Abstract

Experimental observations of spatiotemporal mode-locked multiple-soliton, including harmonic mode locking and multiple pulses, in multimode fiber (MMF) lasers are reported. Numerical simulations are conducted to investigate the nonlinear dynamics of multi-pulsing. The influences of cavity parameters on the spatiotemporal outputs are analyzed by simulations, which agree with the experimental observations qualitatively. This work would contribute to understanding the complex spatiotemporal nonlinear dynamics in MMF lasers.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Mode-locked fiber lasers have attracted much attention due to the advantages of compact configuration, low cost, and insensitivity to the thermal load. Though great progress has been made in single-mode fiber (SMF) lasers, their commercial competition is restricted owing to the confine in pulse energy, and this is where the multimode fiber (MMF) come to people’s sight. In addition, MMF based space-division multiplexing technique is also a hot topic in optical fiber communication to enhance the transmission capacity [1]. The study of MMF dates back to 1974 [2]. Recently, some nonlinear propagation phenomena in MMF has been observed experimentally [3–10]. Owing to the unique features of graded-index (GRIN) MMF, i.e. relatively small modal dispersion which is comparable to chromatic dispersion, GRIN fiber is generally employed in the investigations of nonlinear pulse propagation with strong multimode nonlinear interactions. Numerous remarkable research of nonlinear physics has been investigated in passive GRIN multimode fibers, including modeling of nonlinear pulse propagation [11,12], observation of multimode soliton [3,4], spatiotemporal instability [5–7], controlling nonlinear multimode interactions [8,9], spatial beam cleaning [10,13], spatiotemporal light beam compression [14], etc. In addition, optical sources with supercontinuum generation [15] and wavelength detuning [16] were generated by use of GRIN fiber. Due to the much more complex nonlinear behavior compared to SMF, ultrafast optical sources with MMF have been achieved with unprecedented performance. In 2017, a successful demonstration of spatiotemporal mode-locked (STML) MMF laser was reported [17]. Simultaneously locking of multiple transverse and longitudinal modes was achieved, which improves the pulse energy of fiber laser and provides a new opportunity to study nonlinear wave propagation.

It is well-known that passively mode-locked fiber lasers could also serve as good platforms for investigating complex nonlinear science [18]. In SMF based lasers, plenty of dissipative structures and self-organization effects have been reported and investigated, such as soliton explosions [19,20], dark solitons [21], rogue waves [22], multiple-soliton [23], etc. Among them, multiple-soliton (also called multipulsing) attracts lots of attention since it is often observed in experiments and there are many types of multiple-soliton. Usually, only one pulse travels inside the cavity, and the repetition rate of output pulses is inversely proportional to the cavity round-trip time. However, in high pump power regime, multiple-soliton tends to be generated. Operations of multiple-soliton have been intensively reported and analyzed in SMF lasers [18]. Depending on the parameters of fiber laser, there are different distributions of multiple solitons. For instance, in the case of soliton molecules (in other words, bound state of solitons), the distance of multiple solitons is usually small (in the order of pulse duration) so that the direct interaction among them cannot be neglected [18]. For the state with longer distance between the multiple solitons [24], herein we call it “multiple pulses”. Especially, a regime of multiple pulses, where all the output pulses are equally separated and the pulse repetition rate is multiple of the fundamental repetition rate, is called harmonic mode locking [25]. To date, multipulsing dynamics have renewed attention due to the improvement of single-shot measurement techniques. With the assistant of these methods, e.g. dispersive Fourier transform (DFT) technique, the transient and non-stationary properties of the multiple-solitons in ultrafast lasers can be observed. The rapid internal dynamics of soliton molecules were tracked [26,27], and the birth of soliton molecules was uncovered in single-shot manner [28]. The dynamics of soliton molecular complexes was unveiled with the help of DFT technique and numerical simulations [29]. The birth and dynamic behaviors of multiple pulses in passively mode-locked fiber laser was experimentally investigated by use of the DFT technique [30]. Combining DFT and time-lens techniques, real-time observation of both the spectral and temporal evolutions of multiple-soliton was achieved [31].

While most investigations into multiple-solitons were achieved in SMF lasers, only soliton molecule was experimentally observed by us in STML MMF lasers [32]. It is worthwhile to identify whether other types of STML multiple-soliton states, e.g. multiple pulses and harmonic mode locking, can be formed in MMF lasers. We note that multi-pulsing has been observed in mode-locked lasers with MMF [33]. However, our platforms are different from that of [33]. The MMF in [33] supports only a few waveguide modes, and the higher-mode content is much smaller than the fundamental mode. Whereas in our lasers the MMF can tolerate dozens of transverse modes, and the higher-order modes are conspicuous. In addition, due to the modal dispersion, the nonlinear intermodal interaction in the step-index MMF of [33] is much smaller than that of GRIN MMF used in our cavity. More importantly, the spatiotemporal characteristics of these multipulsing states in STML MMF lasers need to be investigated.

In this paper, the multiple-soliton states in STML MMF lasers are investigated. We report on the experimental observation of harmonic mode locking and multiple pulses in STML MMF lasers. The multipulsing dynamics in MMF cavity are investigated by numerical simulations, which agree with experimental observations. This work would contribute to further understanding the complex spatiotemporal nonlinear dynamics in STML MMF lasers, and bridge the two active areas of multipulsing dynamics in mode-locked lasers and the spatiotemporal nonlinear optical dynamics in MMFs.

