## 4 Results and Discussion
### The Raman Response Function of Ge-Sb-Se Glass
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mode** & **Component** & **Peak Intensity** & **Gaussian** & **Lorentzian** \\
**Number** & **Position** & **A\({}_{1}\)** = **A\({}_{1}\)/\(w_{\text{e}\text{i}}\)** & **FWHM** & **FWHM** \\
**i** & **(cm\({}^{-1}\))** & **(10\({}^{-12}\) m/w/THz)** & **(cm\({}^{-1}\))** & **(cm\({}^{-1}\))** \\ \hline
1 & 49.63 & 4.14 & 10.07 & 3.36 \\ \hline
2 & 59.55 & 7.20 & 16.35 & 5.54 \\ \hline
3 & 75.90 & 10.86 & 27.30 & 9.10 \\ \hline
4 & 103.96 & 17.61 & 43.48 & 14.49 \\ \hline
5 & 198.10 & 26.79 & 27.99 & 9.33 \\ \hline
6 & 212.44 & 22.58 & 84.78 & 28.26 \\ \hline
7 & 257.42 & 4.87 & 16.74 & 5.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the parameters used in the multiple–vibrational–mode model. Figure 3: The Raman gain spectrum of Ge\({}_{12.5}\)Sb\({}_{15}\)Se\({}_{72.5}\) glass calculated using Raman spectra (solid line) and seven different Gaussian functions (dashed line). structure Raman gain materials. In addition, the presence of multiple peaks in the Raman gain spectrum suggests the contribution of multiple vibrational modes to the Raman gain, thereby supporting the superiority of the multiple vibrational mode model in accurately representing the actual response behavior of Ge\({}_{12.5}\)Sb\({}_{15}\)Se\({}_{72.5}\) glass. The Raman gain spectrum is proportional to the imaginary part of the transfer function as reported by [PERSON] [41], which is given as follows. \[g_{R}(\omega)=\sum_{i=1}^{N}\frac{A_{i}}{2}\int\limits_{0}^{\infty}\{\cos[( \omega_{v,i}-\omega)t]-\cos[(\omega_{v,i}+\omega)t]\}\exp(-\gamma_{i}t)\exp(- \Gamma_{i}^{2}t^{2}/4)dt \tag{5}\]  
There are no reports on experimental or accurate theoretical simulations about the Raman response function of Ge\({}_{12.5}\)Sb\({}_{15}\)Se\({}_{72.5}\) glass. Therefore, to verify the accuracy of the Raman response function solved using the multiple vibrational mode model, we inversely deduce the Raman gain spectrum using Equation (5) and compare it with the experimentally measured data obtained from Equation (4), and the result is shown in Figure 5.  
Figure 4: The Raman response function of Ge\({}_{12.5}\)Sb\({}_{15}\)Se\({}_{72.5}\) glass calculated using the multiple-vibrational–mode model. Figure 5: The Raman gain spectrum obtained from the multiple–vibrational–mode model (red) and the Raman spectra (black). Figure 5 demonstrates that the curve profiles generated using the multiple vibrational mode model and the Raman spectra are consistent with only a small difference observed around 150 cm\({}^{-1}\). Hence, we conclude that the multiple vibrational mode model is plausible. The Raman fraction can be obtained from the Raman response function using Equation (6) [42]. \[g_{R}(\omega)=\frac{2\omega_{0}}{c}n_{2}f_{R}\text{Im}[\hat{t}_{R}(\omega)] \tag{6}\]  
Here, the transfer function \(\hat{t}_{R}(\omega)\) is the Fourier transform of the Raman response function, \(\omega_{0}\) is the center frequency, \(c\) is the speed of light in vacuum, and \(n_{2}\) is the nonlinear refractive index of the Ge\({}_{12.5}\)Sb\({}_{15}\)Se\({}_{72.5}\) glass. The Raman fraction \(f_{R}\) of the SIF fiber is obtained as 0.4615.