## 2 Methods  
The experimental setup is presented in Figure 1. A Ti:sapphire femtosecond laser amplifier system (Legend Elite HE+, Coherent Inc., Santa Clara, CA, USA) was employed to generate 800 nm, 35 fs, 500 Hz laser pulses with a horizontal linear polarization and a single-pulse energy up to 4 mJ. The combination of the half-wave plate (HWPI) and the linear polarizer (LP1) was used to continuously adjust the laser power. A Ronchi phase grating (RG1) with a period of 64 \(\upmu\)m was loaded on the spatial light modulator (SLM, PLUTO-NIR-015, Holeeye Inc., Berlin, Germany) to split the incident laser into \(\pm 1\) diffraction orders. The total power of the \(\pm 1\) diffraction orders contained more than 80% of the incident laser power. Two plano-convex lenses (L1 and L2) with focal lengths of 30 cm formed a 4\(f\) system. A pinhole filter (F) was located at the Fourier plane of the 4\(f\) system to block other diffraction orders (except \(\pm 1\) orders) of RG1. The fast axes of the two quarter-wave plates (QWP1 and QWP2) behind the pinhole filter were perpendicular to each other, making the linearly polarized \(\pm 1\) diffraction orders convert to the left- and right-handed circularly polarized beams. Another Ronchi grating (RG2) made of fused silica and with the same period as that loaded on the SLM was conjugated with RG1 relative to the 4\(f\) system, acting as a beam combiner. When the grating vectors of RG1 and RG2 were conjugated relative to the 4\(f\) system, we found that the combined femtosecond laser beam after RG2 was a scalar beam with horizontally linear polarization. When rotating RG2 around its normal direction, we generated a non-cylindrical vector beam with spatially changing linear polarization. The polarization distribution of the non-cylindrical vector beam was investigated by the setup shown in the dashed frame of Figure 1. When LP2 with horizontal transmission axis was employed, the beam intensity distribution recorded by the CCD camera (LU135M,  
Figure 1: Experimental setup of fabricating the compound subwavelength surface patterns by femtosecond vector beam irradiation of the tungsten target (top view). HWP: half-wave plate, LP: linear polarizer, M: high reflectance mirror, SLM: spatial light modulator, RG: Ronchi grating, L: plano-convex lens, F: pinhole filter, QWP: quarter-wave plate, ND: neutral density filter, Obj: 10\(\times\) objective, W: tungsten sample, TS: electric controlled translation stage. Lumenera Inc., Ottawa, ON, Canada) was shown, as presented in Figure 2a. During the experiments, as the angle \(\theta\) between the transmission axis of LP2 and the \(x\)-axis increased (see Figure 2a), the periodic intensity distribution in Figure 2a gradually moved towards the negative direction of the \(y\)-axis. The dependence of the dark stripe's position in Figure 2a on \(\theta\) is presented in Figure 2b, which quantitatively shows the polarization state of the femtosecond vector beam. The goodness of fit (\(R^{2}\)) was 0.99, showing that a perfect linear relationship existed between the dark stripe position and \(\theta\). Thus, this concludes that the local polarization direction of the femtosecond vector beam can be indicated by the tangential direction of the blue solid curve shown in Figure 2a. The polarization variation speed along the vertical direction increased with the increase in the rotation angle of RG2. A further explanation of the formation of the vector beam by the setup in Figure 1 can be found in [29,30]. When we removed the setup shown in the dashed frame in Figure 1, the apparatus in Figure 1 was able to be used to fabricate LIPSSs patterns on the tungsten surface. A 10\(\times\) objective (NA = 0.25) was employed to focus the femtosecond vector beam. After taking the dispersion of the optical elements in the experimental setup and the dispersion in air into account, we calculated the pulse duration at the target surface to be 55 fs. The tungsten sample had a purity of 99.95% and was polished by the abrasive paper of #1200.