1. Introduction

Ultrafast mode-locked fiber lasers as efficient pulse sources have been applied in diverse fields from metrology to material fabrication [1–3], and are also interesting nonlinear systems with rich dynamics during their static or transient pulse generation processes. Their output pulses often have the same temporal and spectral profile after each roundtrip when the laser operating in a stable regime. However, under certain conditions, various types of complex soliton nonlinear phenomena have been demonstrated [4–6]. While in many earlier experiments only the pulse energy variations of these dynamic phenomena could be investigated, more recently dispersive Fourier-transform (DFT) as an ultrafast real-time technique can be leveraged to resolve the spectro-temporal dynamics in lasers [7,8]. This relatively simple but powerful technique has been applied to the study of various phenomena, such as soliton explosion [9,10], build-up and transient dynamics [11,12], soliton molecules [13,14] and dissipative soliton resonance [15].

Pulsating solitons as one of the nonlinear dissipative phenomena have brought about numerous theoretical investigations [5,6,16]. Based on those theoretical predictions, a variety of pulsating solitons, such as plain pulsating, period doubling, extreme soliton pulsations, could appear in a mode-locked laser. The pulse energy modulations of several pulsating soliton states had been clearly observed [17–20]. Recently, the transient spectral dynamics of single- and double-periodic pulsating solitons have been observed using DFT [21], as well as ‘invisible soliton pulsation’ with no obvious pulse energy variation [22]. A few mechanisms had been attributed to the generation of some of these states, such as the intracavity energy and nonlinear effect [21], gain dynamics [19,20], and relaxation of gain and saturable absorption [17]. However, systematic experimental studies of the spectro-temporal dynamics of other pulsating soliton states as well as their controllable means of generations through schemes like spectral filtering [5] are still needed.

In this paper, we present the experimental generation and observation of various types of soliton pulsation from a dispersion-managed hybrid mode-locked fiber laser. By leveraging the DFT technique, the dynamics of different pulsating soliton states are revealed through the recorded ultrafast real-time spectra evolutions of the pulses.

2. Experimental setup

Figure 1 shows the schematic of the hybrid mode-locked Er-doped fiber laser. The all-fiber laser consists of a hybrid wavelength division multiplexer/isolator (WDM/ISO), a piece of 0.46-m-long EDF forward pumped by a 980 nm laser diode, a polarization controller (PC, PLC-202), a 40% fiber output coupler (OC), an in-line polarizer (ILP) with 35 cm-long polarization-maintained fiber (PMF) pigtails, a single-wall carbon nanotube (SWNT) modelocker, and a section of dispersion compensating fiber (DCF). Due to presence of the potential nonlinear polarization rotation (NPR) effect in fiber and the polarizer, the pulse formation process in the cavity could be simultaneously affected by the NPR effect and the physical mode-locker. The birefringence of PMF introduces spectral filtering effect with a ∼ 15.9 nm period for intracavity spectral control [23]. The cavity length is ∼ 4.32 m and the total group velocity dispersion (GVD) is tuned to −0.05 ps²/km by tailoring the length of DCF.

 

Fig. 1. Schematic of the fiber laser and the experimental setup.

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As depicted in Fig. 1, the characterization of the laser output is performed by using an optical spectrum analyzer (OSA), an RF spectrum analyzer (RSA) as well as a DFT setup. While the OSA and RSA give the conventional, averaged optical spectral and RF spectral information, the DFT setup directly monitor the ultrafast spectral and temporal evolution of the pulse sequence. It is implemented by temporally stretching the ultrafast pulse in a spool of 14-km-long single-mode fiber (SMF) with a total dispersion of ∼231 ps/nm, and the pulses are then fed into a photodetector (PD) with 18 GHz bandwidth followed by a high-bandwidth, real-time oscilloscope (Tekronix MSO72004). The sampling rate of the oscilloscope is set to 50 Gs/s with its maximum record length under a measurement time window. The shot-to-shot spectral resolution is estimated to be ∼0.26 nm for this DFT configuration.

