# Progress and Summary of Photodarkening in Rare Earth Doped Fiber
## 6 Theoretical Study
When \(t\) tends to infinity, the real equilibrium value (\(dC_{B}/dt=0\)) will be achieved and depends on the discussed transition rates only,  
\[\frac{C_{B}^{eq}}{C_{A0}}=\frac{k_{+}k_{R}}{k_{R}(k_{-}+k_{T}+k_{+})+k_{+}k_{ T}} \tag{10}\]  
According to Formula (10), when \(k_{R}=0\), all color centers would vanish after longtime pumping. [PERSON] [34] showed that the color centers were generated by irradiation of different wavelengths, but they could also be eliminated by optical or thermal treatment. PD irradiated by a single pump was also a process of simultaneous generation and annihilation of color centers, resulting in a balanced loss of \(\alpha_{eq}\) after a long time constant power pump. They proposed that the probability of a complex-forming a color center was \(p\). The measurable PD loss was caused by the absorption of these color centers. The temporal development of this loss was determined by the statistical distribution of the transition rates and the pump power (\(P\)). Furthermore, the PD effect during pump light irradiation was understood as a reversible process of activation (creation) and deactivation (destruction) of color centers. Therefore, we use the activation (formation) rate \(k_{+}\)(\(s\),\(P\)) \(\geq 0\), the deactivation (erasure) rate \(k_{-}\)(\(s\),\(P\)) \(\geq 0\), and the master equation \(\delta p/\delta t=k_{+}(1-p)-k_{-}p\) to determine the temporal evolution of the PD state in a relaxation process. This results in [34]  
\[p(t,s,P)=p_{eq}+\big{\{}p_{0}(s)-p_{eq}\big{\}}e^{-\kappa(s,P)t} \tag{11}\]  
\[p_{eq}=\frac{k_{+}(s,P)}{k_{+}(s,P)+k_{-}(s,P)} \tag{12}\]  
\[-\Delta\alpha(t)=\alpha_{0}+\alpha_{1}\cdot\exp(-t/\tau_{1})+\alpha_{2}\cdot \exp(-t/\tau_{2}) \tag{4}\]0  
Here \(0\leq p_{0}(s)\leq 1\) is the initial state at \(t=0\). \(\kappa(s\),\(P)\) is the relaxation (transition) rate, depending on the \"speed\" parameter \(s\) and the pump power \(P\). The final state, \(p_{eq}=p(t\)\(\rightarrow\)\(\infty\), \(s\), \(P\)), is the attractor of the relaxation process. It represents the balance of the activation and deactivation processes. This result allows \(p_{eq}\) depending on both \(s\) and \(P\). [PERSON] et al. [96] proposed a formula to describe the time-dependent power loss along the fiber caused by competing PD and PB processes. \[-\Delta\alpha(t)=\alpha_{0}+\alpha_{1}\cdot\exp(-t/\tau_{1})+\alpha_{2}\cdot \exp(-t/\tau_{2}) \tag{4}\]1  
where \(\alpha_{s}\) and \(\alpha_{E}\) are intrinsic absorption and additional absorption, \(P_{s}\) is the signal power distribution along the fiber, \(t\) is time, and \(Z\) is the ordinate in the optical fiber. This formula describes the influence of intrinsic absorption \(\alpha_{s}\) and additional absorption \(\alpha_{E}\) causedby PD on the propagation power in the fiber. time-dependent excess loss coefficient is written as  
\[-\Delta\alpha(t)=\alpha_{0}+\alpha_{1}\cdot\exp(-t/\tau_{1})+\alpha_{2}\cdot \exp(-t/\tau_{2}) \tag{4}\]2  
where \(\sigma_{c}\) is the absorption cross-section of the color center at the signal wavelength, \(N_{c}\) is the color center density, \(\psi_{s}\) is the normalized signal profile, and \(\alpha\) is the core radius. In 2003, he proposed a formula to describe the change of color center density when PD and PB were carried out simultaneously using the form of the stretch index with a time-dependent rate coefficient [97]. \[-\Delta\alpha(t)=\alpha_{0}+\alpha_{1}\cdot\exp(-t/\tau_{1})+\alpha_{2}\cdot \exp(-t/\tau_{2}) \tag{4}\]3  
where \(\beta\) and \(\kappa\) are respectively the dispersion parameters for PD and PB, \(\
u_{d}\) and \(\
u_{b}\) are the PD and PB rate coefficients, and \(N_{\rm C,max}\) is the maximum color center density.