## 3 Results and Analysis
### Numerical Analysis of Hybrid EYDF Amplifier
According to the fibers' specification, the signal overlap factor \(\Gamma_{s}\) of segment 1 and segment 2 were calculated to be 0.83 and 0.96, respectively, and the pump overlap factor \(\Gamma_{p}\), which can be approximated as the area ratio of core to cladding, was 0.0023 and 0.0092, respectively. The pump absorption of these fibers was experimentally measured as mentioned above. The \(\sigma_{65}\) and \(\sigma_{56}\) are the emission and absorption cross-section of Yb-ions, respectively, and \(\sigma_{21}\) and \(\sigma_{12}\) are those of Er-ions. \(N_{2}\) and \(N_{1}\) are the population of the upper and lower energy levels of Er-ions, respectively. In the steady-state equations, the population of metastable levels \(N_{3}\approx 0\). The Er-ion concentration approaches \(N_{1}+N_{2}\). \(N_{5}\) and \(N_{6}\) are the population of the upper and lower energy levels of Yb-ions, respectively, and their summation is the Yb-ions concentration. According to the absorption coefficient and absorption cross-section, the Er-ions and Yb-ions concentration of SM-EYDF-6/125 in the simulation are 1.36 \(\times\) 10\({}^{24}\) and 1.29 \(\times\) 10\({}^{25}\), respectively. The Er-ions and Yb-ions concentration of MM-EYDF-12/125 are 3.55 \(\times\) 10\({}^{24}\) and 3.83 \(\times\) 10\({}^{25}\), respectively. The output power properties can be obtained in the simulation by substituting the fiber parameters at the corresponding position. It should be noted that since the signal light can pass through the SMS structure with almost no insertion loss (less than 0.1 dB), and because the inner cladding has the same diameter, the insertion loss in the SMS structure to the pump light is relatively small and was ignored. The additional loss caused by the MMI effect is marked as \(\alpha_{\text{SMS}}(\lambda)\), which was obtained above by BPM analysis of the SMS structure. It is assumed that the MMI loss \(\alpha_{\text{SMS}}(\lambda)\)affects the forward ASE power at the end of the multimode fiber position \(z=z_{2}\), and reacts to the backward ASE power at \(z=z_{1}\). So, the boundary condition of Yb-ASE and Er-ASE with MMI loss is shown in Equation (4). \(dz\) is the length unit that divides the fiber length. \[\left\{ \begin{array}{l}P_{\textit{E}rasE}^{+}(z_{2}+dz,\lambda)=\alpha_{ \textit{S}\textit{MS}}(\lambda)\cdot P_{\textit{E}rasE}^{+}(z_{2},\lambda)\\ P_{\textit{Y}\textit{W}ase}^{+}(z_{2}+dz,\lambda)=\alpha_{\textit{S}\textit{MS} }(\lambda)\cdot P_{\textit{Y}\textit{W}ase}^{+}(z_{2},\lambda)\\ P_{\textit{E}rasE}^{-}(z_{1}-dz,\lambda)=\alpha_{\textit{S}\textit{MS}}( \lambda)\cdot P_{\textit{E}rasE}^{-}(z_{1},\lambda)\\ P_{\textit{Y}\textit{W}ase}^{-}(z_{1}-dz,\lambda)=\alpha_{\textit{S}\textit{MS} }(\lambda)\cdot P_{\textit{Y}\textit{W}ase}^{-}(z_{1},\lambda)\end{array} \right. \tag{4}\]  
In Equation (4), the wavelength \(\lambda\) range of Er-ASE and Yb-ASE are 1000-1100 nm and 1500-1600 nm, respectively. The output property of the hybrid fiber amplifier can be obtained by adding the MMI loss \(\alpha_{\textit{S}\textit{MS}}(\lambda)\) to the equations and solving them with the finite difference method [28]. The pump and signal power evolution in the hybrid and uniform fiber amplifier are plotted in Figure 3a. The launched signal power of the hybrid fiber amplifier with or without considering the MMI effect are both higher than the 4.52-W signal power obtained by the uniform fiber amplifier with the same fiber length and pump conditions due to its high pump absorption. Considering the spectral modulation effect of the SMS structure, the signal power in the hybrid amplifier (marked as hybrid fiber with MMI) was 5.51 W under 20 W pump power. However, the power calculated considering only the fiber size and doping concentration of the hybrid fiber, which is marked as the hybrid fiber without MMI in the figure, was 5.25 W, which is less than that with MMI by 0.26 W. The optical conversion efficiency of the hybrid fiber amplifier is 27.5%, which is 4.9% more efficient than that of the uniform one.