2. Experimental setup

The experimental scheme and measurement setup are shown in Fig. 1, which are similar as those of [17,32]. As illustrated in Fig. 1(a), the ring cavity comprises 1.5-m double-cladding gain fiber (Liekki YB1200-10/125DC with 10 μm core), 1.5-m passive GRIN MMF (YOFC OM4 with 50 μm core), and free-space optical components for pump incident and polarization controlling. The gain fiber supports about 3 spatial modes, and several tens of spatial modes can be supported in the GRIN MMF [17]. To excite multiple transverse modes, the GRIN MMF is aligned offset from gain fiber’s center by ~10 μm during the fiber splicing. The gain fiber is pumped by a 980 nm laser diode. A shortpass dichroic mirror (SPDM, cut-off wavelength of 1000 nm) is used for signal feedback, assisted by a mirror M1. The signal feeds back into the gain fiber, the core size of which is much smaller than that of GRIN MMF. Thus most transverse modes are eliminated, serving as a virtual spatial filter. An intracavity spectral filter (SF, bandpass filter with center wavelength of 1030 nm and 3-dB bandwidth of 10 nm) and the above-mentioned virtual spatial filter form a spatiotemporal filter, compensating the modal- and chromatic-dispersion of fibers [17]. Two quarter-wave plates, one half-wave plate and one polarization beam splitter (PBS) are used for nonlinear polarization rotation mode-locking. The cavity round-trip time of the laser is 16.5 ns, and the corresponding fundamental repetition rate is 60.8 MHz.

 figure: Fig. 1

Fig. 1 Experimental setup. (a) The MMF laser (SPDM, short pass dichroic mirror; L1 and L2, lens; OM4, passive GRIN multimode fiber; HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarization beam splitter; BS, beam splitter; ISO, isolator; SF, spectral filter; M1-3, mirrors); (b) Measurement setup. Method 1: spectral filtering; Method 2: spatial sampling (BS: beam splitter; OSA: optical spectrum analyzer; PD, photodetector).

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The output, ejected by the PBS, is directly measured by an optical spectrum analyzer (OSA) with 0.06 nm resolution, a 5 GHz photodetector recorded by an 8 GHz real-time oscilloscope (Keysight Infiniium DSOS804A) and a radio frequency (RF) signal analyzer, and a beam profiler. To further verify the spatiotemporal mode-locking, two measurement methods based on spectral filtering and spatial sampling are adopted [17,32], as illustrating in Fig. 1(b). For method 1, a tunable bandpass spectral-filter is used to get different spectrum regions of the output, which is then measured by the OSA, oscilloscope/RF spectrum analyzer (via photodetector), and beam profiler. For method 2, the output is spatially sampled through directly coupling into an MMF which is placed on a movable stage (moving in x and z directions), then is measured by the OSA and oscilloscope/RF spectrum analyzer (via photodetector).

3. Experimental observation of STML multiple-soliton states

Similar to our previous report [32], soliton molecule could be generated in the present MMF laser. More important, other STML states of multiple-soliton could also be observed in this MMF laser, and the typical experimental observations are given in this section.

3.1 Harmonic mode locking

Self-start single-pulse mode locking can be readily achieved by increasing the pump power to 6.48 W with suitable set of waveplates. Then we adjust the waveplates and/or increase the pump power, and harmonic mode locking occurs. It is noteworthy that, there are sudden transitions in the output beam profile when the operation states alter from single-pulse operation to multipulsing mode locking, indicating sudden changes of the transverse mode composition. This spatiotemporal transition is also observed in our laser, as well as in [17], when the laser starts (single-pulse) mode-locking from continuous-wave (CW) lasing operation as the pump power increases.

A typical output of harmonic mode locking is given in Fig. 2, showing a state of second-order harmonic mode locking with 6.48 W pump power. As shown in Fig. 2(a), the interval of output pulses is 8.2 ns, and the measured repetition rate is 121.6 MHz [shown in Fig. 2(b)], which is twice of the fundamental repetition rate. Figure 2(b) gives RF spectrum at 121.6 MHz with 1.5 kHz span and 1 Hz resolution, and the signal to noise ratio at the repetition rate is measured to be around 50 dB. The fitting curve of autocorrelation trace in Fig. 2(c) shows a 2.47 ps pulse duration with a Gaussian shape assumption. The beam profile is given in Fig. 2(e), which indicates that there exist many high-order transverse modes. The corresponding optical spectrum is given in Fig. 2(d) (the solid curve), with a bandwidth of 27 nm.

 figure: Fig. 2

Fig. 2 Second-order harmonic mode locking state. (a) Pulse train measured by oscilloscope; (b) RF spectrum of pulse train with a resolution bandwidth of 1 Hz (inset: RF spectrum over a 1 GHz span with a resolution bandwidth of 2 kHz); (c) Autocorrelation trace of output pulses; (d) Spectra of the whole output (solid curve) and the spectral-filtered outputs (dashed curves), with the corresponding beam profiles shown in (e)-(h); (e) Beam profile of the whole output, with black circles represented the spatial sampling positions (one sampler fixed at the center, and another sampler moving along the direction of arrow); (f)-(h) Beam profiles corresponding to different spectrum regions by tunable spectral-filtering of the output; (i) Pulse intensities and pulse intervals of the pulse trains sampled by the moving sampler at different position (the intensity corresponding to the fixed sampler is 53); (j) pulse train sampled by the fixed sampler; (k) a typical pulse train sampled by the moving sampler; (l) and (n) are the corresponding optical and RF spectra of (j) and (k), respectively; (m) RF spectra of the spectral-filtered outputs corresponding to (f)-(h).

