# A Multi-Layer Erbium-Doped Air-Hole-Assisted Few-Mode Fiber with Ultra-Low Differential Modal Gain
## 2 Theory and Model  
The amplification of the FM-EDFA is based on a quasi-three-level erbium-ion system [26] and two groups of differential equations are used in the simulation: rate and propagation equations [8]. \(N_{0}\) is the dopant density of the erbium-doped fiber (EDF), and \(\tau\) is the spontaneous emission lifetime of the erbium ions. As the erbium ions in the excited state is negligible, the erbium-ion system under 980 nm pump can be regarded as a two-level system. The populations in upper and lower energy levels at the position \((r,\varphi,z)\) are \(N_{1}(r,\varphi,z)\) and \(N_{2}(r,\varphi,z)\), given by \[N_{1}(r,\varphi,z)=\frac{\left[\frac{1}{\tau}+\frac{1}{nv_{s}}\sum\limits_{i=1}^{m_ {g}}[P_{ASE,i}(z)+P_{s,i}(z)]\sigma_{es}\Gamma_{s,i}(r,\varphi)\right]N_{0}(r, \varphi)}{\left[\frac{1}{nv_{s}}\sum\limits_{i=1}^{m_{g}}[P_{ASE,i}(z)+P_{s,i} (z)](\sigma_{es}+\sigma_{as})\Gamma_{s,i}(r,\varphi)+\frac{1}{\tau}+\frac{1}{nv _{g}}\sum\limits_{j=1}^{m_{g}}P_{p,j}(z)\sigma_{ap}\Gamma_{p,j}(r,\varphi) \right]}, \tag{1}\]  
\[N_{2}(r,\varphi,z)=N_{0}(r,\varphi,z)-N_{1}(r,\varphi,z), \tag{2}\]  
where \(\Gamma_{s,i}(r,\varphi)\) and \(\Gamma_{p,j}(r,\varphi)\) are the normalized intensity patterns of the \(i\)-\(th\) signal mode and \(j\)-\(th\) pump mode of the active fiber, \(h\) is Planck constant, \(\sigma_{as}\) and \(\sigma_{ap}\) are the absorption cross-section areas at signal and pump wavelengths, \(\sigma_{es}\) and \(\sigma_{ep}\) are the emission cross-section areas, \(m_{s}/m_{p}\) and \(v_{s}/v_{p}\) are the total number of guided modes and optical frequency at \(\lambda_{s}\) and \(\lambda_{p}\), respectively. The power in signal mode \(i\) (\(P_{s,i}\)), the amplified spontaneous emission (ASE) power in signal mode \(i\) (\(P_{ASE,i}\)), and pump power in mode \(j\) (\(P_{p,j}\)) evolve as below, where \(\Delta v\) is the equivalent amplifying bandwidth and \(a\) stands for the radius of uniform doping region. \[\frac{dP_{s,i}}{dz}=P_{s,i}\int\limits_{0}^{2\pi}\int\limits_{0}^{a}\Gamma_{s, i}(r,\varphi)[N_{2}(r,\varphi,z)\sigma_{es}-N_{1}(r,\varphi,z)\sigma_{as}] rdrd\varphi, \tag{3}\]  
\[\frac{dP_{ASE,i}}{dz}=P_{ASE,i}\int\limits_{0}^{2\pi}\int\limits_{0}^{a}\{ \Gamma_{s,i}(r,\varphi)\left[N_{2}(r,\varphi,z)\sigma_{es}-N_{1}(r,\varphi,z) \sigma_{as}\right]+2\sigma_{es}hv_{s}\Delta vN_{2}(r,\varphi,z)\Gamma_{s,i}( r,\varphi)\right\}rdrd\varphi, \tag{4}\]  
\[\frac{dP_{p,j}}{dz}=-P_{p,j}\int\limits_{0}^{2\pi}\int\limits_{0}^{a}\Gamma_{ p,j}(r,\varphi)N_{1}(r,\varphi,z)\sigma_{ap}rdrd\varphi, \tag{5}\]  
The gain of the signal and the DMG of the amplifier is defined as  
\[Gain(dB)=10\log_{10}\frac{P_{s,i}(z=l)}{P_{s,i}(z=0)}, \tag{6}\]  
\[DMG(dB)=\max(Gain)-\min(Gain). \tag{7}\]  
From the former equations, the modal gain of different modes in an FM-EDFA is affected by \(\Gamma_{s,i}(r,\varphi)\), \(\Gamma_{p,j}(r,\varphi)\) and \(n_{0}(r,\varphi)\), which can be expressed by the overlap integral, where \(n_{0}(r,\varphi)\) is the doping concentration of erbium-ion in the core. \[\eta_{s,i}^{p,j}=\int\limits_{0}^{2\pi}\int\limits_{0}^{a}\Gamma_{s,i}(r, \varphi)\Gamma_{p,j}(r,\varphi)n_{0}(r,\varphi)rdrd\varphi, \tag{8}\]  
Due to the pump mode selection independence, we analyze the characteristics of FM-EDFA under cladding pumping condition. Thus, \(\Gamma_{p,j}(r,\varphi)\) is a constant, and the overlap integral factors are only determined by the normalized intensity of the signal modes and the doping profile. The normalized intensity of different guided modes can be adjusted according to the introduction of the central air hole in the AH-FM-EDF. However, since the differences still exist only considering the RI, multi-layer doping designs are taken into account to further balance the overlap integral factors, which have different concentrations in different regions. The design of the erbium-ion doping profile is usually non-coincident with the fiber RI profile, as it can further eliminate the gain differences.