## 2 RSOA Direct Modulation Assisted by MRR-Based Notch Filtering
### Modeling  
The performance of the setup in Figure 1 is characterized by the power that emerges at the output of the directly modulated RSOA, \(P_{RSOA}(t)\), and of the cascaded MRR-based notch filter, \(P_{MRR}(t)\). This means that in order to simulate the operation of the scheme, it is necessary to know and calculate at each one of these points the lightwave-encoded signal electric fields, since they are normalized so that their squared modulus represents power, that is, \(P_{RSOA,MRR}(t)=\left|E_{RSOA,MRR}(t)\right|^{2}\)[32]. For the RSOA case  
\[E_{RSOA}(t)=\sqrt{P_{CW}}\exp\left[(1-j\alpha_{LEF})h(t-2 Ln_{g}/c)\right], \tag{1}\]  
where \(\alpha_{LEF}\) is the RSOA linewidth enhancement factor, \(n_{g}\) is the group refractive index of the semiconductor material, \(c\) is the speed of light in vacuum, and the RSOA gain response integrated over its length is denoted by \(h(t)\), which is shifted in time by twice the RSOA one-way transit time so as to account for the double pass taken by the lightwave signal within the RSOA optical cavity. Now, function \(h(t)\) obeys the following one-dimensional differential equation [27]  
\[\frac{dh(t)}{dt}=-\frac{h(t)-\Gamma aN_{0}L\Bigg{[}\frac{I(t)}{I_{0}}-1\Bigg{]} }{T_{car}}-\frac{\exp\left[2h(t)\right]-1}{E_{sat}}P_{CW}, \tag{2}\]  
Figure 1: Conceptual view of reflective semiconductor amplifier (RSOA) direct modulation assisted by serially connected microring resonator (MRR). FSR: free spectral range, RF: radio frequency. where \(\Gamma\) is the RSOA confinement factor, \(a\) is the RSOA differential gain, \(N_{0}\) is the RSOA carrier density at transparency, \(T_{car}\) is the RSOA carrier lifetime, \(I_{0}=qALN_{0}/T_{car}\) is the RSOA current at transparency (where \(q\) is the electron charge and \(A\) is the area of the semiconductor active region), and \(E_{sat}\) is the RSOA saturation energy. Equation (2) has been obtained after making the following valid assumptions and simplifications: (a) The RSOA internal losses are neglected, as they can be well compensated by the RSOA CW gain, \(G_{CW}\), through the adjustment of the RSOA bias current, since these two parameters are directly related [27]; (b) the RSOA reflectivity is considered perfect, that is, 100%, as in practice this is a basic feature of the specific devices that favors their use [33], in particular as intensity modulators [10]; and (c) the RSOA round-trip propagation time, which is determined by the RSOA active region length, is smaller than the pulse repetition interval of the applied excitation. This condition is satisfied for the direct modulation rates pursued in RSOAs being less than 1 mm long [34]. The profile of each pulse inside the \(N=2^{7}-1\) bit-long NRZ pseudorandom binary sequence (PRBS) of the injection current is described in compact form [35; 36] as the sum of a DC term and a modulation term  
\[I(t)=\overbrace{(I_{bias}-I_{m})}^{\text{DC term}}+\overbrace{2I_{m}\left\{ H(t)\left[1-\mathrm{e}^{-t^{2}/t_{T}^{2}}\right]-H(t-T)\left[1-\mathrm{e}^{-(t-T)^{2}/t_{ T}^{2}}\right]\right\}}^{\Delta\left\{\left\{\left\{\left\{\left\{\left\{\left\}{ \right\}\right\}\right\}\right\}\right\}\right\}\text{,}} \tag{3}\]  
where \(H(t)\) is Heaviside step function, \(I_{bias}\) and \(I_{m}\) are the RSOA bias current and induced RF modulation current, respectively-which are a small fraction of the current at transparency-and \(t_{r}\) is the pulse rise time, which occupies a small portion of the pulse repetition period, \(T\). For the MRR case, obtaining the electric field at its output involves the following steps. First, the RSOA temporal response is extracted by numerically solving for \(h(t)\) from (2). This is done by following the numerical method formulated in [37] but adapted to account for a temporal-dependent total injection current [38].