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To further confirm the spatiotemporal mode-locking state, we carry out the two measurements based on spectral filtering and spatial sampling [17,32], with the setup shown in Fig. 1(b). For method 1, by tuning the spectral-filter, three different spectrum regions (i.e. different transverse mode components) are selected from the whole output spectra, as shown in Fig. 2(d) [dashed curves labeled by (f), (g), and (h)]. The corresponding beam profiles for these sliced spectra are different, as shown in Figs. 2(f)-2(h); while the corresponding RF spectra overlap, as shown in Fig. 2(m). This indicates that the transverse modes are synchronously locked. For method 2, the output is spatially-sampled at different positions, illustrated by the black circles in Fig. 2(e), by directly coupling part of the output beam into two MMFs simultaneously. One MMF sampler is fixed at the center of the beam profile where the fundamental mode dominated, while another sampler collecting varied combination of modes by moving the MMF aside the center axis from bottom to top, recording pulse trains every 0.5 mm. The average powers sampled at different positions are presented in Fig. 2(i), reflecting the intensity distribution of the beam. In addition, the pulse intervals of the pulse trains sampled by the moving sampler are plotted in Fig. 2(i), which stay unchanged with 8.2 ns. The data sampling from the center of beam by the fixed sampler is marked as (j), and the data sampling from the 4th position by the moving sampler are marked as (k). The corresponding pulse trains, optical and RF spectra of (j) and (k) are shown in Figs. 2(j)-2(l) and (n). Similar to the first measurement, the optical spectra are distinct to each other, inferring that the sampled transverse mode components are different; while the RF spectra are almost the same, confirming the spatiotemporal harmonic mode locking.

3.2 Multiple pulses

Moreover, by carefully rotating the waveplates and/or increasing the pump power, STML multiple pulses can also be generated. As depicted in Fig. 3(a), slight waveplates adjustments is employed here with fixed 6.48 W pump power, there are two pulses coexisting in the cavity with separation of 2.7 ns. The optical spectra and beam profile are given in Fig. 3(b) with a bandwidth of 28 nm. The experiment data similar to Fig. 2(i) is given in Fig. 3(c), indicating that the pulse intervals of the pulse trains sampled at different positions of the beam are the same. In both sections 3.1 and 3.2, the results of spatial sampling measurement show that the multiple pulses are multimodal. Furthermore, the results infer that, for each transvers mode, the mode-resolved multiple pulses are the same. In another word, the multiple pulses have the same modal composition. The following numerical simulations support this tentative conclusion.

 figure: Fig. 3

Fig. 3 Multiple pulses state. (a) Pulse train measured by oscilloscope; (b) Optical spectrum (inset: beam profile); (c) Pulse intensities and pulse intervals of the pulse trains sampled at different position.

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3.3 Soliton molecule

In addition, keeping rotating the waveplates, STML soliton molecule in the MMF cavity is also observed with the same pump power, as illustrated in Fig. 4. From Fig. 4(a), the time interval agrees with the fundamental repetition. The modulation spectrum in Fig. 4(b) reveals the existence of soliton molecular with a bandwidth of 25 nm and modulation period of 0.350 nm. Spatial-sampling method is also adopted, and the autocorrelation traces of two samplings overlap each other, as shown in Fig. 4(c) with inserted beam profile. The autocorrelation trace shows the time separation between the adjacent peaks is 10 ps, corresponding the spectrum modulation period of 0.354 nm in theory. More experimental observations of soliton molecule have been discussed in our previous work [32].

 figure: Fig. 4

Fig. 4 Soliton molecule state. (a) Pulse train measured by oscilloscope; (b) Optical spectrum (inset: beam profile); (c) Autocorrelation traces of different spatial samplings, S1 and S2.

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4. Numerical simulations

4.1 Simulation model

To give the insight into the multiple-soliton in STML MMF laser, we further numerically investigate the evolution of multiple-soliton by the method described in [17,34]. For the pulse propagation in the passive MMF, the following generalized multimode nonlinear Schrödinger equation (GMMNLSE) is used [34], neglecting the self-steepening and higher-order dispersion:

zAp(z,t)=i(β0(p)[β0(0)])Ap(β1(p)[β1(0)])Aptiβ2(p)2t2Apin2ω0cl,m,n{(1fR)SplmnKAlAmAn*+fRAlSplmnRtdτAm(z,tτ)An*(z,tτ)hR(τ)}
where Ap(z,t) is the electric field temporal envelop for spatial mode p, βm(p) is the the m-st order dispersion for mode p, SK plmn and SR plmn are the nonlinear coupling coefficients for the Kerr and Raman effect, respectively. hR is the Raman response of the fiber medium, fR (set to be 0.18) is the Raman contribution to the Kerr effect, ℜ denotes the real part only, and n2 is the nonlinear index of refraction. The dispersion and mode-coupling tensor of MMF are calculated based on the GRIN MMF. The few-mode gain fiber is approximately treated as a single-mode fiber [17]. The effect of nonlinear polarization rotation is modelled by an ideal saturable absorber (SA) with absorption function q=q0/(1+|Ap|2/Psat)+1-q0 for each transverse mode Ap, where q0 is the modulation depth and Psat is the saturation power.

In the simulations, we use the cavity parameters according to the experimental conditions. The length of GRIN fiber LGRIN=1.5m, for the gain fiber, LGain=1.5m, β2,Gain=20fs2/mm,SK=SR=1.3W-1km-1, the saturation energy ESat=1.5nJ, the bandwidth of gain Ωg=40nm, for the SA q0=0.7, a lumped loss (including the loss due to output, the coupling loss, etc.) of 80% is inserted between the MMF and SA, and other parameters will be given for specific simulation examples. The computation of multimode pulse propagation is massive. Thus in our simulations, the number of modes considered is limited, the time window cannot be too large (hence the harmonic mode locking is not simulated), and the simulations start with small pulses. The 6 lowest-order spatial modes of the passive MMF are considered and the coupling coefficients of each mode with the gain fiber are set to be in proportion [2,4,2,2,1,1], and then normalized to 1. We note that the adopted values of the simulation parameters may affect some details of the simulation output; however, the general conclusions are independent of these specific values, confirming by numerous simulations.

