## 2 Design of Single-Grating Monochromators for Ultrashort Pulses
Equation 7 can be compared to Equation 2 to find the condition when the bandwidth at the output of a monochromator with \(\Delta\tau_{\mathrm{G}}\) pulse-front tilt is equal to that of a Fourier-limited pulse of duration \(\Delta\tau\), that is:  
\[\mathsf{D}\,S=\frac{\lambda}{2} \tag{8}\]  
If the condition stated in Equation 8 is verified, the monochromator is defined to be _time-preserving_, since the minimum bandwidth achievable at the output is almost the same as the bandwidth of a Fourier-limited pulse of a duration equal to the grating pulse-front tilt. A monochromator for XUV ultrashort pulses is here designed in both the geometries, to outline the main characteristics of the two configurations. HH emission from a Ti:Sa laser is considered, with a half-width divergence of 2 mrad. The entrance and exit arms are assumed to be \(p=q=400\) mm, that is, a safe distance to operate the first toroidal mirror that is illuminated also by the intense IR generating laser. The interval of operation is the 15-55 nm wavelength band, _i.e.,_ between harmonics H15 and H53. We analyze two different temporal regimes: 100 fs or 10 fs half-width pulse-front tilt at the central wavelength of the range of interest, \(\lambda_{\mathrm{c}}=35\) nm. The grating groove density is calculated from Equation 5 and is reported in \[\mu=\arcsin\Biggl{[}\frac{m\lambda\sigma_{o}}{2\sin\gamma}\Biggr{]} \tag{4}\]0 In the case of the CDM, as the included angle increases, the corresponding groove density decreases. In the case of the OPM, the groove density is almost independent from the choice of the altitude and azimuth angles. The monochromator pulse-front tilt as a function of the wavelength is shown in \[\mu=\arcsin\Biggl{[}\frac{m\lambda\sigma_{o}}{2\sin\gamma}\Biggr{]} \tag{4}\]1 For low-resolution configurations, such as those required for ultrashort pulses, the number of illuminated grooves is almost constant with the wavelength in both geometries; consequently, the pulse-front tilt 2 \(\Delta\tau_{\mathrm{G}}=\lambda\)N/\(c\) increases linearly and is almost the same for the two configurations. \begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline \multicolumn{3}{|c|}{**Classical diffraction mount**} & \multicolumn{3}{|c|}{**Off-plane mount**} \\ \hline
**Included angle (\({}^{\circ}\))** & **Groove density** & **Blaze angle at** & **Altitude (\({}^{\circ}\))** & **Groove density** & **Blaze angle at** \\  & **(gr/mm)** & **35 nm (\({}^{\circ}\))** & & **(gr/mm)** & **35 nm (\({}^{\circ}\))** \\ \hline \multicolumn{3}{|c|}{**100 fs half-width pulse-front tilt at 35 nm**} & \multicolumn{3}{|c|}{**100 fs half-width pulse-front tilt at 35 nm**} \\ \multicolumn{3}{|c|}{(0.11 nm-115 meV bandwidth on a 50-um slit)} & \multicolumn{3}{|c|}{(0.11 nm-115 meV bandwidth on a 50-um slit)} \\ \hline
155 & 255 & 1.2 & 4 & 1100 & 16.0 \\ \hline
160 & 210 & 1.2 & 7 & 1100 & 9.1 \\ \hline
165 & 165 & 1.2 & 10 & 1100 & 6.4 \\ \hline \multicolumn{3}{|c|}{**10 fs half-width pulse-front tilt at 35 nm**} & \multicolumn{3}{|c|}{**10 fs half-width pulse-front tilt at 35 nm**} \\ \multicolumn{3}{|c|}{(1.1 nm-1.15 eV resolution on a 50-um slit)} & \multicolumn{3}{|c|}{(0.11 nm-1.15 eV bandwidth on a 50-um slit)} \\ \hline
155 & 24 & 0.1 & 4 & 110 & 1.6 \\ \hline
160 & 19 & 0.1 & 7 & 110 & 0.9 \\ \hline
165 & 15 & 0.1 & 10 & 110 & 0.6 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of a plane-grating monochromator with a half-width pulse-front tilt of 10/100 fs at the wavelength \(\lambda_{\mathrm{c}}=35\) nm. The half-width beam section on the grating is 0.8 mm. The grating in the CDM has been assumed to be operated in the external order, _i.e.,_\(m=-1\). The grating efficiency curves, as resulting from simulations, are shown in Figure 4. For saw-tooth profiles, the efficiency is expected to be theoretically high in both configurations. Nevertheless, there are deep differences between the CDM and the OPM when the actual shape of the groove profile is analyzed.