## 3 Modeling
In the context of this research, we adopt the assumption that the input optical power (P\({}_{\text{in}}\)(t)) for data streams A, B, and CLK is represented by Gaussian-shaped RZ pulses, as outlined in previous studies [8; 9; 13; 18]:  
\[P_{A,\;B,\;CLK}\;(t) \equiv P_{in,\;CR-SOA}(t)=\sum_{n=1}^{K}a_{n(A,\;B,\;CLK)}\;\frac{2 \sqrt{\pi}\;\overline{\tau_{FWHM}}}{\overline{\tau_{FWHM}}}\quad exp\left[- \frac{4\ln[2]\;(t-\;n\;T)^{2}}{\overline{\tau_{FWHM}^{2}}}\right] \tag{5}\]  
Here, \(\alpha_{n(\text{A, B, CLK})}\) denotes the n-th pulse within the A, B, or CLK data streams. K (=\(2^{7}-1\)) refers to the length of a pseudorandom binary sequence (PRBS). It represents the number of bits in the sequence before it repeats or reaches a defined endpoint. Power P\({}_{\text{A}}\)(t) from data A and half of the continuous wave (CW) beam power are merged in the initial stage and introduced into the upper arm of CR-SOAs-MZII. Simultaneously, power P\({}_{\text{B}}\)(t) from data B and the remaining half of the CW beam are combined and injected into the lower arm of CR-SOAs-MZII. Consequently, the input optical powers for the XOR1 function are explicitly defined as per the arrangement described in [8; 18]:  
\[P_{in,\;CR-SOA_{1}}(t) = P_{A}(t)\;\;+\;\;0.5\;P_{CW} \tag{6}\] \[P_{in,\;CR-SOA_{2}}(t) = P_{B}(t)\;\;+\;\;0.5\;P_{CW} \tag{7}\]  
In the configuration of CR-SOAs-MZII2, the power P\({}_{\text{C}}\)(t) obtained from data C is intricately combined with half of the CW beam power, guiding it into the upper arm of CR-SOAs-MZII2. Concurrently, the power P\({}_{\text{D}}\)(t) derived from data D, along with the \begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbol** & **Definition** & **Value** & **Unit** \\ \hline E\({}_{0}\) & Pulse energy & 0.5 & p] \\ T\({}_{\text{FWHM}}\) & Pulse width & 1 & ps \\ T & Bit period & 8.33 & ps \\ K & PRBS length & 127 & - \\ \(\lambda_{\text{A}}\) & Data A wavelength & 1547 & nm \\ \(\lambda_{\text{B}}\) & Data B wavelength & 1540 & nm \\ \(\lambda_{\text{CW}}\) & CW wavelength & 1550 & nm \\ P\({}_{\text{CW}}\) & CW power & 0.3 & mW \\ I & Injection current & 300 & mA \\ P\({}_{\text{sat}}\) & Saturation power & 30 & mW \\ \(\tau_{\text{c}}\) & Carrier lifetime & 200 & ps \\ \(\tau_{\text{t}}\) & Transition lifetime from CR to AR & 5 & ps \\ \(\eta\) & Population inversion factor & 0.3 & - \\ \(\alpha\) & \(\alpha\)-factor & 4 & - \\ \(\alpha_{\text{CH}}\) & CH linewidth enhancement factor & 1 & - \\ \(\alpha_{\text{SHB}}\) & SHB linewidth enhancement factor & 0 & - \\ \(\varepsilon_{\text{CH}}\) & CH nonlinear gain suppression factor & 0.1 & W\({}^{-1}\) \\ \(\varepsilon_{\text{SHB}}\) & SHB nonlinear gain suppression factor & 0.1 & W\({}^{-1}\) \\ \(\tau_{\text{CH}}\) & Temperature relaxation rate & 0.3 & ps \\ \(\tau_{\text{SHB}}\) & Carrier-carrier scattering rate & 0.1 & ps \\ \(\Gamma\) & Confinement factor & 0.3 & - \\ \(\alpha\) & Differential gain & 10\({}^{-20}\) & m\({}^{2}\) \\ N\({}_{\text{tr}}\) & Transparency carrier density & 10\({}^{24}\) & m\({}^{-3}\) \\ L & AR length & 500 & \(\mu\)m \\ d & AR thickness & 0.3 & \(\mu\)m \\ w & AR width & 3 & \(\mu\)m \\ G\({}_{0}\) & Unsaturated power gain & 30 & dB \\ N\({}_{\text{SP}}\) & Spontaneous emission factor & 2 & - \\ u & Optical frequency & 193.55 & THz \\ B\({}_{0}\) & Optical bandwidth & 2 & nm \\ \(\hbar\) & Reduced Planck’s constant & 1.05 \(\times\) 10\({}^{-34}\) & J-s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Default values for numerical parameters [7; 8; 9; 13; 18]. remaining half of the CW beam, is seamlessly merged and introduced into the lower arm of CR-SOAs-MZI2.