Solving the mystery of the walk-off soliton

Pioneering works on multimode fiber transmission [1,2], dated 40 years ago, predicted the existence of multimode solitons, providing conditions for the temporal trapping of the input optical modes to form a spatiotemporal soliton [3-5]. Only recently [6-8], multimode solitons were experimentally investigated in graded-index multimode fibers (GRIN), unveiling the complexity of a new, uncharted field. In our work, we experimentally and numerically investigated the propagation of ultrashort pulses over long distances of GRIN fiber. We discovered a new class of spatiotemporal solitons with surprising properties: basically single-mode, they cannot be described by the variational theory; their pulsewidth and energy are independent of the input pulse duration, and appear to depend only on the fiber dispersive parameters and, therefore, the wavelength. The new solitons are promising for the delivery of high-energy laser beams, for high-power spatiotemporal mode-locked multimode fiber lasers, and for high-bit rate multimode fiber networks.


Main
A mysterious object came to our attention, when testing the transmission of ultrashort pulses (input pulsewidth and wavelength were 60 fs to 240 fs, and 1300 nm to 1700 nm respectively) over long spans of graded-index (GRIN) optical fiber (see Methods -Experiments). When coupling exactly on the fiber axis, with 15 µm input beam waist, we could excite 3 axial-symmetric modes, that will be addressed from now on as LP01, LP02, LP03, for familiarity with multimodal step-index fibers. The fraction of power carried by these modes was calculated by a specific software [9] to be 52%, 30%, and 18%, respectively. The input laser pulse energy ranged between 0.1 and 20 nJ.
What we observed did not seem to obey predictions of the variational theory for spatiotemporal solitons [10][11][12]: Fig. 1 resumes the experimental evidence after 120 m of GRIN fiber. By testing different input wavelengths (1420 to 1550 nm) and input pulse widths (67 to 245 fs), spatiotemporal solitons with a common minimum pulsewidth of 260 fs were observed at the fiber output, for values of the input pulse energy ranging from 2 nJ up to 4 nJ (Fig. 1a). The case of a 1300 nm input wavelength represented an exception, which resulted into a minimum pulsewidth of 200 fs at higher energy.
The output beam waist (Fig. 1b) was severely reduced from its input value (down to a value of 8.5 µm, which is close to the theoretical value of 7.7 µm for the fundamental fiber mode), in correspondence of the input energy leading to minimum output pulsewidth. In this regime, the output beam shape was substantially monomodal, with a measured ! =1.45, against the value of 1.3 of the input beam. The curve of the beam waist vs. energy was much narrower for input short (67 fs) pulses than for long (235 fs) pulses. Also for the beam waist, the case of 1300 nm represented an exception, with no significant beam reduction.
In all cases, numerical simulations performed with a coupled-mode equations model (see Methods -Simulations) fully confirmed the experimental results (empty dots).
Experimental evidence with shorter and longer spans of GRIN fiber (Fig. Supplementary   1) provided less stringent requirements for the optimal soliton energy when the fiber length is reduced. In particular, for short spans (e.g., 2 m or 10 m) the output pulsewidth remained constant for input pulse energies greater than 2.5 nJ.
What was this strange object? The minimum output pulsewidth, at 120 m distance, appeared to be independent on the input pulsewidth and wavelength, and occurred at comparable input energies in all cases. Why was the case at 1300 nm different? We decided to investigate this, starting from simulations. Fig. 2 is a numerical representation of the evolution of an input 235 fs pulse, composed of 3 axial modes, propagating over 120 m of GRIN fiber, at the optimal input energy of 2.5 nJ. During their propagation, the 3 non-degenerate modes remain temporally trapped; the mode LP01 acts as an attractor for other modes, owing to non-phase matched, asymmetrical inter-modal four-wave mixing (IM-FWM), and to inter-modal stimulated Raman scattering (SRS) [14]; at the output, a monomodal bullet remains. The pulse carried by the fundamental or LP01 mode experiences SSFS, while it traps and captures a large portion of energy carried by higher-order modes (HOMs).  