# High-Harmonic and Terahertz Spectroscopy (HATS): Methods and Applications
## 3 HATS for Molecules
### The Alignment Dependent Electron Dynamics for HHG with THz Phase Reference
As discussed in Section 2.1, laser-controlled motion of slow electrons for THz emission can be strongly distorted by the Coulomb potential in the forward-rescattering process, but it seems counter-intuitive that this distortion exhibits scarcely any dependence on the specific Coulomb fields from different gas types or the molecular alignments. If the two-color fields are linearly polarized, the electronic motion is mainly along the direction of laser polarization (with the transverse momentum distribution being neglected). When varying the Coulomb potential, the forward-rescattered electrons should be affected as observed by the holography of photoelectrons [93; 94]. However, the experimental results in Figure 9 and Ref. [82] exhibit a trivial dependence on Coulomb potential. It should be noted that the trivial dependence here does not mean the Coulomb potential is not important, but means the difference of the optimal phases between different atoms or molecules is not significant. The possible reason for the trivial dependence on gas types or molecular alignments could be attributed to the less spatial resolution of the slow electrons for THz emissions because the de Broglie wavelength of electron with 1 THz kinetic energy is about 220 A, which is much larger than the typical size of atoms or molecules. To examine the laser field dependence of the optimal phases of THz yield, the laser intensity of the fundamental pulse (ranging from \(0.8\times 10^{14}\) to \(1.8\times 10^{14}\) W/cm\({}^{2}\)) and the intensity ratiosbetween the second harmonic and the fundamental pulse (ranging from 0.001 to 0.1) are varied [67]. The nearly unchanged value of the optimal phases indicates that the THz optimal phase is a robust phase reference for the laser fields. Note that here the laser intensity variation is limited to make sure that the tunneling ionization still dominates the ionization of electrons. Therefore, under the typical experimental conditions for THz generation from the dual-color laser fields, the optimal phase of the slow electron for THz emission will not change under different Coulomb potential or laser conditions, which makes the modulation of THz yield a well-defined reference for the dual-color relative phase. Furthermore, HATS can help to investigate the participation of inner molecular orbital for HHG from the aligned N\({}_{2}\) by applying the THz phase reference. As illustrated in Figure 10, THz yield modulations exhibit no significant dependence on the molecular alignment, making the THz optimal phase a straightforward phase reference for different harmonics. By varying the aligned angle \(\Theta\) and the dual-color relative phase, two-dimensional modulations of even harmonics from the 28 th to the 32 nd harmonics can be traced with the THz phase reference. To give a better visualization of the relative phase dependence, the modulations of each horizontal lines are normalized to their maxima. For instance, the top horizontal line of the 28 th harmonic yields is normalized to the maximum value of the 28 th harmonic under different dual-color relative phases with the parallel alignment. Optimal phases for the 28 th and the 30 th harmonics are almost the same at different aligned angles. However, for the 32 nd harmonic, the optimal phases varies about 0.2\(\pi\) from parallel to perpendicular alignment. The modulation of the 32 nd harmonic is not as clear as the modulations of the other harmonics, due to its proximity to the cutoff and its low intensity. The difference between optimal phases for different alignments could be interpreted as the participation of the lower-lying molecular orbital (i.e., the HOMO-1) at the cut off region of high harmonic spectrum. The HOMO and HOMO-1 of N\({}_{2}\) exhibit the \(\sigma_{g}\) and \(\pi_{u}\) symmetries, respectively. Therefore, if the polarization of the driving laser is parallel to the molecular axis, a strong dipole will be induced in the HOMO, whereas no dipole is induced in the HOMO-1.