4.2. Evolutions of multiple-soliton

With the simulation model and parameters above, a typical example of multiple-soliton evolutions is shown in Fig. 5. Figures 5(a) and 5(b) present the mode-resolved and total soliton molecule evolutions with roundtrips, respectively. The calculated output beam profile is inserted. In the simulations, we input two identical small pulses (but out of phase) with 1 ps pulse duration, 0.005 nJ energy and 6 ps separation. The soliton molecule approaches the steady states at ~600th roundtrip, though the pulse energy approaches a steady state very quickly [Fig. 5(e)]. This is because the pulses interaction is weak in each round, and the soliton molecule spend many roundtrips to adapt it. The intracavity pulse evolution in the last roundtrip is plotted in Fig. 5(c), labeled with intracavity position. The mode-resolved output of the last roundtrip (i.e. the steady state) is given in Fig. 5(d), with pulse separation of 7.5 ps. The durations of output pulses are about several picoseconds, and the calculated spectral bandwidth is about 20 nm, indicating that the output is highly chirped. Intracavity mode-resolved evolutions (data not shown) of the pulse energy indicate that there are periodical energy fluctuations for all modes when propagating through the MMF, induced by nonlinear interactions (thus energy transfers) among the transverse modes. Note that we cannot obtain steady state of soliton molecule with close-separated, in-phase pulses input by the current simulation model, and the reason is described in part 1 of section 4.3. Simulations of multiple pulses are also carried out, started by two small pulses similar to the case of soliton molecule but with much larger pulse separation and different phase relationship. The evolution of multiple pulses approaches the steady state much more quickly, without the direct pulses interaction. Detailed numerical results and the discussion of our simulation model are given in section 4.3

 figure: Fig. 5

Fig. 5 Mode-resolved simulation results for soliton molecule in the MMF laser. Esat = 2 nJ, small signal gain g0 = 15 m−1 and Psat of the SA is 180 W. (a) Mode-resolved pulse evolution with roundtrip (inset: beam profile); (b) Pulse evolution with roundtrip; (c) Intracavity pulse evolution in the last roundtrip; (d) Temporal output of the last roundtrip; (e) Pulse energy evolution with roundtrip. In (a), the initial pulses intensities are enlarged 200 times for clarity. In mode-resolved figures, the modes are color-coded. Among those modes, mode 2 is the highest, modes 1, 3 and 4 are almost overlapped, modes 5 and 6 are almost overlapped. OC: output coupler, SA: saturable absorber, BF: bandpass filter, SF: spatial filter.

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4.3 Discussion

  • (1) In the simulations, with different initial input pulses condition (pulse energy, pulse width), the steady output could be the same. This suggests that the solutions of GMMNLSE converge to the attractor of a fixed point, which determined by the inherent properties of dissipative systems [35]. As an example, evolutions of multiple pulses with different initial conditions are illustrated in Fig. 6. For the simulation of Figs. 6(a) and 6(b), the initial two in-phase pulses are different in term of energy. The temporal separation of the two pulses is 900 ps, which is far enough to neglect the inter-pulse interaction. The evolution of the two pulses is shown in Fig. 6(a), which approaches the steady state quickly. The corresponding stable output, two identical pulses, is presented in Fig. 6(b). Furthermore, Fig. 6(c) shows the evolution where the initial input is random noise. Figure 6(d) is the steady output, the same as Fig. 6(b) expect for time separation. Numerous numerical simulations confirm that, multiple-soliton is an intrinsic feature of STML MMF lasers. Moreover, the simulation results of the output multiple pulses support our conclusion drawn from the experiment observations, i.e. the multiple pulses at steady state have the same modal components.
     figure: Fig. 6

    Fig. 6 Evolution of multiple pulses with different initial conditions. Small signal gain g0 = 20 m−1 and Psat of the SA is 180 W. (a) Evolution with two unequal pulse input, and (b) is the corresponding mode-resolved steady output; (c) Evolution with random noise input, and (d) is the corresponding steady output.

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    For large pulse separation considered here, the direct interaction between multiple pulses can be neglected. Thus for the simulation given in this part, it can represent the case with larger temporal separation. In our simulation model, long-range interactions, e.g. soliton-CW radiation interactions gain depletion and recovery [36], are not taken into account. Thus in the case of input pulses with large separation, the separation of pulses does not alter during evolution, and the separation at steady states depends on the initial assumed separation of pulses. In addition, the model of SA is simplified in our simulations, which may neglect some SA induced attractive or repulsive effects among pulses with short separation [37]. In the future, a more accurate simulation model, taking into account more effects is required for the STML MMF lasers.

  • (2) The excitation of high-order modes is an important feature of STML MMF lasers. MMF cavity with different condition of mode excitations is also investigated by numerical simulations, by use of different sets of the coupling coefficient. For three different coupling coefficients (CC I, II, and III), all the evolutions converge to steady states quickly. The mode components of the stable outputs, as well as the corresponding beam profiles, are different for the three cases, as shown in Fig. 7. This phenomenon is also observed in the experiments, i.e. the beam profile changes when adjusting the coupling between the GRIN fiber and the gain fiber.
     figure: Fig. 7

    Fig. 7 Mode-resolved output intensities and the corresponding beam profiles with various coupling coefficients (CCs) operating in multiple pulses state. CC I = [4,2,2,1,1,1], CC II = [2,4,2,2,1,1], CC III = [1,1,1,2,2,4], and other parameters are the same as Fig. 6.

