1. Introduction

Harmonic mode-locking (HML) is a fundamental phenomenon occurring in ultrafast lasers [1–3]. It forces a laser radiating multiple solitons of identical parameters per round trip and thus can easily boost the laser repetition rate, which is attractive to many applications including laser sensing and ranging, optical communication, and frequency comb generation [1,4–8]. On the other hand, it offers an important platform for studying complex dynamics of solitons inside a laser cavity, such as soliton formation, evolution and interactions [9–12]. Over the past decades, a variety of theories have been developed regarding multi-soliton formation and evolution into HML. For instance, under strong pumping strength, multiple solitons could be formed inside a laser cavity, along with non-soliton components (dispersive waves), due to the soliton energy quantization effect [13] which was further explained by Tang et al. as a consequence of intracavity peak power clamping and soliton gain competition [14]. In some other cases [15–17], pulse splitting is responsible for the multiple-soliton generation. For instance, a pulse splitting process was recently observed in the buildup of mode locking [18,19]. In the absence of a significant dispersive wave, the solitons repel each other as a result of time-dependent gain depletion and recovery [20] which impose phase shifts and consequently relative group velocity changes on the solitons [16]. This long-range repulsive interaction of the solitons may eventually lead to HML. However, unveiling the fast-evolving dynamics of a HML laser requires ultra-rapid diagnostic tools especially in the spectral domain

The recent development of the time-stretched dispersive Fourier transform (TS-DFT) technique makes it possible to capture transient moments in soliton lasers [18,21] and has led to a number of intriguing phenomena being observed, such as dissipative rogue waves [22], soliton explosions [23–25], soliton rains [26], breather wave molecules [27], etc. Most recently, with this technique Liu et al. has investigated the buildup dynamics of a long-fiber-cavity HML laser [28], showing a complex lasing process including relaxation oscillation, beating dynamics, single pulse formation, self-phase modulation induced instability, pulse splitting, repulsion and separation, and finally HML pulse locking. Furthermore, the HML pulses are found originating from a giant pulse whose temporal evolution trajectory shows a turning point (or a big corner) as a consequence of intensity reduction induced pulse group velocity change. However, with different laser parameters (e.g., fiber cavity length), the soliton buildup dynamics may appear differently. For instance, it has been reported that the pulse splitting plays a key role in multi-soliton formation in ultra-long fiber lasers, while the interactions between a soliton and dispersive waves are found crucial for generation of multiple solitons in short-cavity (e.g., a few meters long) fiber lasers [14,17]. Thus, it would be interesting to explore the differences of HML in a short-cavity laser and the long-ring-cavity fiber laser in [28].

In this paper, we investigate buildup dynamics of a self-started HML soliton fiber laser using TS-DFT. With a short-cavity laser configuration involving a semiconductor saturable absorber mirror, we observe transient HML dynamics rather different from that in [28]. In particular, we observe Q-switching mode locking, transient stable mode locking, and transient multiple pulses ahead the formation of HML. Besides, we believe that the stable HML solitons in our fiber laser are formed from a transient multi-pulse state rather than a giant pulse in the case of a long-cavity fiber laser [28]. Our results are new to the existing demonstrations of HML lasers and may enrich the understanding of soliton dynamics in ultrafast lasers.

2. Experimental setup

The experimental setup is schematically illustrated in Fig. 1(a). The ring cavity comprises 0.8 m of erbium-doped fiber (Er80-4/125-HD-PM, Liekki) with second-order dispersion of 20.0 ps²/km at 1550 nm and 1.2 m of polarization maintain fiber (PMF, dispersion of −25 ps²/km), resulting in net cavity dispersion of −0.014 ps² (at 1550 nm). The laser with a fiber-coupled semiconductor saturable absorber mirror (SESAM) produces sub-ps pulses at a fundamental repetition frequency of 100 MHz (a cavity round-trip time of 10 ns) under a pump power of 100 mW. When the pump power is increased to 350 mW, the laser repetition frequency is locked to its 4th–order harmonic at 400 MHz. The average output power is 28 mW. The HML emission spectrum is measured with an optical spectrum analyzer (OSA, Yokogawa AQ6370D) at a resolution of 0.2 nm as displayed in Fig. 1(b). The fiber solitons are confirmed with the characteristic Kelly sidebands. The radio-frequency (RF) spectrum of the HML laser is recorded using a RF analyzer (NSA1036-TG, OWON; 100-kHz resolution bandwidth) and shown in Fig. 1(c). The fundamental repetition frequency modes are suppressed by more than 40 dB. In order to regularly study the build-up process of the HML laser, the pump power is modulated with an acoustic-optic modulator (AOM) driven by a square wave generated by a RF signal generator (R&S SMC100A). Meanwhile, real-time spectral acquisition is realized by time-domain stretching the fiber laser pulses in a single-mode fiber (SMF) of 5 km (dispersion of −23 ps²/km at 1550 nm). To avoid nonlinear effects in SMF, the light is attenuated to less than 1 mW before being launched into the fiber. The laser emission is characterized by both direct photodiode detection and TS-DFT with a high-speed digital oscilloscope (Keysight DSAV334A, 33 GHz).