3. Experimental results on different pulsing dynamics

3.1 Stable mode-locking state

By properly adjusting the pump power to the mode-locking threshold of ∼190 mW and the position of the intracavity PC, a mode-locked optical spectrum with an output power of 1.42 mW is observed and shown in Fig. 2(a). Note that the pump power is then fixed at ∼190 mW in the subsequent experiments. The corresponding full-width at half maximum (FWHM) of the spectrum is ∼41 nm. The fundamental repetition frequency of the laser cavity is ∼46.63 MHz, as seen in Fig. 2(b). The RF spectrum shows no noticeable side peaks other than those at the harmonics of the fundamental frequency. This indicates a stable, time-invariant mode-locking state. To further validate the shot-to-shot behaviors of the mode-locked operation, the real-time DFT measurement is performed. Thirty shot-to-shot spectrum profiles are presented in Fig. 2(c), and it can be seen that each of them have the similar shape. Figure 2(d) also illustrates the shot-to-shot spectra over 150 roundtrips, and the pulse energy retrieved from the spectral profile remains relatively stable as seen in Fig. 2(e). The averaged DFT spectrum over 1800 consecutive single-shot measurements shown in the inset of Fig. 2(a) matches the result from the OSA very well, which confirms the accuracy of our DFT setup. These results show that the solitons have uniform intensity without any pulsating behavior, indicating that the fiber laser is operating in a stationary mode-locking regime.

 

Fig. 2. Output characteristics of the stable mode-locking state. (a) Log-scale, time-averaged mode-locked spectrum. Inset, comparison between the linear OSA spectrum and the averaged DFT spectrum; (b) RSA output; (c) 30-roundtrip shot-to-shot spectral profiles; (d) 150-roundtrip shot-to-shot spectra; (e) the pulse energy evolution of that in (d).

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3.2 Pure pulsating soliton and pulsating soliton with period-doubling state

It had been suggested that various pulsating soliton states could be generated by changing the intracavity spectral filtering effect in previous theoretical studies [5]. In the laser configurations similar to this one, it had been shown that, with the presence of the ILP with PMF, the lasing spectral profile can be tuned by adjusting the intracavity-loss-dependent gain spectrum [24] and the spectral filtering effect [23]. Therefore, in our experiment, as shown in Fig. 3(a), when the knob of intracavity PC is further rotated by over 15° that varies the incident angle between the input polarization and the slow axis of the fiber waveplate, a different mode-locked state can be observed. The averaged spectrum with a ∼38 nm 3-dB bandwidth can be observed at an output power of ∼1.38 mW. Several peaks are symmetrically located around the fundamental repetition frequency with a separation of 1.7 MHz in the RSA spectrum, indicating an operation in the ‘pure’ pulsating soliton regime [21]. In the shot-to-shot spectra of 30 roundtrips in Fig. 3(c), for the first few ones, the spectrum shape is relative stable, and then the soliton expands its spectral width into a significantly different shape with much increased pulse energy, before it gradually breathes and self-reshape back to the original state. Compared to the stable mode-locking state, the soliton’s spectrum has a peak at a longer wavelength and a weaker peak on the shorter wavelength side. There is a slight overall shift to the longer wavelength too. Figures 3(d)–3(e) show the spectrum evolution over 150 roundtrips and the corresponding pulse energy, respectively. The soliton energy varies with a ∼27-roundtrip period, consistent with the 1.7 MHz spacing in Fig. 3(b). The large (∼3 dB) variation in the pulse energy is also in line with the theoretical predictions [6].

 

Fig. 3. Output characteristics of pure pulsating soliton state and pulsating soliton with period- doubling state. (a) and (f) Log-scale, time-averaged spectrum; (b) and (g) RSA spectrum; (c) and (h) 30-roudtrip shot-to-shot spectral profiles; (d) and (i) 150-roundtrip shot-to-shot spectra; (e) and (j) the pulse energy evolution of that in (d) and (i), respectively.

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Furthermore, by subtly further adjusting the PC by a few more degrees, a mode-locking state with a slightly different averaged spectrum with 3-dB bandwidth of 38 nm can occur, as shown in Fig. 3(f). Though the averaged optical spectrum looks very similar to what is shown in Fig. 3(a), the pulse RF spectrum shows a significant change. Sidebands with ∼ 2.1 MHz equal spacing around the fundamental frequency are present in Fig. 3(g). Furthermore, frequency components with the same frequency spacing are located exactly around the half of the fundamental frequency. To illustrate the spectro-temporal dynamic of the pulses, 30- and 150-roundtrip DFT spectral profiles are shown in Figs. 3(h)–3(i), respectively. It can be seen that, periodically, the soliton gradually gets stronger and then returns back to its original state in ∼ 22 roundtrips, in accordance with the 2.1 MHz sideband spacing in the RF spectrum. With a similar overall trend to what is discussed in the previous paragraph, what is noticeably different is that the constant switching in the position of the maximal spectral peak between the shorter and longer wavelength relative peaks after each roundtrip. Figure 3(j) shows that, besides the large, slow modulation similar to that in Fig. 3(e), there is also a smaller and faster modulation after each roundtrip. We note that a similar soliton energy modulation pattern of pulsating soliton with period-doubling had been observed before [5]. Yet, the corresponding spectro-temporal dynamics had not been experimentally measured, to our best knowledge.