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  • (3) In the simulations, stable state with only one single pulse can also be achieved, by decreasing the small signal gain g0 or increasing the saturation power of SA Psat, as seen in Fig. 8. This conclusion matches our experimental observations, where the operation state depends on the pump power and the set of waveplates. Furthermore, the relative intensity of each mode is also calculated, and the results of two of the modes are also given in Fig. 8. As shown in Fig. 8, the distribution of these modes suddenly changes when the operation state alters. This conclusion also matches with our experimental observations mentioned above, where the beam profile suddenly changes as the state alters. The transitions in both temporal and spatial domains probably result from the complex pulses interact, in both their transverse and temporal degrees of freedom. In addition, the spatial SA may also affect the multimode pulse evolution, thus the beam profile of steady state, when the pulse intensity alters due to the change of operation states. Our observations suggest that the multipulsing dynamics in MMF laser are spatiotemporal.
     figure: Fig. 8

    Fig. 8 The intensities of modes 2 and 3, normalized to that of mode 1, of the steady output versus (a) small signal gain g0 and (b) Psat of the SA. Regimes in different colors represent different operation states of the STML MMF laser.

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5. Conclusion

Unlike SMF lasers, there exists nonlinear intermodal interaction, thus energy transfer, among the transverse modes in the STML MMF lasers. Therefore, the spatiotemporal nonlinear dynamics in STML MMF lasers are multifarious and complex. In this paper, we report on the experimentally demonstration of two typical STML states of multiple-soliton in an MMF laser, i.e. harmonic mode locking and multiple pulses. Numerical simulations are carried out to investigate the evolution and steady output of multiple-soliton. The experimental observation and numerical simulation show that, the mode components, thus the beam profile, of output multiple-soliton could be different for a same MMF cavity with different coupling coefficient or even different operation states.

More importantly, from the present results we can draw a conclusion that, when multiple pulses are formed, they all appear to be multimodal, and have the same modal composition. When the laser transits from one regime to another, including within the multipulsing regime, this conclusion remains true at steady state, though the modal composition of the steady-state pulse changes. Our investigations suggest that the multipulsing dynamics in MMF laser are spatiotemporal, i.e. the pulses evolve in both their transverse and temporal degrees of freedom. We should point out that, the conclusions above are drawn from our current findings in the established MMF oscillator. In this laser, the gain fiber is a few-mode fiber since Yb-doped GRIN MMF is not commercially available. Accordingly, the gain fiber is simplified as an SMF in the simulations. Nevertheless, the passive fiber used in the oscillator is a highly-multimode GRIN fiber, and this simplified STML laser could provide a platform for studying some spatiotemporal pulse dynamics of the nonlinear systems with MMF. More investigations into the multipulsing dynamics in MMF lasers, including lasers with complete highly-multimode fibers, are required in the future.

Funding

National Key Scientific Instrument and Equipment Development Project of China under Grant 2014YQ510403, National Natural Science Foundation of China (NSFC) under Grant 51527901.

Acknowledgments

The authors wish to thank the anonymous reviewers for their constructive suggestions and comments.

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21. H. Zhang, D. Y. Tang, L. M. Zhao, and R. J. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express 18(5), 4428–4433 (2010). [CrossRef]   [PubMed]  

22. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007). [CrossRef]   [PubMed]  

23. Y. Meng, S. Zhang, X. Li, H. Li, J. Du, and Y. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012). [CrossRef]   [PubMed]  

24. A. Grudinin, D. Richardson, and D. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28(1), 67–68 (1992). [CrossRef]  

25. Z.-C. Luo, M. Liu, H. Liu, X.-W. Zheng, A.-P. Luo, C.-J. Zhao, H. Zhang, S.-C. Wen, and W.-C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based topological insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013). [CrossRef]   [PubMed]  

26. G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017). [CrossRef]   [PubMed]  

27. K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118(24), 243901 (2017). [CrossRef]   [PubMed]  

28. X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018). [CrossRef]   [PubMed]  

29. Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019). [CrossRef]   [PubMed]  

30. Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017). [CrossRef]  

31. P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018). [CrossRef]  

32. H. Qin, X. Xiao, P. Wang, and C. Yang, “Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser,” Opt. Lett. 43(9), 1982–1985 (2018). [CrossRef]   [PubMed]  

33. E. Ding, S. Lefrancois, J. N. Kutz, and F. W. Wise, “Scaling fiber lasers to large mode area: an investigation of passive mode-locking using a multi-mode fiber,” IEEE J. Quantum Electron. 47(5), 597–606 (2011). [CrossRef]   [PubMed]  

34. L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018). [CrossRef]  

35. P. Grelu, J. Béal, and J. Soto-Crespo, “Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime,” Opt. Express 11(18), 2238–2243 (2003). [CrossRef]   [PubMed]  

36. J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998). [CrossRef]  

37. M. Olivier and M. Piché, “Origin of the bound states of pulses in the stretched-pulse fiber laser,” Opt. Express 17(2), 405–418 (2009). [CrossRef]   [PubMed]  

