## 4 Applications
### High-Order Harmonic Generation Spectroscopy
Direct ionization is detrimental for rescattering-based experiments since it is responsible for a reduction of the efficiency by depleting the ground state population and lowering the amount of possibly recolliding electrons. In Figure 10a, the maximum ponderomotive energy acquired by electrons undergoing interaction with an external field is shown as a function of the radiation wavelength (for a fixed beam intensity). In this sense, a remarkable result was that achieved in 2012 by [PERSON] et al. [PERSON] et al. (2012). Bright high-harmonic X-ray radiation with a spectrum extending up to 1.6 keV (\(\lambda=7.7\)A) was demonstrated by focusing 10-mJ mid-IR pulses at 3.9 \(\upmu\)m into a hollow core fiber filled with He gas (see Figure 10b). Due to the broad spectrum produced, extending up to the XUV and X-ray, the IR-driven HHG process can be used as a source for spectroscopic applications with elemental sensitivity [PERSON] et al. (2012). We mention here the very recent achievement by [PERSON] and coworkers [PERSON] and coworkers [PERSON] and others (2012). In their experiment, the high-order harmonics spectrum generated with a few-cycle 1.8-\(\upmu\)m driving field was focused into a polythiophene sample, and the X-ray absorption near edge structure (XANES) response of the sample was acquired. X-ray spectroscopy is nowadays mainly performed at synchrotron and free-electron laser facilities, where an extremely bright radiation in the picosecond and femtosecond time-scale is produced. In this sense, the HHG-based setup currently provides an unsurpassed tabletop alternative to these large-scale infrastructures. Besides providing a source of radiation for spectroscopy, HHG is in turn a spectroscopy technique itself, as the harmonics spectrum contains information about the structure and the dynamics occurring in the emitter. As mentioned above, the use of IR pulses enables the increase of the electron energy up to the keV level. An electron energy of a few hundreds eV corresponds to a De Broglie wavelength of the order of a few A, on the same scale of the internuclear distances between atoms in molecules and solids. As a result, the spectrum of high-order harmonics is strongly affected by the structural properties of the emitter. Under appropriate inversion algorithms, an HHG-based tomographic reconstruction of the emitter can be performed with a sub-nanometer resolution [PERSON] and others (2012). Indeed, the structure of the Highest Occupied Molecular Orbital (HOMO) has been successfully probed in simple molecules by driving HHG with ultrashort IR light pulses [PERSON] and others (2012); [PERSON] and others (2012); [PERSON] and others (2012). As pointed out above, different components of the HHG spectrum are emitted at different times within a temporal window of the order of half optical cycle of the carrier frequency. A time-to-energy mapping is thus possible, and dynamics occurring on a sub-cycle time scale can potentially be recorded in the harmonics spectrum. Figure 10: **(a)** Maximum electron return energy as a function of the wavelength for an intensity of 1.0 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\). In the inset, an electron energy distribution is shown, as measured after ionization of the C\({}_{2}\)H\({}_{2}\) molecule using the 3.1 \(\upmu\)m radiation. Adapted with permission from [PERSON] et al. (2012), Copyright APS, 2015; **(b)** High-order harmonics spectral yield in xenon at different driving laser wavelengths (0.8 \(\upmu\)m, 1.3 \(\upmu\)m, 2 \(\upmu\)m and 3.9 \(\upmu\)m). In the inset, the Fourier transform–limited pulse duration for the High-order Harmonic Generation (HHG) radiation driven by the 3.9-\(\upmu\)m pulses is shown, corresponding to 2.5 attoseconds. Adapted with permission from [PERSON] et al. (2012), Copyright AAAS, 2012. Moreover, fingerprints of ultrafast electronic processes have been observed in HHG measurements.