## 2 Simulation and Experimental Procedure
### Theoretical Model  
The theoretical model is based on the differential equations underlying the multilevel rate equations describing the ion populations of the different erbium energy levels, as well as the power propagation equations describing the longitudinal evolution of pump, signal and amplified spontaneous emission (ASE) power. A schematic of the Er\({}^{3+}\) energy level system illustrating the transitions considered in the model is shown in Figure 1.  
Specifically, in our simulations, the following phenomena were taken into account: (1) spontaneous decay from \({}^{4}\)I\({}_{13/2}\), \({}^{4}\)I\({}_{11/2}\) and \({}^{4}\)I\({}_{9/2}\) energy levels; (2) pump absorption and stimulated emission between \({}^{4}\)I\({}_{15/2}\) and \({}^{4}\)I\({}_{11/2}\) energy levels; (3) signal absorption and stimulated emission between \({}^{4}\)I\({}_{15/2}\) and \({}^{4}\)I\({}_{13/2}\) energy levels; (4) uniform cooperative upconversion beetween a pair of excited Er\({}^{3+}\) ions: \({}^{4}\)I\({}_{13/2}\) + \({}^{4}\)I\({}_{13/2}\)\(\rightarrow\)\({}^{4}\)I\({}_{9/2}\) + \({}^{4}\)I\({}_{15/2}\) and \({}^{4}\)I\({}_{11/2}\)\(+\)\({}^{4}\)I\({}_{11/2}\)\(\rightarrow\)\({}^{4}\)I\({}_{15/2}\) + \({}^{4}\)S\({}_{3/2}\); (5) cross-relaxation process between two Er\({}^{3+}\) ions: \({}^{4}\)I\({}_{9/2}\) + \({}^{4}\)I\({}_{15/2}\)\(\rightarrow\)\({}^{4}\)I\({}_{13/2}\). The rate Equations (1)-(5) are reported below and are solved with the simulation code using the Newton-Raphson method:  
\[\frac{dN_{1}}{dt}=-(W_{12}+W_{13})N_{1}+W_{31}N_{3}+W_{21}N_{2}+\frac{N_{2}}{ \tau_{21}}+C_{up}N_{2}^{2}+C_{3}N_{3}^{2}-C_{14}N_{1}N_{4} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c} \hline
**Core/Cladding** & **Erbium Ions** & **Losses** & **Losses** & **Background** & **Core** \\
**Diameter** & **Concentration** & **@1530 nm** & **@980 nm** & **Losses** & **Dopants** \\ \hline
2.8 \(\upmu\)m/127 \(\upmu\)m & 1.21 \(\times\) 10\({}^{19}\) ions/cm\({}^{3}\) & 24 dB/m & 14 dB/m & 6.2 dB/km & Al-Er-Ce \\ \hline \end{tabular}
\end{table} 
Table 1: Optical fiber parameters. \begin{table}
\begin{tabular}{c c c c} \hline
**Fiber Length** & **Signal–Pump** & **Signal Power** & **Pump Power** \\
**Configuration** & **at 1550 nm** & **at 980 nm** \\ \hline
4.3 m & Co-propagating & 1 \(\upmu\)W & 50 mW \\ \hline \end{tabular}
\end{table} 
Table 2: EDFA parameters. \[\frac{dN_{2}}{dt}=W_{12}N_{1}-W_{21}N_{2}-\frac{N_{2}}{\tau_{21}}-2C_{up}N_{2}^{2 }+C_{14}N_{1}N_{4} \tag{2}\]  
\[\frac{dN_{3}}{dt}=W_{13}N_{1}-W_{31}N_{3}+A_{43}N_{4}-A_{32}N_{3}-C_{3}N_{3}^{2} \tag{3}\]  
\[\frac{dN_{4}}{dt}=-A_{43}N_{4}+A_{32}N_{3}+C_{up}N_{2}^{2}-C_{14}N_{1}N_{4} \tag{4}\]  
\[\sum_{i=1}^{4}N_{i}=N_{Er} \tag{5}\]  
In these equations, \(N_{i}\) is the population of the \(i\)-th energy level; \(\tau_{21}\) corresponds to the radiative lifetime of the \({}^{4}\)I\({}_{13/2}\) energy level; \(W_{ij}\) is the absorption/stimulated emission rate between energy levels \(i\) and \(j\); \(A_{ij}\) is the non-radiative decay rate from levels \(i\) to level \(j\); \(C_{up}\) and \(C_{3}\) are the homogeneous upconversion coefficients related to \({}^{4}\)I\({}_{13/2}\) and \({}^{4}\)I\({}_{11/2}\) energy levels, respectively; \(C_{14}\) is the cross-relaxation coefficient between Er\({}^{3+}\) ions; and \(N_{Er}\) is the total Er\({}^{3+}\) population [35; 36; 37]. The total distribution of Er\({}^{3+}\) ions is assumed to be uniform over the cross section of the fiber core and along the whole fiber length. The optical power propagation equations for the pump, signal and ASE, along the forward (+) and backward (\(-\)) directions are reported in (6)-(8). They form a two-boundary values problem which is solved using an iterative forward and backward integration technique, suitably developed, based on the Runge-Kutta method.