References

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  4. Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
    [Crossref] [PubMed]
  5. K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
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  6. L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
    [Crossref]
  7. U. Teğin and B. Ortaç, “Spatiotemporal Instability of Femtosecond Pulses in Graded-Index Multimode Fibers,” IEEE Photonics Technol. Lett. 29(24), 2195–2198 (2017).
    [Crossref]
  8. O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
    [Crossref]
  9. L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
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  10. Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
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  13. K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
    [Crossref]
  14. K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
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  15. G. Lopez-Galmiche, Z. Sanjabi Eznaveh, M. A. Eftekhar, J. Antonio Lopez, L. G. Wright, F. Wise, D. Christodoulides, and R. Amezcua Correa, “Visible supercontinuum generation in a graded index multimode fiber pumped at 1064 nm,” Opt. Lett. 41(11), 2553–2556 (2016).
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    [Crossref] [PubMed]
  17. L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
    [Crossref] [PubMed]
  18. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).
    [Crossref]
  19. A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015).
    [Crossref]
  20. P. Wang, X. Xiao, H. Zhao, and C. Yang, “Observation of duration-tunable soliton explosion in passively mode-locked fiber laser,” IEEE Photonics J. 9(6), 1–8 (2017).
    [Crossref]
  21. H. Zhang, D. Y. Tang, L. M. Zhao, and R. J. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express 18(5), 4428–4433 (2010).
    [Crossref] [PubMed]
  22. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
    [Crossref] [PubMed]
  23. Y. Meng, S. Zhang, X. Li, H. Li, J. Du, and Y. Hao, “Multiple-soliton dynamic patterns in a graphene mode-locked fiber laser,” Opt. Express 20(6), 6685–6692 (2012).
    [Crossref] [PubMed]
  24. A. Grudinin, D. Richardson, and D. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28(1), 67–68 (1992).
    [Crossref]
  25. Z.-C. Luo, M. Liu, H. Liu, X.-W. Zheng, A.-P. Luo, C.-J. Zhao, H. Zhang, S.-C. Wen, and W.-C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based topological insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013).
    [Crossref] [PubMed]
  26. G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
    [Crossref] [PubMed]
  27. K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
    [Crossref] [PubMed]
  28. X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018).
    [Crossref] [PubMed]
  29. Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019).
    [Crossref] [PubMed]
  30. Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
    [Crossref]
  31. P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
    [Crossref]
  32. H. Qin, X. Xiao, P. Wang, and C. Yang, “Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser,” Opt. Lett. 43(9), 1982–1985 (2018).
    [Crossref] [PubMed]
  33. E. Ding, S. Lefrancois, J. N. Kutz, and F. W. Wise, “Scaling fiber lasers to large mode area: an investigation of passive mode-locking using a multi-mode fiber,” IEEE J. Quantum Electron. 47(5), 597–606 (2011).
    [Crossref] [PubMed]
  34. L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
    [Crossref]
  35. P. Grelu, J. Béal, and J. Soto-Crespo, “Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime,” Opt. Express 11(18), 2238–2243 (2003).
    [Crossref] [PubMed]
  36. J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998).
    [Crossref]
  37. M. Olivier and M. Piché, “Origin of the bound states of pulses in the stretched-pulse fiber laser,” Opt. Express 17(2), 405–418 (2009).
    [Crossref] [PubMed]

2019 (1)

Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019).
[Crossref] [PubMed]

2018 (6)

P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
[Crossref]

H. Qin, X. Xiao, P. Wang, and C. Yang, “Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser,” Opt. Lett. 43(9), 1982–1985 (2018).
[Crossref] [PubMed]

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018).
[Crossref] [PubMed]

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
[Crossref]

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]

2017 (9)

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref] [PubMed]

M. Conforti, C. Mas Arabi, A. Mussot, and A. Kudlinski, “Fast and accurate modeling of nonlinear pulse propagation in graded-index multimode fibers,” Opt. Lett. 42(19), 4004–4007 (2017).
[Crossref] [PubMed]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

U. Teğin and B. Ortaç, “Spatiotemporal Instability of Femtosecond Pulses in Graded-Index Multimode Fibers,” IEEE Photonics Technol. Lett. 29(24), 2195–2198 (2017).
[Crossref]

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
[Crossref]

P. Wang, X. Xiao, H. Zhao, and C. Yang, “Observation of duration-tunable soliton explosion in passively mode-locked fiber laser,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[Crossref] [PubMed]

2016 (5)

2015 (2)

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-locked fiber laser,” Optica 2(1), 36–39 (2015).
[Crossref]

2013 (3)

Z.-C. Luo, M. Liu, H. Liu, X.-W. Zheng, A.-P. Luo, C.-J. Zhao, H. Zhang, S.-C. Wen, and W.-C. Xu, “2 GHz passively harmonic mode-locked fiber laser by a microfiber-based topological insulator saturable absorber,” Opt. Lett. 38(24), 5212–5215 (2013).
[Crossref] [PubMed]

W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
[Crossref] [PubMed]

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

2012 (3)

2011 (1)

E. Ding, S. Lefrancois, J. N. Kutz, and F. W. Wise, “Scaling fiber lasers to large mode area: an investigation of passive mode-locking using a multi-mode fiber,” IEEE J. Quantum Electron. 47(5), 597–606 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

2007 (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

2003 (1)

1998 (1)

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998).
[Crossref]

1992 (1)

A. Grudinin, D. Richardson, and D. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28(1), 67–68 (1992).
[Crossref]

1974 (1)

Akhmediev, N.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).
[Crossref]

Amezcua Correa, R.

Andral, U.

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[Crossref] [PubMed]

Antonio Lopez, J.

Barthelemy, A.

Barthélémy, A.

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

Béal, J.

Bendahmane, A.

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

Bergman, K.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998).
[Crossref]

Billet, C.

P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
[Crossref]

Broderick, N. G. R.

Caravaca-Aguirre, A. M.

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
[Crossref]

Christodoulides, D.

Christodoulides, D. N.

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref] [PubMed]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref] [PubMed]

L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

Coillet, A.

Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019).
[Crossref] [PubMed]

Collings, B. C.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998).
[Crossref]

Conforti, M.

Couderc, V.

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

Cui, Y.

X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018).
[Crossref] [PubMed]

Ding, E.

E. Ding, S. Lefrancois, J. N. Kutz, and F. W. Wise, “Scaling fiber lasers to large mode area: an investigation of passive mode-locking using a multi-mode fiber,” IEEE J. Quantum Electron. 47(5), 597–606 (2011).
[Crossref] [PubMed]

Du, J.

Dudley, J. M.

P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
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Dupiol, R.

Eftekhar, M. A.

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

G. Lopez-Galmiche, Z. Sanjabi Eznaveh, M. A. Eftekhar, J. Antonio Lopez, L. G. Wright, F. Wise, D. Christodoulides, and R. Amezcua Correa, “Visible supercontinuum generation in a graded index multimode fiber pumped at 1064 nm,” Opt. Lett. 41(11), 2553–2556 (2016).
[Crossref] [PubMed]

Erkintalo, M.

