Hybrid soliton dynamics in liquid-core fibres

The discovery of optical solitons being understood as temporally and spectrally stationary optical states has enabled numerous innovations among which, most notably, supercontinuum light sources have become widely used in both fundamental and applied sciences. Here, we report on experimental evidence for dynamics of hybrid solitons—a new type of solitary wave, which emerges as a result of a strong non-instantaneous nonlinear response in CS2-filled liquid-core optical fibres. Octave-spanning supercontinua in the mid-infrared region are observed when pumping the hybrid waveguide with a 460 fs laser (1.95 μm) in the anomalous dispersion regime at nanojoule-level pulse energies. A detailed numerical analysis well correlated with the experiment uncovers clear indicators of emerging hybrid solitons, revealing their impact on the bandwidth, onset energy and noise characteristics of the supercontinua. Our study highlights liquid-core fibres as a promising platform for fundamental optics and applications towards novel coherent and reconfigurable light sources.

: Refractive index fit and model comparison. Experimental data points of the refractive index of bulk neat carbon disulfide measured by different groups. The solid black line refers to the fit obtained when using the two-oscillator Sellmeier equation (Eq. 1 in the main paper), whereas the two colored lines are calculated from either a one-oscillator model or a Cauchy equation (green: Samoc [4], blue: Kedenburg [5]). The names in the legend refers to the first author of the respective reference: Pfund [1], Wilhelmi [2], and Ghosal [3].

Supplementary Note 2: Impact of Model Uncertainties on Simulation Results
In Suppl. Fig. 2a we compare our model with the two other models using supercontinuum simulation (incl. loss) with identical input parameters (460 fs sech 2 pulse, 7 nJ) and realistic nonlinearity of CS 2 (85% molecular fraction, i.e. f m = 0.85). The results reveal a non-negligible difference between bandwidth and fission length between the models, whereas our model is closest to the early supercontinuum onset and the narrow bandwidth measured in the experiment.
In Suppl. Fig. 2b we investigated the impact of small uncertainties of the dispersion on our system indicators (i.e. bandwidth and supercontinuum onset energy) by comparing the output spectrum for three different core diameters. The results show only weak variations in the spectral bandwidth and small, but notable, differences in the measurable fission length.  (Samoc et al. [4], Kedenburg et al. [5], and our model from the main paper) and for (b) our dispersion model for three different diameters.
In Suppl.

Supplementary Note 3: Liquid-core Fibre Properties
Suppl. Fig. 5 shows the waveguide properties of each of the seven transversal modes supported in a CS 2 /silica step-index fibre with 4.7 µm core diameter. The V parameter is a measure of the guidance with V = 1 being the empiric limit where lossless propagation can be assumed.

Supplementary Note 4: Ultrafast Laser System and Transmission Limits
In the experiment a table-top fibre laser with two amplification stages is used similar to that described in [7] (Suppl. Fig. 6a). The thulium-doped fibres are pumped at 790 nm and the amplification fibres are constantly water cooled at a temperature of 20 °C. A grating compressor is used to compensate the second-order (β 2 ) phase of the output pulses. An acousto-optical modulator allows step-wise reduction of the pulse repetition rate starting at 11.6 MHz. The system features an output spectrum with a 20 dB bandwidth of 26 nm (Suppl. Fig. 6b) and near-transform-limited optical pulses with a FWHM width of 460 fs (Suppl. Fig. 6c). The pulse shape was reconstructed from the Fourier transform of the output spectrum and a third-order phase offset (β 3 = -0.025 ps 3 ) was added to match the recorded auto-correlation.

Output spectrum and (c) pulse shape I(t) reconstructed from the measured autocorrelation (AC) and spectrum of the Tm-based laser source. (d) Microscope image of the capillary input after transmission drop was measured (average power limit). A yellowish residue around the capillary hole is visible.
Supplementary Table 1 shows the experimental damage thresholds of the liquid-core fibre indicated by a sudden transmission drop observed for various input pulse parameters and pulse repetition rates ν rep . The pulse peak intensity I 0 is calculated in the focus assuming 50 % coupling efficiencies (as measured for low power coupling). Comparing the three parameter sets, we find that there are two origins for the drop: (a) thermal load for high average power (Suppl. Tab. 1, row 3, i.e., high repetition rate, thus low pulse energy), and (b) for high pulse energy (Suppl. Tab. 1, row 1, i.e., low repetition rate, low average power). In case (a) a yellow precipitation debris on the input side was observed (Suppl. Fig. 6d Figure 7 shows the power characteristic of the liquid-core fibre used in the supercontinuum experiments, i.e. input versus output pulse energy. It correlates well with the our simulation results revealing two domains of linear loss. The slope of the curve changes at the supercontinuum onset energy (E p < 2.5 nJ), after which the broadened spectra approach the strong absorption at 2.25 µm. The slope difference between experiment and simulation in the second domain originate from inaccuracies of our fit model for the CS 2 absorption, which cannot be improved on the basis of the old absorption data. However, the data reveal that the absorption of the system is known and dominantly linear.
Supplementary Fig. 7: Comparison of the input/output power characteristic of our system between experiment and simulation. Two linear absorption regimes are distinguishable by two different slopes of the linear power characteristics. Nonlinear absorption does not appear to be dominant.