# Temporal Response of Ultrafast Grating Monochromators
## 2 Ultrafast Response of Grating Monochromators
The main advantage of the OPM when compared to the CDM is the much higher efficiency (up to a factor three), 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, that makes the OPM suitable for the design of TDC monochromators with high throughput [19; 20]. It is worth to mention here that the double-grating configuration is presently applied in the XUV, for wavelength that are typically longer than \(\approx\)10 nm. For shorter wavelengths, i.e., entering the soft X-ray region, the losses due to the decrease in reflectivity should be considered when balancing the advantages of the TDC configuration in terms of temporal response with the low photon throughput. In the following, the paper will be focused on the OPM. The geometry is shown in Figure 1. The direction of the incoming rays is described by two parameters: the altitude and the azimuth. The altitude \(\gamma\) is defined as the angle between the direction of the incoming rays and the direction of the grooves. The azimuth \(\mu\) of the incoming rays is defined to be zero if they lie in the plane perpendicular to the grating surface and parallel to the rulings. Let \(\
u\) define the azimuth of the diffracted light at wavelength \(\lambda\) and order \(m\). The grating equation is \(\sin\!\gamma\) (\(\sin\!\mu\) + \(\sin\!\
u\)) = \(m\lambda\delta\), where \(\delta\) is the groove density. For a monochromator, the grating is operated in the condition \(\mu\) = \(\
u\) and the resulting grating equation is \(\mu\) = \(\arcsin(\frac{1}{2}m\lambda\delta/\sin\!\gamma)\). A schematic of a TDC monochromator with plane gratings in the OPM is shown in Figure 2. The gratings are used in collimated beams [21], therefore each of the two sections of the monochromator consists of three optical elements at grazing incidence: the collimating toroidal mirror, the plane grating and the focusing toroidal mirror. Figure 1: Off-plane mount (OPM) geometry. Two are the effects to be considered when evaluating the temporal response of the TDC configuration: (1) the compensation of the pulse-front tilt and (2) the GD introduced within the bandwidth. The pulse-front tilt is ideally fully compensated for a double-grating configuration, although residual aberrations introduced by the toroidal mirrors may give some distortion of the pulse-front, which has to be evaluated through ray-tracing simulations. With reference to Figure 2, in the following the term \"input arm\" will refer to the two collimators T1 and T3 and will indicate the distance between the source and the center of T1 or the distance between the intermediate slit and the center of T3; the term \"output arm\" will refer to the two focusing elements T2 and T4 and will indicate the distance between the center of T2 and the intermediate slit or the distance between the center of T4 and the output focal point. For the TDC configuration, the number of grooves illuminated on G2 has to be the same as G1. Since the gratings are equal and are operating at the same altitude angle, this condition is fulfilled if the output arm of T2 is equal to the input arm of T3. Furthermore, in order to minimize the aberrations, each of the two sections is operating with unity magnification, i.e., the input arm of the collimator is equal to the output arm of the successive focusing element. Therefore, all the arms are the same and the configuration is fully symmetrical. As shown in the following simulations, the pulse-front distortion due to aberrations in the symmetric TDC configuration is expected to be in the few-femtoseconds or even sub-femtosecond time scale, therefore it can be usually neglected.