**1. Introduction**

XUV and X-ray radiations have been used for many fundamental discoveries and outstand‐ ing applications in natural sciences. It has played a crucial role in basic research, medical diagnostics, and industrial development. In particular, the impressive developments in laser technology over the last decades lead to the generation of XUV and X-ray coherent ultra‐

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short and ultra-intense pulses in the femtosecond and sub-femtosecond time scale (1 fs = 10−15 s) [1–3]. Ultrafast short-wavelength radiation offers the unique capability to access and measure the structural arrangement and electronic structure inside the nucleus [4, 5]. The main available tools to generate ultrashort coherent pulses are presently high-order laser harmon‐ ics generated in gas and free-electron lasers [6].

High-order harmonics (HHs) are generated through the interaction between an ultrashort laser pulse and a gas in a cell or in a jet. Because of the strong peak power of the femtosecond laser pulse, a nonlinear interaction with the gas takes place and produces odd laser harmonics that may easily extend well above the order of several tens. The HH spectrum is described as a sequence of peaks corresponding to the odd harmonics of the fundamental laser wavelength and having an intensity distribution characterized by a plateau whose extension is related to pulse intensity and frequency. The use of advanced phase-matching mechanisms and interaction geometries has made possible the generation of HHs in the water window region between 2.3 and 4.4 nm, while still using a table-top laser source [7–9]. The radiation generat‐ ed with the scheme of the HHs using few-optical-cycle laser pulses is currently the main tool for the investigation of matter with attosecond resolution (1 as = 10−18 s) [10–13]. Both trains [14, 15] and isolated [16–19] bursts of attosecond pulses have been experimentally demonstrated. The physical background of ultrashort pulses originates from the model of HH generation, i.e., the phase-matched emission of radiation results from the recombination of a tunnel-ionized electron with its parent ion. Once the conditions for such recombination are realized in only one occurrence per laser pulse, an isolated pulse is generated. Both trains and isolated attosecond pulses are positively chirped, resulting from the different duration of the quan‐ tum paths that contribute to the emitted spectrum [20]. Due to the nonzero chirp, the pulse temporal duration is longer than the Fourier limit. Positively chirped pulses may be tempo‐ rally compressed by introducing a system that gives a compensating negative chirp. The compression has been achieved using a thin metallic filter with negative group-delay dispersion (GDD) as discussed in Refs. [15, 17] or broadband multilayer-coated optics with aperiodic layers [21].

Free-electron laser (FEL) sources generate radiation in the XUV and X-ray spectral regions with high spatial coherence, ultrashort time duration, and an increase of 6–8 orders of magnitude on the peak brilliance with respect to third-generation synchrotrons. FEL operation relies on a relativistic electron beam as the lasing medium which moves freely through a periodic magnetic structure (i.e., the undulator) that induces radiation [22]. There are presently four FEL user facilities operated at short wavelengths and dedicated to user-defined experiments: FLASH in Germany, SACLA in Japan, LCLS in USA, and FERMI in Italy. Presently, FEL pulses as short as 3 fs have been characterized at LCLS [23]. In order to increase the temporal resolution in pump-probe experiments, several approaches have been proposed for the generation of ultrashort FEL pulses in the femtosecond and sub-femtosecond regime, as the time slicing [24–26] or the reduction of the electron bunch charge [27]. Most of these meth‐ ods rely on the selection of a small portion of the electron beam which undergoes FEL amplification, with a reduction of the amount of charge that contributes to the light amplifi‐ cation. A different possibility is the optical compression of the radiation pulse generated by the whole electron beam that is required to have a nonzero energy chirp in order to generate a chirped pulse. As for optical lasers, where frequency chirping is introduced to stretch the pulse before its amplification and then compensated after amplification to recover the ultrashort duration and high peak power, chirped-pulse amplification (CPA) may also be applied to FELs [28, 29]. In case of seeded FELs, the seeding laser pulse has to be stretched in time before interacting with the electron beam. This solution allows the use of the whole electron beam charge obtaining a significantly higher number of photons.

short and ultra-intense pulses in the femtosecond and sub-femtosecond time scale (1 fs = 10−15 s) [1–3]. Ultrafast short-wavelength radiation offers the unique capability to access and measure the structural arrangement and electronic structure inside the nucleus [4, 5]. The main available tools to generate ultrashort coherent pulses are presently high-order laser harmon‐

