# Design and Analysis of an O+E-Band Hybrid Optical Amplifier for CWDM Systems
## 2 Theoretical Background  
### Spectroscopic Properties and Rate Equations of Pr\({}^{3+}\)  
Figure 1 exhibits the absorption and emission cross-sections and the simplified energy level diagram of Pr\({}^{3+}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Study** & **Technology** & **Operating Band** & **Gain Bandwidth** & **Gain Flatness** & **NF** \\ \hline
[13] & HBEDF-ZEDF & C+L & 70 nm & 1.8 dB & 4.3–7.9 dB \\
[18] & SOA-EDFA-Raman & L & 9 nm & 0.8 dB & 5–5.9 dB \\
[19] & EYDWA-Raman & C & 23 dB & 2.78 dB & 5–6 dB \\
[20] & EYDFA-Raman & C & 20 nm & 1.37 dB & 5–5.6 dB \\
[21] & Raman-EDFA & C+L & 70 nm & - & 3 dB \\
[22] & HDFA-TDFA & 2000 nm & 89 nm & - & 6.5–7.5 dB \\
[24] & BDFA-EDFA & E+S & 90 nm & - & 7 dB \\
[25] & EDFA-OPA & S & 60 nm & 1.2 dB & 5.1–6.1 dB \\
[26] & EDFA-Raman & C+L & - & 1.9 dB & - \\ Proposed & PDFA-SOA & O+E & 180 nm & 1 dB & 4–5.9 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Literature review of HOAs. It is evident from Figure 1a that Pr\({}^{3+}\) has two pump absorption bands with their peaks centered at 1010 nm and 1400 nm. The pump wavelengths in the 1000-1040 nm wavelength range are widely used to excite the Pr\({}^{3+}\)[6,27]. The emission starts at 1220 nm but the emission cross-section is maximum at around 1300 nm. The simplified energy level diagram of Pr\({}^{3+}\) based on four-level absorption and radiative transitions is shown in Figure 1b. It can be observed that the main energy levels are labeled as \({}^{1}\)G\({}_{4}\), \({}^{3}\)P\({}_{0}\), \({}^{1}\)D\({}_{0}\), \({}^{3}\)F\({}_{4}\), \({}^{3}\)F\({}_{3}\), \({}^{3}\)H\({}_{5}\), and \({}^{3}\)H\({}_{4}\)[29]. The \({}^{1}\)G\({}_{4}\leftrightarrow{}^{3}\)H\({}_{4}\) transitions enable the pump ground state absorption (GSA) and emission. Similarly, the \({}^{3}\)H\({}_{4}\rightarrow{}^{3}\)F\({}_{4}\) and \({}^{1}\)G\({}_{4}\rightarrow{}^{3}\)H\({}_{5}\) transitions hold the signal GSA and signal emission. Moreover, the carrier density of the \({}^{1}\)G\({}_{4}\) level can be decreased by the up-conversion effect due to the \({}^{1}\)G\({}_{4}\rightarrow{}^{1}\)D\({}_{2}\) and \({}^{1}\)G\({}_{4}\rightarrow{}^{3}\)H\({}_{5}\) transitions [29] because the energy difference between the \({}^{1}\)G\({}_{4}\) and \({}^{1}\)D\({}_{2}\) levels is equal to the energy difference between the \({}^{1}\)G\({}_{4}\) and \({}^{3}\)H\({}_{5}\) levels [6,27,29]. The carrier densities at each level are represented by \(n_{1}\), \(n_{2}\), \(n_{3}\), \(n_{4}\), and \(n_{5}\) and the total density \(n_{t}\) is given by [27]:  
\[n_{t}=n_{1}+n_{2}+n_{3}+n_{4}+n_{5} \tag{1}\]  
The rate equations for the energy level diagram of Pr\({}^{3+}\) shown in Figure 1b can be written as [27]:  
\[\frac{dn_{3}}{dt}=\gamma_{13}n_{1}-\left(\gamma_{35}+\gamma_{34}+\gamma_{32}+ \gamma_{31}+\frac{1}{\tau_{3}}+cn_{3}\right)n_{3}+\frac{B_{43}}{\tau_{4}}n_{4 }+\frac{B_{53}}{\tau_{5}}n_{5} \tag{2}\]  
\[\frac{dn_{4}}{dt}=\left(\gamma_{35}+\frac{c}{2}n_{3}\right)n_{3}-\frac{n_{4}} {\tau_{4}} \tag{3}\]  
\[\frac{dn_{5}}{dt}=\gamma_{35}n_{3}-\frac{n_{5}}{\tau_{5}} \tag{4}\]  
In the above expressions, the transition rates \(\gamma_{13}\), \(\gamma_{31}\), \(\gamma_{32}\), \(\gamma_{34}\), and \(\gamma_{35}\) are given by \(\frac{P_{\rm e}^{2}\gamma_{3}\eta_{\rm g}}{A_{\rm c}\hbar n_{\rm p}}\), \(\frac{P_{\rm e}\gamma_{3}\eta_{\rm g}}{A_{\rm c}\hbar n_{\rm p}}\), \(\frac{P_{\rm e}\gamma_{3}\eta_{\rm g}}{A_{\rm c}\hbar n_{\rm s}}\), \(\frac{P_{\rm e}\gamma_{3}\eta_{\rm g}}{A_{\rm c}\hbar n_{\rm s}}\), and \(\frac{P_{\rm e}\gamma_{3}\eta_{\rm g}}{A_{\rm c}\hbar n_{\rm s}}\), respectively. Similarly, \(B_{53}\) and \(B_{43}\) are the branching ratios for the \({}^{3}\)P\({}_{0}\)-\({}^{1}\)G\({}_{4}\) and \({}^{1}\)D\({}_{2}\)-\({}^{1}\)G\({}_{4}\) transitions, which are evaluated by the Judd-Ofelt analysis as having numerical values of 2 % and 9 %, respectively.