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Abstract
Multimode fiber lasers have become a new platform for investigating nonlinear phenomena since the report on spatiotemporal mode-locking. In this work, the multimode soliton pulsation with a tunable period is achieved in a spatiotemporal mode-locked fiber laser. It demonstrates that the pulsation period drops while increasing the pump power. Moreover, it is found that different transverse modes have the same pulsation period, asynchronous pulsation evolution and different dynamical characteristics through the spatial sampling technique and the dispersive Fourier transform technique. To further verify the experimental results, we numerically investigate the influences of the gain and the loss on the pulsation properties. It is found that within a certain parameter range, the pulsation period drops and rises linearly with the increase of the gain and the loss, respectively. The obtained results contribute to understanding the formation and regulating of soliton pulsations in fiber lasers.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Because of the nonlinear interactions between the transverse modes, multimode fibers (MMFs) are ideal medium to study 3D nonlinear phenomena, such as beam self-cleaning [1,2], multimode solitons [3–5], spatiotemporal modulation instability [6,7], and spatiotemporal light beam compression [8]. As a result, MMFs have attracted extensive attention in recent years. Particularly, spatiotemporal mode-locked (STML) fiber lasers constructed by MMFs are thriving for their potential to generate high-energy pulses [9]. Since the first demonstration of STML fiber laser by Wright et al. in 2017 [10], various nonlinear phenomena have also been reported including multimode soliton molecule generation [11,12], self-similar pulses [13], solitary vortices [14], and spatiotemporal period-doubling [15]. Among them, spatiotemporal period-doubling reveals that pulse-train modulations are inconsistent in different transverse modes, and the output beam profile fluctuates rapidly and periodically. Nevertheless, there are still many intriguing nonlinear phenomena worth investigating, such as multimode soliton pulsation.
Soliton pulsation is a local oscillation state where the pulse width and amplitude change periodically [16]. Many complicated soliton pulsating behaviors have been theoretically predicted, for instance, creeping and erupting, based on the 1D complex cubic-quintic Ginzburg-Landau equation [17]. However, due to the lack of high-resolution real-time measurement methods, the detailed characterization of the pulsation behavior is difficult. With the development of dispersive Fourier transform (DFT) technique for real-time spectral measurement of ultra-short pulses [18–22], the experimental study of soliton pulsation is promoted and various soliton pulsation dynamics have been revealed in single-mode fiber lasers [23–27]. Very recently, diverse multimode soliton pulsations have also been preliminarily observed in an STML MMF laser [28]. However, the pulsation period, as one of the important features of soliton pulsation, has been rarely studied. Leo et al. theoretically investigates dynamical instabilities of the 1D soliton pulsation to analyze the pulsation period variation within a certain parameter region [29]. Wu et al. adopt an evolutionary algorithm to obtain single breathers with controllable oscillation period [27] and Wang et al. observe that the pulsation period decreases as the pump power increases in a single-mode fiber laser [30]. It remains to be further revealed whether the pulsation period can be controlled continuously or discretely, and which cavity parameters affect it.
In this work, multimode soliton pulsation in an STML fiber laser is investigated. Under the stable STML state, multimode soliton pulsations are achieved by properly manipulating the polarization controller (PC) and the pump power. Using the spatial sampling technique and DFT technique, it is found that the pulsation period is not affected by the transverse modes, but there is a pulsation asynchronous phenomenon between the transverse modes. Furthermore, different transverse modes have different dynamical characteristics. Particularly, the pulsation period is linearly tunable by adjusting the pump power. To further investigate the influence of cavity parameters on the pulsation properties, a series of numerical simulations are performed. It is found that within a certain parameter range, the pulsation period changes linearly with the increase of the gain and the loss, confirming the experimental results. The obtained results are conducive to understanding the formation mechanism of soliton pulsations and building the groundwork for regulating the pulsation characteristics.
2. Experimental setup
The schematic of the experimental setup is depicted in Fig. 1, which includes two parts: an MMF laser and a spatial sampling system. The MMF laser shown in Fig. 1(a) has a linear structure by setting two semiconductor saturable absorber mirrors (SESAMs, BATOP, SAM-1064-21-3ps-1.3b-0) at both ends of the cavity. These two SESAMs are not only served as mode-locked devices to provide a saturable absorption effect, but also regarded as two mirrors that reflect light to oscillate back and forth in the cavity. Besides, the gain medium is a 0.83 m long ytterbium-doped fiber (YDF, Coherent, LMA-YDF-20/125-9 M, NA: 0.08), supporting six transverse modes, and pumped by a 980 nm multimode laser diode through a matched fiber combiner. To adjust the polarization states and control the excited light field, a PC is placed in the cavity. The laser is output by the optical coupler 1 (OC1), where the pigtails are graded-index MMF (Corning 62.5/125 µm, NA: 0.275). The total length of MMF is ∼3.7 m. The passive fiber (20/125 µm DC, NA: 0.08) matched to the YDF has a length of ∼2 m.
