## 3 Modeling  
The intricate dynamics governing carrier reservoirs in CR-SOAs are aptly encapsulated by a set of coupled differential equations. These equations intricately depict the nuanced interactions between carriers, photons, and other relevant parameters. Specifically, they account for the nonlinear intraband effects of spectral hole burning (SHB) and carrier heating (CH) [7; 8; 9]:  
\[\frac{dh_{AR}(t)}{dt}\ =\ \frac{h_{CR}(t)-h_{AR}(t)}{\tau_{t}(1+\eta)}\ \ +\ \ \frac{\eta h_{0}}{\tau_{c}(1+\eta)}\ \ -\ \ \frac{h_{AR}(t)}{\tau_{c}}\ -\ (exp\left[h_{AR}(t)\ \ +\ h_{CH}(t)\ \ +\ h_{SHB}(t)\right]\ \ -\ \ 1)\ \ \frac{P_{in}(t)}{E_{sat}} \tag{1}\]  
\[\frac{dh_{CR}(t)}{dt}\ \ =\ \ -\ \frac{\eta(h_{CR}(t)-h_{AR}(t)}{\tau_{t}(1+\eta)} \ +\ \ \frac{h_{0}-h_{CR}(t)}{\tau_{c}(1+\eta)}\ \ -\ \ \frac{h_{CR}(t)}{\tau_{c}} \tag{2}\]  
\[\frac{dh_{CH}(t)}{dt}\ \ =\ \ -\ \frac{h_{CH}(t)}{\tau_{CH}}\ \ -\ \ \frac{\varepsilon_{CH}}{\tau_{CH}}(exp\left[h_{AR}(t)\ \ +\ h_{CH}(t)\ \ +\ h_{SHB}(t)\right]\ \ -\ \ 1)\ P_{in}(t) \tag{3}\]  
\[\frac{dh_{SHB}(t)}{dt}\ \ =\ \ -\ \frac{h_{SHB}(t)}{\tau_{SHB}}\ \ -\ \frac{ \varepsilon_{SHB}}{\tau_{SHB}}(exp\left[h_{AR}\left(t\right)\ \ +\ h_{CH}(t)\ \ +\ h_{SHB}(t)\right]\ \ -\ \ 1)\ P_{in}(t)\ \ -\ \ \frac{dh_{AR}(t)}{dt}\ \ -\ \ \frac{dh_{CH}(t)}{dt} \tag{4}\]  
The functions \(h_{\text{AR}}\), \(h_{\text{CR}}\), \(h_{\text{CH}}\), and \(h_{\text{SHB}}\) represent the CR-SOA's gain integrated across its length concerning carrier recombination between AR and CR, carrier heating (CH), and spectral hole burning (SHB). The factor (\(\eta\)), defined as \(\text{N}_{\text{AR}}/\text{N}_{\text{CR}}\), signifies the population inversion factor of the CR-SOA. Additionally, \(\text{G}_{0}=\exp[h_{0}]=\alpha\Gamma(\tau_{c}/\text{eV}-\text{N}_{\text{tr}})L\) denotes the unsaturated power gain of the CR-SOA, while \((\text{E}_{\text{sat}})=\text{P}_{\text{sat}}\ \tau_{c}=\text{wd}\hbar\omega_{0}/\alpha\Gamma\) characterizes its saturation energy [7; 8; 9]. The information on these parameters, including their respective values and definitions, is provided in Table 2 [7; 8; 9; 13; 18]. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & & & & & & \\ \hline
**Case** & **A** & **B** & **C** & **D** & \(\text{P}_{\text{e}}^{\ast}\) & **Ce** & \(\text{P}_{\text{o}}^{\ast}\) & **Co** & **Result** \\ \hline
**1** & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \\ \hline
**2** & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & \\ \hline
**3** & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & **Error** \\ \hline
**4** & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & \\ \hline
**5** & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & \\ \hline
**6** & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & **Error** \\ \hline
**7** & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & \\ \hline
**8** & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & \\ \hline
**9** & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & **Error** \\ \hline
**10** & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \\ \hline
**11** & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ \hline
**12** & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & \\ \hline
**13** & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ \hline
**14** & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ \hline
**15** & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ \hline
**16** & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Truth table for 4-bit word parity checking. In the context of optical communication or signal processing, using intensity-modulated return-to-zero (RZ) data format with a Gaussian form helps optimize certain aspects of the signal, such as bandwidth efficiency and signal-to-noise ratio. The Gaussian shape contributes to a controlled rise and fall time of the pulse, minimizing potential distortions and facilitating efficient transmission and detection of information [21; 22].