# Temporal Response of Ultrafast Grating Monochromators
## 1 Introduction  
Since its discovery, the process of high-order harmonic (HH) generation in gas has been at the center of experimental and theoretical investigations for studies of electronic structure and ultrafast dynamics of atoms, molecules and condensed matters [1]. Moreover, the radiation generated with the scheme of the HHs using few-optical-cycles laser pulses is at present the central tool for the investigation of matter with attosecond resolution [2]. HH generation occurs when an electron, liberated from an atom or molecule by an intense ultrashort laser fields, gains energy from the field and recombines with the parent ion [3]. The HH spectrum is described as a sequence of peaks corresponding to the odd harmonics of the fundamental laser wavelength with an intensity distribution characterized by a plateau spanning into extreme-ultraviolet (XUV) and soft X-rays whose extension is related to the pulse intensity. Thus, HH radiation can be used to perform a large variety of time-resolved experiments without the need for big facilities like synchrotrons or free-electron lasers.  
The selection of a single harmonic from consecutive multiple harmonics is required for many spectroscopic applications. Various methods are used, such as multilayer mirrors [4], selected enhancement by laser pulse shaping [5], zone-plate monochromators [6], gratings at grazing incidence [7].  
In particular, the use of gratings to realize monochromatic beamlines with ultrafast time response is well established not only for HHs [8], but also for intense free-electron laser pulses [9]. Gratings give broad tunability, covering the whole XUV spectral region, and high spectral selectivity. Different configurations have been already realized: single-stage grating monochromators with high efficiency and response in the few tens of femtoseconds time-scale [10; 11; 12] and double-stage reflective-grating monochromators with ultrafast response below ten femtoseconds [13; 14; 15; 16]. In the first case, when using a single grating, a residual pulse-front tilt due to diffraction has to be accepted at the output of the monochromator. It can be minimized by choosing a suitable geometry to obtain temporal responses in the range of few tens of femtoseconds [17]. In the second case, two consecutive gratingsare employed: the first one performs the spectral selection on an intermediate slit while the second one corrects for the pulse-front tilt introduced by the diffraction.  
In this paper, the temporal response of double-grating monochromators is analyzed considering two effects on the ultrafast pulse given by the configuration. The first effect is the compensation of the pulse-front tilt, i.e., all the rays emitted by the source in different directions at the same wavelength have to travel the same optical path. Ideally the compensation is perfect for a double-grating configuration, although aberrations may give a residual distortion of the pulse-front. The second effect is the group delay (GD) introduced by the two gratings, i.e., different wavelengths within the bandwidth transmitted by the slit travel different paths. Similarly to grating pulse shapers for optical laser pulses, the pair of gratings in the time-delay-compensating configuration can be considered as a XUV pulse shaper, capable to introduce a controllable GD. The methodology to calculate the GD introduced by the double-grating configuration is presented in Section 2. In terms of temporal response, the best operative condition for a monochromator is found when the GD and the Fourier limit for the given bandwidth are almost equivalent. For a narrower bandwidth, the GD decreases but the Fourier limit increases, giving a longer temporal response. For a broader bandwidth, the Fourier limit decreases but the GD increases, giving again a longer temporal response. Some practical design examples are provided to show the design rules and the expected performances.