## 3 Results  
We used the pulse energy of only \(\sim\)17 mJ in this experiment, because the CaF\({}_{2}\) windows affect the frequency spectrum of the pump pulse greatly if the pulse energy is higher. Owing to the loss from multiple cutout of the center beam and other optical elements, the final peak power of driving laser for HHG was about 0.57 TW. The intensity at the focus point was 8 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\). Ar gas was filled in the gas cell with a 100-mm medium length. Figure 2 presents the intensities of the 29 th harmonic with respect to various pressures in the gas cell and the relative positions between the laser focus and the gas cell, respectively. Obviously, the intensity of the 29 th harmonic reached the maximum around the pressure of 12 Torr and the relative position was 0 mm, which is the optimized phase-matching condition. Figure 3 shows the harmonic spectra in the optimized phase-matching condition. These harmonics were accumulated with 10 shots and only one Al filter in the beam path. It is worth noting that the other filter was mounted on a motorized translation stage. By comparing the harmonic spectra detected with one filter or two, we could obtain the transmission ratios of the aluminum filters for individual harmonics. We did not use the transmission ratio in a database, because the transmission ratio of the Al filter can gradually change due to oxidation. By using this method, we estimated the transmission ratio of a 500 nm-thick aluminum filter was 5% for the 29 th harmonic. Figure 1: Schematic of the experimental high-order harmonic generation (HHG) setup. Spherical mirror (SM): gold-coated spherical mirror; Al: 500 nm-thick aluminum filters; Si: silicon reflecting mirror. The equipment marked with a double-headed arrow is mounted on the motorized translation stage. Figure 3: Measured harmonics in a 100-mm medium length with Ar gas. (**a**) Spatial (y-axis) and spectral (x-axis) profiles of HHG in the optimized conditions. (**b**) Spatial integration of HHG in the optimized condition. Figure 2: The 29 th harmonic intensity obtained in the experiment. (**a**) Intensity of the 29 th harmonic as a function of the relative position between the laser focus and the gas cell; (**b**) intensity of the 29 th harmonic as a function of the pressure in the gas cell. When the SM is removed from the beam path, the harmonics illuminate the photodiode detector for the absolute output energy measurement. Here, three Si mirrors were mounted at the Brewster's angle of 75\({}^{\circ}\) (light grazing incidence angle, \(\sim\)15\({}^{\circ}\)), acting as beam splitters that can efficiently reflect the harmonics, but absorb most of the fundamental pulses. [PERSON] et al. [18] has determined that the reflectivity of Si mirror is \(\sim\)0.56 for a 30-nm spectral region, when it is at the Brewster's angle for 800 nm. In this section, the IR pulses were completely blocked, and only the remaining XUV signal was sent into the photodiode detector (AXUV100G). By comparing the HHG energy obtained with one or three Si mirrors, the measured reflectivity of a Si mirror was 0.42, lower than that measured by [PERSON] et al. [18], which might be influenced by the roughness of Si mirrors. Thus, the obtained total energy of HHG was estimated to be \(\sim\)300 nJ/pulse in the optimized phase-matching condition. Considering the spectral distribution in Figure 3b, every individual harmonic energy was separately calculated. Therefore, the 29 th harmonic energy was estimated to be \(\sim\)100 nJ by a single shot. By considering the distance between the gas cell and the XUV-CCD camera, the spatial profile of harmonics was converted to the emission angles. The spatial profile of the 29 th harmonic in the optimized condition is shown in Figure 4. The 29 th harmonic beam divergence was estimated to be \(\sim\)0.3 mrad (FWHM). We did not measure the pulse duration of the harmonic.