## 2 Design of Single-Grating Monochromators for Ultrashort Pulses
This makes the OPM much more complex to be adopted than the CDM for high-resolution monochromators. The realization of monochromators for ultrashort pulses introduces an additional and essential parameter to be considered in the comparisons of the two configurations, namely the effect of the grating geometry on the pulse-front tilt. Since the gratings for OPM monochromators are operated in parallel light, we will compare the two configurations with plane gratings, as shown in Figure 2.  
Figure 1: The grating geometry: (**a**) classical-diffraction mount (CDM); (**b**) off-plane mount (OPM). The configuration has three optical elements, namely two mirrors and the grating. The first mirror (M\({}_{1}\)) acts as the collimator, the second mirror (M\({}_{2}\)) as the condenser. The magnification is in unity to minimize the aberrations, _i.e._, \(p=q\), where \(p\) is the input arm of the collimator, that is, the distance between the entrance source and the center of M\({}_{1}\), and \(q\) the output arm of the condenser, that is, the distance between the center of M\({}_{2}\) and the exit slit. The number of illuminated grooves in the two geometries is, respectively, \(N_{\rm c}=2\ D\ p\ \sigma_{\rm o}/{\rm cos}a\) and \(N_{\rm o}=2\ D\ p\ \sigma_{\rm o}\), where \(D\) is the half-width beam divergence at the input. The half-width pulse-front tilt is \(\Delta\tau_{G}\approx\
icefrac{{1}}{{2}}{\rm N}m\lambda c^{-1}\), that is:  
\[\Delta\tau_{G,c}=c^{-1}m\lambda\mathbb{D}p\,\frac{\sigma_{c}}{\cos\alpha},\ \ \Delta\tau_{G,o}=c^{-1}m\lambda\mathbb{D}p\,\sigma_{o} \tag{5}\]  
where \(\Delta\tau_{G,c}\) and \(\Delta\tau_{G,o}\) refer, respectively, to the CDM and the OPM. The two configurations give the same tilt for \(\sigma_{\rm o}/{\rm cos}a=\sigma_{\rm o}\). Therefore, once the maximum pulse-front tilt has been fixed, the groove density in the CDM is lower than in the OPM, _i.e._, \(\sigma_{\rm c}<\sigma_{\rm o}\). The condition to maintain the pulse-front tilt below the input half-width pulse duration \(\Delta\tau\) after the diffraction from N grooves is, respectively, \(\Delta\tau_{G,c}\leq\Delta\tau\) and \(\Delta\tau_{G,o}\leq\Delta\tau\) The shorter the time response, the lower the requested groove density, as stated in Equation 5.  
Let us suppose that the width of the exit slit, W, is equal to the size of the XUV source, \(S\), as imaged at the output of the monochromator. This is the minimum width that has to be kept to guarantee the complete transmission of the diffracted beam. The OPM does not give any magnification of the source since \(p=q\), therefore, \(W_{\rm o}=S\). Instead, the source magnification in the CDM is equal to the anamorphic factor of the grating, \({\rm cos}\alpha/{\rm cos}\beta\), therefore, \(W_{\rm c}=S\ {\rm cos}\alpha/{\rm cos}\beta\). By using the formula of the grating dispersion, the half-width output bandwidth results:  
\[\Delta\lambda_{c}=W_{c}\,\frac{\cos\ \beta}{\left|m\right|q\,\sigma_{c}}= \frac{S\cos\alpha}{\left|m\right|q\,\sigma_{c}},\ \ \Delta\lambda_{o}=W_{o}\,\frac{1}{\left|m\right|q\,\sigma_{c}}= \frac{S}{\left|m\right|q\,\sigma_{o}} \tag{6}\]  
where \(\Delta\lambda_{\rm c}\) and \(\Delta\lambda_{\rm o}\) refer, respectively, to the CDM and the OPM. By putting together Equations 5 and 6 we finally obtain:  
\[\Delta\lambda=\frac{\mathbb{D}S\lambda}{c}\frac{1}{\Delta\tau_{G}} \tag{7}\]  
Figure 2: Monochromator with plane gratings: (**a**) CDM; (**b**) OPM. as the minimum spectral bandwidth at the output of a single-grating monochromator with pulse-front tilt equal to \(\Delta\tau_{\mathrm{G}}\). Once the tilt has been defined, the minimum bandwidth at wavelength \(\lambda\) depends on the characteristics of the source, namely the size and the divergence.