Fig. 1. Schematic of the experimental setup. (a) The MMF laser (SESAM, semiconductor saturable absorber mirror; YDF, ytterbium-doped fiber; PC, polarization controller; OC, optical coupler); (b) The spatial sampling system (C1, C2 and C3, collimators; BS, beam splitter; A1 and A2, apertures; S1and S2, sampled signals; DCF, double-clad fiber).
The laser output is imported into the spatial sampling system, as illustrated in Fig. 1(b). In this system, 10% light is output through OC2 for measuring the fundamental characteristics of the laser. The remaining 90% light is transmitted from the collimator 1 (C1) to the beam splitter (BS) with a splitting ratio of 50%, dividing the light to two spatial sampling channels. These channels take two 1 mm diameter apertures (A1, A2) to spatially sample the input signal. Then two sampled signals (S1, S2) are caught by the collimator 2 (C2) and the collimator 3 (C3), respectively. The characteristics of the laser cavity output and these two sampled signals are measured by an optical spectrum analyzer (OSA, Yokogawa, AQ6317C), an oscilloscope (Tektronix, DSA70804, 8 GHz) with a photodetector (PD, Newport, 1623 InGaAs nanosecond optical detector), an autocorrelator (AC, Femtochrome, FR-103WS), a radio frequency (RF) spectrum analyzer (Agilent, E4407B ESA-E SERIES, 26.5 GHz), and a charge-coupled device (CCD) camera (Goldeye G-033SWIRTEC1). In addition, the DFT technique is adopted to realize single-shot spectral measurements, which is implemented by temporally stretching the output pulse at the OC2 in a 15-km-long single-mode fiber (Corning SMF-28e).
3. Experimental results
In the experiment, stable STML state is obtained at a pump power of 666 mW, the results are shown in Fig. 2. In order to further confirm the STML state, the spatial sampling system is used to detect different spatial positions of the output laser [31]. The spectra corresponding to the whole output and two sampled signals are displayed in Fig. 2(a). The whole spectrum demonstrates a central wavelength of 1071.6 nm and a 3-dB bandwidth of 0.31 nm. The spectra of two sampled signals (S1, S2) are different since they correspond to different spatial sampling positions as presented in the inset of Fig. 2(a). Note that the STML beam profile is not a Gaussian distribution due to the coherent superposition of multiple transverse modes. Different spatial sampling positions contain different transverse mode components. The pulse-trains before and after sampling are exhibited in Fig. 2(b), where the interval of two pulses is 63.2 ns, matching well with the fundamental repetition-rate of 15.83 MHz in the RF spectrum of Fig. 2(c). According to the spectra and pulse-trains of two sampled signals, we believe that multiple transverse modes are synchronously locked, achieving STML operation. Moreover, the RF spectrum displays a signal-to-noise ratio of up to 65 dB, which indicates the good stability of the STML pulse. The pulse duration is 8.33 ps as shown in the autocorrelation trace of Fig. 2(d).
By adjusting the PC, multimode soliton pulsation can be observed. The typical pulse evolution is shown in Fig. 3(a) at the pump power of 786 mW, where the pulsation period is 500 roundtrips (RTs). The spectra corresponding to the whole pulse and two sampled signals are displayed in Fig. 3(b). The whole spectrum displays the central wavelength locating at 1070.7 nm with a 3-dB bandwidth of 0.36 nm. The inset of Fig. 3(b) demonstrates the beam profile and different spatial sampling positions, illustrating that the soliton pulsation also contains multiple transverse modes. Particularly, the RF spectrum in Fig. 3(c) shows multiple sidebands around the main peak. The main peak represents the fundamental repetition-rate with a signal-to-noise ratio of 71 dB. The offset between the first side peak and the main peak is 31.65 kHz, which corresponds to the pulsation period. The corresponding real-time spectrum evolution of S1 and S2 is depicted in Fig. 3(d) and Fig. 3(g), respectively, which is captured by a 15 km long single-mode fiber as a DFT device. It presents a periodic modulation consistent with the pulse evolution in Fig. 3(a), where the period is also 500 RTs. The normalized energy evolution periodically fluctuates as exhibited in Figs. 3(h)-(j). As can be seen from the energy curves, the two sets of sampling signals have the same pulsation period, but their pulsation evolution is shifted by 37 RTs. It indicates that in the pulsation, although different transverse mode components are contained in different sampling signals, the pulsation period is not affected by the transverse modes, but there is a pulsation asynchronous phenomenon between the transverse modes. As shown in Fig. 3(e) and Fig. 3(f), the pulse spectra of different sampling channels are different at RT numbers of maximal and minimal spectrum extent within one period of S1 and S2. This stems from the fact that different transverse modes interact and transfer energy with each other during the propagation process, resulting in different real-time spectra for different sampling signals. This further confirms that different transverse modes exhibit distinct dynamical characteristics.
