## Abstract

Multimode interference (MMI) has been considered to be critical and investigated extensively in mode-locked laser based on single transverse mode systems, whereas there are few researches related to three-dimensional nonlinear dynamics within lasers. In this paper, we demonstrate all-fiber high-power spatiotemporal mode-locked (STML) laser by optimizing MMI filtering, where we find that the MMI filtering plays an important role in counteracting the coupling of high-order modes and improving output power of STML laser. The results under weak coupling condition when the length of graded-index multimode fiber (GIMF) is integral multiple of beat length show that the oscillator generates dissipative soliton pulses at 1036.86 nm with pulse width of 5.65 ps, and the slope efficiency of pump-signal is up to 10.3% with average power/energy of 215 mW/6 nJ, which is the highest among all-fiber STML lasers in normal dispersion regime. Besides, the multiple-soliton of STML, including multiple pulses and harmonic mode-locking can be observed in the experiment. Our work significantly broadens the dimensions of design for all-fiber high-power STML and makes them much more accessible for being put into applications.

© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

How to improve the energy/average power of ultrafast lasers has attracted extensive interests among scholars due to their wide application in optical communication [1,2], biomedical field [3,4], manufacturing [5] and material processing [6,7]. Dissipative soliton based on single-mode fiber provides a simple platform to perform all laser parameters virtually, such as ultrashort pulse duration [8], ultralow noise [9], and ultrahigh stability [10] in addition to high average power. However, the single-mode fiber is limited in improving power and space-division multiplexing [11] due to the restriction of mode area [12]. Obviously, using multimode fiber is an effective way to overcome above limitations because of its large mode area and the existence of multiple modes.

By contrast, operation in multiple transverse modes can lead to enormous complexity of ultrafast dynamics, such as high-order mode coupling and intermodal dispersion [13]. And various spatiotemporal nonlinear dynamics in multimode fibers (MMF) have been studied, such as self-focusing, self-cleaning and self-organized instability [14–17]. Nevertheless, the spatiotemporal mode-locking technology based on multiple transverse modes is promising to be employed in high-power lasers [18], optical frequency comb [19] and LiDAR [20], etc. In normal region, spectral filtering is considered as an effective approach to reduce the periodic spectral expansion caused by linear chirp in a form of cutting the edge of spectrum [21]. However, some factors must be considered in mode-locking of higher dimensions, in which weak coupling to high-order modes is essential [22–24]. In 2017, Wright et al. demonstrated the first STML fiber laser [25], in which the multiple transverse modes and longitudinal modes can lock synchronously. Until 2020, the STML laser with all-fiber ring structures has been first reported [26], with the help of multimode fiber based filter, the average power/pulse energy of 12 mW/0.5 nJ can be realized. It is worth noting that, once high-order mode dimension in mode-locking is considered, it is of great necessity to conduct spatial filtering regardless of whether the STML is achieved in the normal or negative dispersion regime. Besides, MMI by introducing MMF that sandwiches between few-mode fibers with different parameters has been widely used in spatial filtering at present [27,28]. However, the modulation of MMI and the influence of its parameters on STML formation are still received little attention and poorly understood in multimode cavities with immense spatiotemporal complexity.

In this paper, the key to achieve high-power STML is precisely controlling the self-imaging effect which is caused by MMI filtering. When optimizing the MMI effect in FMF structure, the self-imaging plays two critical roles. One is filtering, passively modulating the spatial and spectral filtering by controlling the length of GIMF, the other is to improve the coupling efficiency between GIMF and few-mode fiber, which can improve the average power and slope efficiency of STML. The experimentally demonstrated all-fiber STML laser is self-starting and generates pulse of 215 mW/6 nJ average power/energy with the slope efficiency of 10.3$\%$. Besides, the pulse duration is 5.65 ps and the central wavelength is 1036.86 nm. To the best of our knowledge, the dissipative spatiotemporal soliton has the highest output pulse energy and average power among the reported all-fiber ytterbium-doped STML lasers. The principle of MMI optimization we obtained for self-starting spatiotemporal soliton can be widely extended to high-power STML laser.

