## 2 Numerical Simulator  
We create an optical field, \(E\), that is normalised such that \(|E|^{2}\) represents the optical power. The four level 28 Gbaud PAM4 signal is encoded within the optical power domain. We set the extinction ratio between the strongest and weakest levels of the PAM4 signal to be 8 dB. Before describing the SOA model we first describe the linear photo-detector model as depicted in Figure 3a. The received photo-current is given as  
\[i_{RX}=R|E_{rec}(t)|^{2} \tag{1}\]  
where \(E_{rec}\) is the optical field incident on the photodiode, the detector responsivity \(R\) = 1 A/W throughout this paper. In the detector, electrical thermal noise is added to the current generated by the signal, the thermal noise \(n_{RX}\) is considered to be white Gaussian noise with autocorrelation [16]  
\[\int\limits_{-\infty}^{\infty}n_{RX}(\tau)n_{RX}(\tau+t)d\tau=\frac{4 kT}{R_{ RX}}\delta(t). \tag{2}\]  
With \(k\) being Boltzmann's constant, \(T\) (K) the temperature, and \(R_{RX}\) = 50 \(\Omega\) is the impedance of the receiver, and \(\delta\) is the Dirac delta function. The combined signal and noise is low-pass filtered and then a decision is made on the transmitted PAM4 symbol. Error counting after comparing the received and transmitted bits determines the BER. In the simulator the current waveform is generated by generating a random waveform with values taken from a zero-mean, unity variance Gaussian distribution,  
\[\overline{n}_{RX}=\sqrt{\frac{4 kT}{R_{RX}}B_{sim}}e_{dec}. \tag{3}\]  
\(e_{dec}\) is a random noise waveform whose values are taken from a zero-mean, unity-variance Gaussian distribution, and \(B_{sim}\) is the simulation bandwidth (i.e., inverse sampling time) and equals 1 THz throughout this paper. A different value for \(e_{dec}\) is generated for each sampling point in the waveform for the PAM4 signal \(E\). Figure 2: (a) Simplified schematic of a travelling wave single-section SOA where the population inversion is created by biasing the whole amplifier with a single current; (b) Multi-section SOA, the bias currents to the smaller amplifier sections are independently set. The multi-section biasing arrangement creates an SOA with improved noise and gain saturation performances compared to the case in (a). The amount of thermal noise remains fixed and the BER changes by changing the received signal power \(\left|E_{rec}\right|^{2}\). We calculate the BER for different values of received optical signal power with the results shown in Figure 4. We indicate the sensitivity to achieve a BER of \(3.16\times 10^{-4}\) because this is the BER at which the forward error correction algorithms can effectively yield error-free performance [2]. The received power that attains a BER of \(3.16\times 10^{-4}\) is estimated to be \(-11.4\) dBm, which is in line with an experimentally obtained sensitivity of \(-10.8\) dBm [15], and within an acceptable range for receiver sensitivity. Reducing the responsivity of the photodiode to \(0.7\) would add \(+1.5\) dB to our calculated receiver sensitivity. The detector model is central to all of our simulations and we use the detector model after amplifying the signal in an SOA. The simulation models for the SOAs are shown in Figure 3b,c. Typically, SOAs are travelling wave models and consideration for the spatially-distributed gain should be considered, though in the absence of internal scattering losses the SOA can be equivalently treated as a lumped device [17] without needing to consider the propagation of the optical field in the SOA. The lumped SOA model approach  
Figure 4: Receiver bit-error rate (BER) performance for 28 Gbaud PAM4. The ER is \(7.6\) dB. The sensitivity at BER \(3\times 10^{-4}\) occurs at \(-11.8\) dBm. Figure 3: Simulation scenarios performed in this submission. (**a**) Linear receiver with PIN photodiode including thermal noise. LPF: Lowpass filter; (**b**) Pre-amplified PAM4 signal using an SOA.