# A Multi-Layer Erbium-Doped Air-Hole-Assisted Few-Mode Fiber with Ultra-Low Differential Modal Gain
## 3 Results and Discussions  
Besides the RI of the fiber, the DMG correlates with the erbium doping profile. Normally, more accurate amplification can be obtained between different modes when the number of doping-layers increases. It is worth mentioning that the PSO algorithm can achieve the concentration optimization under any number of layers. Considering the fabrication in practice, we analyzed three types of dopants profiles in the following section, including single layer (uniform doping), double-layer, and triple-layer doping. The three types of doping profiles are illustrated in Figure 2. The doping boundaries are set as \(r_{1}\), \(r_{2}\), and \(r_{3}\). The corresponding doping concentration from the inside out is \(N_{1}\), \(N_{2}\), and \(N_{3}\). In the simulation, the degenerate states of the signal mode are not considered. Each signal input power is set as \(-10\) dBm. The pump power is set as \(3\) W. The PSO algorithm is applied to determine the doping concentration and boundaries. Figure 1: (**a**) Schematic diagram and the refractive index distribution of the AH-FM-EDF; (**b**) Normalized intensities and (**c**) effective refractive indices of the guide modes. In the first uniform doping profile, the erbium ion concentration is 4.4 \(\times\) 10\({}^{25}\) m\({}^{-3}\). Then in the double and triple-layer cases, the doping radius and concentration are iteratively optimized by the PSO algorithm. The results are shown as below. \[\text{Double}-\text{layer:}\left\{ \begin{array}{l}N_{1}=2.5\times 10^{25}\text{ m}^{-3},2\leq r\leq 7.72\\ \\ N_{2}=~{}6.1\times 10^{25}\text{ m}^{-3},7.72<r\leq 9\\ \end{array} \right.\]  
\[\text{Triple}-\text{layer:}\left\{ \begin{array}{l}N_{1}=4.84\times 10^{25} \text{ m}^{-3},2\leq r\leq 4.5\\ N_{2}=1.13\times 10^{25}\text{ m}^{-3},4.5<r\leq 7.36\\ N_{3}=~{}6.39\times 10^{25}\text{ m}^{-3},7.36<r\leq 9\\ \end{array} \right.\]  
The gains and DMG of the three doping cases are shown in Figure 3, where Figure 3a-c corresponds to the gain characteristics under uniform-, double-layer, and triple-layer doping, and Figure 3d illustrates the DMG results. It can be found from Figure 3a that at uniform doping, there is a big difference in gains and the DMG is more than 4 dB (LP\({}_{01}\) = 24.79 dB, LP\({}_{11}\) = 23.45 dB, LP\({}_{21}\) = 22.39 dB, LP\({}_{31}\) = 20.12 dB) at the fiber length of 1 m. Then, with the PSO algorithm, the signal gains in the double- and triple-layer cases are illustrated in Figure 3b,c. Under the same conditions, the gains are all higher than 20 dB in double- and triple-layer doping cases. Comparing the DMG results under different doping cases as shown in Figure 3d, the DMG of double-layer doping is effectively reduced to 0.55 dB and a DMG of 0.14 dB is further obtained in the triple-layer doping cases. Actually, the simulation results under four-layer doping cases have also been carried out. However, the DMG cannot be further reduced through the increase in the number of doping layers. The triple-layer doping case is considered the optimal solution for the proposed AH-FM-EDF. Due to the low DMG of 0.14 dB at the wavelength of 1550 nm, the following discussion is implemented in the triple-layer doping cases. The gain characteristics over the whole C-band is illustrated in Figure 4. All the signal gains are higher than 19 dB. In the wavelength range of 1530-1550 nm, the DMG keeps lower than 0.15 dB. At the wavelength 1525 and 1565 nm, the DMG are 0.21 dB and 0.26 dB, respectively. Consequently, at each wavelength, the DMG is lower than 0.3 dB. Moreover, due to the similar signal gains, there is an excellent gain flatness within 1 dB over the C-band. As we all know, it is impossible to maintain a stable gain value at any signal input power level for an FM-EDFA. It provides more gains to the small signals and less to the large ones, which is called the gain saturation effect [31]. The maximum output capability of the FM-EDFA is characterized by the saturated signal output power.