 

Fig. 1 (a) Experimental setup. AOM, acoustic-optic modulator; WDM/OI, wavelength division multiplexer integrated with an optical isolator; LD, laser diode; SMF, single mode fiber; PD, photo-detector; SESAM, semiconductor saturable absorber mirror; OSA, optical spectrum analyzer; OC, output coupler. (b) The output spectrum of the harmonically mode-locked fiber laser and (c) the corresponding RF inter-mode beat notes.

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3. Results and discussion

Figure 2 shows the time trace of the laser output when the pump power is modulated at 0.4 Hz (a duty cycle of 50%). As the pump power being switched off and on, the self-starting fiber laser undergoes a transient stage of giant bursts and then a steady HML state. The lasting time of the transient regime varies from a few milliseconds to several hundreds of milliseconds. Figure 3(a) shows a separate measurement of a portion of this transient regime using a 20-GHz photodetector (VPDV2120, Finisar). An expanded view plotted in Fig. 3(b) reveals that under each burst Q-switched mode-locking states and transient stable mode-locking states evolve alternatively. In the latter states, the laser emits stable pulses (at 10-ns period) only in a short time window. Based on a statistical study of over 100 bursts, we find that the durations of the transient-mode-locking states follow Poisson distribution (peaked at 25 μs, corresponding to 2500 round trips). Interestingly, despite the stochastic nature of the bursts, they are separated by a regular period of 10( ± 0.1) μs, possibly determined by the recovery process of the entire lasing system [20].

 

Fig. 2 Recorded laser output signals using direct photo-diode detection (bandwidth of 1 GHz). The giant Q-switching bursts and the steady HML states are identified.

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Fig. 3 (a) A transient regime recorded in the self-starting process of HML and (b) zoom-in of a single burst in this regime (marked with the red dash box in (a)). (c) TS-DFT results displayed in a two-dimensional view corresponding to the part in the red box of (b).

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Moreover, the rapidly-evolving pulses (marked in red in Fig. 3(b)) with transient spectral dynamics simultaneously characterized with TS-DFT are shown in Fig. 3(c). Complex spectral interference patterns caused by multi-soliton evolution in the short-lived mode-locking state are identified. One of the pulses beats the others and grows into a broadband pulse with a comparatively stable intensity. This competition process resembles that of soliton build-up in a commonly mode-locked fiber laser where only one pulse survives eventually from the multiple pulses [19]. However, after a short while (~10 μs) it suddenly grows into a giant pulse accompanied by rapid spectral broadening, and then decreases with reduced spectral width, and finally vanishes. Such phenomenon presenting in the transient mode-locking state prior to the onset of a HML state has not been reported previously. The first turning point (around the 1700th round trip in Fig. 3(c)) happens possibly because the saturable absorber is bleached by the evolving pulse, which results in a sudden decrease of intracavity loss. The unbalanced gain and loss make the pulse growing rampantly until it saturates the gain and causes gain depletion at the second turning point (~100 round trips after the first one). Without sufficient gain, the pulse dies out quickly.

Figure 4 depicts the formation and temporal evolution of the HML solitons measured by direct photodiode detection. We record the detection signals in a single-shot time window of ~10 ms (corresponding to ~1 million round trips) at a sample rate of 40 GS/s. Because of our limited computation power, we select only parts of the data and plot them in two-dimensional views (Figs. 4(a)-4(e)). As shown in Fig. 4(a), at the beginning of the HML build-up process several pulses arise and unevenly distribute in the cavity; and then one of the pulses outbursts from background fluctuations and evolves into a transient multi-pulse bound state, but appearing as a high-energy pulse in this measurement due to the limited temporal resolution, i.e. ~30 ps. This multi-pulse state is confirmed with spectral interferences (shown in Fig. 5). As the round trips number increases, we record two (Fig. 4(b)), and then three (Fig. 4(c)), and eventually four low-energy solitons (Fig. 4(d)). After about 8 million round trips (~80 ms), the solitons become evenly-spaced inside the cavity (Fig. 4(e)). Note that Fig. 4(e) is recorded in a separate measurement, demonstrating the time evolution of stabilized HML solitons.

