## 3 Discussion
As will be shown below, the SH signal measured from the entrance facet of SiO\({}_{2}\) in an air atmosphere \(S_{entr}^{SiO_{2},air}(\tau)\) has the main contribution from the gas (\(S_{entr}^{SiO_{2},air}(\tau)\sim\left|A_{ins}^{air}(\tau)\right|_{2}\sim \left(E_{av}^{ins}(\tau)\right)^{2}\)). As seen from Figure 3b, \(S_{entr}^{SiO_{2},air}(\tau)\) is shifted to the right by \(\sim\)120 fs relative to \(\left(E_{ THz}^{ins}(\tau)\right)^{2}\). This confirms the proposed model. Additionally, it is interesting that the shape of \(S_{entr}^{SiO_{2},air}(\tau)\sim\left(E_{av}^{ins}(\tau)\right)^{2}\) is rather close to \(\left(E_{ THz}^{ins}(\tau)\right)^{2}\) (with a slight deviation of the lateral maxima). This allows us to assume that \(E_{av}^{ins}(\tau)\) and \(E_{ THz}^{ins}(\tau)\) have similar profiles. In the framework of the model, taking into account that \(n_{2\omega}^{gas}-n_{\omega}^{gas}<<1\) in gas, we obtain [31]:  
\[A_{ins}^{gas}(\tau)=16\pi\chi_{gas}^{(3)}E_{av}^{ins}(\tau)\left(E_{\omega}^{ ins}\right)^{2}\frac{L_{c}^{gas}}{\lambda_{\omega}n_{2\omega}^{gas}}\exp \left(-i\frac{\pi}{2}\left(1+\frac{D}{L_{c}^{gas}}\right)\right)\sin\left( \frac{\pi}{2}\frac{D}{L_{c}^{gas}}\right), \tag{10}\]  
where \(L_{c}^{gas}=\lambda_{\omega}/\left(4\left(n_{2\omega}^{gas}-n_{\omega}^{gas} \right)\right)\) is coherent length (\(L_{c}^{air}\approx 2.5\) cm in air and \(L_{c}^{He}\approx 60\) cm in helium), and \(\chi_{gas}^{(3)}\) is the effective third-order nonlinear susceptibility of gas (\(\chi_{air}^{(3)}/\chi_{He}^{(3)}\approx 20\)[47, 48]). Assuming D-1 cm, the phase of the peak complex amplitude in air and in helium can be estimated to be \(\arg(A_{ins}^{air})\approx-1.4\pi/2\) and \(\arg(A_{ins}^{He})\approx-\pi/2\), respectively. Now let us use the proposed model to describe the SH signals shown in Figure 3 and to find \(\chi_{simp}^{(3)}(-2\omega;\omega,\omega,0)\). For the simplicity of our presentation, we will take \(C_{entr}=1\) (complex SH amplitudes will be in arb.u). Since helium makes a significantly smaller additional contribution to the SH signal compared to air, the measurements in the helium atmosphere are our major interest. Using Equations (6) and (8), it is possible to establish the relationship between the nonlinear susceptibilities of the studied (tellurite and chalcogenide glasses) and the reference (SiO\({}_{2}\)) samples from the corresponding known SH signals. However, contributions of SiO\({}_{2}\) and gas (helium) towards the reference SH field amplitude should be also known. (Note that the absolute value of nonlinear susceptibility \(\chi_{simp}^{(3)}(-2\omega;\omega,\omega,0)\) can be also obtained from the magnitude of the SH signal, the terahertz field magnitude and the intensity of the fundamental optical pulses using Equations (6) and (8), with an accounting of the quantum efficiency of the PMT and transmission coefficients of the optical filters (BG39 and BP, see Figure 1b). This was earlier performed in Ref. [34] and close values of \(\chi^{(3)}_{SiO_{2}}(-2\omega;\omega,\omega,0)\) and \(\chi^{(3)}_{SiO_{2}}(-\omega;\omega,-\omega,\omega)\) were shown for SiO\({}_{2}\).) From the comparison of the SH signal magnitudes in Figure 3a,b, we conclude that only the material (SiO\({}_{2}\)) makes a substantial contribution to the SH signal from the exit facet of SiO\({}_{2}\) in He (red points in Figure 3a), whereas the contribution from He gas is negligible. This agrees with the fact that the SH signal profile is well fitted by the profile of the terahertz electric field squared (dotted line in Figure 3a). Thus, Equation (7) is simplified to \(S^{SiO_{2},He}_{exit}(\tau)=C^{SiO_{2}}_{exit}R^{SiO_{2}}T^{SiO_{2}}\Big{|} A^{SiO_{2}}_{tr}(\tau)\Big{|}^{2}\), which, taking into account the peak value of the SH signal \(S^{SiO_{2}}_{exit}\)\(\sim\)3000 counts, gives the peak amplitude \(A^{SiO_{2}}_{tr}=146\).