for different input wavelengths (1350, 1550 or 1680 nm). In each case, the optimal input energy that produces a minimum pulsewidth at 120 m was chosen; in all cases, it ranges between 2 nJ and 3 nJ when going from shorter to longer wavelengths. This energy remains the same for both initial pulse widths of 67 fs or 235 fs, respectively. In all cases, two pulses with the same wavelength and different initial pulsewidth form a soliton with identical initial pulsewidth "# . The necessary propagation distance for soliton formation is 1 m at 1350 nm (and, not shown, at 1300 nm and 1420 nm), and 6 m for both 1550 nm and 1680 nm. As the soliton propagates, its pulsewidth increases because of SSFS, which increases its wavelength, and local fiber dispersion, so that the soliton condition is maintained [3,15,16]. For longer distances, the pulsewidth differences, which are observed in Fig. 3 for the several input wavelengths, approach to a common value, as experimentally confirmed at 120 m (see Fig. 1b). The mystery was beginning to reveal itself. Input pulses with the same wavelength, comparable energies, and different pulse widths, all generate after a few meters of propagation a spatiotemporal soliton with common pulsewidth "# . "# increased for growing values of the input wavelength. Once that the spatiotemporal soliton was formed, a slow energy transfer into the LP01 mode was experimentally observed, while it simultaneously experiences SSFS (Fig. 1c). For distances larger than 100 m, the generated soliton appeared to be intrinsically monomodal, with a near-field waist approaching that of the fundamental mode of the MMF (Fig. 1b). For all tested wavelengths and pulse widths, a long-distance soliton was always observed for comparable input energies (i.e., between 2 nJ and 3 nJ in the case of initial excitation of 3 axial modes). Numerical simulations (Supplementary Fig. 2)  We recall now the following characteristic lengths: i) The mean modal walk-off length of the forming soliton & = # $ ,,,,, ⁄ , with # = "# 1.763 ⁄ , defined as the distance where, in the linear regime, the modes separate temporally.
iii) The random mode coupling and birefringence correlation lengths, *+ and *, , which are the characteristic length scales associated with linear coupling between degenerate modes or between polarizations, respectively [17][18][19].
In order to explain the mystery, we may assume that, when nonlinearity acts over distances shorter than those associated with random mode coupling and birefringence, i.e., for '( < *+ , *, , it is possible to observe a spatiotemporal soliton which is attracted into an effectively single-mode soliton [14]. Therefore, Eq. 1 confirms that a spatiotemporal soliton may be formed from the initial pulse, eventually leaving behind a certain amount of energy in dispersive waves. The soliton initial pulsewidth, and therefore energy, depends on the fiber dispersion parameters. Our mysterious solitonic object appears to be clamped to the fiber walk-off length & . The soliton pulsewidth "# , at its formation distance, varies with the wavelength of the input pump pulse, but its value turns out to be independent of input pulse duration. From Fig. 4, we find that a soliton with duration longer than few hundreds of femtoseconds cannot arise from non-degenerate modes. For relatively long input pulses (e.g., a 10 ps input pulse carried by 15 modes, as in the example of Supplementary Fig. 3), groups of modes separate temporally. However, in this case, it is still possible to inject the proper energy in each group of degenerate modes, in order to obtain the generation of several independent spatiotemporal solitons.

Simulations
Numerical simulations are based on a coupled-mode equations approach [20,21], which requires the preliminary knowledge of the input power distribution among fiber modes.

Experiments
The experimental setup included an ultra-short pulse laser system, composed by a Care was taken during the input alignment in order to observe, in the linear regime, an output near-field that was composed by axial modes only; this could be particularly appreciated for long lengths of GRIN fiber (120 m and more). The used fiber was a span (from 1 m to 850 m) of parabolic GRIN fiber, with core radius * = 25 µm, cladding radius 62.5 µm, cladding index *6<= = 1.444 at 1550 nm, and relative index difference ∆= 0.0103.
At the fiber output, a micro-lens focused the near field on an InGaAs camera (Hamamatsu C12741-03); a second lens focused the beam into an optical spectrum analyzer (Yokogawa AQ6370D) with wavelength range 600-1700 nm, and to a real-time multiple octave spectrum analyzer (Fastlite Mozza) with a spectral detection range of 1100-5000 nm. The output pulse temporal shape was inspected by using an infrared fast photodiode, and an oscilloscope (Teledyne Lecroy WavePro 804HD) with 30 ps overall time response, and an intensity autocorrelator (APE pulseCheck 50) with femtosecond resolution.