spectral intensity maps of these three states, from IDS-N1 to IDS-N3, the shape of the positive-correlation region switches from spindle-shaped to almost square-shaped. In addition, the partially mode-locked states gradually gain a higher shot-to-shot stability. The 3D spatio-spectral evolution of the state IDS-N1 and IDS-N3 and their corresponding single-shot spectra are plotted in Figs. 4(g) and 4(h), respectively. The state IDS-N1 is characterized by a high spectral breathing ratio of larger than four and strong explosion instabilities, resulting in highly structured spectra during the whole evolution. In comparison, for IDS-N3, the explosion instabilities still exist while the spectral breathing is less pronounced. As a result, the spectra during these dissipative soliton explosions are more confined compared to the state IDS-N1. This explains the different patterns in their corresponding spectral intensity correlation maps. Another significant difference between the two states is that the explosion instabilities in IDS-N1 are almost consecutive without any sign of transient stability, similar to ‘successive soliton explosions’ [43]. However, IDS-N3 can frequently recover to a quasi-stable state with an unstructured and smooth spectrum, as presented in Fig. 4(h) from the 1620^th to the 1720^th roundtrip. In addition, the spectra of IDS-N3 state undergoes red shifting during the onset of explosions, featuring a higher level of asymmetry.

It has been numerically verified that the quintic nonlinear gain/loss coefficient plays an important role in soliton pulsations in dissipative systems [44]. Experimentally, this suggests that laser cavities with a NPE mode-locker can generate pulsating solitons as reverse saturable absorption (the quintic nonlinear loss term) can be engineered with the adjustment of the polarization states in the cavity [45]. In our cavity, at a fixed pump level, by rotating the polarization controller, nonlinear polarization phenomena can lead to different reverse saturable absorption, such that strength of the pulse breathing dynamics is adjusted. The transition from consecutive soliton explosions with strong breathing (IDS-N1) to intermittent explosion instabilities with transient ordering (IDS-N3) can thus be explained. When the pulse breathing ratio is high, the development of strong self-phase modulation (SPM) leads to a broader spectrum with a high level of symmetry and spectral structures, arising from soliton explosion instabilities [17]. When the pulse breathing is weaker or fully suppressed, the spectral broadening is reduced and the spectra during the onset of the soliton explosions are more confined spectrally and asymmetric. Such a difference in the spectral symmetry during soliton explosions is expected when different types of perturbations are present in the laser. More specifically, the quasi-symmetric explosion instabilities of state IDS-N1 can be induced by pulse collapse due to SPM while the asymmetric explosion instabilities of state IDS-N3 can be triggered by multi-pulsing instabilities since the explosion events are accompanied by patterns of transient dissipative soliton molecules [14,46], as shown by the dynamics from 1740^th to 1900^th roundtrips in Fig. 4(h). This first demonstration of the real-time characterization of these two distinctive types of dissipative soliton explosions in our Tm-fiber laser cavity agrees well with a numerical model of explosion dynamics for an Er-doped laser [47].

4.3 Dissipative soliton explosions

Incoherent states dominated by dissipative soliton explosion dynamics are obtained when the mode-locking states approach the steady-state DSN regime for a PC angle range of 72° to 76°.

The dynamics of three representative dissipative soliton explosion states (EXPL1, EXPL2, EXPL3) are shown in Fig. 5 for the corresponding PC angles of 73°, 74° and 76°. As shown in Figs. 5(a)-(c), the spatio-spectral dynamics of these EXPL states are characterized by quasi-periodic dissipative soliton explosions with different explosion rates. Different from the IDS-N states with composite dynamics, no other instabilities exist in the EXPL regime and the quasi-stable states in between the dissipative soliton explosions feature a much longer lifetime.