 

Fig. 4 Selected views of temporal evolution of the HML solitons. The entire measurement lasts ~10 ms (~1 million round trips). (a) 2000 round trips in the buildup process of a HML state. For the rest, each contains 40000 round trips, starting at a different round-trip number: (b) 130000, (c) 180000, and (d) 380000. (e) A separate measurement for the stabilized HML solitons.

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Fig. 5 Real-time spectral characterization of HML soliton dynamics. (a)-(e) Selected plots of the HML soliton spectral evolution. (f) Spectral comparison of the four evenly distributed HML solitons.

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Further investigations of the transient spectral dynamics of the evolving solitons are carried out with TS-DFT in a separate measurement. Parts of the data are computerized and displayed in separated views (Figs. 5(a)-5(e)). The spectral interference at the beginning of Fig. 5(a) agrees with the region of multiple pulses in Fig. 4(a). However, the interference patterns after 500 round trips in Fig. 5(a) indicate that a multi-pulse bound state instead of a single giant pulse is formed and eventually evolves into the equidistant HML solitons. Similar patterns are observed in different measurements. The spectral evolution of the fiber laser from the multi-pulse state to the HML state is pictured in Figs. 5(b)-5(e). We compare the spectral profiles of different solitons at different round trips and find that the solitons share the same spectral profiles (as shown in Fig. 5(f)). Also, the spectral shape of a soliton keeps the same as it evolves during the self-ordering process. Note that the fast time axis (0-10 ns) in Figs. 5(b)-5(e), in principle, should reflect the wavelength axis in TS-DFT in one-to-one correspondence but not the case here. The additional temporal drifts of the solitons in the cavity caused by their phase shifts complicate the relationship between the fast time and the wavelength in TS-DFT. Therefore, the absolute wavelengths of the transient spectra of the evolving solitons are not known in our case. However, once the HML solitons are stabilized (e.g., in Fig. 5(e)), their spectra can be calibrated to the one in Fig. 1(b). Also, the Kelly sides are not observed in Fig. 5(e) due to the limited dynamic range of the oscilloscope.

In general, the evenly-spaced HML solitons are believed to be a result of the soliton energy quantization effect; and their evolution with relative motions between them has been explained by relative group velocity changes caused by phase shifts of the solitons [16]. The temporal separation of the soliton S1 and S4 as a function of round trips is measured in a larger time scale using direct detection of reduced bandwidth (5 GHz) and plotted in Fig. 6. Note that S1 and S4 are the two solitons with the largest intracavity separation in the self-ordering process. As shown in Fig. 6, the relative movement of the solitons slows down as the separation increases; and eventually it stops at a fixed separation point (~7.5 ns, about three fourth of a round trip), indicating that the self-ordering process has ended. Such behavior is consistent with the theoretical predictions [16,17]. Note that in our SESAM mode-locked fiber laser, other multi-soliton phenomena [29], such as soliton bunching and disordered multiple-solitons, have also been observed.

 

Fig. 6 Temporal separation of two solitons (red) and its first-order derivative with respect to round trips (blue).

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The HML buildup processes here show some different dynamics from that presented in [28]. For example, we observe Q-switching mode locking, transient stable mode locking, and transient bound states of multiple pulses (Fig. 4(a)) during the buildup of HML. These differences could arise from rather different laser configurations between these two works. For instance, the cavity length in our case is rather short (~2 m) comparing to that (~24.3 m) in [28], which as a result induces largely different lasing parameters for the two lasers including intracavity dispersion, nonlinearity, as well as gain and loss competition [14–17]. Besides, the different types of saturable absorbers (the carbon nanotube saturable absorbers used in [28] and the semiconductor saturable absorber in our work) may also contribute to the different HML buildup dynamics observed in the two works. Unfortunately, our investigation is so far limited to the 4th order HML due to the limited pump power. However, very similar lasing behaviors have also been found in the self-starting processes of the 2nd and 3rd order HML states of the fiber laser. The results based on our short-cavity fiber laser indicate that the buildup dynamics of HML may appear differently in varied laser configurations.

4. Summary

In conclusion, we demonstrate spectral and temporal dynamics of a HML fiber laser from the moment of pump light on to the establishment of evenly spaced HML solitons. We record a bunch of stochastic transient mode-locking states before the laser entering a HML state. In each transient pulsing state, the pulses periodically alter from a metastable state to a rapidly-changing state, which is likely caused by bleaching of the saturable absorber and gain depletion. In the HML buildup process, after a short stage of multi-soliton competition, a transient multi-pulse state is formed and then gradually evolves into the steady HML state. Our results show different HML buildup dynamics in a short-cavity fiber laser comparing to that in a long-cavity laser (e.g. in [28]), which could be useful for further theoretical and experimental studies of soliton behaviors in ultrafast lasers.