## 4 Results and Discussion  
### The Raman Response Function of Ge-Sb-Se Glass  
According to previous work of the researchers in this paper, when fused silica is used as a reference standard, the Raman gain spectrum of Ge-Sb-Se or other glasses can be calculated by comparing spontaneous Raman spectra measured under the same experimental conditions, as shown in Equation (4) [40]. \[ \begin{split}\frac{g_{8 RG\beta-sample}^{k\gamma}(\omega_{p}- \Omega_{\beta-sample}^{k})}{g_{8 RG\beta-si\beta 2}^{k\gamma}(\omega_{p}-\Omega_{-si\beta 2}^{k})}=&\frac{K_{GR-sample}^{k \gamma}K_{SR-si\beta 2}^{k\gamma,\gamma}}{K_{SR-sample}^{k\gamma}K_{GR-si\beta 2}^{k\gamma,\gamma}}\cdot\frac{(\omega_{p}-\Omega_{\beta-si\beta 2}^{k})^{3}n_{si\beta 2}( \omega_{p})n_{sample}(\omega_{p}-\Omega_{\beta-si\beta 2}^{k})}{(\omega_{p}- \Omega_{\beta-si\beta 2}^{k})^{3}n_{sample}(\omega_{p})n_{si\beta 2}( \omega_{p}-\Omega_{\beta-si\beta 2}^{k})}\\ &\cdot\frac{\left[1-R\big{(}\omega_{p-si\beta 2}\big{)}\right]\left[1-R\big{(}\omega_{p}- \Omega_{\beta-si\beta 2}^{k}\big{)}\right]}{\left[1-R\big{(}\omega_{p}- \Omega_{\beta-sample}^{k}\big{)}\right]}\cdot\frac{I_{g-sample}^{k\gamma, \gamma}(\omega_{p}-\Omega_{\beta-sample}^{k})}{I_{g-si\beta 2}^{k\gamma,\gamma 2}( \omega_{p}-\Omega_{\beta-si\beta 2}^{k})}\end{split}  \tag{4}\]  
where \(\Omega_{\beta}^{k}\approx\Omega=\omega_{p}-\omega_{s}\) is the frequency shift of the Raman peak from the laser frequency \(\omega_{p}\), and \(R(\omega)=\frac{[n(\omega)-1]^{2}}{[n(\omega)+1]^{2}}\) is the reflectance coefficient at normal incidence. The parameters \(K_{SR}^{k\gamma}\) and \(K_{GR}^{k\gamma}\) are constants that contain all the phonon and electromagnetic constant parameters, and the difference between them is \(K_{SR}^{k\gamma,\gamma}\), which contains the Bose-Einstein correction factor \(F_{BE}(v,T)=1+\left[\exp(hv/kT)-1\right]^{-1}\)(\(T\), \(h\) and \(k\) refer to temperature, Planck constant, and Boltzmann constant, respectively) that accounts for the thermal  
Figure 2: The comparison of Raman response functions of fused silica. The blue curve is the single Lorentzian model [38], the red curve is the multiple–vibrational–mode model [31], and the black curve is the experimental result [39]. statistical fluctuation for each individual mode with an energy \(kT\). From Equation (4), it is evident that the ratio of the Raman gain coefficient of other materials to that of fused silica can be determined by comparing the ratio of their respective refractive indices, reflectivity coefficients, Raman spectral intensities, and Raman frequency shifts. Generally, it is widely acknowledged in the field that the Raman gain coefficient of fused silica remains constant when the pump wavelength is held constant. If we employ fused silica as the reference standard, the Raman gain coefficients of Ge-Sb-Se or other glass materials can be determined through a comparative analysis of spontaneous Raman spectra acquired under identical experimental conditions. By leveraging this approach, we not only ensure precise calculations but also establish a robust framework for assessing and contrasting the Raman gain coefficients of various glass compositions. The Raman gain spectrum of Ge\({}_{12.5}\)Sb\({}_{15}\)Se\({}_{72.5}\) glass obtained using this method is shown in Figure 3. Here, the dashed line represents the spectrum modeled using seven Gaussian functions, and each Gaussian peak corresponds to a vibrational mode of a unit composing the glass structure. Table 1 provides the parameters used in the multiple vibrational mode model, and the ratio of the Gaussian linewidth to the Lorentzian linewidth is 3.0 in this work. Based on Equation (3) and data in Table 1, the Raman response function of Ge\({}_{12.5}\)Sb\({}_{15}\)Se\({}_{72.5}\) glass is depicted in Figure 4. It is noteworthy that the Raman response function of Ge\({}_{12.5}\)Sb\({}_{15}\)Se\({}_{72.5}\) glass is characterized by a sum of multiple damped harmonic oscillations, in contrast to the simple damped harmonic oscillation observed in the single Lorentzian