The zoom-in plot of one complete explosion events of EXPL3 from the 500^th roundtrip to the 4500^th roundtrip is shown in Fig. 5(g). The explosion dynamics can be divided into three representative evolution stages: I) The pulse gets destabilized and soliton explosions occur, while the spectrum becomes highly structured featuring higher peak intensities. II) The pulse gradually builds up again and undergoes damped relaxation oscillation dynamics. III) The pulse gets stabilized and the spectrum recovers. This quasi-stable state is maintained until the next explosion occurs. The corresponding ACF in Fig. 5(h) shows the temporal evolution during the soliton explosion. Consistently, during the onset of explosion, the pulse abruptly collapses, sheds energy, followed by an oscillation recovery process until a quasi-stable state is reached.

During stage I), a spectral interference pattern can be observed clearly, suggesting that the dissipative soliton explosions are also induced by multi-pulsing instabilities, in the transition regime between a stable single-pulsing and a multi-pulsing state. Here, we demonstrate that the rate of dissipative soliton explosions can be adjusted by the modulation depth of the intracavity filter. The modulation depth of the Lyot filter can be tuned by rotating the intracavity PC, as clearly presented in Fig. 2(a). From EXPL1 to EXPL3, the average interval between adjacent dissipative soliton explosion events increases while the spectra keep narrowing, as shown in Figs. 5 (a-c). To quantify the effect of the filter on explosion instabilities, a dissipative soliton explosion rate ${R_{expl}}$ is defined as the inverse of the average roundtrips between adjacent explosions. The filtering strength in the cavity is represented by the inverse of 10-dB spectral width $\lambda $. As shown in Fig. 5(i), the explosion rate decreases linearly with respect to $1/ \lambda $. It is expected that a stable DSN state can be obtained with a stronger filtering effect by further increasing the modulation depth. For a PC angle of 76.2­°, any instabilities can be fully suppressed in the cavity and stable single-pulsing DSNs are generated with the spectral width reaching the narrowest point.

The effect of a filter on the dissipative soliton explosion instabilities is further confirmed with numerical simulations based on the generalized complex cubic-quintic Ginzburg-Landau equation (CQGLE) [48]. This continuous model has been widely used to search for pulse solutions of passively mode-locked lasers and other complex pulse dynamics and has an analog in the corresponding lumped laser model [49]. For example, the parameter $\beta $ can be directly related to the intracavity spectral filtering strength. The spectral filter transmission T in the CQGLE model is $T(\omega )= {e^{ - \beta {\omega ^2}}}$ without considering the linear net gain/loss, where $\omega $ is the angular frequency of the light. With other parameters of the simulation model fixed, $\beta $ is adjusted such that different dissipative soliton explosions states induced by multi-pulsing instabilities can be generated (See Supplement 1 for more details). Figure 5(i) summarizes the evolution of the explosion rate ${R_{expl}}$ with respect to the filter strength in the simulation. ${R_{expl}}$ decreases fairly linearly and is suppressed from 0.13 to 0.02 as the filtering strength increases, consistent with our experimental findings. Thus, the theoretical model can be used to analyze the evolution of the explosion dynamics in the anomalous dispersion regime.

Due to the suppressed explosion instabilities and increased lifetime of quasi-stable states, the spectral intensity correlation maps, see Figs. 5(d)-(f), show an increasing positive correlation across the spectral range from EXPL1 to EXPL3. At the same time, the spectral range of the correlation map also becomes narrower. Since the spectra during the explosion events are fairly confined within the spectral range of the quasi-stable state, the intensity correlations maps feature sharp edges, in contrast to that of the IDS-N1 states.

 

Fig. 5. Characterization of dissipative soliton explosions. (a)-(c) Spatio-spectral dynamics and (d)-(f) corresponding spectral intensity correlation maps of three states EXPL1, EXPL2 and EXPL3. (g) Zoom-in plot of the area surrounded by the blue box in (c), showing details of one complete explosion cycle. (h) The corresponding field autocorrelation function of (g). (i) Experimental dissipative soliton explosion rates with respect to the inverse of pulse spectral width $\lambda $ (top) and simulated explosion rates with respect to the parameter $\beta $ in the CQGLE model (bottom). $\beta > 0$ accounts for the spectral filtering effect.

