## 2 Monochromator Design and Realization  
### Optical Design  
The monochromator covers a broad spectral range through the selection of one among two or more interchangeable plane gratings (PGs), illuminated along the direction perpendicular to the grooves by the converging rays coming from a focusing cylindrical mirror (CM). The diffracted light is then reflected by a plane mirror (PM) toward a vertical exit slit acting as the filtering element. The PM optic can translate and rotate--this is required to maintain the spectral focus at the same position of the slit plane when the grating subtended angle is changed. A schematic layout of the design is shown in Figure 1. The realized configuration only cares for the tangential focus (i.e., on the plane of the spectral dispersion) provided by the CM. In the sagittal direction (i.e., perpendicular to the tangential one) the rays are assumed to be collimated by a preliminary mirror, as explained in Section 4. For each wavelength of interest, tangential aberrations on the focal plane must be minimized to obtain a high resolution. The condition to have the spectral focus on the slit plane is  
\[\frac{cos\alpha_{G}}{cos\beta_{G}}=const., \tag{1}\]  
where \(\alpha_{G}\) and \(\beta_{G}\) are the grating incidence and diffraction angles, respectively. This condition must be fulfilled for each wavelength within the spectral interval of operation and can be expressed in terms of the grating subtended angle \(k=\alpha_{G}-\beta_{G}\) as  
\[\frac{\cos\left\{\arcsin\left[\frac{m\lambda\sigma}{2\cos\left(\frac{\pi}{2} \right)}\right]+\frac{k}{2}\right\}}{\cos\left\{\arcsin\left[\frac{m\lambda \sigma}{2\cos\left(\frac{\pi}{2}\right)}\right]-\frac{k}{2}\right\}}=\left( \frac{p_{G}}{q_{G}}\right)^{\frac{1}{2}}, \tag{2}\]  
where \(\sigma\) is the groove density, m is the internal diffraction order (\(m\geq 0\)), and \(p_{G}\) and \(q_{G}\) are, respectively, the entrance and exit arms of the PG, as reported in Figure 1.  
From Equation (2), the subtended angle \(k\) is calculated for each wavelength of interest and changed by translating the PM along an axis parallel to the beam output direction and rotating it around an axis passing through its center, as reported in Figure 2b. The variation of the length of \(q_{G}\) due to the translation of the plane mirror, i.e., the variation of the distance between the grating and the exit slit is almost negligible at grazing incidence. Tangential coma is the second main undesired aberration once the spectral defocusing has been corrected. This specific design allowed us to select the entrance arm \(p_{M}\) of the CM to have the coma  
Figure 1: Tangential layout of the monochromator referred to the dispersion (spectral plane). The cylindrical mirror focus is placed after the grating. In order to maintain the final focal point at the same position, a variation of the grating subtended angle and the related translation and rotation of the plane mirror are required. Figure 2: (**a**) External and (**b**) internal view of the three-elements monochromator vacuum chamber. given by the grating compensated by the coma that is specifically introduced by the mirror. Reference [21] shows that the condition for coma compensation at the specific wavelength \(\lambda_{C}\) is given by  
\[p_{M}=\left(q_{G}\frac{\cos^{2}\alpha_{G,\lambda_{C}}}{\cos^{2}\beta_{G, \lambda_{C}}}+D\right)\left[1+A_{\lambda_{C}}\left(q_{G}\frac{\cos^{2}\alpha_{G,\lambda_{C}}}{\cos^{2}\beta_{G,\lambda_{C}}}+D\right)^{-1}\right]^{-\frac{1}{ 2}}, \tag{3}\]  
where \(D\) is the mirror-to-grating distance, and \(\alpha_{G,\lambda_{C}}\) and \(\beta_{G,\lambda_{C}}\) are the grating incidence and diffraction angles at the wavelength \(\lambda_{C}\), respectively.