High-order harmonics (HHs) are generated through the interaction between an ultrashort laser pulse and a gas in a cell or in a jet. Because of the strong peak power of the femtosecond laser pulse, a nonlinear interaction with the gas takes place and produces odd laser harmonics that may easily extend well above the order of several tens. The HH spectrum is described as a sequence of peaks corresponding to the odd harmonics of the fundamental laser wavelength and having an intensity distribution characterized by a plateau whose extension is related to pulse intensity and frequency. The use of advanced phase-matching mechanisms and interaction geometries has made possible the generation of HHs in the water window region between 2.3 and 4.4 nm, while still using a table-top laser source [7–9]. The radiation generat‐ ed with the scheme of the HHs using few-optical-cycle laser pulses is currently the main tool for the investigation of matter with attosecond resolution (1 as = 10−18 s) [10–13]. Both trains [14, 15] and isolated [16–19] bursts of attosecond pulses have been experimentally demonstrated. The physical background of ultrashort pulses originates from the model of HH generation, i.e., the phase-matched emission of radiation results from the recombination of a tunnel-ionized electron with its parent ion. Once the conditions for such recombination are realized in only one occurrence per laser pulse, an isolated pulse is generated. Both trains and isolated attosecond pulses are positively chirped, resulting from the different duration of the quan‐ tum paths that contribute to the emitted spectrum [20]. Due to the nonzero chirp, the pulse temporal duration is longer than the Fourier limit. Positively chirped pulses may be tempo‐ rally compressed by introducing a system that gives a compensating negative chirp. The compression has been achieved using a thin metallic filter with negative group-delay dispersion (GDD) as discussed in Refs. [15, 17] or broadband multilayer-coated optics with

Free-electron laser (FEL) sources generate radiation in the XUV and X-ray spectral regions with high spatial coherence, ultrashort time duration, and an increase of 6–8 orders of magnitude on the peak brilliance with respect to third-generation synchrotrons. FEL operation relies on a relativistic electron beam as the lasing medium which moves freely through a periodic magnetic structure (i.e., the undulator) that induces radiation [22]. There are presently four FEL user facilities operated at short wavelengths and dedicated to user-defined experiments: FLASH in Germany, SACLA in Japan, LCLS in USA, and FERMI in Italy. Presently, FEL pulses as short as 3 fs have been characterized at LCLS [23]. In order to increase the temporal resolution in pump-probe experiments, several approaches have been proposed for the generation of ultrashort FEL pulses in the femtosecond and sub-femtosecond regime, as the time slicing [24–26] or the reduction of the electron bunch charge [27]. Most of these meth‐ ods rely on the selection of a small portion of the electron beam which undergoes FEL amplification, with a reduction of the amount of charge that contributes to the light amplifi‐ cation. A different possibility is the optical compression of the radiation pulse generated by

ics generated in gas and free-electron lasers [6].

226 228High Energy and Short Pulse Lasers

aperiodic layers [21].

Indeed, for both HH and FEL facilities, the availability of a compressor tunable in the spectral band of operation of the source, capable of changing the spectral phase of the pulses, is particularly attractive, either to compensate for the intrinsic chirp or to realize CPA.

Here we discuss the use of gratings at grazing incidence to realize a tunable device to manipulate the spectral phase of XUV and X-ray–chirped pulses. In designing instruments for photon handling in the XUV ultrafast domain, there are some basic differences with respect to the traditional optical schemes in the visible and infrared domain [30, 31]. The first is to exploit the very short duration of the pulse, the study of the optical length of the rays gathered by the pupil and their equalization is mandatory. Moreover, the very extended bandwidth of operation may be exploited only if the instrument has a rather flat spectral response. Finally, an overall high throughput of the instrument is often a crucial feature to maintain high peak intensity.

The use of gratings at grazing incidence to realize tunable monochromators for XUV ultra‐ fast pulses is well established. Several monochromatic beamlines are presently in operation both with HHs [32–35] and FELs [36–39].

When using a grating for ultrafast pulses, the main problem faced with is the pulse-front tilt that is introduced by diffraction [40]. Indeed, each ray that is diffracted by two adjacent grooves is delayed by *mλ*/*c*, where *m* is the diffraction order, *λ* is the wavelength, and *c* is the speed of light in vacuum. The pulse-front tilt is given by the total difference in the optical paths of the diffracted beam, that is *ΔτG= mλN*/*c*, where *N* is the total number of the illuminated grooves. This effect, that is totally negligible for picosecond or longer pulses, is noticeable in the femtosecond time scale, since it can dramatically degrade the instrumental ultrafast re‐ sponse. To overcome this effect, a double-grating configuration has to be adopted to compen‐ sate for the pulse-front tilt [41, 42]. The first grating is demanded to spectrally disperse the beam on an intermediate focal plane, where a slit performs the spectral selection, while the second grating compensates for the pulse-front tilt of the diffracted beam by equalizing the length of the optical paths. Double-grating instruments have been demonstrated to be very effective for HHs, with time resolution well below 10 fs [43–47]. Double-grating configura‐ tions, which are able to preserve the ultrafast duration of the pulse, have been also proposed as beam splitters for ultrafast intense pulses [48] and as infrared (IR)-XUV beam separators for HHs [49].

Here, we focus on the use of gratings to realize devices able to manipulate the spectral phase of chirped pulses, in particular to be used as XUV compressors.
