# Temporal Response of Ultrafast Grating Monochromators
## 2 Ultrafast Response of Grating Monochromators
### Temporal Response of TDC Monochromators with High Spectral Resolution  
In case of high resolution, the output bandwidth is limited by the monochromator and not by the intrinsic bandwidth of the single harmonic. Once the slit aperture \(W\) is fixed, the bandwidth transmitted through the slit is:  
\[\Delta\lambda=\frac{W}{q\sigma} \tag{5}\]  
In terms of temporal response, the best trade-off is found when the GD and the Fourier limit are almost equivalent. For a narrower bandwidth, the GD decreases but the Fourier limit increases. For a broader bandwidth, the Fourier limit decreases but the GD increases. The condition GD = Fourier is obtained combining Equations (1), (2) and (5) and, for Gaussian pulses, is expressed by:  
\[W=q\sqrt{\frac{0.22\lambda}{q-p}} \tag{6}\]  
Aim of the optical design is to find the parameters that fulfill Equation (6) to find the best trade-off between energy and temporal resolution.  
A test case is here discussed for sub-150 meV energy bandwidth and sub-50 fs temporal response in the 20-120 nm (62-10 eV) interval, that are typical values required for ultrafast photoelectron spectroscopy. The requirements are resumed in Table 1. A set of parameters, in terms of groove density and lengths of the arms, is also listed in Table 1.  
Figure 3: Performance of a low-resolution time-delay-compensated (TDC) monochromator for high-order harmonics (HHs). (**a**) Bandwidth on 100-um slit, compared to HH separation. The pulse bandwidth for 10-fs Fourier-limited Gaussian pulses is also shown. (**b**) Pulse-front tilt at the intermediate slit, 4-mrad extreme-ultraviolet (XUV) divergence. (**c,d**) group delay (GD) and group delay dispersion (GDD) at the output. Monochromator parameters: \(q\) = 500 mm, \(p\) = 150 mm.  
The performances, in terms of energy bandwidth and temporal response, are resumed in Figure 4. The slit width calculated by Equation (6) is shown in Figure 4a. It is the width giving an output bandwidth in which the GD is equal to the Fourier limit. The slit aperture is in the range 75-160 \(\upmu\)m, that is a typical interval for practical operations. The corresponding bandwidth is shown in Figure 4b. By choosing the appropriate grating, the bandwidth is anyway below 100 meV for wavelengths longer than 32 nm. For shorter wavelengths, the bandwidth increases up to 150 meV at 25 nm. The pulse-front tilt given by the first grating stage is plotted in Figure 4c for 4-mrad XUV divergence. The tilt is definitely above 100 fs in the full range of operation. After the second grating, the residual pulse-front distortion, as simulated through a ray-tracing procedure, is lower than 2 fs within the whole interval of operation, i.e., fully negligible. The corresponding Fourier limit and GD are shown in Figure 4d,e and are in the range 10-30 fs in the whole interval of operation. The GDD is definitely higher than the low-resolution case, typically of the order of 100 fs\({}^{2}\). The GD can be reduced if the grating-to-mirror distance \(p\) is kept higher, that is, for a lower \(q\)-\(p\) value, at the expense of a larger size of the monochromator. In this case, the use of a second grating stage increases the temporal resolution by almost one order of magnitude when compared to a monochromator with a single grating for the same output bandwidth.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Requirements** & \\ \hline Wavelength range & 25–115 nm (10–50 eV) \\ Energy bandwidth & \textless{}150 meV \\ Temporal response & \textless{}50 fs \\ \hline
**Parameters** & \\ \hline Gratings & 300/600/1200 grooves/mm \\ Exit arm \(q\) & 800 mm \\ Mirror-to-grating \(p\) & 150 mm \\ Slit aperture & 75–160 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Requirements and parameters for a TDC monochromator with sub-150 meV energy bandwidth.  
Figure 4: _Cont._