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Higher-order mode supercontinuum generation in dispersion-engineered liquid-core fibers

Supercontinuum generation enabled a series of key technologies such as frequency comb sources, ultrashort pulse sources in the ultraviolet or the mid-infrared, as well as broadband light sources for spectroscopic methods in biophotonics. Recent advances utilizing higher-order modes have shown the potential to boost both bandwidth and modal output distribution of supercontinuum sources. However, the strive towards a breakthrough technology is hampered by the limited control over the intra- and intermodal nonlinear processes in the highly multi-modal silica fibers commonly used. Here, we investigate the ultrafast nonlinear dynamics of soliton-based supercontinuum generation and the associated mode coupling within the first three lowest-order modes of accurately dispersion-engineered liquid-core fibers. By measuring the energy-spectral evolutions and the spatial distributions of the various generated spectral features polarization-resolved, soliton fission and dispersive wave formation are identified as the origins of the nonlinear broadening. Measured results are confirmed by nonlinear simulations taking advantage of the accurate modeling capabilities of the ideal step-index geometry of our liquid-core platform. While operating in the telecommunications domain, our study allows further advances in nonlinear switching in emerging higher-order mode fiber networks as well as novel insights into the sophisticated nonlinear dynamics and broadband light generation in pre-selected polarization states.

. Spectral distribution of the difference of the effective indices of the various modes ∆neff for a CS2-core fiber with a diameter Øcore = 3.9 µm. At 2.1 µm the effective indices of TM01 and HE21 mode match. Figure S2 demonstrates the influence of the soliton wavelength on both phase-matched DWs in the ND regimes of a system with double ZDWs using the phase-matching equation (Equ.1 of main text), which includes the nonlinear phase of the soliton. A red-shifted soliton wavelength causes both phase-matched DWs to blue-shift and vice versa. In contrast to the short-wavelength DW1, which can be already generated during the initial fission process by a higher-order soliton at pump wavelength ( Fig. S2(a)), the wavelength of DW2 obtained by nonlinear simulations (purple line) is phase-matched only when considering the soliton at around 1.85 µm, Fig. S2(c), which corresponds to the split-off fundamental soliton generated after the initial fission process as detected in nonlinear simulations and experiments.

Supplementary information III: Stokes parameters of injected beam and theoretical fiber modes
To determine the Stokes parameters S0, S1, S2 and S3 of the fiber input beam, four intensity images are taken after the initially Gaussian beam is converted to ring-shaped beams with radial/azimuthal polarization by the s-wave plate (and an additional half-waveplate for the HE21-like polarization). Three images are taken behind a linear polarizer at 0°, 45° and 90°, while for the fourth image an additional quarter-wave plate is inserted (polarizer at 45°); both components are removed after characterization [3,4]. The measured spatial distributions of the first three Stokes parameters (S0, S1 and S2) of the input beam ( Fig. S3(a)) are in good agreement with the Stokes parameters calculated from the simulated fiber modes (Fig. S3(b)). The last Stokes parameter S3 characterizing circular polarization does not vanish in experiments as predicted by simulations. This discrepancy is caused by the quarter-wave plate, which is only inserted for the measurement of S3. The concentric lopes present in all measurements result from diffraction as the beam was slightly bigger than the s-wave plate aperture. Figure S3. Stokes parameters S0, S1, S2 and S3 of (a) measured input beams before the coupling lens and (b) simulated higherorder TM01, TE01 and HE21 modes. The appropriate scale bars are shown in the top left plots of (a) (2 mm) and (b) (2 µm). The color scale in the lower right corner applies for all images.

Supplementary information IV: Overlap calculations and focusing of fields
To estimate the power distribution across the different higher-order modes at the input of the sample, the electric field of the experimental beam Eexp is extracted from the Stokes parameter measurements after the s-waveplate. To account for the focusing of the incoupling lens, the fields are numerically focused by geometric refraction on a reference sphere while maintaining the intensity law, as explained in the book Principles of Nano Optics [5]. The modal amplitude amode is a measure of the fraction of power in a particular fiber mode and was calculated by the 2D overlap integral of the cross-product of the electric field vector Eexp of the numerically focused beam and the magnetic field vector Hmode of the fiber modes taking into account its total power Pmode [6] . (S1)

Supplementary information V: Coherence properties and example evolution calculated by multimode nonlinear pulse propagation simulations
In order to quantify the spectral coherence (exemplarily for the HE21 mode), 100 output spectra were simulated independently with random shotnoise distributions and an ensemble average was calculated to obtain the first-order coherence according to reference [7]. The average output spectrum (red curve, right axis) and the coherence (blue curve, left axis) are shown in Fig. S4 and reveal a very high spectral coherence. Figure S4. Simulated coherence (blue curve) and averaged output spectrum (red curve) from 100 independent numerical simulations of the HE21 mode.
Both, the temporal and spectral evolution as a function of the propagation distance of the TM01 mode (Fig. S5), show typical soliton dynamics during SCG. By self-phase modulation in combination with anomalous dispersion the pulse spectrally broadens and temporally shortens in the first millimeter of propagation. The maximum spectral extent is reached at the fission point of about 1 cm. At this point the higher-order soliton decays into fundamental solitons accompanied by dispersive wave generation. After 2 cm of propagation, no significant changes in the spectrum are observed as only dispersive effects occur because there is no significant temporal overlap between the spectral components. Figure S5. (a-b) Temporal and (c-d) spectral evolution of the TM01 mode (99%) and HE21 mode (1%) as a function of the propagation distance, excited simultaneously in a CS2-core fiber with Øcore= 3.9 µm and in-fiber pulse energy of 403 pJ. Note that the temporal and spectral fields are normalized independently for both modes to their respective maximum for better visibility of the weak HE21 mode. Figure S6 shows simulated spectra of individual modes excited simultaneously with different modal amplitudes (a) 99 % TM01, 1 % HE21, (b) 96 % TE01, 3 % TM01, 1 % HE21 in the CS2-core fiber (Øcore = 3.9 µm with a total in-fiber energy of 403 pJ. The individual spectra correspond to the spectra shown in Figs. 3b-1 and 3b-2 where the sum of the spectral power densities of all contributing modes is plotted. In Fig. S6 the generation of new frequency components in the weakly excited modes is clearly visible, which is a result of nonlinear intermodal coupling. Calculations of the intra-modal phase-matching between soliton and DWs (triangles in Fig. S6) match well with the simulated spectral positions of both short-and longwavelength DWs of the strongly excited mode. Figure S6. Simulated spectra of strongly and weakly excited higher-order modes for a total in-fiber energy of 403 pJ injected into a CS2-core fiber with Øcore= 3.9 µm. The input power ratios are (a) 99 % TM01, 1 % HE21, (b) 96 % TE01, 3 % TM01, 1 % HE21. The triangles mark the calculated phase-matching wavelengths for the prominent mode.