# High-Harmonic and Terahertz Spectroscopy (HATS): Methods and Applications
### High-Order Harmonics and Terahertz Wave: Up- and Down-Conversion of Laser Frequency  
The interaction between atoms or molecules and the intense laser fields induces a dipole moment that leads to the emissions of high-order harmonics, which can be considered as the up-conversion of the fundamental laser frequency. In 1992, [PERSON] [6] and [PERSON] [30] interpreted this generation process by the classical three-step model: an electron is freed from the laser dressed Coulomb field, accumulates energy \(E_{k}\) within the propagation and is finally driven back to the nuclei to emit photons with energy \(\Omega\) equalling \(E_{k}+I_{p}\). This simple man model can give an intuitive interpretation to some experimental observations, such as the cut-off energy of the harmonic spectroscopy, the long and short trajectories of electrons and the chirp between different orders of high harmonics [31]. In terms of quantum mechanics, the freed electron can be viewed as a spreading wave, and the re-collision of electron can be treated as the coherent superposition of the recombined and the remaining electronic wave packets [32]. Typically, only the odd harmonics of the fundamental laser pulses can be generated, due to the coherence between the consecutive periodic emissions. The even harmonics [22] or the supercontinuum [33; 34] can also be generated by breaking the symmetry of the electric field via adding a weak second harmonic pulse.  
Atoms or molecules in strong laser fields can radiate THz wave with the photon energy at meV, which is the down-conversion of the fundamental laser frequency. The first experimental observation on the far infrared/THz emissions from the interaction between laser and rare gas atoms was carried out by [PERSON] in 1993 [35], reflecting the pondermotive force on electrons. In 2000, [PERSON] and his fellows developed the dual-color method to dramatically enhance the THz output [16]. Meanwhile, the dual-color relative phase dependence of THz yield is also investigated, attempting to clarify the generation mechanism [36]. There was a long-existing debate on the generation mechanism, i.e. the third-order nonlinearity parameter from the viewpoint of nonlinear optics, and the current model from the viewpoint of strong field dynamics. [PERSON] firstly explained the enhancement of THz output by considering the third-order nonlinearity parameter \(\chi^{(3)}\), corresponding to the process of \(\Omega_{\text{THz}}=\{\omega_{\text{FP}}+\omega_{\text{FP}}^{\prime}\}- \omega_{\text{SH}}\) and \(\Omega_{\text{THz}}=\omega_{\text{SH}}-\{\omega_{\text{FP}}+\omega_{\text{FP }}^{\prime}\}\). Here, the subscript indexes of frequency \"FP\" and \"SH\" indicate the photons from the fundamental laser pulses and the second harmonics, respectively. If the fundamental laser pulse is monochromatic, the THz output will vanish. Otherwise, the femtosecond laser pulse shares the band width at THz frequency range (for instance, 100 fs in time domain corresponds to 10 THz in frequency domain under the Fourier transformed limit), which gives birth to the non-zero THz wave generation from the four-wave mixing mechanism. In 2004, [PERSON] found that ionization was a prerequisite for THz generation under the dual-color fields by checking the laser intensity dependent THz output [37]. Bound electrons of atoms or molecules in air undergo tunneling ionization from strong laser fields, which occurs mostly near the laser peaks. When only the fundamental laser is present, the net current is zero summed over one laser period. However, by adding a second harmonic with a non-zero relative phase, the asymmetrical ionization from the alternating laser field leads to the nonvanishing directional current between the two consecutive half cycles. Based on the nonvanishing net current of electrons in air-plasma, [PERSON] proposed the plasma current model [38]. This model was verified by solving the time dependent Schrodinger Equation (TDSE) of the electron dynamics within the THz emission process by [PERSON] and [PERSON] [39; 40].