Fabert, M.

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

Fini, J.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
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Genty, G.

P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
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Gloge, D.

Grelu, P.

Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019).
[Crossref] [PubMed]

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
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P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).
[Crossref]

P. Grelu, J. Béal, and J. Soto-Crespo, “Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime,” Opt. Express 11(18), 2238–2243 (2003).
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Grudinin, A.

A. Grudinin, D. Richardson, and D. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28(1), 67–68 (1992).
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Hao, Y.

Herink, G.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

Jalali, B.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
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Kibler, B.

Knize, R. J.

Knox, W. H.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998).
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Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
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Krupa, K.

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[Crossref] [PubMed]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
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Kudlinski, A.

Kurtz, F.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

Kutz, J. N.

E. Ding, S. Lefrancois, J. N. Kutz, and F. W. Wise, “Scaling fiber lasers to large mode area: an investigation of passive mode-locking using a multi-mode fiber,” IEEE J. Quantum Electron. 47(5), 597–606 (2011).
[Crossref] [PubMed]

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998).
[Crossref]

Lefrancois, S.

E. Ding, S. Lefrancois, J. N. Kutz, and F. W. Wise, “Scaling fiber lasers to large mode area: an investigation of passive mode-locking using a multi-mode fiber,” IEEE J. Quantum Electron. 47(5), 597–606 (2011).
[Crossref] [PubMed]

Li, B.

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
[Crossref]

Li, H.

Li, M.-J.

L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

Li, X.

Liu, H.

Liu, M.

Liu, X.

X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018).
[Crossref] [PubMed]

Liu, Z.

L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref] [PubMed]

Lopez-Galmiche, G.

Luo, A.-P.

Luo, Z.-C.

Lushnikov, P. M.

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

Mafi, A.

Mas Arabi, C.

Meng, Y.

Merolla, J.-M.

P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
[Crossref]

Millot, G.

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

Modotto, D.

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]

Mussot, A.

Närhi, M.

P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
[Crossref]

Nelson, L.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Nithyanandan, K.

Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019).
[Crossref] [PubMed]

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[Crossref] [PubMed]

Nolan, D. A.

L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

Olivier, M.

Ortaç, B.

U. Teğin and B. Ortaç, “Spatiotemporal Instability of Femtosecond Pulses in Graded-Index Multimode Fibers,” IEEE Photonics Technol. Lett. 29(24), 2195–2198 (2017).
[Crossref]

Payne, D.

A. Grudinin, D. Richardson, and D. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28(1), 67–68 (1992).
[Crossref]

Piché, M.

Piestun, R.

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
[Crossref]

Qin, H.

Renninger, W. H.

W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
[Crossref] [PubMed]

Richardson, D.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

A. Grudinin, D. Richardson, and D. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28(1), 67–68 (1992).
[Crossref]

Ropers, C.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Runge, A. F. J.

Ryczkowski, P.

P. Ryczkowski, M. Närhi, C. Billet, J.-M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018).
[Crossref]

Sanjabi Eznaveh, Z.

Shalaby, B. M.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

Solli, D. R.

G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017).
[Crossref] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Soto-Crespo, J.

Sylvestre, T.

Tang, D. Y.

Tchofo-Dinda, P.

Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019).
[Crossref] [PubMed]

K. Krupa, K. Nithyanandan, U. Andral, P. Tchofo-Dinda, and P. Grelu, “Real-time observation of internal motion within ultrafast dissipative optical soliton molecules,” Phys. Rev. Lett. 118(24), 243901 (2017).
[Crossref] [PubMed]

Tegin, U.

U. Teğin and B. Ortaç, “Spatiotemporal Instability of Femtosecond Pulses in Graded-Index Multimode Fibers,” IEEE Photonics Technol. Lett. 29(24), 2195–2198 (2017).
[Crossref]

Tonello, A.

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

Tsia, K. K. M.

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
[Crossref]

Tzang, O.

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
[Crossref]

Wabnitz, S.

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A (Coll. Park) 97(4), 043836 (2018).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthelemy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42(7), 1293–1296 (2017).
[Crossref] [PubMed]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

Wagner, K.

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
[Crossref]

Wang, P.

H. Qin, X. Xiao, P. Wang, and C. Yang, “Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser,” Opt. Lett. 43(9), 1982–1985 (2018).
[Crossref] [PubMed]

P. Wang, X. Xiao, H. Zhao, and C. Yang, “Observation of duration-tunable soliton explosion in passively mode-locked fiber laser,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]

Wang, Z. Q.

Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and P. Grelu, “Optical soliton molecular complexes in a passively mode-locked fibre laser,” Nat. Commun. 10(1), 830 (2019).
[Crossref] [PubMed]

Wei, X.

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
[Crossref]

Wen, S.-C.

Wise, F.

Wise, F. W.

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref] [PubMed]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref] [PubMed]

L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
[Crossref] [PubMed]

E. Ding, S. Lefrancois, J. N. Kutz, and F. W. Wise, “Scaling fiber lasers to large mode area: an investigation of passive mode-locking using a multi-mode fiber,” IEEE J. Quantum Electron. 47(5), 597–606 (2011).
[Crossref] [PubMed]

Wong, K. K. Y.

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
[Crossref]

Wright, L. G.

L. G. Wright, Z. M. Ziegler, P. M. Lushnikov, Z. Zhu, M. A. Eftekhar, D. N. Christodoulides, and F. W. Wise, “Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial, and Outlook,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–16 (2018).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref] [PubMed]

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]

L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref] [PubMed]

G. Lopez-Galmiche, Z. Sanjabi Eznaveh, M. A. Eftekhar, J. Antonio Lopez, L. G. Wright, F. Wise, D. Christodoulides, and R. Amezcua Correa, “Visible supercontinuum generation in a graded index multimode fiber pumped at 1064 nm,” Opt. Lett. 41(11), 2553–2556 (2016).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

Xiao, X.

