## 2 Materials and Methods  
In the experiment, the radiation source was a nanosecond capillary discharge EUV laser operating at a 46.9 nm with a pulse width of 1.7 ns. The laser was produced using Ne-like Ar plasma excited with a fast current pulse [14; 15]. The laser was focused into a distorted shape by a toroidal mirror at a grazing incidence angle of 7\({}^{\circ}\). The target was located on a 2D stage to be irradiated at different positions of the surface and beam axis. An X-ray diode (XRD) was used to measure the relative energy of the laser. The target we used in the experiment was copper with a polished surface. The roughness of the surface was estimated to be less than 100 nm. To avoid the formation of an oxide layer on the copper surface, the targets were preserved carefully and moved into the vacuum chamber as quickly as possible. At the beginning of the irradiation, XRD was moved into the beam path to measure the laser energy. After the laser output was stable, the target was moved in and the irradiation started. The surface of the copper target was irradiated by multiple pulses of focused a 46.9 nm laser at different positions along the beam axis. After the irradiation, the damaged surface was detected by SEM (Scanning Electron Microscope). COMSOL Multiphysics (COMSOL Inc, Stockholm, Sweden), which is based on the finite element algorithm (FEM), was used in this section to calculate the temperature variation of copper in and after the laser irradiation. Since the attenuation length for copper at 46.9 nm is approximately 10 nm [16], the material is very nearly opaque to a 46.9 nm laser. In the simulation, the laser was treated as a surface heat source. The energy of the beam was assumed to be absorbed over a negligibly small distance into the material relative to the size of the object that was irradiated. The simulation was built in the module of 'Heat Transfer in Solids'. Since the critical electron density of a 46.9 nm laser is typically higher than solid, the laser could penetrate the expanding plasma plume and continue to heat the target surface directly throughoutthe duration of the laser pulse. Therefore, the heat source term should be involved in the calculation until the pulse ended. To build the function to describe the focused beam, spatial and temporal distribution of the laser was analyzed. Figure 1 showed the Gaussian fitting of normalized temporal distribution. The full width at half maximum (FWHM) of pulse duration was 1.6 ns. The spatial distribution of the laser was complex since the laser was focused on grazing incidence which caused the optical aberration. Even though, mostly the laser energy was focused into a small area that could be fitted into a Gaussian shape. Approximately, the spatial distribution of the focused laser was simulated as a Gaussian distribution. The function of describing the focused laser spot was as follow:  
\[P_{in}(x,y,t)=P_{0}\exp\left[-A\left(\frac{t-\tau}{\tau}\right)^{2}\right]\exp \left[-\frac{(x-x_{0})^{2}}{2r^{2}}-\frac{(y-y_{0})^{2}}{2r^{2}}\right] \tag{1}\]  
where \(P_{0}\) was the peak power of the laser, \(A\) was the Gaussian parameter for a temporal profile, \(\tau\) was the FWHM of pulse duration and \(r\) was the beam radius. The calculated field was built to be a cylinder with a radius of 1 mm and a height of 0.5 mm, of which the volume was much smaller than the copper target used in the experiment. It was aimed at reducing the amount of calculation while, at the same time, ensuring accuracy. The surface of the sample was set to be heat-insulated to the surrounding environment since the interaction occurred in the vacuum chamber with pressure below 10\({}^{-4}\) Pa. The simulation was built under the condition that the whole pulse energy of the laser with the peak power \(P_{0}\) of \(6\times 10^{4}\) W was contributed to raising the temperature of the surface (in the experiment, the laser with \(\sim\)50 \(\upmu\)J and \(\sim\)100 \(\upmu\)J were both used to irradiate Cu targets. In both cases, the melting trace was detected.