# High-Harmonic and Terahertz Spectroscopy (HATS): Methods and Applications
## 3 HATS for Molecules
### The Alignment Dependent Electron Dynamics for HHG with THz Phase Reference  
Phases for high harmonics are of importance in understanding the electron dynamics and compressing the attosecond pulse (train). The techniques for characterizing the harmonics' phases can be generally classified as ex situ, where the attosecond pulse train photoionizes a target medium with  
Figure 8: The normalized theoretical (dashed line) and experimental (solid line) angle-dependent differential PICS of the 21 st (**a**), 23 rd (**b**), and 25 th (**c**) harmonic orders as functions of alignment. The green arrows indicate the angular minima for clarity. Reproduced with permission from [21]. Copyright APS Publishing, 2015.  
an additional fundamental laser pulse, and in situ, where the measurement of phase takes place in the medium in which the high harmonics are generated. These two approaches form a temporal gating of detection or generation of high harmonics with an additional pulse, which are extensively reviewed in Ref. [9]. The ex situ and in situ approaches have been proven equivalent in most cases for atoms [47], but the in situ approach seems to show less dependence on the recombination phase [91]. For molecules, ex situ detections on harmonics' phases have been carried out by applying the reconstruction of attosecond beating by interference of two-photon transition (RABBIT) technique [46; 49] or generating another high harmonic beam to form a Young's type two-emitter interferometer [72]. Actually, both the ex situ and the in situ approaches measure the derivative of the harmonic phase, not exactly the phase itself [1]. This raises two problems before comparing the experimental results to the theoretical calculations: (1) the absolute phases of different harmonics cannot be directly obtained from measurements; and (2) the absolute phases of high harmonics from molecules cannot be determined under different alignments. Therefore, several assumptions should be made in advance to give a phase reference, for instance, the same values of the optimal phases of the 15 th harmonic from Kr and aligned CO\({}_{2}\)[49], the same values of the optimal phases of the 15 th harmonic under all alignments of N\({}_{2}\) and laser intensities [46]. Applying an additional pulse to form an interferometer is a more direct way to obtain both the amplitudes and phases of high harmonics from aligned molecules [72], but it requires a quite precise control on the temporal and spatial overlapping of high harmonics from two separated emission sources [92]. Since the dual-color laser scheme could be applied as the in situ approach, these two problems can be solved easily by carrying out the HATS detection with the THz phase reference. Firstly, it is necessary to verify the robustness and validity of the THz phase reference when applying the HATS measurements. Hard-collision electrons contribute to high harmonics could be compared with the soft-collision electrons for THz generation, because these emissions are originating from the same driving laser fields but with different birth times of the ionized electrons. Based on this fact, dual-color relative phase dependent THz modulations could serve as a phase reference to calibrate high harmonics, as demonstrated in Section 2.2. Motions of electrons can be viewed as propagation in a combined field of the Coulomb field and the oscillating laser, therefore the robustness and validity of this THz phase reference should be examined by varying the Coulomb potential and the laser fields, respectively. To show the Coulomb potential dependence on the optimal phases, different gas types are compared in Figure 9. The results defy intuition that the optimal phases for THz output from atoms or small molecules keep nearly unchanged. Meanwhile, experiments on random, aligned and anti-aligned nitrogen molecules show nearly no difference on the optimal phases for THz output [82].