H. Qin, X. Xiao, P. Wang, and C. Yang, “Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser,” Opt. Lett. 43(9), 1982–1985 (2018).
[Crossref] [PubMed]

P. Wang, X. Xiao, H. Zhao, and C. Yang, “Observation of duration-tunable soliton explosion in passively mode-locked fiber laser,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]

Xu, W.-C.

Xu, Y.

Y. Yu, B. Li, X. Wei, Y. Xu, K. K. M. Tsia, and K. K. Y. Wong, “Spectral-temporal dynamics of multipulse mode-locking,” Appl. Phys. Lett. 110(20), 201107 (2017).
[Crossref]

Yang, C.

H. Qin, X. Xiao, P. Wang, and C. Yang, “Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser,” Opt. Lett. 43(9), 1982–1985 (2018).
[Crossref] [PubMed]

P. Wang, X. Xiao, H. Zhao, and C. Yang, “Observation of duration-tunable soliton explosion in passively mode-locked fiber laser,” IEEE Photonics J. 9(6), 1–8 (2017).
[Crossref]

Yao, X.

X. Liu, X. Yao, and Y. Cui, “Real-time observation of the buildup of soliton molecules,” Phys. Rev. Lett. 121(2), 023905 (2018).
[Crossref] [PubMed]

Yu, Y.

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Figures (8)

Fig. 1
Fig. 1 Experimental setup. (a) The MMF laser (SPDM, short pass dichroic mirror; L1 and L2, lens; OM4, passive GRIN multimode fiber; HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarization beam splitter; BS, beam splitter; ISO, isolator; SF, spectral filter; M1-3, mirrors); (b) Measurement setup. Method 1: spectral filtering; Method 2: spatial sampling (BS: beam splitter; OSA: optical spectrum analyzer; PD, photodetector).
Fig. 2
Fig. 2 Second-order harmonic mode locking state. (a) Pulse train measured by oscilloscope; (b) RF spectrum of pulse train with a resolution bandwidth of 1 Hz (inset: RF spectrum over a 1 GHz span with a resolution bandwidth of 2 kHz); (c) Autocorrelation trace of output pulses; (d) Spectra of the whole output (solid curve) and the spectral-filtered outputs (dashed curves), with the corresponding beam profiles shown in (e)-(h); (e) Beam profile of the whole output, with black circles represented the spatial sampling positions (one sampler fixed at the center, and another sampler moving along the direction of arrow); (f)-(h) Beam profiles corresponding to different spectrum regions by tunable spectral-filtering of the output; (i) Pulse intensities and pulse intervals of the pulse trains sampled by the moving sampler at different position (the intensity corresponding to the fixed sampler is 53); (j) pulse train sampled by the fixed sampler; (k) a typical pulse train sampled by the moving sampler; (l) and (n) are the corresponding optical and RF spectra of (j) and (k), respectively; (m) RF spectra of the spectral-filtered outputs corresponding to (f)-(h).
Fig. 3
Fig. 3 Multiple pulses state. (a) Pulse train measured by oscilloscope; (b) Optical spectrum (inset: beam profile); (c) Pulse intensities and pulse intervals of the pulse trains sampled at different position.
Fig. 4
Fig. 4 Soliton molecule state. (a) Pulse train measured by oscilloscope; (b) Optical spectrum (inset: beam profile); (c) Autocorrelation traces of different spatial samplings, S1 and S2.
Fig. 5
Fig. 5 Mode-resolved simulation results for soliton molecule in the MMF laser. Esat = 2 nJ, small signal gain g0 = 15 m−1 and Psat of the SA is 180 W. (a) Mode-resolved pulse evolution with roundtrip (inset: beam profile); (b) Pulse evolution with roundtrip; (c) Intracavity pulse evolution in the last roundtrip; (d) Temporal output of the last roundtrip; (e) Pulse energy evolution with roundtrip. In (a), the initial pulses intensities are enlarged 200 times for clarity. In mode-resolved figures, the modes are color-coded. Among those modes, mode 2 is the highest, modes 1, 3 and 4 are almost overlapped, modes 5 and 6 are almost overlapped. OC: output coupler, SA: saturable absorber, BF: bandpass filter, SF: spatial filter.
Fig. 6
Fig. 6 Evolution of multiple pulses with different initial conditions. Small signal gain g0 = 20 m−1 and Psat of the SA is 180 W. (a) Evolution with two unequal pulse input, and (b) is the corresponding mode-resolved steady output; (c) Evolution with random noise input, and (d) is the corresponding steady output.
Fig. 7
Fig. 7 Mode-resolved output intensities and the corresponding beam profiles with various coupling coefficients (CCs) operating in multiple pulses state. CC I = [4,2,2,1,1,1], CC II = [2,4,2,2,1,1], CC III = [1,1,1,2,2,4], and other parameters are the same as Fig. 6.
Fig. 8
Fig. 8 The intensities of modes 2 and 3, normalized to that of mode 1, of the steady output versus (a) small signal gain g0 and (b) Psat of the SA. Regimes in different colors represent different operation states of the STML MMF laser.

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z A p ( z , t ) = i ( β 0 ( p ) [ β 0 ( 0 ) ] ) A p ( β 1 ( p ) [ β 1 ( 0 ) ] ) A p t i β 2 ( p ) 2 t 2 A p i n 2 ω 0 c l , m , n { ( 1 f R ) S p l m n K A l A m A n * + f R A l S p l m n R t d τ A m ( z , t τ ) A n * ( z , t τ ) h R ( τ ) }

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