## 3 Results and Analysis
### Numerical Analysis of Hybrid EYDF Amplifier
The multimode fiber length \(L_{1}\) was selected to be 1 m to obtain sufficient loss and a suitable wavelength interval,  
Figure 2: (**a**) The transmission spectrum of the active SMS with 5-cm multimode fiber; (**b**) the maximum loss in the 1-\(\upmu\)m band and wavelength spacing with variation in multimode fiber length; (**c**) the transmission spectrum of the active SMS with 1-m multimode fiber. (Inset) the transmission loss of a uniform EVDF-6/125 fiber; (**d**) the zoomed transmission spectrum in the range of 1545–1555 nm. and the corresponding wavelength spacing was about 1 nm. The transmission spectrum of the active SMS with 1-m multimode fiber is plotted in Figure 2c. Because the wavelength of Yb-ASE is much shorter than that of Er-ASE, Yb-ASE can stimulate more transverse modes in the multimode fiber, and obtains more significant MMI effects. The maximum loss in the 1-\(\upmu\)m band was 17.7 dB at 1013 nm, while it was only 2.3 dB at 1553 nm. The transmission loss of a uniform EYDF-6/125 was also simulated in the wavelength range of 1-1.6 \(\upmu\)m, which is shown as a red line in the insertion in Figure 2c. The spectrum fluctuations of the uniform fiber were less than 0.04 dB. This comparison of the results demonstrates that the difference in transmission loss between the 1 and 1.5 \(\upmu\)m bands is caused by the effect of multimode interference. The different transmission loss between the 1-\(\upmu\)m and 1.5-\(\upmu\)m band make it possible to suppress Yb-ASE in the amplifier. The transmission spectrum of the SMS is zoomed from 1545 to 1555 nm and plotted in Figure 2d to evaluate the loss of signal at 1550 nm. The transmission loss at 1550 nm is only 0.35 dB, and the full width at half maximum (FWHM) of the transmission peak is near 1 nm, which is equal to the wavelength interval. Although the transmission spectrum was periodically modulated, the loss value at spectral drops was still less than 2.5 dB and can be reduced by the gain property of the active fiber and the suppression of Yb-ASE. It can be inferred that this structure can not only be applied in narrow linewidth laser amplification, but also it could support laser amplification at the 1.55-\(\upmu\)m band. Based on the BPM analysis of the SMS structure, the output properties of the hybrid fiber amplifier were further simulated. Besides the mentioned pump absorption and core diameter, the steady state rate equations of EYDF and other parameters are referred to the literature [28; 29]. The model of the simulation is plotted in Figure 1. The propagation equations for the signal power \(P^{+}s(z)\), pump power \(P^{+}p(z)\), Yb-ASE power \(P^{\pm}\gamma_{\text{YASE}}(z,\lambda)\) and Er-ASE power \(P^{\pm}\varepsilon_{\text{E}A\text{E}}(z,\lambda)\) are shown as follows [29]:  
\[\left\{ \begin{array}{l}\frac{dP_{s}^{+}(z)}{dz}=\{\Gamma_{s}[N_{2}(z)\sigma_{ 21}(\lambda_{s})-N_{1}(z)\sigma_{12}(\lambda_{s})]-\alpha(\lambda_{s})\}P_{s} ^{+}(z)+\frac{2 hc^{2}}{\lambda^{3}}\Gamma_{s}N_{2}(z)\sigma_{21}(\lambda_{s})m _{s}\Delta\lambda\\ \frac{dP_{s}^{+}(z)}{dz}=\{\Gamma_{p}[N_{6}(z)\sigma_{65}(\lambda_{p})-N_{5}(z )\sigma_{65}(\lambda_{p})-N_{1}(z)\sigma_{12}(\lambda_{p})]-\alpha(\lambda_{p} )\}P_{p}^{+}(z)\\ \pm\frac{dP_{EASE}^{+}(z,\lambda)}{dz}=\{\Gamma_{s}[N_{2}(z)\sigma_{21}( \lambda)-N_{1}(z)\sigma_{12}(\lambda)]-\alpha(\lambda)\}P_{E,ASE}^{\pm}(z, \lambda)+\frac{2 hc^{2}}{\lambda^{3}}\Gamma_{s}N_{2}(z)\sigma_{21}(\lambda)m_{s }\Delta\lambda\\ \frac{\pm dP_{\gamma_{\text{YASE}}}^{+}(z,\lambda)}{dz}=\{\Gamma_{s}[N_{6}(z) \sigma_{65}(\lambda)-N_{5}(z)\sigma_{56}(\lambda)-N_{1}(z)\sigma_{12}(\lambda )]-\alpha(\lambda)\}P_{\gamma_{\text{YASE}}}^{+}(z,\lambda)+\frac{2 hc^{2}}{ \lambda^{3}}\Gamma_{s}N_{6}(z)\sigma_{65}(\lambda)m_{s}\Delta\lambda\end{array} \right. \tag{3}\]  
Here the symbol \"\(\pm\)\" represents forward (+) and backward (-) transmitted lights. The rate equations of the ytterbium and erbium energy system are referred to in the literature [29]. All the parameters in segment 1 are the same to those in segment 3.