# Progress and Summary of Photodarkening in Rare Earth Doped Fiber
## 6 Theoretical Study  
The theoretical formula can be a useful tool to study the phenomenon of PD, the most widely used formula is the classical tensile exponential function, which makes the data fit. The formula is as follows [12; 25; 29; 33; 34; 35; 50; 67; 93; 94; 95]. \[\alpha_{\rm PD}(t)=\alpha_{\rm max}[1-\exp(-(t/\tau)^{\beta})] \tag{3}\]  
where \(\alpha_{\rm PD}(t)\) represents the loss induced at a time \(t\) after the pump is switched on, \(\alpha\)max the loss at the final equilibrium state, \(\tau^{-1}\) is the time scale, and \(\beta\) is a parameter that ranges between 0 and 1 and is referred to as the stretching parameter. This equation is the most commonly used empirical formula, which can describe the relaxation rates of many physical properties of complex systems such as polymers and glasses, and the fitting result is the same as the exponential attenuation distribution of the experimental results. The time constant \(\tau^{-1}\) is determined as a benchmarking method for PD rate comparison. [PERSON] et al. [25] used a double exponential function to fit the equation as follows. \[-\Delta\alpha(t)=\alpha_{0}+\alpha_{1}\cdot\exp(-t/\tau_{1})+\alpha_{2}\cdot \exp(-t/\tau_{2}) \tag{4}\]  
The constants \(\alpha_{0}\) are saturation parameters; \(\alpha_{1}\) and \(\alpha_{2}\) are amplitude constants; \(\tau_{1}\) and \(\tau_{2}\) are PD time constants of fast exponential decay and slow exponential decay, respectively. The normalized probe transmission intensities could be fitted with a stretched-exponential function of the form [3; 7; 18; 27]  
\[T(t)=A\cdot\exp[-(t/\tau)^{\beta}]+(1-A) \tag{5}\]  
where \(T\) is the normalized transmission, \((1-A)\) is a steady-state transmission of a fit, \(\tau\) is a characteristic time constant for a fit, and \(\beta\) is a dispersive parameter in a range between 0 and 1. The fitting of the curves was done using the least square method. [PERSON] et al. [29; 94] described the conversion of the PD precursor \(A\) to the color center \(B\) by optical pumping. The transition from a precursor \(A\) to a chromocentric \(B\) is described by reversible first-order reaction at the micro-level. Based on the experimental phenomena of TB, they also described the non-absorbent state \(C\) as the intermediate state of TB. \[A\xrightarrow[k_{-}]{k_{+}}B\xrightarrow[k_{T}]{k_{T}}C\xrightarrow[k_{R}]{k _{R}}A \tag{6}\]  
with the transition rates \(k_{+}\) and \(k_{-}\) of excitation and de-excitation of color centers, respectively. \(k_{T}\) is the TB rate and \(k_{R}\) is the recovery rate. Rate constant \(\tau^{-1}=k_{-}+k_{+}+k_{T}+k_{R}\)  
From the reactions (6), it is assumed that \(C_{A}(t)+C_{B}(t)+C_{C}(t)\equiv C_{A}(0)+C_{B}(0)+C_{C}(0)\) is a constant, hence, the whole process can be described by the following two linear rate equations  
\[\frac{dC_{A}}{dt}=k_{R}C_{A0}-(k_{+}+k_{R})C_{A}+(k_{-}-k_{R})C_{B} \tag{7}\]  
\[\frac{dC_{B}}{dt}=k_{+}C_{A}-(k_{-}+k_{T})C_{B} \tag{8}\]\(C_{A}\), \(C_{B}\), and \(C_{C}\) are the densities of precursor \(A\), color center \(B\), and nonabsorbable State \(C\), respectively. With the special initial conditions \(C_{A}(0)=C_{A0}\) and \(C_{B}(0)=C_{C}(0)=0\), the explicit solution for the density of color centers normalized to the initial density of precursors is  
\[\frac{C_{B}(t)}{C_{A0}}=\frac{k_{+}}{2 GW}[Z(e^{-\beta t}-e^{-\alpha t})+k_{R}W( 2-(e^{-\beta t}+e^{-\alpha t}))] \tag{9}\]  
with the abbreviations \(G=k_{R}(k_{-}+k_{T}+k_{+})+k_{+}k_{T}\), \(Z=k_{R}(k_{-}+k_{T}+k_{+})+2k_{+}k_{T}+k_{R}^{2}\), \(W=\sqrt{S^{2}-4G}\), \(S=k_{-}+k_{+}+k_{R}\), \(\alpha=(S+W)/2\) and \(\beta=(S-W)/2\). The quantity \(C_{B}(t)/C_{A0}\) is considered to be proportional to the PD loss. According to Formula (9), the first term in square brackets describes the rapid rise (with \(\alpha\)) of PD loss after the pump is turned on and slow decline (with \(\beta\)) after passing the maximum. The second term describes the conversion of intermediate state \(C\) to precursor \(A\).