## 2 Design of Single-Grating Monochromators for Ultrashort Pulses  
Grazing-incidence diffraction gratings may be used in two different geometries: the classical-diffraction mount (CDM) and the off-plane mount (OPM). The CDM is shown in Figure 1a. The grating equation is \(\mathrm{sin}\alpha-\mathrm{sin}\beta=m\lambda\sigma_{\mathrm{c}}\), where \(\alpha\) and \(\beta\) are, respectively, the incidence and diffraction angles (both defined with positive signs), \(\lambda\) is the wavelength, \(m\) the diffracted order and \(\sigma_{\mathrm{c}}\) the groove density. In the case of a monochromator, the grating equation is expressed as:  
\[\alpha=\frac{K}{2}+\mathrm{arcsin}\bigg{[}\frac{m\lambda\sigma_{ \mathrm{c}}}{2\cos(K/2)}\bigg{]} \tag{3}\]  
where \(K\) is the subtended angle and \(\beta=K-\alpha\). The number of illuminated grooves is \(N_{\mathrm{c}}=S\sigma_{\mathrm{c}}/\mathrm{cos}\alpha\), where \(S\) is the beam section on the grating measured in the direction normal to the light propagation. The illuminated grooves are lower in the case \(m=-1\), _i.e._, the external diffracted order, since \(\alpha\) is smaller. In the following, the grating in the CDM will be assumed to be operated in the external diffracted order, to maintain the number of illuminated grooves as small as possible. The OPM is shown in Figure 1b. The grating equation is \(\mathrm{sin}\gamma\) (\(\mathrm{sin}\mu+\mathrm{sin}\
u\)) = \(m\lambda\sigma_{\mathrm{o}}\), where \(\sigma_{\mathrm{o}}\) is the groove density, \(\gamma\) is the altitude defined as the angle between the direction of the incoming rays and the direction of the grooves and \(\mu\) and \(\
u\) are, respectively, the azimuth of the incoming and diffracted rays. The azimuth is defined to be zero if the rays lie in the plane perpendicular to the grating surface and parallel to the rulings. In the case of a monochromator, the grating is operated in the condition \(\mu=\
u\) and the grating equation is:_Appl. Sci._**2013**, \(3\)  
\[\mu=\arcsin\Biggl{[}\frac{m\lambda\sigma_{o}}{2\sin\gamma}\Biggr{]} \tag{4}\]  
The number of illuminated grooves is \(N_{\rm o}=S\ \sigma_{\rm o}/\cos\mu\simeq S\ \sigma_{\rm o}\). Here, and in the following, it will be assumed \(\mu\leq 15^{\circ}\), therefore \(\cos\mu\simeq 1\). This approximation is fully verified for low-resolution configurations, as those that will be presented below. The grating in the OPM can be used indifferently at the orders \(m=\pm 1\), since this does not affect the illuminated area. In both configurations, the wavelength selection is performed by rotating the grating around the axis that is tangent to the surface, passes through the grating center and is parallel to the grooves. For saw-tooth groove profiles, the blaze condition is the configuration for which the diffraction efficiency is maximized, that is, when the diffracted light leaves the grating in such a way to perform a specular reflection on the groove surface. This is verified for \(\delta=(\alpha\beta)/2=(\alpha-K/2)\), in the case of the CDM, and for \(\delta=\mu\), in the case of the OPM, where \(\delta\) is the grating blaze angle. The main advantage of the OPM when compared to the CDM is the higher efficiency, since it has been theoretically demonstrated and experimentally measured that the peak diffraction efficiency in the OPM is close to the reflectivity of the coating at the altitude angle [17, 18]. Therefore, the OPM is a good candidate for the design of XUV monochromators with high efficiency [19]. On the contrary, the OPM has two main drawbacks for the design of XUV monochromators. First of all, the OPM monochromators require plane gratings [20, 21], so they need two additional optical elements, namely two grazing-incidence mirrors, to collimate and focus the beam. Furthermore, an effect of the OPM is the distortion and the rotation of the image after the diffraction, due to the highly nonspecular reflection from the grating, especially for the large azimuth angles that are required to achieve high spectral resolution [22].