## 2 Experimental Measurements
The IKS23 sample exhibited a significantly higher nonlinear response considering that the intensity used was 15 times less. The nonlinear coefficients calculated using Equations (1)-(5) are listed in Table 1 (see below). The magnitudes of \(n_{2}\) and \(\beta\) for tellurite glasses are about (5-6) \(\times\) 10\({}^{-15}\) cm\({}^{2}\)/W and (6-8) \(\times\) 10\({}^{-2}\) cm/GW, respectively, and lie in a reasonable range of values [39; 40]. The nonlinear coefficients of IKS23 (\(n_{2}\) = 1.45 \(\times\) 10\({}^{-13}\) cm\({}^{2}\)/W, and \(\beta\) = 7.8 cm/GW) are more than 2-3 times higher than those of the literature data for As\({}_{2}\)S\({}_{3}\) at a wavelength of \(\sim\)0.8 \(\upmu\)m (\(n_{2}\) = 0.65 \(\times\) 10\({}^{-13}\) cm\({}^{2}\)/W [41], and \(\beta\) = 1-4 cm/GW [42]) despite their similar compositions (see also the review in Ref. [43]). The main explanation for this discrepancy is the expected dependence of the nonlinearity in IKS23 on the probe laser intensity. Indeed, for As\({}_{2}\)S\({}_{3}\) and As\({}_{2}\)Se\({}_{3}\), a significant increase in \(n_{2}\) and \(\beta\) with decreasing probe intensity below 1 GW/cm\({}^{2}\) and their saturation for the intensity above 1 GW/cm\({}^{2}\) was discovered in Ref. [44] and proved in Refs. [45; 46]. In our experiment, the incident optical intensity on IKS23 was \(\sim\)0.8 GW/cm\({}^{2}\), while in Ref. [41] the probe intensity exceeded several tens of GW/cm\({}^{2}\) (in Ref. [42], the probe intensity varied from 0.8 to 14 GW/cm\({}^{2}\)). This fact explains the higher nonlinear coefficients measured in our experiment. A small addition of Se in IKS23 should also somewhat increase \(n_{2}\) and \(\beta\) due to the nonlinear response in As\({}_{2}\)Se\({}_{3}\) being about 10 times stronger than that in As\({}_{2}\)S\({}_{3}\)[44]. Figure 1: (a) Z-scan scheme for \(\chi^{(3)}(-\omega;\omega,-\omega,\omega)\) measurement. (b) TFISHG scheme in reflection geometry for \(\chi^{(3)}(-2\omega;\omega,\omega,0)\) measurement. The left inset shows the arrangement of optical elements and the sample inside a box filled with air or helium. The right inset shows SH detection setup with a diaphragm for selective measurement of SH signals reflected from the entrance and exit surfaces of the fused quartz (SiO\({}_{2}\)). To measure the nonlinear susceptibility \(\chi^{(3)}(-2\omega;\omega,\omega,0)\) responsible for the SHG, we used the TFISHG technique in reflection geometry [34] (in this part of the experiment, we used all samples). The schematic diagram of the experimental setup is shown in Figure 1b (for more detail, see Ref. [34]). The probe femtosecond laser pulses of the fundamental harmonic (with energy \(\sim\)650 nJ) were weakly focused on the sample at the angle \(\alpha=45^{\circ}\) collinear with the strongly focused terahertz pulses, which were generated by the tilted-pulse-front technique in a LiNbO\({}_{3}\) crystal (not shown in Figure 1b). The diameter of the optical beam near the sample was \(\sim\)160 \(\upmu\)m and the peak intensity of the pulse was \(\sim\)25 GW/cm\({}^{2}\). The sample was placed near the focus of the terahertz beam (the size of the beam was \(\sim\)500 \(\upmu\)m and the peak amplitude of terahertz electric field was \(\sim\)170 kV/cm). The duration of the terahertz pulse was about 1 ps, so the terahertz field can be considered as quasi-static relative to the femtosecond fundamental optical pulse (terahertz field waveform was measured in ambient air by a conventional time-domain spectroscopy technique using a 200 \(\upmu\)m GaP crystal). The time delay between the optical and terahertz pulses was controlled by a motorized delay line placed in the part of the scheme responsible for the terahertz generation. As a result of the nonlinear interaction of the femtosecond optical radiation with the glass sample in the presence of the terahertz field, the pulses of second harmonic optical radiation at \(\lambda_{2\omega}=395\) nm wavelength were generated in the transmission and reflection directions.