# L-Band Erbium-Doped Fiber Optimization and Transmission Investigation +
## 2 Fiber Optimization and Verification  
### The Structure Optimization of the FM-EDF  
The few-mode erbium-doped fiber (FM-EDF) serves as a crucial component of the few-mode erbium-doped fiber amplifier (FM-EDFA). Therefore, we initially optimized the active fiber structure. The optimization encompassed two aspects: the refractive index (RI) profile and the doping concentration distribution. A triple-layered-core trench-assisted structure was selected, given its excellent loss and dispersion performance, as reported in our previous work [18]. The RI profile is illustrated in Figure 1a. This RI profile supports four mode groups in the L-Band; the distributions of the LP\({}_{01}\), LP\({}_{11}\), LP\({}_{21}\), and LP\({}_{02}\) mode groups are demonstrated in Figure 1b. Following the RI profile illustrated in Figure 1a, the doping concentration of erbium ions was carefully optimized. As shown in Figure 1a, the core radius was 10 \(\upmu\)m, and as shown in Figure 1b, the mode fields were mainly distributed within a radius of 10 \(\upmu\)m. Consequently, the erbium ions were doped within this 10 \(\upmu\)m radius region. Meanwhile, to simplify the fiber manufacturing, the doping area was divided into four distinct parts (Figure 2), with each region maintaining a constant doping concentration. The length of the EDF is another crucial parameter that influences the modal gain. Thus, the active fiber length (\(L_{EDF}\)) was optimized with the doping profile. In addition, to obtain a higher modal gain in the L-band, 1480 nm was selected to be the pump wavelength. A bi-directional pump scheme was selected, and the forward and backward pump powers were set to 400 mW and 450 mW, respectively. These values were determined based on the characteristics of the two pump lasers used in the actual system. The optimization contained five parameters, which posed a challenge due to performance limitations of the PC. Searching the whole solution space was impractical. To address this challenge and reduce the optimization complexity, the particle swarm optimization (PSO) algorithm [19] was applied to optimize the fiber structure, including the length of the EDF and the four doping concentrations. The optimization platform is Matlab R2021a. Figure 1: (**a**) The refractive index (RI) profile. (**b**) The mode distributions of the four mode groups (1600 nm). Figure 2: Schematic of the different doping regions and doping concentrations. The PSO realizes optimization by maximizing or minimizing the fitness (\(F\)). To realize high modal gain and low different modal gain at the same time, the fitness is defined as follows (1):  
\[F=\frac{G_{\text{max}}^{2}}{DMG}, \tag{1}\]  
where \(G_{max}\) and \(DMG\) are the maximum modal gain and differential modal gain (\(DMG\)) among all signal modes, respectively. By maximizing fitness, the modal gain and \(DMG\) can be optimized simultaneously. The iterative solution method for every single particle is shown in Equations (2) and (3):  
\[x_{i+1,j}=v_{i+1,j}+x_{i,j}, \tag{2}\]  
\[v_{i+1,j}=v_{i,j}(0.9-i/2N_{i})+c_{1}r_{1}\left(x_{p,best,j}-x_{i,j}\right)+c_ {2}r_{2}\left(x_{g,best,j}-x_{i,j}\right), \tag{3}\]  
where \(x_{i,j}\) is the \(j\)-th component of the solution at generation \(i\); \(v_{i,j}\) is the \(j\)-th component of the particle velocity at generation \(i\); \(N_{i}\) is the total number of iterations; \(x_{p,best,j}\) and \(x_{g,best,j}\) represent the best solution of the single particle and the best solution among all the particles, respectively; constants \(c_{1}\) and \(c_{2}\) are set to 0.5; and \(r_{1}\) and \(r_{2}\) are two random numbers ranging from 0 to 1. For this optimization, the total number of iterations was set to 100, and the particle number was set to 20. The iteration curves of \(G_{max}\) and \(DMG\) are illustrated in Figure 3a; as the iteration progresses, the \(DMG\) rapidly decreases to a low level of 0.2 dB. Meanwhile, the modal gain reduces at first, and then gradually grows from 32 dB to 38 dB.