## 2 Numerical Model  
Transitions at DWP have been well studied in previous work [13; 14; 15]. Those transition processes can be classified as pump absorption, lasing, relaxation, and interionic processes. All processes except relaxation are shown in Figure 1. \(P_{976}\) excite ions from level \({}^{4}\)I\({}_{15/2}\) to \({}^{4}\)I\({}_{11/2}\), known as ground state absorption (GSA). The lifetime of level \({}^{4}\)I\({}_{11/2}\) is relatively long and allows GSA to create a virtual ground state (VSA); thus, the transition from level \({}^{4}\)I\({}_{11/2}\) to level \({}^{4}\)F\({}_{9/2}\) pumped by \(P_{1976}\) is called virtual ground state absorption (VGSA). Three excited state absorption (ESA) processes are involved, including ESA\({}_{1a}\) and ESA\({}_{1b}\) related to \(P_{976}\), and ESA\({}_{2}\) related to \(P_{1976}\). SE\({}_{1}\) and SE\({}_{2}\) stand for stimulated emission at 2.8 \(\upmu\)m and 3.5 \(\upmu\)m, respectively. Interionic processes are marked by \(W_{ijkl}\) and represent the energy transfer between ions from energy level \(i\) and \(j\) and the transition to energy level \(k\) and \(l\). Four interionic processes have been reported up to now, which are non-radiative, but play a key role in the laser operation. The transition rate from level \(i\) to level \(j\) during pump absorption processes can be described as:  
\[R_{ij}(z,t) =\Gamma\cdot v\cdot G_{ij}\cdot F(z,t) \tag{1}\] \[=\Gamma\cdot v\cdot\left(\sigma_{ij}N_{i}(z,t)-\sigma_{ji}N_{j}(z, t)\right)\cdot F(z,t),\]  
where \(z\) is the position along the fiber, \(t\) represents time, \(v=c_{0}/n_{core}\) is the speed of light inside the fiber core, \(c_{0}\) is the speed of light in vacuum, \(n_{core}\) is the core reflective index, \(\sigma_{ij}\) is the effective cross sections of absorption or emission of a transition from level \(i\) to level \(j\), \(N_{i}\) is the particle population of level \(i\), and \(F\) is the density of laser photons inside the fiber core. \(\Gamma\) is the mode overlap, representing the power in the mode that overlaps the core. Only the overlapped part of the laser can interact with the Er\({}^{3+}\) ions. The differences between a core pump and a clad pump are mainly attributed to the different in \(\Gamma\). The mode overlap of a gaussian beam is given by [16]:  
\[\Gamma=1-\exp\left(-2\frac{a^{2}}{\omega^{2}}\right), \tag{2}\]  
\[\omega\approx a\cdot\left(0.65+\frac{1.619}{V^{3/2}}+\frac{2.879}{V6}\right), \tag{3}\]  
\[V=\frac{2\pi a}{\lambda}\cdot NA, \tag{4}\]  
where \(a\) is the core radius of the fiber, and \(NA\) is the numerical aperture of the fiber. However, for clad pumping, the mode overlap can be estimated by \(\Gamma=A_{core}/A_{clad}\), where \(A_{core}\) and \(A_{clad}\) are the cross-sectional areas of the fiber core and clad, respectively. The photon density inside the fiber core can be calculated by:  
\[F(\
u)=\Gamma(\
u)\frac{P(\
u)n_{core}(\
u)}{A_{core}h\
u c_{0}}, \tag{5}\]  
Figure 1: The energy levels of erbium ions and energy transfer processes. where \(P\) is the laser power, \(h\) is the Planck's constant, and \(\
u\) is the photon frequency. The relaxation rate \(r_{ij}\) from level \(i\) to level \(j\) is given by:  
\[r_{ij}=\beta_{ij}/\tau_{i}, \tag{6}\]  
\[\sum_{j=1}^{i-1}\beta_{ij}=1, \tag{7}\]  
where \(\beta_{ij}\) is the branching ratio from level \(i\) to level \(j\) and \(\tau\) is the intrinsic lifetime. There have been four interionic processes reported thus far, three energy-transfer up conversions (ETU), \(W_{1103}\), \(W_{2206}\), and \(W_{4251}\), and one cross relaxation (CR) process, \(W_{5031}\).