## 3 Discussion
In our samples, \(d_{THz}>>\lambda_{2\omega}/n_{2\omega}\), which gives:  
\[A_{ref}^{samp}(\tau)=-4\pi\frac{\chi_{eff}^{(3)}E_{THz}(\tau)E_{\omega}^{2}}{[ n_{2\omega}^{samp}\cos(\beta_{2\omega}^{samp})+n_{\omega}^{samp}\cos(\beta_{ \omega}^{samp})][n_{2\omega}^{samp}\cos(\beta_{2\omega}^{samp})+\cos(\alpha)]}, \tag{8}\]  
where \(\chi_{eff}^{(3)}=3/2\chi_{samp}^{(3)}(-2\omega;\omega,\omega,0)\), \(n_{\omega}^{samp}\), and \(\beta_{\omega}^{samp}\) are the sample refractive index and the transmission angle for the fundamental wavelength; \(E_{THz}(\tau)\) is the transmitted terahertz field related to the corresponding incident fields \(E_{THz}^{sins}(\tau)\) through the Fresnel transmission coefficient \(t_{THz}^{sins}\) (note that in general, \(E_{THz}(\tau)\) should not trace \(E_{THz}^{sins}(\tau)\) due to material dispersion); and \(E_{\omega}=t_{\omega}^{samp}E_{\omega}^{ins}\), where \(E_{\omega}^{ins}\) is the amplitude of the incident fundamental optical field and \(t_{\omega}^{samp}\) is the Fresnel transmission coefficient. The amplitude of the SH field generated inside the material \(A_{tr}^{samp}(\tau)\) (in our case this field is propagated only in SiO\({}_{2}\)), in contrast, is rather sensitive to the mutual propagation dynamics of the terahertz and optical pulses [31]. In fused quartz, the coherent length of the fundamental and SH radiation (\(\sim\)10 \(\upmu\)m) is significantly less than the walk-off length of the optical and terahertz pulses: \(\sim\)200 \(\upmu\)m. This corresponds to the adiabatic regime of SHgeneration which, with an allowance for the theory developed in Ref. [31] and generalized for the oblique incident terahertz and optical pulses, gives (see also [37]):  
\[A_{tr}^{SiO_{2}}(\tau)=4\pi\frac{\chi_{eff}^{(3)}F_{\omega}^{2}E_{ THz}(\tau)}{\left(n_{2\omega}^{SiO_{2}}\right)^{2}-\left(n_{\omega}^{SiO_{2}} \right)^{2}}\frac{n_{\omega}^{SiO_{2}}\cos(\beta_{\omega}^{SiO_{2}})+\cos( \alpha)}{n_{2\omega}^{SiO_{2}}\cos(\beta_{2\omega}^{SiO_{2}})+\cos(\alpha)} \approx 4\pi\frac{\chi_{eff}^{(3)}F_{\omega}^{2}E_{ THz}(\tau)}{\left(n_{2\omega}^{SiO_{2}}\right)^{2}-\left(n_{\omega}^{SiO_{2}} \right)^{2}}. \tag{9}\]  
Note that Equation (9) is valid only for \(\tau\) less than the time delay required for terahertz and optical pulses to overlap on the exit facet of the SiO\({}_{2}\) layer. (For the 2 mm thick SiO\({}_{2}\) layer used in the TFISHG experiment, this condition gives \(\tau<2\) ps; for \(\tau>2\) ps, there appears to be an SH signal from the exit facet of the SiO\({}_{2}\) layer [31, 32].) The complex amplitudes of the SH field generated in the gas \(A_{ins}^{gas}(\tau)\) are mainly determined by the spatiotemporal dynamics of the terahertz pulse during the joint propagation with the fundamental optical pulse after the parabolic mirror PM2 (see the inset in Figure 1b). Optical and terahertz pulses propagate in gas with approximately the same velocities. However, due to strong terahertz beam-focusing (with NA-0.4) and having a near-single cycle duration, not only the amplitude but also the terahertz field waveforms change significantly (see Figure 4). Thus, \(A_{ins}^{gas}(\tau)\) should not trace the local in space \(E_{ THz}^{ins}(\tau)\) measured near the focus of the parabolic mirror PM2 (where the sample was placed). On the other hand, the strongest terahertz field and, as a consequence, a more efficient SH emission occurs in a definite region near the focus. Thus, we can assume that the SH is generated in some effective gas layer, which, according to our estimates, has a thickness \(D\approx 1\) cm. Additionally, we can consider average terahertz fields \(E_{av}^{ins}(\tau)\) with a constant profile in the effective gas layer. According to Figure 4, \(E_{av}^{ins}(\tau)\) should be shifted forward for some time delay \(\Delta\tau_{ins}\) relative to the terahertz field in focus \(E_{ THz}^{ins}(\tau)\). This time shift was directly observed in the experiment.