Fig. 2. Characteristics of stable STML pulse: (a) the average spectrum before and after sampling (inset: the beam profile with two sampling positions); (b) the pulse-train before and after sampling; (c) the RF spectrum; and (d) the autocorrelation trace.
Fig. 3. Characteristics of multimode soliton pulsation: (a) the pulse evolution; (b) the average spectra before and after sampling (inset: the beam profile with two sampling positions); (c) the RF spectrum; (d) and (g) the real-time spectra from S1 and S2, respectively; (e) the real-time spectra at 46 RT and 178 RT from S1; (g) the real-time spectra at 9 RT and 141 RT from S2; (h)-(j) the energy evolution curves of the whole pulse, S1 and S2, respectively.
In general, for soliton pulsation, variation in pump power directly influences the intracavity gain and nonlinear effects, which will significantly change the pulsation characteristics [16]. Here, we investigate the variation of pulsation period with the pump power, while fixing the PC. It is found that the multimode soliton pulsation period decreases linearly when the pump power rises. The results are summarized in Fig. 4. Figure 4(a) records the corresponding RF spectra. The pulsation period can be calculated from the interval between the first side peak and the main peak in the RF spectrum, and presents in Fig. 4(b). It can be considered that the multimode soliton pulsation period varies linearly with the pump power. In addition, we note that in the process of increasing the pump power, the degree of energy fluctuation in the pulsation hardly changes. It should be pointed out that the multimode soliton pulsation disappears when the pump power rises over 866 mW. It is due to the fact that the excessive energy accumulation disturbs the dynamic balance of gain, loss, nonlinearity and dispersion in the cavity, resulting the disappearance of the soliton pulsation.
Fig. 4. Pulsation period varies with the pump power: (a) RF spectra under different pump powers; (b) pulsation periods (blue line) and frequency intervals between the first side peak and the main peak in RF spectra (red line).
4. Numerical simulations and discussions
To provide further insight into the characteristics of the multimode soliton pulsation, numerical simulations based on the generalized multimode nonlinear Schrödinger equations (GMMNLSE) are performed [32]. Here, simulation parameters of the cavity are determined to approximate the above experimental conditions: YDF length ${L_{YDF}} = 0.83m$, matching passive fiber ${L_{GDF}} = 2.1\; m$, MMF length ${L_{MMF}} = 3.7\; m$, and total cavity length $L = 6.63\; m$. For the YDF, we set gain-related parameters as follows: the small-signal gain coefficient $g = 2.17$, the gain saturation ${G_{sat}} = 2.1\; nJ$, and the gain bandwidth ${\Omega _g} = 40\; nm$. Besides, only the first ten and six transverse modes are considered in the MMF and the YDF, respectively. The soliton pulsation can be achieved. Figure 5 demonstrates the characteristics of a multimode soliton pulsation with a 350 RTs pulsation period. The pulse-train is shown in Fig. 5(a). The inset of Fig. 5(a) displays the normalized energy evolution, which indicates that the output pulse energy fluctuates with the same period of the pulse-train. To further investigate the evolution characteristics of different transverse modes in the pulsation, the normalized energies of the LP01, LP11a, LP21a and LP02 modes are extracted and compared. As shown in Fig. 5(b), different modes have the same pulsation period of 350 RTs. In addition, the pulsation evolution of LP11a, LP21a and LP02 modes is shifted by 139 RTs, 300 RTs, and 266 RTs, respectively, relative to the LP01 mode. This is in good agreement with the above experimental results. It further confirms that in the pulsation, different transverse modes have the same pulsation period and asynchronous pulsation evolution characteristics. The pulse and beam profile at the 1500 RT are presented in Fig. 5(c), and the corresponding spectra are given in Fig. 5(d). Note that although multiple transverse modes are locked together, they have inconsistent pulse durations, spectral shapes, and intensities due to their spatial distribution and gain competition in the gain medium.
Fig. 5. Numerical simulation results of the stable multimode soliton pulsation: (a) the pulse-train (inset: the energy evolution); (b) the energy evolution of the LP01, LP11a, LP21a and LP02; (c) the pulse at the 1500 RT (inset: the corresponding beam profile); (d) the spectra at the 1500 RT.
