## 2 Numerical Simulator
The lower power limit of amplifying signals in SOAs is determined by the NF (or effective NF for the case of multi-section SOAs) and mainly independent of the SOA gain. For specific parameter selection, we aim to replicate SOA saturation characteristics of an actual multi-section SOA [10]. We set both (multi-section and single section) SOAs to have a total unsaturated gain of 13 dB. For the single section SOA: \(h_{0}=6\), \(\alpha_{loss}=3\), this implies that the unsaturated power gain is  
\[\int\limits_{-\infty}^{\infty}n_{RX}(\tau)n_{RX}(\tau+t)d\tau=\frac{4 kT}{R_{ RX}}\delta(t). \tag{2}\]1  
The NF is large at 8 dB, \(\tau_{S}=200\) ps, \(P_{sat}=20\) mW. Note, that for the single section SOA, only a single differential equation of the type in (4) is solved. The breakdown of the parameters for each section in the multi-section SOA is given in \[\int\limits_{-\infty}^{\infty}n_{RX}(\tau)n_{RX}(\tau+t)d\tau=\frac{4 kT}{R_{ RX}}\delta(t). \tag{2}\]2 We assume that that the biasing arrangement allows for the first section to have modest gain with a small NF of 4 dB, and the third section to have the largest gain with a large NF of 9 dB. The multi-section SOA has an overall net gain of  
\[\int\limits_{-\infty}^{\infty}n_{RX}(\tau)n_{RX}(\tau+t)d\tau=\frac{4 kT}{R_{ RX}}\delta(t). \tag{2}\]3  
The saturation power \(P_{sat}\) is not a unique quantity to the SOA though rather the saturation energy which is the product \(P_{sat}\tau_{S}\) remains constant [17]. Under strong bias currents, the carrier density increases and the carrier lifetime \(\tau_{S}\) falls due to increased bimolecular and Auger recombination, therefore we adjust the saturation powers such that the product \(P_{sat}\tau_{S}=4\) pJ for the single-section SOA and for all subsections of the multi-section SOA. The SOA gain saturation provides much insight into the improvements offered by the multi-section SOA. We will calculate the SOA gain-output power relations for both SOAs. We create continuous wave fields without any modulation and calculate the steady-state optical power gain. Note that stochastic effects are avoided by setting the NF = \(-50\) dB in order to get clear and precise gain calculations. The results are shown in \[\int\limits_{-\infty}^{\infty}n_{RX}(\tau)n_{RX}(\tau+t)d\tau=\frac{4 kT}{R_{ RX}}\delta(t). \tag{2}\]4 For low output powers below \(-5\) dBm the unsaturated gain for both SOAs is 13 dB. The SOA gain falls as the output increases and we note that \(P_{\text{3 dB}}\) is the output power at which the gain has fallen from the unsaturated value by 3 dB. The single section SOA has a \(P_{\text{3 dB}}\) of 8.28 dBm, whereas the \(P_{\text{3 dB}}\) for the multi-section SOA is 10.74 dBm. We also plot on Figure 5 the gain saturation performance when the biasing arrangements are swapped between the first and third sections, the output saturation power reduces to 7.1 dBm. Such a biasing arrangement might be favourable for enhancing the nonlinear behaviour of the SOA and could be exploited for all-optical signal processing purposes [19], though amplification using this configuration is not explored here. Experimental results taken from a multi-section SOA show a \(P_{\text{3 dB}}\) of 9.3 dBm for the high saturation power configuration and a \(P_{\text{3 dB}}\) of 6.3 dBm for the swapped biasing arrangement [10]. We next examine how the SOAs perform when amplifying PAM4 signals. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Multi Section SOA,** & **Multi Section SOA,** & **Multi Section SOA,** & **Single Section** \\  & **Section 1** & **Section 2** & **Section 3** & **SOA** \\ \hline \(h_{0}\) & 1.5 & 2 & 2.5 & 6 \\ \(\alpha_{loss}\) & 1 & 1 & 1 & 3 \\ \(\alpha_{H}\) & 3 & 3 & 3 & 3 \\ \(P_{sat}\) & 15 mW & 20 mW & 45 mW & 20 mW \\ \(\tau_{S}\) & 260 ps & 200 ps & 88 ps & 200 ps \\ NF & 4 dB & 8 dB & 9 dB & 8 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of SOA parameters.