## 4 Applications
### High-Order Harmonic Generation Spectroscopy  
The HHG process is a highly non-linear light-matter interaction involving upconversion of the driving photon energy into higher values. In particular, emission of high-order harmonics of the fundamental one is observed. The discretization of the spectrum into harmonics derives from an interference process involving photon emission events occurring at different temporal windows. Indeed, light bursts of high energy photons are emitted by the target exposed to the radiation every half optical cycle of the laser field. The newly-generated spectral components provides coherent radiation in the extreme ultraviolet (XUV) and in the X-ray spectral region. An easy-to-understand explanation of the harmonics emission process is provided by the so-called three-step model [55; 56]. The three-step model is a semi-classical model that depicts HHG as a sequence of three steps: as a first step, tunnel ionization takes place, resulting in the ejection of an electron. The ejected electron is then accelerated by the external laser field, gaining kinetic energy. Because of the oscillating nature of the laser electric field, the accelerated electron has a non-negligible probability of recombining with the parent ion. Recombination is associated with a photon emission. According to this picture, the maximum energy (\(\hbar\omega_{m}\)) of the emitted photons is provided by the following expression: \(\hbar\omega_{m}=I_{p}+3.17U_{p}\), where \(I_{p}\) is the ionization potential of the target and \(U_{p}\) is the ponderomotive energy gained by the electron in the external field. The ponderomotive energy depends on the wavelength of the driving radiation, as \(\lambda^{2}\). The increase in the electron energy comes at the cost of a dramatic reduction of the HHG efficiency due to a reduced recombination probability associated with the spreading of the electron wave-function. The HHG yield scales as \(\approx\)\(\lambda^{-6}\)[57; 58]. As a result, in HHG experiments driven by IR pulses, the acquisition process takes longer since integration over a larger number of shots is required for achieving a suitable signal to noise ratio. In order to reduce measurement times in the long-wavelength regime, high repetition rate sources (\(>\)1 kHz) have to be used. Another factor affecting the HHG spectrum extension is phase matching. HHG takes place in an extended medium made up of a collection of dipoles, all acting as a source of XUV radiation. The spectrum of radiation emitted at a macroscopic level corresponds to that of a single dipole only if all emitters are in phase. Macroscopic effects in HHG are related to several factors: the focusing geometry of the driving pulses; the dispersion due to the plasma and neutral target; the nature of the sample (dipole contribution). Due to dispersion, phase-matching strongly depends on the wavelength of the driving radiation. When an unfavorable dispersion regime for the driving wavelength is attained, a suppression of the harmonics intensity and a reduction of the spectrum extension is observed [59]. HHG driven by near-IR and mid-IR pulses has been extensively studied in gas targets made up of atoms, molecules and clusters. In this spectral region, HHG does not suffer from strong macroscopic effects. Therefore, the advantage of generating an extended harmonics spectrum can be fully exploited, without incurring strong phase-matching limitations [60; 61; 62]. Besides spectral extension, few-cycle IR pulses are particularly useful for studying atoms, molecules and solids with transitional energies on the order of a few eV. In the HHG process, the excitation induced in the target involves a field-driven transition between bound and unbound states. When an exciting radiation with a photon energy much lower than the energy of the bound-unbound transition is used, direct ionization is less likely to occur.