## 2 Numerical Simulator
ASE is added before SOA, the receiver is as in (**a**); (**c**) multi-section SOA, the SOA is split into three separate SOA subsections, ASE noise is added before each SOA subsection. Adjusting the bias can alter the NF and saturation power of each subsection. The detector is as in (**a**). avoids the need to subdivide the SOA into smaller subsections in order to compute the propagation of the optical field. Additionally, the lumped SOA model approach requires that only a single differential equation needs to be numerically solved to estimate the SOA gain. The computational savings by adopting the lumped SOA model approach makes that approach attractive for use within communication system simulators, where tens of thousands of data symbols need to be simulated in order to gauge meaningful system performance. An example of using the lumped SOA model approach for communication system simulators is given in [18]. To address the issue of including non-negligible internal scattering losses we developed an improved lumped SOA model [14] that is accurate to \(\sim\)1 dB and we have used this model to estimate the wavelength conversion performance of optical signals in SOAs [19]. The SOA dynamics are described by a single differential equation for each subsection [14]:  
\[\frac{dh_{i}}{dt}=\frac{h_{0,i}-h_{i}}{\tau_{S}}-\frac{h_{i}}{h_{i}-\alpha_{ loss,i}}\frac{\exp(h_{i}-\alpha_{loss,i})-1}{P_{sat}\tau_{S}}|E_{in}(t)|^{2}. \tag{4}\]  
With \(h\) representing the integrated gain-coefficient over the length of the SOA section, \(h_{0}\) is the unsaturated value of the integrated gain coefficient, \(\tau_{S}\) is the carrier lifetime, \(\alpha_{loss}\) is the integrated scattering losses, \(P_{sat}\) is the saturation optical power at which the gain coefficient reaches half of the unsaturated value, and \(|E_{in}(t)|^{2}\) is the incident optical power to the SOA. The second term on the right hand side of (4) describes stimulated emission with the first term on the right hand side describing gain recovery due a replenishment of the gain via the bias current. The output field from the SOA subsection is  
\[E_{out,i}(t)=E_{in,i}(t)\exp\biggl{(}\frac{1}{2}\{(1-j\alpha_{H})h_{i}(t)- \alpha_{loss,i}\}\biggr{)}. \tag{5}\]  
\(\alpha_{H}\) being the gain-phase coupling parameter. The '\(i\)' index in (4) and (5) refer to a subsection in the multi-section SOA. To account for the ASE generated within the SOA we add an equivalent ASE noise to the optical field before passing the field through the SOA [18]. The equivalent optical ASE noise \(n_{ASE}\) is assumed to be white Gaussian noise with the following autocorrelation:  
\[\int\limits_{-\infty}^{\infty}n_{ASE}(\tau)n_{ASE}^{*}(t+\tau)d\tau=n_{sp}h \upsilon\frac{\exp(h_{0,i}-\alpha_{loss,i})-1}{\exp(h_{0,i}-\alpha_{loss,i})} \delta(t). \tag{6}\]  
With \(h\upsilon\) being the photon energy, \(n_{sp}\) is the gain inversion parameter and is related to the NF by NF = \(\log_{10}(2n_{sp})\). There is a correction factor to ensure that the ASE noise power equals \(n_{sp}h\upsilon(\exp[h_{0,i}-\alpha_{loss,i}]-1)\) at the output [18]. This correction factor is accurate while the SOA gain remains unsaturated, though the ASE noise only affects the performance of low-power input signals up to a few tens of \(\mu\)W. In the numerical simulator we generate two random waveforms with samples taken from two uncorrelated, zero-mean, unity-variance Gaussian distributions \(e_{ASE-I,i}\) and \(e_{ASE-Q,i}\) to represent the in-phase (\(I\)) and quadrature (\(Q\)) components of the optical field  
\[\int\limits_{-\infty}^{\infty}n_{RX}(\tau)n_{RX}(\tau+t)d\tau=\frac{4 kT}{R_{ RX}}\delta(t). \tag{2}\]0  
For the multi-section SOA, the ASE generated within each section is uncorrelated. We consider the SOA as a pre-amplifier, and typically SOA pre-amplifiers have moderate gains of 13 to 15 dB amplification [8; 9; 10; 11] in order to avoid deleterious gain saturation. Selecting a device with a larger gain will only reduce the input optical power at which gain saturation occurs.