## 2. Numerical simulation of STML

#### 2.1 Spatial-spectral filtering

A schematic diagram of the FMF filter is showed in Fig. 1(a), which consists of a sandwiched graded-index multimode fiber (GIMF, OM4, YOFC) section, the length of which is $L$, fusion splices to gain fiber and few-mode fiber (LMA-GDF-10-125, Nurfern) at both end-facets of GIMF, we choose the shortest possible FMF filter to simplify the computational complexity on the premise that the self-imaging effect can be observed. The input field from few-mode fiber excites a number of guided modes as it propagates along GIMF [29,30]. Notably, the periodic self-imaging points occurs in few-mode fiber, and the difference in fiber parameters only changes the self-imaging period in GIMF, as depicted in Fig. 1(a). The beat length is consistent with the calculation result obtained from the equation $L_s=\pi R/\sqrt {2}\triangle$, where $\triangle$ is relative refractive index difference of fiber, $R$ is the core radius of fiber. To investigate the influence of GIMF length on FMF filter, we select GIMF with integral multiple of beat length $(n+1)L_s$, odd multiple of half beat length $(n+1/2)L_s$, and the length of $(n+3/4)L_s$, the intensity distribution at the fusion point between GIMF and few-mode fiber is showed in the bottom of Fig.1(a). Figure 1(b) shows the beam profile changes in a period of self-imaging in GIMF, which are caused by the energy coupling between low-order modes and high-order modes. The expansion of beam profile proves that the enhancement of coupling to high-order modes, and the shrinkage of beam profile indicates the weakening of coupling to high-order modes. In order to offset the strong coupling from low-order modes to high-order modes, the fusion point between GIMF and few-mode fiber should coincide with the self-imaging point in GIMF, which can maximize the transmittance of FMF filter and boost the output power as well as slope efficiency of the fiber laser. Figure 1(c) shows the normalized power distribution of FMF filter with various lengths of GIMF. What can be observed is that the output power of FMF filter reaches the maximum (almost no inset loss) when the length of GIMF is an integer multiple of the beat length $(n+1)L_s$. The wavelength sensitivity of FMF filter is also an important factor that needs to be considered in the experiment. Figure 1(d) illustrates the transmission of FMF filter that is plotted as a function of the wavelength with three different cases of GIMF length. The transmittance of FMF oscillates sinusoidally as a function of wavelength, and as the length of GIMF is increasing, the sinusoidal-like transmission curve presents the characteristic of red-shift. Compared the length of GIMF in Fig. 1(c) with Fig. 1(d), resolution of GIMF length with respect to transmittance is one order of magnitude higher than the other, which indicates that it is feasible to balance the requirement of spatial filtering and spectral filtering in FMF structure with proper length of GIMF. It also shows that there is no need to apply another segment of GIMF in cavity to serve as bandpass filter compared with previous reported one.

#### 2.2 Numerical results of STML dynamics

The schematic of all-fiber ytterbium-doped STML laser is illustrated in Fig. 2(a). The numerical simulations are performed based on the cavity to investigate the dynamic process of dissipative spatiotemporal soliton formation. The generalized multimode nonlinear Schrödinger equation is used to describe the pulse propagation in multimode fiber [31,32]. We take self-imaging effect into account when pulse propagates in GIMF. Thus, a periodic modulation equation is taken into account and described as $\gamma (z)=n_2w_0/cA_{eff}(z)$ [33,34]. And the beam radius of pulse can be described as $a^2 (z)=a_0^2 [cos^2(\sqrt {k} z)+sin(\sqrt {k}z \beta _0^2 a_0^4 g]$, where $a_0$ is initial spot radius, $k=2\triangle /r_{MMF}^2, \beta _0= \omega _0 n_0/c$. $r_{MMF}$, $\omega _0$ and z is core radius, central frequency and position of GIMF respectively. The spatial and spectral filtering is introduced by the FMF structure, and nonlinear polarization rotation (NPR) technology is deemed as a fast saturable absorber.

