## 2 Experimental Setup
### Generation and Isolation of the 9 th Harmonic  
The 270-nm pulse was loosely focused with an \(f\) = 2000-mm lens into a gas flow cell (10 cm long, \(z\)1.3 mm holes on both sides) in a vacuum chamber. The beam waist size and focused intensity at the gas cell are estimated to be 140 \(\upmu\)m and 2 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\), respectively. The entrance window to the chamber was a 1-mm-thick CaF\({}_{2}\) plate, and the gas cell was filled with rare gas. Though the 9 th harmonic (= 3 rd \(\times\) 3 rd) is the lowest order obtained from the 270-nm driving pulse and thus is generated most efficiently, higher-order harmonics (15 th (= 3 rd \(\times\) 5 th, 54 nm), 21 st (= 3 rd \(\times\) 7 th, 39 nm), etc.) can also be generated in the gas cell. Since the 9 th harmonic is well separated in energy (>9 eV) from the higher-order harmonics, it can be isolated quite easily using aluminium mirrors, as described below. Among several target rare gas species (Ar, Kr, and Xe), the highest conversion efficiency for the 9 th harmonic generation was achieved in Kr [5].  
The generated 9 th harmonic was sent to the next chamber through an aperture (AP) with a diameter of \(z\)2 mm for differential pumping. To separate the 9 th harmonic from the 270-nm driving pulse, it was reflected twice by SiC Brewster-angle beam splitters [19]. The incident angle to the SiC beam splitters was set at 71.8\({}^{\circ}\), at which the reflectivity at 270 nm is minimized. The calculated reflectivities at 270 and 90 nm per bounce are <0.1% and 40%, respectively. Since the epitaxially grown SiC wafer (Cree Inc., Durham, NC, USA) was sufficiently smooth, we used the wafer without further polishing. We found that the SiC beam splitters degrade due to VUV irradiation; after 20-30 h of operation, the reflectivity at 90 nm was reduced by a factor of two, whereas at 270 nm it increased. Since the reflectivities could be fully restored to their initial values by changing the irradiated position, it appears that a layer of contamination builds up on the SiC surface.  
After a double bounce by bare aluminum mirrors (Al) and passing through another aperture (AP) for differential pumping, the 90-nm pulse was sent to the time-of-flight (TOF) chamber. As shown in Figure 3a, aluminum is highly reflective for the 9 th harmonic (\(\sim\)90% for normal incidence), while its reflectivity for the 15 th harmonic (and even higher harmonics) is two orders of magnitude smaller than that for the 9 th harmonic. Figure 3b shows a spectrum of the 9 th harmonic acquired with a VUV spectrometer (VM502, Roper Technologies, Lakewood Ranch, FL, USA) after a couple of bounces by bare aluminum mirrors. For the acquisition of this spectrum, the target gas medium was Kr and the back pressure was 15 Torr. It is clear that a single-order 9 th harmonic pulse at 90 nm was obtained. In actuality, no evidence of the higher-order harmonics manifested itself in our TRPES spectra [11; 12; 13; 14; 15; 16; 17]. As shown in Figure 3c, the spectral width of the 9 th harmonic was \(\sim\)0.5 nm, and TL pulse width calculated from the spectrum was \(\sim\)30 fs. Although we have not measured the pulse width of the 9 th harmonic directly, its actual pulse width appears somewhat longer than the TL width; the cross-correlation time between the 90-nm and 205-nm pulses estimated using (1 + 1') photoionization of nitric oxide [14] was 80-90 fs.  
Figure 2: Schematic illustration of the experimental setup for vacuum-ultraviolet time-resolved photoelectron spectroscopy (VUV TRPES) using 205-nm pump and 90-nm probe pulses. THG: Single-pass third-harmonic generation stage; FHG: Fourth-harmonic generation using nonlinear crystals; AP: Apertures for differential pumping; SiC: SiC Brewster-angle beam splitters; Al: Bare aluminum mirrors; TOF: Magnetic-bottle time-of-flight photoelectron spectrometer. The inset shows the THG spectrum.