## 1 Introduction
On the other hand, for other promising glass families with large nonlinearity, for example, tellurite and chalcogenide glasses, the use of all-optical \(\chi^{(3)}\) in electric-field-induced \(\chi^{(2)}_{eff}\) remains open. Moreover, the coefficient \(\chi^{(3)}(-2\omega;\omega,\omega,0)\) inherent for \(\chi^{(2)}_{eff}\) has not been measured for tellurite and chalcogenide glasses up until now. A possible advancement of the EFISHG technique is to use terahertz electromagnetic pulses instead of a DC electric field. This so-called terahertz-field-induced SHG (TFISHG) technique was used for the investigation (primarily the transmission geometry) of the nonlinear response of different materials, including ferroelectric layers [27; 28], electro-optic crystals [29; 30], SiO\({}_{2}\), and optical adhesives [31; 32]. Superstrong (above the DC-field breakdown threshold) terahertz fields (\(\sim\)20 MV/cm) were employed for studying SHG in p-doped Si under below-band-gap optical excitation [33]. TFISHG in reflection geometry was used to characterize the Si(111) surface under above-band-gap optical excitation with a moderate terahertz field (\(\sim\)300 kV/cm) [34]. The specific feature of the TFISHG measurements, shown in Refs. [31; 34], is the influence of ambient gas (air) on the detected second harmonic (SH) signal (this influence was also discussed in regards to EFISHG [35]). In the previous studies, due to a rather strong SH signal from the sample or/and some overlapping geometry of the fundamental optical and terahertz pulses, the SHG from ambient gas could be neglected or strongly suppressed by using helium gas instead of air [31; 34]. However, in the case of a small SH signal from the sample (due to its small nonlinearity), even rather weak generation from the ambient gas may be significant. In this paper, third-order nonlinear susceptibilities \(\chi^{(3)}(-2\omega;\omega,\omega,0)\) and \(\chi^{(3)}(-\omega;\omega,-\omega,\omega)\) of different samples of tellurite and chalcogenide glass families were compared using the TFISHG and z-scan [36] techniques. All measurements were carried out using femtosecond laser pulses at the 790 nm fundamental wavelength. In view of the opacity of the samples for SH radiation (at a 395 nm wavelength) as well as of the strong absorption of terahertz radiation (especially in tellurite glasses), the TFISHG measurements were performed in reflection geometry. The generated SH radiation was detected in the reflection direction with respect to the incident fundamental optical pulses. Terahertz pulses were also assumed to be incident on the sample from the side of the laser pulses. In such a setup, the SH radiation generated from the sample can be rather small and comparable to (or even smaller than) the SH generated before the sample in the ambient gas. By changing the time delay between the femtosecond optical and terahertz pulses and using reference signals from fused quartz, we investigated the relationship between the gas and sample contributions to the total SH signals. As a result, the nonlinear susceptibilities \(\chi^{(3)}(-2\omega;\omega,\omega,0)\) (responsible for SHG) for tellurite and chalcogenide glasses were measured for the first time to the best of our knowledge. Their direct comparison with \(\chi^{(3)}(-\omega;\omega,-\omega,\omega)\) (responsible for the nonlinear self-action) allowed us to assess the applicability of the commonly used assumption of Kleiman symmetry for glasses [18].