## 2 Experimental Measurements  
An amplified Ti:sapphire laser system at \(\lambda_{\omega}=790\) nm central wavelength (1 kHz, 70 fs, 1 mJ) was used for measuring nonlinear coefficients by the TFISHG and z-scan techniques. We used the following as samples: fused quartz (SiO\({}_{2}\)), two tellurite glasses doped with oxides of tungsten, lanthanum and bismuth (or aluminum) with the composition (mol%) 71.2 TeO\({}_{2}\)-23.7 WO\({}_{3}\)-4 La\({}_{2}\)O\({}_{3}\)-1.1 Bi\({}_{2}\)O\({}_{3}\) (TWLB) and 70.6 TeO\({}_{2}\)-23.5 WO\({}_{3}\)-4 La\({}_{2}\)O\({}_{3}\)-1.9 Al\({}_{2}\)O\({}_{3}\) (TWLA), and three chalcogenide glasses: As\({}_{2}\)S\({}_{3}\) and commercially available infrared optical glass IKS23 and IKS25 (produced by the S.I. Vavilov State Optical Institute). IKS23 has a composition of As-S-Se, in which Se is a small additive and its optical transparency lies in the range of 0.6-9 \(\upmu\)m. IKS25 has a composition of As-Se-Sb-Sn, in which Sb and Sn are small additives, and its optical transparency lies in the range of 1-17 \(\upmu\)m.  
The z-scan scheme had a classical configuration (see Figure 1a). The optical radiation attenuated by two wedges and calibrated neutral filters was focused (by a lens with F = 40 cm) on a sample placed on a motorized stage. The transmitted radiation was collected into a photodiode, which produced a signal registered by a lock-in amplifier. A variable diaphragm (aperture) was placed in front of the photodiode to perform open- and close-aperture z-scan measurements (in the latter case, the aperture was set to cut about 50% of the transmitted optical energy). For each sample, a series of measurements was recorded for different incident intensities. All were normalized and then a \"background\" trace with small optical power (which contained linear distortions introduced by the sample) was subtracted from the \"signal\" traces with high power. After division of the closed aperture by the open aperture traces, the nonlinear phase \(\delta\varphi_{NL}\) was extracted from the difference in peak and valley normalized transmittances \(\delta T_{pv}\) using Equation (1) [36]:  
\[\delta\varphi_{NL}=\frac{\sqrt{2}\cdot\delta T_{pv}}{0.406\cdot(1-s)^{0.25}}, \tag{1}\]  
where \(s=0.5\). A nonlinear refractive index was then obtained from Equation (2):  
\[n_{2}=\frac{\lambda_{\omega}}{2\pi}\frac{\delta\varphi_{NL}}{L_{eff}I}, \tag{2}\]  
where \(L_{eff}=(1-\exp(-\alpha L))/\alpha\), \(L\) is sample thickness (in our samples \(L_{eff}\approx L\)), \(\alpha\) is linear absorption coefficient, and \(I\) is the intensity of laser radiation in the material placed into focus. A two-photon absorption (2 PA) coefficient \(\beta\) was calculated from the valley of normalized transmittances \(\delta T\) of an open aperture z-scan trace by solving Equation (3):  
\[\delta T\approx-\sum_{m=1}^{\infty}\frac{(-\beta L_{eff}I)^{m}}{(1+m)^{3/2}}, \tag{3}\]  
With known \(n_{2}\) and \(\beta\), real and imaginary parts of the nonlinear susceptibility \(\chi^{(3)}(-\omega;\omega,-\omega,\omega)\) were obtained from Equations (4) and (5) [37]:  
\[Re\chi^{(3)}(-\omega;\omega,-\omega,\omega) = \frac{cn_{\omega}^{2}}{12\pi^{2}}n_{2}, \tag{4}\] \[Im\chi^{(3)}(-\omega;\omega,-\omega,\omega) = \frac{\lambda_{\omega}cn_{\omega}^{2}}{48\pi^{3}}\beta, \tag{5}\]where \(n_{\omega}\) is the sample refractive index. To ensure accuracy of our measurements, we used as a reference signal a close aperture z-scan trace in SiO\({}_{2}\) for which the nonlinearity \(n_{2}\) is known to be 2.5 \(\times\) 10\({}^{-16}\) cm\({}^{2}\)/W [25; 26; 38]. Figure 2 shows z-scan traces for two tellurite glasses (TWLA and TWLB) and IKS23 (in this part of the experiment, IKS25 was not considered, as it is opaque at the 790 nm wavelength, the As\({}_{2}\)S\({}_{3}\) sample was rather thick for our experimental setup, and we used data from the literature). In the TWLB sample, we observed a somewhat stronger nonlinear effect than that in TWLA, both in open- and closed-aperture z-scan traces.