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5. Coherence evolution of mode-locking states

Compared to other cavities with large polarization-dependent losses, the Lyot filtering effect in our cavity is moderate due to the lack of a real polarizer, which can be confirmed by the modulation depth of the output power evolution in Fig. 2(a). This leads to a unique feature of our cavity – complete mode-locking or partial mode-locking can be sustained for most angles of the PC, enabling the generation of diverse steady-state and intermediate pulsing regimes. Typically, it can be challenging to reach many non-stationary states predicted numerically by the CQGLE model in experiments due to their sensitivity to the various cavity parameters [50].

As presented in the above sections, the different transitionary states between stable mode-locking of DSA and DSN show quite distinctive pulse dynamics and degrees of coherence. Although the corresponding mechanisms of these states have been analyzed in detail, a full picture of the transition process from one steady-state regime to another in the same cavity is incomplete. Especially, a question arises naturally: how does the coherence in the cavity evolve during the transition? In previous studies on supercontinuum generation, spectral intensity correlation maps were used to reflect the degree of coherence degradation during noise-driven supercontinuum generation [39]. Here, we analyze the coherence evolution during the transition between DSA and DSN pulses by an average spectral correlation ${r_{ave}}$. This is defined as the mean value of the spectral intensity correlation function $\rho ({{\lambda_1},{\lambda_2}} )$ in the 15 dB spectral-bandwidth range from the peak of the optical spectrum. The average spectral correlation can reveal significant variations of the coherence between different states. When a state is completely stochastic, the average spectral correlation is zero. For states with higher partial coherence, the average spectral correlation increases correspondingly since the dynamics are not completely driven by noise. As shown in Fig. 6, during the full transition process from a DSA state (PC angle of 0°) to a DSN state (PC angle of 76.2°), the evolution of the spectral correlation is clustered into two main groups: (I) Low coherence region for a PC angle range between 0 to 36°. A correlation degradation occurs when the mode-locking deviates from DSA to IDS-A (color-coded in red). (II) High coherence region for a PC angle range of 40° to 76.2° accompanied by a correlation recovery process, since the mode-locking is restored from IDS-N states (color-coded in light blue) and EXPL states (color-coded in darker blue) to a DSN state.

 

Fig. 6. Evolution of average spectral correlation with respect to the intracavity PC angle. Color coded regions: A low coherence region with degradation in correlation during the transition from DSA to IDS-A (red). A high coherence region with correlation increase within the IDS-N states (light blue) and correlation recovery to DSN state with the suppression of dissipative soliton explosion instabilities (blue).

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This analysis confirms the qualitative results observed directly from the real-time measurements and in addition reveals the coherence evolution (de-coherence and coherence-recovery) during the transition process between two stationary pulses of the cavity. The stable multi-pulsing DSA state features a high coherence with fixed pulse separations. For the IDS-A states, a pulse bunch with different degrees of randomness is formed in each roundtrip. The sub-pulses within the pulse bunch gradually experience stronger instabilities like soliton collapse and soliton collisions, leading to an increasing stochasticity and almost complete degradation of coherence, as reflected by the average correlation value of only 0.11 for the state IDS-A3. For the IDS-N states, as the pulse breathing becomes weaker and the explosion events become less frequent, the nature of the instabilities turns from almost consecutive pulsation and explosions to intermittent explosions with transient quasi-stable states. During this transition, the spectral correlation gradually increases from 0.47 (IDS-N1) to 0.55 (IDS-N3). The following coherence recovery can be summarized as the suppression of dissipative soliton explosions by a stronger spectral filtering effect. From EXPL1 to EXPL3, the pulses acquire a higher coherence rapidly (from 0.53 to 0.63) in a narrow PC angle range until the stable single-pulsing DSN state is reached.

