# High-Harmonic and Terahertz Spectroscopy (HATS): Methods and Applications
## 3 HATS for Molecules  
### THz Reference for Molecular Photoionization  
In this section, we introduce two examples of HATS applications for molecules. In general, the arbitrary pointing direction of a molecule obstructs the direct observation of the angular structures of molecular orbital. Thanks to the development of molecular non-adiabatic alignment techniques [69], tomography of the highest occupied molecular orbital (HOMO) was demonstrated on nitrogen molecule in 2004 [70]. Since then, high harmonics from aligned molecules are applied to image the HOMO of carbon dioxide [71; 72], the lower occupied molecular orbitals of nitrogen [46; 73] and carbon dioxide [49; 74; 75]. Based on the quantitative re-scattering (QRS) model on HHG, alignment-dependent HHG from molecules can be viewed as the product of the alignment-dependent ionization rates and the recombination matrix elements, while the recombination of electron and the subsequent emissions of the harmonics is the inverse process of photoionization [76]. Therefore, high harmonics from aligned molecules can be applied to deduce the molecular photoionization cross section (PICS). However, previous research required the separated calculation or experiment on the angle-dependent ionization of molecules. On the one hand, the ionization theories of molecules predict the wrong angular dependence when comparing to the experimental observations, such as the molecular strong field approximation (MO-SFA) [77] and the molecular Ammosov-Delone-Krainov (MO-ADK) theory [78]. For nitrogen or oxygen, there is only discrepancy in the absolute size of the angle-dependent ionization rates, while, for carbon dioxide, the calculated angle of peak ionization appears at around 30 degrees aligned angle between the molecular axis and the driven laser field polarization, but experimental results support the 45-degree alignment [79; 80]. On the other hand, the source density for harmonic emission is about \(10^{17}\) cm\({}^{3}\), which is below the typical detection limit of electrons or ions. This low density makes the joint measurement on high harmonics and electrons/ions from aligned molecules a challenging task. Figure 5: (**a**) Normalized THz waveform as a function of the dual-color relative phase and the position of gas jet. (**b–d**) The harmonic yields (with the background signals removed) as functions of the dual-color relative phase and the position of gas jet relevant to the laser focus. For a clear visualization of the intensity modulations of the even harmonics, the intensity at and beyond \(3\times 10^{6}\) photon counts is represented by the same color scale. The solid (dashed) white lines indicate the short (long) trajectory contributions. Instead, THz wave generation from aligned molecules can be treated as a calibration of the angular dependent ionization. According to the rescattering process discussed in Section 2.1, THz generation is mainly caused by the directly escaping and the forward-rescattered electrons. This makes the alignment dependent THz yield a suitable benchmark to calibrate the angular dependent ionization of molecules. Alignment dependent THz yield of molecules is proportional to the square of angular ionization probabilities [81]. Thus, by carrying out the joint detection of HATS from aligned molecules and using the alignment dependent THz yields, the angular differential PICS could be retrieved [21; 82]. The light path for HATS measurement on molecules is analogous to the one in Figure 2a, with an additional beam for molecular alignment [21]. As shown in Figure 6, normalized modulations of HATS are compared under different alignments. Note that the modulations are obtained from the aligned yields subtracted by the random aligned yields, and then normalized to the maximum value for the 21 st harmonic and THz wave, respectively. There is a positive correlation between the THz generation and the the 21 st harmonic generation from nitrogen.