In order to investigate the evolution of spatiotemporal soliton in time and spectral domain in our cavity, the pulse propagation in one roundtrip is calculated and presented in Fig. 2(b). In our simulation, the parameters we used are as follow: the core diameter, core index and cladding index of GIMF is 50 $\mathrm{\mu}$m, 1.464 and 1.458 at 1030 nm, respectively. The bandwidth of gain is 50 nm. The modulation depth of artificial saturable absorber (SA) is 50$\%$ with the lump loss of 10$\%$, and the saturable energy is 6 nJ. The group velocity dispersion of GIMF is 19.6 $ps^2/km$ with $n^2$ of 3.2$\times 10^{-20}$ $m^2/W$. Other parameters can be seen in the results. The few-mode fiber is treated as single mode fiber and the step in GIMF is taken as $L_s/4$ to decrease computational time and present more details thoroughly. We take 6 modes in our simulation, the excitation coefficients of which is chosen as [35$\%$ 20$\%$ 20$\%$ 10$\%$ 10$\%$ 5$\%$], corresponding to LP01, LP11a, LP11b, LP21a, LP21b, and LP02, respectively. In the gain few-mode fiber, the spectral broadening can be observed due to its high nonlinearity. And then, it reaches a steady state with 10.1 nm spectral bandwidth at the end face of GIMF. This is mainly owing to the graded-index distribution of GIMF inhibits the intermodal dispersion, which is the main reason for the spectral broadening. Considering the central wavelength of oscillation and the multimode coupling of different types of fibers, we select the length of GIMF segment used for spatial-spectral filtering to be 2.7 m which yields an 8 nm bandwidth bandpass filtering. After filter and SA, the spectral bandwidth of the pulse decrease from 10.63 nm to 9.81 nm. The temporal profile of the pulse can achieve 16.66 ps before the filter and 14.76 ps after the SA, respectively. Figure 2(c) shows the corresponding stable spatiotemporal soliton pulse shape and spectrum after 200 round-trips. The evolutions of pulse and spectrum in the form of total energy can be visualized (See the evolutions in Visualization 1). Overall, the combination of intracavity spatial-spectral filtering and SA is strong enough to counteract the linear chirp and strong coupling to high-order modes in normal dispersion regime.

## 3. Experimental setup and result

Based on the simulation results, we establish an all-fiber ytterbium-doped STML laser as shown in Fig. 2(a). NPR is used to perform the role of an artificial ultrafast SA, a 1 m heavily doped ytterbium step-index fiber (Yb1200-10/125DC, nlight) serves as gain medium, and pumped by a 976 nm diode laser. The 30/70 output coupler (OC) with graded-index multimode pigtail (OM4 with 50 $\mathrm{\mu}$m core, YOFC) sandwiches between gain fiber and few-mode fiber (LMA-GDF-10/125, nurfern), which is used as a FMF spatial-spectral filter. Figure 3 records the gradual transition to STML in the multimode fiber cavity with increasing pump power (the measured step is 0.1 W). Amplified spontaneous emission (ASE) is the dominant state of the laser when the pump power is less than 0.7 W, and the power distribution of the beam profile is relatively dispersive. When the pump power increases up to 0.7 W, the multimode continuous wave (MM CW) replaces ASE, lasing peak at 1032.91 nm can be observed and the beam profile shrinks comparatively. Keep on increasing pump power, STML operation can be obtained at the pump power from 1.6 W to 2.8 W, the contraction takes place in the beam profile without any precursory as shown in Fig. 3(a) and 3(h). Compared with the MM CW operation, the spectrum of STML state is obviously broadening, there is a steady-state pulse with fixed sequences in the time domain, which indicates that the oscillator reaches a stable STML state, the shrinkage of beam profile in mode-locking state shows that the coupling effect from low-order modes to high-order modes is weakening. As the pump power exceeds 2.8 W, the collapse suddenly occurs in time and frequency domain of STML, and the reason for beam profile extension presents in Fig. 3(a) and 3(k) results from the strengthening coupling towards high-order modes. In STML operation, the spectral bandwidth broadens to the maximum and the beam profile shrinks to the minimum, the power distribution of modes is highly concentrated. The spectra and corresponding beam profiles at different pump power are illustrated in Fig. 3(d)-(k).