Aside from the above transition dynamics, another interesting evolution occurs at the boundary between the IDS-A and IDS-N states. As shown in Fig. 6, a large gap in the spectral correlation is observed around a PC angle of 38°. Such an abrupt transition in the pulse coherence indicates a change in the dominant pulse-shaping mechanism in the cavity. This is confirmed by the DFT results: The IDS-A states feature a stochastic but relatively stationary profile while IDS-N states are characterized by complex dissipative dynamics. It is well known that different pulse-shaping mechanisms exist in various dispersion regimes of mode-locked fiber lasers. Typically, soliton pulse shaping dominates in a net-anomalous dispersion regime [3] while dissipative dynamics can be more pronounced in a net-normal dispersion regime [4]. In our cavity, the mode-locking states can switch seamlessly between two steady-state pulses with high coherence dominated by distinctive pulse shaping mechanisms typically obtained in the two different dispersion regimes. During the mode-locking evolution, it is expected that the transitionary states will also experience different characteristic pulse shaping mechanisms. Actually, the dynamics of the IDS-A and IDS-N states are very similar to those of incoherent dissipative solitons reported in a net-anomalous and net-normal dispersion, respectively [22]. In general, due to soliton pulse-shaping in the anomalous dispersion regime, a chaotic bunch of soliton-like pulses can form a pulse complex while they also experience stochastic interactions like soliton collapse and collisions. In the normal dispersion regime, during the pulse build-up process, instead of forming multiple soliton-like pulses, only one pulse is amplified into an incoherent dissipative soliton which experiences strongly turbulent dissipative dynamics. These distinctive features can be used to identify the dominant pulse shaping mechanism in our laser cavity.

Thus, we suggest that the coherence difference arises from two distinct pulse-shaping mechanisms of the partially mode-locked states. When soliton pulse-shaping dominates (IDS-A), due to the stochastic feature of the sub-pulses within the pulse bunch, there is only a weak correlation between different wavelengths. In contrast, only one pulse with random spikes growing on the same base will propagate in the cavity for the incoherent dissipative solitons with a pulse-shaping mechanism dominated by dissipative effects (IDS-N). As a result, although the IDS-N states experience complex dissipative dynamics, their spectral correlation is higher than that of the IDS-A pulses. It is believed that the abrupt coherence change corresponds to the transition point where the complex of soliton-like sub-pulses converts into a single incoherent pulse dominated by dissipative effects.

6. Conclusion

To conclude, by virtue of SHG-DFT, we investigated the diverse pulse-by-pulse partial mode-locking dynamics in a versatile mode-locked Tm-doped fiber laser. By solely rotating the intracavity polarization controller, three groups of incoherent dissipative solitons were formed during a continuous transition between two steady-state dissipative soliton solutions (DSA and DSN) of the cavity. We studied the effect of the intracavity polarization controller on the cavity parameters to gain deeper insights into the transition dynamics of different mode-locking states, including: I) soliton-like pulse bunches with varying degrees of randomness affected by the linear cavity phase delay; II) recovery of transient coherence from consecutive pulse instabilities by adjusting the reverse saturable absorption; III) suppression of dissipative soliton explosions induced by a birefringent spectral filter. With the statistical analysis of the spectral intensity correlation, we experimentally resolved the complete pulse coherence evolution between two steady-state pulses in a single ultrafast fiber laser for the first time, including coherence degradation followed by a recovery process. Aside from the steady-state operation of the laser, the intermediate transitionary states are also dominated by distinctive pulse shaping mechanisms typically reserved for different dispersion regimes. We believe the universal concept of spectral intensity correlation can be extended to analyze other laser dynamics. The results here can be of great significance to understand the role of intracavity polarization controller and to guide the future design of versatile mode-locked fiber laser sources. By providing deeper insights into the transitionary laser physics, our findings can also stimulate the discovery of novel mode-locking dynamics in ultrafast fiber lasers.