## 4 XPW Generation Experiment  
In order to further examine the gain narrowing pre-compensation effect with the WP, we conducted an XPW generation experiment, and the layout is shown in Figure 8. A zero-order half waveplate was inserted before the compressor of the laser system, and the half waveplate and compressor were combined as an energy attenuator to reduce the output energy to \(\sim\)250 \(\upmu\)J and to reduce the nonlinear effect in the first Glan laser polarizer of the XPW generation system [22]. Those output pulses went through the XPW generation system. The input beam firstly went through an energy attenuation system, a zero-order half waveplate and a Glan laser polarizer with a diameter of 12 mm in order to finely adjust the input beam energy and to obtain a purely vertical polarization laser beam. Secondly, a Galilean beam reducer, including a concave mirror with f = 1000 mm and a convex mirror with f = \(-\)350 mm, was used to focus the input beam in the air. Thirdly, two BaF\({}_{2}\) crystals with crystallographic orientations of [011] were both placed after the focal position to generate a high efficiency XPW [23] by optimizing the crystal position, separately. Fourthly, the XPW was collimated with a focal lens f = 1800 mm, and the second Glan laser polarizer was used to eliminate the fundamental beam. Finally, the generated XPW generation was reflected by a 45\({}^{\circ}\) ultrafast laser mirror to the spectrometer.  
Figure 6: The spectra of the amplified beam for different rotation angles of the WP.  
Figure 7: The measured pulse duration and spectral phase after optimizing the compressor, without WP and with WP at an angle of 34\({}^{\circ}\).  
The input laser energy was kept at 220 uJ, and the beam diameter was \(\sim\)600 um (at 1/\(e^{2}\) of the intensity profile) at the focal position. Since the XPW conversion efficiency and spectral broadening effect depend on the laser intensity on the BaF\({}_{2}\) crystal [24], we chose the same XPW conversion efficiency with BaF\({}_{2}\) crystals to test the gain narrowing pre-compensation effect, with the WP at angle of 34\({}^{\circ}\) and without the WP. The first BaF\({}_{2}\) crystal with a thickness of 1 mm was inserted behind the focal position until the XPW energy reached \(\sim\)10 uJ (efficiency \(\sim\)4.6%) to reduce the nonlinear effect and to avoid crystal damage. In addition, the second BaF\({}_{2}\) crystal with a thickness of 2 mm was placed behind the first BaF\({}_{2}\) crystal, with a total XPW energy of 33 uJ (efficiency 15%) to avoid damaging the crystal. The typical spectra of the input laser beam, the XPW with the 1 st BaF\({}_{2}\) crystal (XPW1) and the XPW with the 1 st and 2 nd BaF\({}_{2}\) crystals (XPW2) are shown in Figure 9, without the WP and with the WP at an angle of 34\({}^{\circ}\), respectively. The ultrafast reflection mirror supports the wavelength range from \(\sim\)750 nm to \(\sim\)850 nm for P-polarization, which induced the measured XPW spectra with the WP at an angle of 34\({}^{\circ}\) to have a dip at 860 nm. The FWHM of the generated XPW spectra was 45 nm without the WP, and the XPW spectra broadening was 1.8 times compared with the input spectra of 25 nm. The FWHM of the generated XPW spectra was 68 nm with the WP at an angle of 34\({}^{\circ}\), and the XPW spectra broadening was 1.7 times compared with the input spectra of 40 nm. The XPW spectra was broadened \(\sim\)1.7 times, which is consistent with normal XPW experiment results [25,26]. The XPW experiment indicated that gain narrowing pre-compensation with a WP works well, as the compressed pulse was close to being transform-limited and was consistent with the measured pulse duration. The much greater broadband XPW with a WP would be a benefit for high-temporal-contrast ultra-fast ultra-intensity laser amplification.