# A Comprehensive Study on EDFA Characteristics: Temperature Impact
## 2 Analytical Model
### Rate Equations  
We take into account the three-level model, as shown in Figure 1. The population density in energy level 1 (ground state), energy level 2 (metastable state), and energy level 3 (pumping state) are N\({}_{1}\), N\({}_{2}\), and N\({}_{3}\), respectively. The system of differential rate equations is driven as follows [2; 3; 29]:  
\[\frac{\mathrm{d}\mathrm{N}_{1}}{\mathrm{d}t}=-\mathrm{R}_{13}\mathrm{N}_{1}+ \mathrm{R}_{31}\mathrm{N}_{3}-\mathrm{W}_{12}\mathrm{N}_{1}+\mathrm{W}_{21} \mathrm{N}_{2}+\mathrm{A}_{21}\mathrm{N}_{2} \tag{2a}\] \[\frac{\mathrm{d}\mathrm{N}_{2}}{\mathrm{d}t}=\mathrm{W}_{12}\mathrm{N}_{1}- \mathrm{W}_{21}\mathrm{N}_{2}-\mathrm{A}_{21}\mathrm{N}_{2}+\mathrm{A}_{32} \mathrm{N}_{3}\] (2b) \[\frac{\mathrm{d}\mathrm{N}_{3}}{\mathrm{d}t}=\mathrm{R}_{13}\mathrm{N}_{1}- \mathrm{R}_{31}\mathrm{N}_{3}-\mathrm{A}_{32}\mathrm{N}_{3} \tag{2c}\]  
Assuming the steady-state condition (\(\frac{\mathrm{d}\mathrm{N}_{1}}{\mathrm{d}t}=\frac{\mathrm{d}\mathrm{N}_{2}}{ \mathrm{d}t}=\frac{\mathrm{d}\mathrm{N}_{3}}{\mathrm{d}t}=0\)) and solving the three differential rate equations yields [3; 30]:  
\[\mathrm{N}_{2}=\frac{\mathrm{R}_{13}+\mathrm{W}_{12}}{\frac{1}{\varepsilon}+ \mathrm{R}_{13}+\mathrm{R}_{31}+\mathrm{W}_{12}+\mathrm{W}_{21}}\mathrm{N} \tag{3a}\]  
Figure 1: Energy level illustration for Er-doped silica fiber. where N represents the total E\({}_{\rm r}^{+3}\) ion density. Then:  
\[\rm{N_{1}}=\ N-\ N_{2} \tag{3b}\]  
where N\({}_{3}\) is negligible due to the very fast life time for the energy level 3 (~1 \(\rm{\upmu s}\)) as compared to that of the energy level 2 (~10 ms) [2; 3; 16]. The rates W\({}_{12}\), W\({}_{21}\), R\({}_{13}\), and R\({}_{31}\) are related to the absorption and emission cross-sections through the following equations [3; 29; 30]:  
\[\rm{W_{12}}=\frac{\sigma_{s}^{a}\Gamma_{s}\left(P_{s}+P_{ASE}^{+}+P_{ASE}^{-} \right)}{\rm{h}\
u_{s}A_{eff}} \tag{4a}\] \[\rm{W_{21}}=\frac{\sigma_{s}^{e}\Gamma_{s}\left(P_{s}\ +P_{ASE}^{+}+P_{ASE}^{-} \right)}{\rm{h}\
u_{s}A_{eff}}.\] (4b) \[\rm{R_{13}}=\frac{\sigma_{P}^{a}\Gamma_{P}P_{P}}{\rm{h}\
u_{p}A_{eff}}\] (4c) \[\frac{\mathrm{d}\mathrm{N}_{1}}{\mathrm{d}t}=-\mathrm{R}_{13}\mathrm{N}_{1}+ \mathrm{R}_{31}\mathrm{N}_{3}-\mathrm{W}_{12}\mathrm{N}_{1}+\mathrm{W}_{21} \mathrm{N}_{2}+\mathrm{A}_{21}\mathrm{N}_{2} \tag{2a}\]0  
where h is [PERSON]'s constant, and A\({}_{\rm eff}\) is the fibereffective area doped with E\({}_{\rm r}^{+3}\) ions. P\({}_{\rm s}\) and P\({}_{\rm P}\) are, respectively, the signal power transmitted through the fiber and the pump power driven into the fiber along its length. P\({}_{\rm x}^{+}\) and P\({}_{\rm ASE}^{-}\) represent the amplified spontaneous emission (ASE) power in both forward and backward directions, respectively. \(\rm{\upnu_{s}}\) and \(\rm{\upnu_{p}}\) represent the signal and pump frequencies, respectively. \(\rm{\upsigma_{s}^{a}}\), \(\rm{\upsigma_{s}^{e}}\), \(\rm{\upsigma_{p}^{a}}\) and \(\rm{\upsigma_{p}^{e}}\) represent the absorption (a) and emission (e) cross-sections at the signal (s) and pump (p) wavelengths, respectively. The overlap factors at the signal and pump wavelength are, respectively, \(\Gamma_{s}\) and \(\Gamma_{P}\). They can be obtained through using the following equation [3]:  
\[\frac{\mathrm{d}\mathrm{N}_{1}}{\mathrm{d}t}=-\mathrm{R}_{13}\mathrm{N}_{1}+ \mathrm{R}_{31}\mathrm{N}_{3}-\mathrm{W}_{12}\mathrm{N}_{1}+\mathrm{W}_{21} \mathrm{N}_{2}+\mathrm{A}_{21}\mathrm{N}_{2} \tag{2a}\]1  
where b represents the erbium ion E\({}_{\rm r}^{+3}\) doping radius, and w represents the mode radius, which can be calculated using several models.