# Design and Analysis of an O+E-Band Hybrid Optical Amplifier for CWDM Systems
## 2 Theoretical Background
### Spectroscopic Properties and Rate Equations of Pr\({}^{3+}\)
The small-signal gain of the PDFA completely depends upon the transitions between the \({}^{3}\)H\({}_{4}\) and \({}^{1}\)G\({}_{4}\) levels having carrier densities of n\({}_{1}\) and n\({}_{3}\), respectively. When a signal passes through the gain medium of the PDFA having a thickness of d\(\chi\), then the propagation equation of the signal can be written as [27,29]:  
\[\frac{dP_{\rm s}}{dz}=\left[n_{3}(\sigma_{32}-\sigma_{34})-n_{1}\sigma_{13} \right]P_{\rm s}\Gamma_{s} \tag{5}\]  
Figure 1: (a) Pr\({}^{3+}\) absorption and emission spectra. (b) Energy level diagram. In the above expression, \(\sigma_{34}\) is the excited state absorption (ESA) cross-section at the signal wavelength, which is not considered by OptiSystem. Therefore, the small-signal gain achieved is given by [27; 29]:  
\[G=\frac{P_{s}(L)}{P_{s}(0)}=exp\Big{[}L\Gamma_{s}\big{\{}n_{3}(\sigma_{32}- \sigma_{34})-n_{1}\sigma_{13}\big{\}}\Big{]}, \tag{6}\]  
where \(L\) is the length of the PDF. The OptiSystem model of traveling-wave SOA performs lumped amplification, which is applicable to describe the amplification of both the CW and the optical pulsed signals. The coefficient of material gain \(g_{m}\) and carrier density \(N(t)\) are interrelated by the expression [30]  
\[g_{m}(t)=A_{g}\Big{[}N(t)-N_{o}\Big{]} \tag{7}\]  
Similarly, the net gain coefficient \(g\) and material gain \(g_{m}\) are also related to each other by [30]  
\[g(t)=\Gamma g_{m}(t)-\alpha \tag{8}\]  
Neglecting the group velocity dispersion (GVD) in the SOA and taking into account the amplified spontaneous emission (ASE), the gain \(G\) for a traveling-wave SOA at a distance \(z\) is given by [30]  
\[n_{t}=n_{1}+n_{2}+n_{3}+n_{4}+n_{5} \tag{1}\]0  
The carrier and optical intensities are related by the following rate equation [30]:  
\[n_{t}=n_{1}+n_{2}+n_{3}+n_{4}+n_{5} \tag{1}\]1  
where \(\beta\) is the spontaneous emission coefficient, which characterizes the part of the total spontaneous emission coupled to the guided wave and is given by [30]:  
\[n_{t}=n_{1}+n_{2}+n_{3}+n_{4}+n_{5} \tag{1}\]2  
The recombination rate is generally assumed to be linearly proportional to the carrier density in order to obtain almost accurate results; therefore, \(R(n)=\frac{n}{r}\). Assuming \(\alpha=0\)[30], the steady-state expression becomes  
\[n_{t}=n_{1}+n_{2}+n_{3}+n_{4}+n_{5} \tag{1}\]3  
The various symbols used in Equations (1)-(12) are described in \[n_{t}=n_{1}+n_{2}+n_{3}+n_{4}+n_{5} \tag{1}\]4 \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(n_{1},n_{2}\) & Ion densities at ground and excited manifolds \\ \(n_{t}\) & Total number of ions \\ \(I_{p},I_{s}\) & Intensities of pump and signal \\ \(P_{p},P_{s}\) & Power of pump and signal \\ \(P_{p}(0),P_{p}(z)\) & Power of pump at input and a position z along the PDF \\ \(P_{sat}\) & Pump saturation power \\ \(P_{a}\) & Power absorbed \\ \(R_{ij},W_{ij}\) & Transition rates of pump and signal between the \(i^{th}\) and \(j^{th}\) levels \\ \(A_{ij}\) & Rate of spontaneous emission between the \(i^{th}\) and \(j^{th}\) levels \\ \(hv_{p},hv_{s}\) & Photon energies for pump and signal \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Symbols used in Equations (1)–(12) and their descriptions.