## 3 Results and Analysis  
### Numerical Analysis of Hybrid EYDF Amplifier  
The power scaling and ASE of the EYDF amplifier with hybrid or uniform fiber were numerically analyzed. The simulation model of the hybrid EYDF amplifier is shown in Figure 1. The total length \(L\) of the gain fiber is 3 m, for both the hybrid and uniform fiber. The multimode fiber is inserted into two segments of single-mode fiber. The distance between the multimode fiber and \(z=0\) is fixed at 1 m. The length of the multimode fiber is assumed to be \(L_{1}\). The seed and pump laser are forward coupled into the gain fiber at length position \(z=0\), and launched at \(z=L\). The power of the seed and pump in the simulation are assumed to be 30 mW and 20 W, respectively. The output performance of hybrid gain fiber amplifier is affected by two aspects: one is the spectra modulation of the SMS structure, which can be obtained by analyzing the MMI effect; the other is the gain characteristics of the active fiber, which can be obtained by numerically analyzing the steady-state rate equations of the EYDF amplifier. The length of the multimode gain fiber is optimized by simulation. The core diameter and length of the multimode fiber have a significant influence on the transmission spectrum of the SMS structure due to the MMI effect. The modulation depth is related to the number of guiding modes \(N\) in the center multimode fibers. The number of guiding modes \(N\) can be estimated by the core diameter a, the working wavelength \(\lambda\) and the numerical aperture (NA), according to Equation (1) [21]. The \(V\) is normalized frequency. The core NA of MM-EYDF-6/125 is 0.2. If the working wavelengths are 1 and 1.5 \(\upmu\)m, the number of guiding modes participating in multimode coherence are estimated to be 10 and 23, respectively. It can be inferred that the Yb-ASE at the 1-\(\upmu\)m band has higher modulation depth than Er-ASE at the 1.5-\(\upmu\)m band. Figure 1: The schematic diagram of the hybrid EYDF amplifier. CLS, cladding light striper. \[\left\{ \begin{array}{l}N\approx\frac{V^{2}}{2}\\ V=2\pi\frac{\pi}{\lambda}\cdot NA\end{array} \right. \tag{1}\]  
The modulation property of SMS structure fiber was numerically analyzed by the BPM method [27]. Figure 2a is the transmission spectrum of the SMS structure with 5-cm multimode fiber. There is an obvious periodic modulation of the spectrum due to the MMI effect. The modulation of the MMI effect is not obvious in the 1.5-\(\upmu\)m band because of insufficient guiding modes. The maximum modulation depth at the 1.5-\(\upmu\)m band is only 3-4 dB, which is much lower than that of the 1-\(\upmu\)m band where the Yb-ASE located. This phenomenon makes it possible to suppress Yb-ASE in the amplifier. The length of multimode fiber \(L_{1}\) is related to the wavelength spacing \(\Delta\lambda\) of the modulation spectrum. According to the literature [20], the wavelength interval \(\Delta\lambda\) can be calculated by Equation (2):  
\[\Delta\lambda=\frac{16n_{core}a^{2}}{(m-n)[2(m-n)-1]L_{MMF}} \tag{2}\]  
Here, \(m\) and \(n\) are the order number of the linear polarization modes \(LP_{0m}\) and \(LP_{0m}\), respectively. The wavelength interval decreases with the MM fiber length. A sufficiently small wavelength interval can avoid lasing at other wavelengths. The maximum loss in the 1-\(\upmu\)m band and wavelength interval with variations in the multimode fiber length \(L_{1}\) were simulated, which is shown in Figure 2b. When \(L_{1}\) increases from 5 cm to 1.5 m, the wavelength interval decreases rapidly while the maximum loss increases slightly until the length is over 50 cm, then it decreases with the length increment.