# High-Harmonic and Terahertz Spectroscopy (HATS): Methods and Applications
### Rescattering Process and the Joint Measurement of HATS
Therefore, the two emissions could be compared to find new results about the rescattering by monitoring different kinds of electrons (from their emissions) simultaneously. For instance, the THz wave generation, which is a quite slow process compared to the generation of high harmonics (within sub-cycle of laser), can be precisely controlled and understood by tracing the dual-color relative phase dependent yield, which can be found in Section 2.1. THz yield modulations can also be served as a robust ionization reference, which is not critically influenced by other factors such as to the phase matching condition of high harmonics. The above depict of the rescattering process of HATS and LES is based on the time evolution of the electron wavepacket, which is building a bridge between optical emissions and the corresponding electron dynamics. Another perspective is to connect the separated pictures of the electron dynamics and the corresponding photon emission in the frequency domain, following the concept of \"non-linear optics\". As shown in Figure 1a, the generation of the \(q\)th harmonic can be easily understood by the up-conversion of the fundamental laser frequency. It should be noted that here the order of harmonic \(q\) should be odd, and for the even-order high harmonics, the participation of the second harmonic pulse should be considered. For instance, the \(q\)th harmonic (odd) can be combined with one photon of the fundamental pulse and one photon of the second harmonic pulse, and the output will be the \((q+3)\)thharmonic (even). It is tricky to use this viewpoint of energy transform for THz generation due to the failure of the four-wave mixing theory in interpreting the prerequisite of ionization [37]. However, this could be understood by many \"analogous four-wave mixing channels\" taking place among all the laser-dressed states, which could be written as \(\Omega_{\rm THz}=\{q\times\omega_{\rm FP}+\omega_{\rm FP}^{\prime}+\omega_{\rm FP }^{\prime\prime}\}-\{q\times\omega_{\rm FP}+\omega_{\rm SH}\}\), and \(\Omega_{\rm THz}=\{q\times\omega_{\rm FP}+\omega_{\rm SH}\}-\{q\times\omega_{ \rm FP}+\omega_{\rm FP}^{\prime}+\omega_{\rm FP}^{\prime\prime}\}\)[40]. The subscript indexes of FP and SH are similar to the ones applied in Section 1.1, indicating the photons of the fundamental laser pulses and the second harmonics. Note that the THz wave output can be non-zero due to the broad bandwidth of the fundamental pulse. Typically, the second harmonic pulse is considered as a weak-field, thus only one photon with frequency \(\omega_{\rm SH}\) participates in the nonlinear process. \(q\) represents the amount of photons. In this section, a unified physical picture for HATS and rescattering process is presented, including the emissions, the electron dynamics in time domain and the \"extremely non-linear optics\" in frequency domain. Based on this unified picture, the up-conversion of laser energy can be observed by the high-above threshold ionization (electron) and HHG (photon), while the down-conversion can be traced by the LES (electron) and THz wave (photon).