The concept of optical soliton propagation in multimode (MM) fibers was introduced in the literature nearly forty years ago [1]. In 2013, Renninger and Wise observed that MM solitons result from the simultaneous compensation of chro- matic dispersion and modal dispersion [2]. The asso-ciated Raman-induced MM soliton self-frequency shift (SSFS) in GRIN MM fibers was investigated in a low pulse energy (up to 3 nJ) situation, where the MM soli-ton is essentially carried by the fundamental mode of the MM fiber. The spatiotemporal behavior of MM solitons, including their formation and fission, was later investi- gated by Wright et al. with input beams leading to input energy distribution among several guided modes [3-4]. The impact of self-imaging on soliton propagation in GRIN MM fibers was recently investigated using a simplified model, based on the SM generalized nonlinear Schro ̈dinger equation (GNLSE) with a spatially varying effective mode area [5-7], evolving according to the variational approach.

In this work, we theoretically and experimentally study the fission of high input energy (up to 550 nJ), MW peak power femtosecond high-order MM solitons in a GRIN fiber. Input multi-soliton pulses undergo Raman-induced fission into multiple fundamental MM solitons. Our experiments permit to describe the range of validity of the reduced SM soliton description. Moreover, we also reveal several unexpected properties of soliton fission in MM fibers. Specifically, we observed that, in the high input energy regime, nonlinear losses owing to side scattering in the first few centimeters of the fiber lead to output energy clamping and SSFS suppression. MM solitons spontaneously self-organize into a wavelength multiplex, before undergoing SSFS. Moreover, the temporal duration of each MM soliton remains a constant, in spite of their different wavelengths and associated dispersions. The analysis of the order of the generated fundamental solitons reveals their inherent MM nature.

Fig. 1(a) shows the numerically simulated (via the 3D+1 CNLSE) evolution of the intensity spectrum as a function of fiber length, for the input energy of 90 nJ: the fission of the input pulse leads to a complex wavelength multiplex of MM solitons, which is generated at the short distance of about ten centimeters. The resulting soliton wavelength shifts are nonadiabatic, i.e., they occur over a much shorter distance than the length required to obtain an equivalent wavelength shift via the SM SSFS formula.

When displaying the pulse power in the time domain, i.e., by integrating the field intensity across the transverse x, y axes, the simulation of Fig. 1(b) shows that the input pulse undergoes a temporal compression over the first 5 mm of fiber. Correspondingly, it reaches local peak powers up to several MW until it experiences a soliton fission, thus producing a number of solitons, each with a comparable pulse width between 45 fs and 60 fs, and different peak powers, ranging from few tens of kW up to 300 kW.

In our experiments, we observed a strong output energy clamping for an input pulse energy >100 nJ. Nonlinear transmission measurements show that energy losses are limited from zero to below 20% for input energies up to 100 nJ. For energies >150 nJ, the output energy saturates to a nearly constant value. We ascribe the nonlinear loss to multi-photon absorption, leading to the side-emission of broadband blue fluorescence, as well as to scattering from the cladding of NIR radiation, THG, and visible dispersive wave sideband generation. As a result, two energy regimes are observed in our experiments. In the first regime ("low-loss" regime), for energies<100 nJ, nonlinear losses are negligible. Whereas for higher input energies ("high-loss" regime) significant nonlinear loss occurs over the first few cm of fiber, and the output energy is clamped. Eventually, the beam undergoes catastrophic collapse with permanent damages at the input and across the fiber length.

To disclose the details of high-order MM soliton fission, and the subsequent SSFS dynamics, we carried out a cutback experiment. Figure 2 illustrates the length dependence of the output spectrum, for an input pulse energy of 300 nJ. A wavelength multiplex of fundamental Raman MM solitons appears at 7 cm, as an outcome of high-order MM soliton fission.

Figure 3 shows the experimental output spectra in a linear scale (with a GRIN fiber length of 30 cm) for several values of the input energy, and llustrates well the transition from the "low-loss" to the "high-loss" regime. A progressively larger number of MM Raman solitons is generated as the input energy grows larger. Individual MM solitons undergo different amounts of SSFS. In the "high-loss" propagation regime, for input energies >150 nJ, the Raman-induced wavelength shift stops at 2250-2300 nm, owing to the nonlinear losses that clamp the output energy.

The difference between the "low-loss" and the "high-loss" regimes is illustrated by Figure 4, providing experimental data (empty squares and empty circles) and numerical simulations (solid curves) of the wavelength of solitons Raman 1 and Raman 2, respectively, as a function of their output energy E. Here the violet dashed line is analytically obtained from the SM SSFS equation; we inserted the pulsewidth and wavelength of the corresponding output soliton from simulations. As can be seen, in the "low-loss" regime, and up to E=5 nJ, the soliton wavelength increases with soliton energy according to the scalar SSFS formula. For 5<E<15 nJ, that is, in the "low-loss" regime, the soliton wavelength still increases linearly with its energy, but with a slope that is considerably higher from the value predicted by the SSFS. Whereas in the ``high-loss'' regime, i.e., for soliton energies E>15 nJ, experimental data show an irregular distribution of Raman soliton wavelengths, oscillating around 2100 nm for the Raman 1 soliton, and 1900 nm for the Raman 2 soliton, respectively.

The theoretical prediction of the generation of a soliton wavelength multiplex with nearly equal pulse widths is confirmed by the experimental analysis of the temporal duration of the different Raman solitons. Figure 5 compares experimental data and numerical simulations for the pulse width of different output MM Raman solitons, as a function of their energy. In the "low-loss" regime, the pulse width of individual Raman MM solitons is well approximated by the SM analytical soliton formula. A train of pulses with hyperbolic secant temporal shape, different peak powers, and wavelengths, but nearly equal pulse widths (around 45 fs) is generated for soliton energies E>5 nJ, in good agreement with numerical simulations. Whereas in the "high-loss" regime, the black crosses in Fig. 5 show a departure from the SM soliton formula. Experimental spectra and autocorrelations (not shown here) show that all MM Raman-shifted solitons converge to nearly same and constant (with respect to soliton energy) pulse width, with a value between 50 fs and 60 fs.

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