# High-Harmonic and Terahertz Spectroscopy (HATS): Methods and Applications
## 3 HATS for Molecules
### THz Reference for Molecular Photoionization
It has been reported that there also exists a positive correlation between the 23 rd harmonic and the ions output of nitrogen [71], and the pump-probe delay dependences of the harmonic yields from the 21 st to 25 th orders show similar structures [83]. Therefore, since the pump-probe delay dependent modulations of the 21 st harmonic and the THz yields are positively correlated in Figure 6, we can conclude that there exists a positive correlation between THz yields and ionization yields of N\({}_{2}\). The alignment dependent THz result from Y. [PERSON] et al. is also consistent with this conclusion by analyzing the THz yields from laser-aligned air [81]. More interestingly, when the aligned laser is linearly polarized at 50 degrees of the alignment angle, the magnitude of the modulation of high harmonic decreases significantly. This specific angle is called as the \"magic angle\", indicating the combined effect of the evolution of the rotating wavepacket and the angular harmonic yields [84]. HATS method provides an all-optical way in understanding the whole rescattering process from molecules, avoiding an additional ionization experiment. The measured HATS from aligned molecules is the coherent sum of the molecular frame angular emission yields \(S\) with a distribution of rotational wave packet \(\rho\)[85; 86]. It is hard to carry out coherent deconvolution to obtain the single molecular response from the experimental results, which requires the fitting on both the amplitudes and the  
Figure 6: The modulations of the relative yields of the 21 st harmonic (the blue solid lines) and the THz (the red dashed lines) as functions of the time delay between aligning and generation pulses with different aligned angles. The alignment angle is indicated in degrees at the top left corner of each subplot. phases. Therefore, the incoherent summation method is used instead for the small molecules, such as nitrogen and oxygen. By comparing the differences introduced by the coherently and incoherently summation, the validity of the incoherent method was examined for nitrogen [82; 87]. The incoherent sum \(M\) of the molecular frame angular yields from all solid angles takes the form of [86]  
\[M\left(\alpha^{\prime};t_{D}\right)\propto\int_{0}^{2\pi}d\varphi^{\prime}\int_{ 0}^{\pi}d\theta^{\prime}S(\theta,\varphi)\times\rho\left(\theta^{\prime}, \varphi^{\prime};t_{D}\right)\sin\theta^{\prime}, \tag{2}\]  
where \(\theta^{\prime}\) (\(\varphi^{\prime}\)) is the polar (azimuthal) angle about the polarization axis of the aligning pulse, \(\theta(\varphi)\) is the corresponding angle in molecule frame, \(\alpha^{\prime}\) represents the angle between the polarization directions of the aligning and the generation pulses, \(S\left(\theta,\varphi\right)\) is the angular yields of HATS in the molecular frame and \(\rho\left(\theta^{\prime},\varphi^{\prime};t_{D}\right)\) is the angular distribution of the rational wave packet at \(t_{D}\). Note that the primed and unprimed notations of the physical quantities indicate the laboratory frame and molecule frame, respectively. Since the coordinate axis could be transformed from the molecular frame to the laboratory frame [79], the angular yields \(S\left(\theta,\varphi\right)\) could be rewritten as \(S\left(\theta^{\prime},\varphi^{\prime},\alpha^{\prime}\right)\). By using the incoherent deconvolution method, the experimental results and the fittings of the 21 st harmonic are shown in Figure 7b,c, respectively.