To study the influence of the gain on the multimode soliton pulsation period, the gain coefficient is gradually increased in the simulation, while keeping other parameters constant. The results are shown in Figs. 6(a)-(c). Figure 6(a) displays a soliton pulsation with the pulsation period of 241 RTs, which is obtained under the gain coefficient of 2.173. When the gain coefficient increases to 2.174, the period of the soliton pulsation decreases to 202 RTs as presented in Fig. 6(b). Figure 6(c) depicts the pulsation period variation with the gain coefficient. Obviously, the pulsation period decreases linearly with the increase of the gain coefficient. During this process, the total pulse energy rises with the increase of the gain, but the intensity difference between the maximum and minimum pulse energy remains unchanged (0.11 W) within this parameter range. In fact, the gain parameter can reflect the intensity of the pump power. The higher the gain coefficient means the faster the pulse accumulates energy, and the shorter the time for the pulsation to rise from the trough to the peak during the evolution process. Therefore, the pulsation period decreases with the increase of the gain. However, this linear relationship is only valid within this parameter range. If the gain goes beyond this range, the soliton pulsation may become unstable or even disappear. The simulation results agree with the above experimental observation.
Fig. 6. Numerical simulations of the influences of the gain coefficient and the loss ratio on pulsation periods: (a) and (b) the soliton pulsations with the gain coefficient of 2.173 and 2.714, respectively; (c) the pulsation period versus the gain coefficient; (d) and (e) the soliton pulsations with the loss ratio of 0.302 and 0.303, respectively; (f) the pulsation period versus the loss ratio.
Besides the gain, the loss, as another important cavity parameter, also has an impact on the pulsation period. Here, we fix the gain coefficient at 2.174 and study its influence on the pulsation period by changing the loss. Figures 6(d)-(f) shows the results. Figure 6(d) is the soliton pulsation with a pulsation period of 285 RTs under the loss ratio of 0.302. When increasing the loss ratio, the period of the soliton pulsation becomes 315 RTs, as exhibited in Fig. 6(d). Figure 6(f) further illustrates the variation of the pulsation period with the loss. It shows that the pulsation period is approximately linearly affected by the cavity loss within this range. The overall pulse energy decreases with the increase of the loss, but the degree of energy fluctuation remains unchanged (0.11W). Similar to the gain, when the cavity loss increases, the pulse energy accumulation slows down, the time for the pulsation to evolve to the peak increases, and thus the pulsation period increases. If the loss exceeds this range, the soliton pulsation may become stable pulse or unstable and breaks up.
Comparing the experimental results and the numerical simulations, it is found that these multimode soliton pulsations can be maintained in a certain range of gain coefficients (pump powers) and loss. In addition, the pulsation periods vary linearly while increasing the gain and the loss, respectively. Since these multimode soliton pulsations are stable during the process of changing cavity parameters, it can be assumed that in the pulsation state, the cavity parameters such as gain, loss, dispersion and nonlinearity, maintain dynamic balances. Therefore, as the intracavity gain increases, the pulse energy accumulates more quickly, resulting in a faster rise of the pulsation peak and a shorter pulsation period. Similarly, when the intracavity loss increases, the pulse energy accumulates more gently, resulting in a slower rise of the pulsation peak and a longer pulsation period. It means that the pulsation period drops and rises as the increase of the gain and the loss, respectively. Generally, the variation of the pulsation period is hard to achieve linear control [27,29], indicating that the regulation of the pulsation period is very difficult. Here, in the experiment, we achieve a linearly controllable pulsation period by varying the pump power, and in the simulation, by tuning the gain and loss, which is simple and lays the foundation for dynamical regulation of the pulsation period. Certainly, the pulsation period is also related to other parameters such as dispersion and nonlinearity in the cavity, and further research is needed.
5. Conclusion
In conclusion, we experimentally study the multimode soliton pulsation in an STML fiber laser and it is found that the different modes have the same pulsation period, asynchronous pulsation evolution and the different dynamical characteristics by comparing the energy evolution and the real-time spectra of different sampling signals after DFT. Moreover, a periodically tunable multimode soliton pulsation is achieved by tuning the pump power. The simulation results confirm the experimental results. It shows that within a certain range, the pulsation period changes linearly with the increase of the gain and the loss. The obtained results help to understand the formation mechanism of soliton pulsations and lay the foundation for manipulating the soliton pulsations.
Funding
National Natural Science Foundation of China (11974006, 61875058, 92050101); Natural Science Foundation of Guangdong Province (2021A1515011608, 2023A1515011870).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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