## 2 Experimental Section
### Spectral Broadening in a Gas-Filled HCF  
Spectral broadening in a gas-filled HCF is based on self-phase modulation (SPM) due to the intensity dependent refractive index \(n(I)\) of the gas medium:  
\[n(I)=n_{0}+n_{2}\cdot I \tag{3}\]  
Here \(n_{2}\) is the nonlinear refractive index, which is 1.6 \(\pm\) 0.3 \(\times\) 10\({}^{-19}\) cm\({}^{2}\cdot\)W\({}^{-1}\) for SF\({}_{6}\)[20], being comparable to argon, but having a more complex behavior as is outlined as follows. SF\({}_{6}\) is a symmetric molecule and stretching vibrations have to be taken into account with particular importance on the A\({}_{1\mathrm{g}}\) vibrational mode being the strongest Raman active one. The fundamental frequency \(\Omega_{\mathrm{Raman}}\) of 775 cm\({}^{-1}\) corresponds to a temporal period of 43 fs. For the presented experiments, the laser pulse duration is longer than the Raman oscillation period but shorter than the dephasing time \(T_{2}\) of the stretching vibration, which is referred as transient or non-stationary scattering [21] and is limited by  
\[\mathrm{1/\Omega_{\mathrm{Raman}}<\tau_{\mathrm{pulse}}\ll T_{2}} \tag{4}\]  
Figure 2: After spectral broadening in a gas-filled hollow-core fiber (HCF) laser pulses are sent through a prism-compressor with a computer-controlled deformable mirror for pulse shaping. Afterwards, the pulses are focused into an argon-filled capillary where they generate high harmonics. These are detected by an extreme ultraviolet (XUV) spectrometer and its signal is used as a feedback for an evolutionary algorithm. \(T_{{}_{2}}\) is the inverse of the scattering bandwidth (6 ps for SF\({}_{6}\)) [22]. Using fs pulses spectral broadening in SF\({}_{6}\) is governed by an interplay of SPM and stimulated Raman scattering (SRS), especially in an HCF where the intrinsic collinearity of the HCF avoids phase-matched SRS. SRS leads to sidebands in the spectrum at fixed positions (Figure 3b) in contrast to argon (Figure 3a), where only SPM is responsible for spectral broadening (experimental conditions as in Section 2.1.). Since achieving the same spectral broadening in SF\({}_{6}\) allows reduction of either the gas pressure or the input intensity compared to argon, it enables the transfer of this scheme to Yb based fiber lasers delivering rather long pulses [18]. For a better comparison of the applicability for pulse-shaping two broadened spectra (spectral range between 700 nm and 880 nm) and their corresponding Fourier transform assuming a flat spectral phase are shown in Figure 4. For SF\({}_{6}\) two observations should be pointed out: (i) the total spectral width is a few nanometers larger due to the enhanced redshifted part of the spectrum and (ii) the width of the central peak of the spectrum is larger due to the missing dip around 775 nm as for argon. Looking at the Fourier transform of both spectra there is not much difference in terms of pulse duration and shape. However, the presented pulse shaping scheme may benefit from the spectral characteristics of the SF\({}_{6}\)-broadened spectrum due to the more favorable shape: (i) as a minor contribution the increase in total bandwidth directly translates into a better illumination of the deformable mirror surface and (ii) as a major contribution the spectral amplitude distribution is pronounced between 760 nm and 825 nm. It should be emphasized that the pressure of SF\({}_{6}\) is chosen such that the SRS sidebands are weak and cannot be observed in Figure 4a. In this case the spectral phase is still smooth and pulse compression by prisms is possible [18]. However, the cross-phase modulation products of the  
Figure 3: A comparison of spectral broadening in argon-filled **(a)** and SF\(\epsilon\)-filled **(b)** HCF for different gas pressures shows possible benefits for pulse-shaping experiments. For SF\({}_{6}\) the spectral broadening is larger than for argon for otherwise comparable conditions.