## 2 Materials and Methods
### The Experimental Setup
M1, M3: flat multilayer mirrors. M2, M4: spherical concave multilayer mirrors. FFG: flat field grating. T: target. The selection of the wavelengths is performed via two similar pairs of mirrors. Each pair consists of a multilayer flat mirror and a multilayer spherical concave mirror, placed in opposite directions, as shown in Figure 1. The pair of multilayer mirrors reflects only a narrow band (\(\sim\)2 nm) around a central wavelength, while the spherical mirror, having a radius of curvature (ROC) of 1000 mm, focuses the selected XUV beam onto the target at the fourth stage of the setup. Both sets of multilayer mirrors were specially manufactured from SiC/Mg layers (NTT Advanced Technology Corporation) to deliver the central wavelengths of 32.2 nm and 47.6 nm, respectively, at maximum reflectivity for the geometry in use. The pairs of mirrors are fixed on a movable platform placed on a linear rod equipped with a vacuum-compatible motor, which is externally controlled with sub-millimeter accuracy. Thus, our system offers the option of exchanging at will the pairs of mirrors during the measurements in an automated way, which is one of the essential characteristics of our setup. The platform is also equipped with a grazing-incidence flat field diffraction grating (Hitachi 001-0639), which can be automatically inserted into the generated XUV beam path. In this operation option, which may or may not include the insertion of the Al filter, the XUV beam is diffracted towards a high-quantum-efficiency 16-bit XUV vacuum CCD camera (Raptor Photonics, Eagle XO) with a sensor having 2048 pixels (27.65 mm) at the grating diffraction axis and 512 pixels (6.90 mm) at the XUV divergence axis (perpendicular to the grading diffraction axis). The grating supports a wavelength dispersion efficiency ranging from 22 nm up to 124 nm, much broader than the corresponding spectral region supported by the XUV CCD camera. Thus, a selected part of the XUV spectrum, determined by the rotation angle of the grating, also automatically controlled, can be monitored by the XUV CCD camera during the experiment and thus perform an on-line optimization. Moreover, the XUV grating and XUV CCD detector allow for a relatively high spectroscopic resolution from \(\sim\)80 to \(\sim\)100 pixels per 1 nm, on the CCD, for the spectral region of interest. This resolution is a significant experimental tool for optimizing the quality and characteristics of the XUV beam. The fourth stage, D, is an ISO200 six-way cross attached to and pumped by the third stage. It hosts the target, which is mounted on a variable in the three dimensions XYZ of the base, as well as in the polar rotation angle \(\Theta\). Vacuum-compatible steppers for the four degrees of freedom are externally controlled and set with micrometer accuracy. Thus, the position of the target with respect to the laser focal area is accurately controlled. The monochromatic coherent XUV radiation diffracted by the target is recorded at the second vacuum-compatible XUV CCD camera located 9.0 cm after the target at the end of the six-way cross. The XUV CCD camera has identical characteristics as the XUV CCD camera of the third stage, differing only in the sensor size, which is 1024 by 1024 pixels. The software interface for controlling the initial XUV filtering, the selection of recording the XUV spectrum, the selection of the pair of mirrors for selecting the wavelength, and the XYZS position control of the target are shown in Appendix A in Figure A1. In the same figure, a block diagram of the above multi-axis control system is also presented. The positions of the optomechanical devices can be calibrated independently and stored for future use or self-calibrated during the experiments.