# Temporal Response of Ultrafast Grating Monochromators
## 2 Ultrafast Response of Grating Monochromators  
The monochromator demanded for the spectral selection of ultrashort pulses has to preserve the temporal duration as short as in the generation process. The monochromator can be modelled as a filter with a complex frequency response that includes both the non-uniform spectral transmission and the distortion in the spectral phase. Since the XUV pulse at the generation may be produced to be close to its transform limit, any modification of its complex spectrum results in a time broadening as described by its Fourier transform. For a Gaussian profile with no phase or frequency modulation, the half-height duration \(\Delta\tau_{F}\) has a lower limit expressed as:  
\[\Delta\tau_{F}=\frac{2\ln 2}{\pi c}\frac{\lambda^{2}}{\Delta\lambda}\cong \frac{0.44}{c}\frac{\lambda^{2}}{\Delta\lambda} \tag{1}\]  
where \(\lambda\) is the pulse central wavelength, \(\Delta\lambda\) is the half-height spectral width of the pulse and \(c\) is the speed of light in vacuum. In order to preserve the duration of the pulse at the output of the monochromator, the spectral transmission has to be constant within the bandwidth. This condition is always verified if the monochromator is realized by reflecting optics, since the reflectivity variations within the output bandwidth are usually negligible, so that the transfer function can be considered almost constant, although lower than unity. Furthermore, the temporal response of the monochromator has to be shorter than \(\Delta\tau_{F}\). In case of a diffraction grating, the major mechanism that alters the time duration of the pulse is the difference in the lengths of the optical paths of the rays diffracted by different grating grooves. Indeed, a single grating gives inevitably a time broadening of the ultrafast pulse because of the pulse-front tilt due to diffraction, since the total difference in the optical paths of the rays diffracted at first order by N grooves illuminated by radiation at wavelength \(\lambda\) is N\(\lambda\). This effect can be dramatic in the femtosecond time scale. Let us consider a 300 grooves/mm grating illuminated by radiation at 30 nm over a length of 10 mm; the total number of grooves involved in the diffraction is 3000, corresponding to a pulse-front tilt of 90 \(\upmu\)m, i.e., 300 fs. In case of femtosecond pulses, this reduces dramatically both the time resolution capability and the peak intensity at the exit of the monochromator. Nevertheless, it is possible to realize grating monochromators that do not alter the temporal duration of ultrashort pulses by using two gratings in a time-delay-compensated (TDC) configuration: the first grating performs the spectral selection on an intermediate slit, then the second one corrects for the pulse-front tilt introduced by the diffraction. From the point of view of the ray paths, two are the conditions that the design must comply to: (1) the differences in the path lengths of rays having the same wavelength but with different entrance directions within the beam aperture that are caused by the diffraction from the first grating must be compensated by the second grating; (2) two rays at different wavelengths within the spectrum of the pulse to be selected have to be focused on the same point, i.e., the global spectral dispersion has to be zero. Both conditions are satisfied by a scheme with two equal gratings mounted in a symmetrical way. The spectral selection is performed by a slit placed in an intermediate position between the gratings, where the radiation is focused by the first grating. Grazing-incidence gratings can be used in two geometries: the classical-diffraction mount (CDM) and the off-plane mount (OPM). The latter, also called conical diffraction mount, differs from the classical one in that the incident and diffracted wave vectors are almost parallel to the grooves [18].