The typical operation state of STML is experimentally obtained in Fig. 4. The output average power demonstrates a linear increase with the efficiency slope of 10.3$\%$ is presented in Fig. 4(a), and there is no Q-switching or pulse splitting observed when the pump power is up to 2.8 W. The oscillator generates ultrafast pulses with the maximal average power of 215 mW at the fundamental repetition rate of 35.98 MHz, which corresponds to 6 nJ pulse energy. To the best of our knowledge, this is the highest average power and efficiency slope for all-fiber ytterbium-doped STML laser. Figure 4(b) shows the output pulse spectrum at the maximum output with the central wavelength of 1036.86 nm and the bandwidth of pulse is 13.26 nm, the rectangular spectral profile possesses the typical characteristics of dissipative solitons. What needs to be emphasized is that there is not continuous wave component appearing in spectrum indicating that total output power is only contributed by the emission of soliton. The spectral sidebands are caused by periodic perturbations of MMI in the cavity and can be suppressed by controlling the birefringence and the operation of FMF filtering. The corresponding pulse is observed in Fig. 4(c), and the duration of the autocorrelation trace is 5.65 ps by assuming a Gaussian pulse profile. The time-bandwidth product is calculated to be 20.85. Compared with the Fourier limit, the huge gap implies characteristics of dissipative solitons. The 300 ns-span trace recorded pulse train is showed in the inset of Fig. 4(c). Figure 4(d) shows the amplitude stability of STML operation in the frequency domain with a resolution of 1 Hz, the fundamental repetition rate of which is located at 35.98 MHz with a high signal-to-noise of 54 dB. It indicates that the spatiotemporal soliton exhibits relatively high stability. The inset of Fig. 4(d) is the radio frequency (RF) spectrum covers an 800 MHz span, showing that a uniform intensity pattern. Meanwhile, in order to further investigate the stability of the generated spatiotemporal solitons, a long-term stability measurement of the ytterbium-doped spatiotemporal soliton fiber laser has been recorded under the laboratory condition. Figure 4(e) and (f) show the spectrum and average power results over 6 hours, it can be seen that the spectral jitter and average power fluctuation are small (less than 0.041$\%$ in average power fluctuation), such value is acceptable for all-fiber STML laser.

Subsequently, through further adjustment of the pump power and the polarization controllers (PC), we found that the state of STML multiple pulses can also be generated. As depicted in Fig. 5(a), slight adjustment of PCs is employed with fixed 3.6 W pump power, there are two pulses coexisting in the cavity with separation of 3.56 ns, and the period of adjacent pulse pairs is 29.98 ns. The central wavelength is 1036.78 nm with the bandwidths of 9.83 nm, which is illustrated in Fig. 5(c). The beam profile is recorded in Fig. 5(e), multiple peaks in beam profiles indicate that both states are associated with STML state. By finely tuning the PC, the second harmonic mode-locking of spatiotemporal soliton can be obtained with average power of 268 mW, the generated repetition rate of which is 71.96 MHz and the pulse train can be observed in Fig. 5(b). Compared with multiple pulses state, the harmonic mode-locking has little change in spectrum and beam profile as depicted in Fig. 5(d, f).

## 4. Conclusion

In conclusion, we numerically and experimentally demonstrate an all-fiber spatiotemporally mode-locked laser with MMI-based filtering. The segment of GIMF in cavity can play the role of filter, coupler and STML soliton upholder. The integral multiple of $L_s$ is the optimal solution of GIMF length to achieve high coupling efficiency, which can be obtained by solving the intensity distribution and modes evolution in a self-imaging period. Under this condition, not only can the strong coupling towards high-order transverse modes be suppressed, but also the output power of STML laser can be maximized. In the experiment, the spatiotemporal dissipative soliton with 5.65 ps pulse width can be generated. The average power/energy of dissipative soliton is up to 215 mW/6 nJ, and the slope efficiency of pump-signal reaches 10.3$\%$. Moreover, second-order harmonic spatiotemporal dissipative solitons with output power of 268 mW can be realized by finely tuning the PCs. The experimental results show that the optimization principle of filter is of great importance for high-power spatiotemporal soliton generation, and higher power scaling is possibly achieved if the fiber devices with higher damage threshold are employed.

## Funding

National Natural Science Foundation of China (61805023).

## Disclosures

The authors declare no conflict of interest.

## Data availability

Data underlying the results presented in this paper are available from the corresponding authors upon reasonable request.

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