# Gain Equalization for Few-Mode Erbium-Doped Fiber Amplifiers via Strong Mode Coupling
## 2 Numerical Modeling of Strongly Coupled FM-EDFA  
To characterize an FM-EDFA with strong MC, we cannot simply evaluate its modal gain (MG) by launching and receiving each mode one by one as in an uncoupled or weakly coupled situation, because each input would excite all the other guided modes. There is an accurate way to measure MG--namely, calculating singular value decomposition (SVD) of the fiber transfer matrix (TM) [18].  
In our model, an erbium-doped fiber (EDF) comprises multiple gain segments and coupling matrices, as shown in Figure 1. We assume that there are K independent sections, each of which is modeled as an individual amplification gain matrix (\(G_{1}\), \(G_{2}\),..., \(G_{K}\)). Distributed MC can be viewed as a large number of random matrices (\(M_{1}\), \(M_{2}\),..., \(M_{K-1}\)), randomly located before and after each independent section. The TM of the proposed FM-EDFA is given by:  
\[M=G_{K}\cdots G_{i+1}M_{ci}G_{i}\cdots G_{2}M_{c1}G_{1} \tag{1}\]  
where \(M\) is the TM, \(G_{i}\) is the gain matrix in the \(i\)-th section, and \(M_{ci}\) is the random mode-coupling matrix that occurs between two amplification processes (i.e., \(G_{i}\) and \(G_{i+1}\)).  
Figure 1: Multi-segment model of a strongly coupled FM-EDFA.  
Then, we can obtain the overall MG and DMG from M by using the SVD technique at each frequency bin [18], which closely resembles the evaluation of polarization dependent loss [19]. \(M\) can be decomposed into \(N\) spatial channels:  
\[M=V\cdot S\cdot U^{\star} \tag{2}\]  
\[S=\left(\begin{array}{ccc}e^{\frac{1}{2}g_{1}}&&0\\ &\ddots&\\ 0&&e^{\frac{1}{2}g_{N}}\end{array}\right) \tag{3}\]  
where \(U\) and \(V\) represent the equivalent mode mixing at the input and output ends, respectively. We obtain g = (\(g_{1}\), \(g_{2}\),..., \(g_{N}\)), whose elements are the logarithms of the eigenvalues of \(M\cdot M^{\star}\)[20]--i.e., the gain of N spatial channels. Thus, the overall DMG can be written as [3]:  
\[\text{DMG}=max\{g_{n}\}-min\{g_{n}\} \tag{4}\]  
where \(max\{g_{n}\}\) and \(min\{g_{n}\}\) are the maximum and minimum gains in dB.  
Here, EDFs are equally divided into thousands of fiber segments, each of which is modeled using the 4 th-order Runge--Kutta method [21]. MC randomly happens between two fiber segments. Both the number of coupling points and the coupling strength per point determine the degree of mode coupling. Under a 980 nm pump light (LP\({}_{01}\)), the linearly polarized signal modes (LP\({}_{01}\), LP\({}_{11}\), LP\({}_{21}\), LP\({}_{02}\)) are launched with an equal power of 0.1 mW at the end of the fiber input. In this work, EDFs with two different Er-ion distributions are taken into account: Type-A with uniform doping and Type-B with dual-layered doping, as shown in Figure 2 and Table 1. The two typical Er-ion distributions can be obtained by the Modified Chemical Vapor Deposition (MCVD) combined with solution-doping. Other complex doping profiles can also be achieved through experimentation [22; 23].  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Type** & **Core Radius (\(\mu\)m)** & **Cladding Radius (\(\mu\)m)** & **Numerical Aperture** & **Erbium Doping Distribution** \\ \hline Type A & 8 & 62.5 & 0.15 & Uniform doping (Er\({}^{+}\) = 100 ppm) \\ Type B & 8 & 62.5 & 0.15 & Dual-layer doping (\(r<b\): Er\({}^{+}\) = 65 ppm, \(r>c\): Er\({}^{+}\) = 115 ppm) \\ \hline \end{tabular}
\end{table}
Table 1: Fiber parameters of the two different types of EDFs.  
Figure 2: Refractive index profile and Er-dopant distribution of (**a**) the Type-A and (**b**) Type-B 6-mode EDFs (the shaded region represents Er ion doping). (**c**) DMG of Type-B 6-mode EDFA as a function of \(b\) and \(c\).