## 3 Discussion
After substituting this value into Equation (9) and then using Equation (8), we found the peak SH amplitude generated in the reflection direction \(A^{SiO_{2}}_{ref}\approx-0.01A^{SiO_{2}}_{tr}=-1.5\). This value can be considered as a reference in the following calculation. To find the helium contribution \(A^{He}_{ins}(\tau)\), one should consider \(S^{SiO_{2},He}_{entr}(\tau)\) which is shown by the red diamonds in Figure 3b. Due to the low photon numbers, \(S^{SiO_{2},He}_{entr}(\tau)\) has large fluctuations. We averaged several points near the maximum of the SH signal, which gave \(S^{SiO_{2},He}_{entr}\approx 8\) counts. From Equation (6) with known \(A^{SiO_{2}}_{ref}\), taking into account Equation (10) (where, similarly to air, \(\Delta\tau_{ins}\) is assumed to be close to 120 fs and \(E^{ins}_{av}(\tau)\) close to \(E^{ins}_{THz}(\tau)\)), we obtained the SH amplitude generated in helium \(A^{He}_{ins}\approx 9\exp(-i\pi/2)\). Now we can calculate complex SH amplitudes from other samples. In the case of the tellurite samples, the measured SH signal is about 10 times higher than that from SiO\({}_{2}\) (see Figure 3c for the TWLA sample), which is associated with the amplification of the SH signal from the sample due to its larger nonlinearity. However, it should be remembered that the reflection from the sample of the SH field generated in helium also increases (in Equation (6) \(r^{TWLA}_{2\omega}\approx 1.7^{SiO_{2}}_{2\omega}\)). From the peak value of \(S^{TWLA,He}_{entr}\sim 70\) counts, using Equation (6) with known \(A^{He}_{ins}\), we estimated \(\Big{|}A^{TWLB}_{ref}\Big{|}\sim 9\). Note that the profile of \(S^{TWLA,He}_{entr}(\tau)\) diverges noticeably from \(\big{(}E^{ins}_{THz}(\tau)\big{)}^{2}\). This can be explained by partial interference of \(A^{He}_{ins}(\tau)\) and \(A^{TWLA}_{ref}(\tau)\) due to the appearance in the latter amplitude of a nonzero phase (according to our estimates it is about 0.2-0.3 rad; see the dotted curve in Figure 3c). This phase can arise from the complex value of \(n_{2\omega}\) (tellurite glass is not transparent for SH) and the complex value of \(\chi^{(3)}_{TWLA}\) (the latter is supported by the z-scan measurement; see Table 1). The TWLB sample has an SH signal profile similar to TWLA with an amplitude of about 90 counts, which, following the same approach, gives \(\Big{|}A^{TWLB}_{ref}\Big{|}\sim 11\). In the case of chalcogenide glasses, we observed an SH signal that was more than 200 times stronger than that in tellurite glasses (\(S^{Gs_{2}S_{3},He}_{entr}(\tau)\) in Figure 3d for As\({}_{2}S_{3}\)). The time-delay shapes of the SH signals were close to the squared terahertz electric  
Figure 4: Terahertz electric field waveforms at different \(z\) (shown in the inset) along beam path (see Figure 1b). field \(\left(E_{THz}^{ins}(\tau)\right)^{2}\) (dotted curve in Figure 3d). Thus, in this case, the helium contribution was negligible. From the measured peak SH signals, we obtained \(\left|A_{ref}^{As_{2}S_{3}}\right|\approx 152\), \(\left|A_{ref}^{IS23}\right|\approx 148\) and \(\left|A_{ref}^{IS25}\right|\approx 130\). Let us make a brief note about the SH signals measured in air. By comparing \(\left|A_{ref}^{SiO_{2}}\right|_{2}\approx 2\) counts and \(S_{entr}^{SiO_{2}air}\approx 800\) counts (blue points in Figure 3b), we can confirm the above-mentioned statement that \(S_{entr}^{SiO_{2}air}(\tau)\) primarily has a contribution from gas. Then, taking into account the theoretically estimated phase \(-1.4\pi/2\), we have \(A_{ins}^{air}\approx 97\exp(-i0.65\pi)\). By substituting \(A_{tr}^{SiO_{2}}(\tau)\) and \(A_{air}^{ins}(\tau)\) into Equation (7) (with allowance for the \(\Delta\tau_{ins}\approx 120\) fs shift of \(E_{avg}^{ins}(\tau)\) and \(E_{THz}^{ins}(\tau)\)), we obtain the profile (dash-dotted curve in Figure 3a) which agrees rather well with the experimental signal \(S_{exit}^{SiO_{2}air}(\tau)\). This confirms our estimated phase of \(A_{air}^{ins}\).