3.3 Short-period, period-doubling soliton state

When further tuning the PC by more than 10° at the same pump power, another state, the short-period, period-doubling soliton state [5], can be achieved. It is noted that the transition between these four states occur smoothly with the laser kept mode-locked and can be reversed if PC is tuned in the opposite direction. Compared with the stable mode-locked spectrum in Fig. 2(a), the measured OSA spectrum shows some ripples as seen in Fig. 4(a). It has a 3-dB bandwidth of 24 nm and the output power of ∼ 1.34 mW. In the RF spectrum of the laser output, there is a new frequency component of ∼ 23.32 MHz without other side peaks, besides the fundamental one. The new frequency component has 36 dB signal to noise ratio (SNR), comparing with ∼63 dB SNR of the fundamental frequency. This indicates that the laser is in a stable period-doubling state. The transient behaviors in 30 and 150 roundtrips are shown in Figs. 4(c)–4(d). An interesting stable switching between two spectral peaks at 1554.5 nm and 1569.2 nm in every two roundtrips is observed. Their 14.7 nm spacing is quite close to the estimated spectral filtering period of ∼15.9 nm, considering the uneven gain profile, indicating potential influence of the filtering effect on the occurrence of this state. The pulse energy also experiences periodic modulation with a modulation depth of ∼12%, as shown in Fig. 4(e). To our knowledge, this is the first experimental observation of this type of period-doubling dynamics that reveals highly periodic oscillation in both pulse spectrum and energy.

 

Fig. 4. Output characteristics of the short-period, period-doubling state. (a) Log-scale, time-averaged spectrum; (b) RSA output; (c) 30-roundtrip shot-to-shot spectral profiles; (d) 150-roundtrip shot-to-shot spectra; (e) the pulse energy evolution of that in (d).

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4. Numerical simulations and discussions

To provide further insight into the observed dynamics of pulsating solitons and the generation of such states in the presence of the intracavity spectral filtering effect, numerical simulations based on a modified nonlinear Schrödinger equation (NLSE) are performed. Since the pump power of the laser is kept constant in our experiments, the following simulation parameters are used, to be close to the known experimental conditions: cavity length L=4.32 m, small-signal gain coefficient of the gain fiber G=5.32, gain bandwidth Ω_g= 0 nm, nonlinear parameter γ=3 W⁻¹km⁻¹, and the GVD parameters of SMF, EDF and DCF are −22, 16.5 and 52 ps²/km, respectively. The only experimental condition that is varied is the PC position, and therefore the Lyot filtering effect is considered in the model. It is modeled as a sinusoidal filtering function [23], whose modulation depth can be adjusted. Without the filtering effect by setting the modulation depth to 0, our simulation shows that a stable mode-locking state with a spectral width of 43 nm can be achieved. The soliton energy Q [16] by integrating the spectral power remains constant for each cavity roundtrip, as shown in Fig. 5(a). Figure 5(c) shows the simulated spectral dynamics exhibits no changes in each roundtrip as expected.

 

Fig. 5. Simulation of the stable soliton and pulsating soliton states. Soliton energy of (a) the stable and (b) pulsating soliton; spectral dynamics of (c) the stable and (d) pulsating soliton.

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When the modulation depth is set to 3.08%, the soliton pulsating behavior can be observed in the simulation, as shown in Figs. 5(b) and 5(d). To account for the slight change in the cavity loss and the gain tilt [24] with the added filtering effect, the peak of the gain profile is also shifted by ∼1.8 nm in the model. The period of the simulated pulsating solitons is ∼27 roundtrips, similar to our experiments. The periodic switching between two spectral peaks in the spectrum can be seen, which bears similarity to our experimental results. It is noted that the simulations are to qualitatively account for the effect of spectral filtering on the soliton dynamics rather than to provide an accurate quantitative comparison to the experiment results.

5. Conclusions

We experimentally realize the generation of pure pulsating soliton, pulsating soliton with period-doubling and period-doubling soliton from the same laser in a repeatable way. Based on DFT measurements, the ultrafast spectro-temporal dynamics of the latter two states are revealed for the first time, to the best of our knowledge. The preliminary simulation shows that the pulsating soliton with similar characteristics could be generated by intracavity spectral filtering. Our findings could provide additional insights into the complex nonlinear soliton behaviors in lasers, and potential ways to generate and control specific soliton state.