## 2 RSOA Direct Modulation Assisted by MRR-Based Notch Filtering
### Modeling
In this manner, the maximum and average values of the peak power of the marks and spaces within the encoded PRBS can be found so that they can be used to calculate the performance metrics employed in the following section. The operation of the directly modulated RSOA, either alone or assisted by the MRR, is also characterized by the respective modulation responses, which are derived through the process of small-signal analysis [43,44]. Starting with the RSOA, we have  
\[T_{RSOA}\overset{\ref{eq:T_RSOA}}{=}\frac{p_{RSOA}(\Omega_{m})}{p_{RSOA}(0)} \overset{\ref{eq:T_RSOA}}{=}\frac{\Delta h(\Omega_{m})}{\Delta h(0)}\overset{ \ref{eq:T_RSOA}}{=}\frac{\Delta g(\Omega_{m})D(0)}{\Delta g(0)D(\Omega_{m})}, \tag{5}\]  
where the expressions that follow after steps 1, 2 and 3 have been derived by combining relevant information from [18,27] and [45] (c.f. Appendix). More specifically, in 1, \(p_{RSOA}(\Omega_{m})\) is the small-signal power, which is produced at the RSOA output due to the modulation of its current at a given frequency, \(f_{m}=\Omega_{m}/2\pi\), and normalized over its unmodulated counterpart [46]; in 2, \(\Delta h(\Omega_{m})\) is the concomitant small-signal deviation from steady state of the RSOA-integrated gain given by [27]\(\Delta h(\Omega_{m})=[L/D(\Omega_{m})|\Delta g(\Omega_{m})\), where \(\Delta g(\Omega_{m})=TaN_{0}\Delta I(\Omega_{m})/I_{0}\) is the corresponding gain coefficient perturbation incurred by the modulation current of complex envelope \(\Delta I(\Omega_{m})\); and in 3, \(D(\Omega_{m})=1-j\Omega_{m}T_{car}+W\bigg{[}\frac{2T_{car}P_{CW}}{E_{sat}} \exp\left(2g_{0}L+\frac{2T_{car}P_{CW}}{E_{sat}}\right)\bigg{]}\)[27], where \(W[\cdot]\) is [PERSON]'s 'W' function [47]. Function \(D(\Omega_{m})\) links in a compact manner the perturbation of the RSOA-integrated gain to the current modulation from which it has been incurred. Thus, when the RSOA is directly modulated by a sinusoidal electrical excitation, it allows, through first-order approximations, one to analytically express the 3 dB angular frequency of the RSOA modulation response as [27]\(\Omega_{3 dB}=\frac{\sqrt{3}}{T_{car}}\Bigg{\{}1+W\bigg{[}\frac{2T_{car}P_{CW}}{E_{ sat}}\exp\left(2g_{0}L+\frac{2T_{car}P_{CW}}{E_{sat}}\right)\bigg{]}\Bigg{\}}\). This formula provides an estimate of the maximum RSOA modulation frequency and hence of its direct modulation capability at a given data rate [28]. In general, however, the modulation current may be in digital form and have any shape. Then, the 3-dB modulation bandwidth can be derived from (5) by taking the Fourier transform of the modulation current in the time domain, which in our case is given by the second term in the right-hand side of (3). By using Fourier transform formulas and properties of the functions involved therein, and after tight algebraic manipulations, in the course of which the complementary error function and the imaginary error function come into being, we find the explicit expression  
\[ \begin{split}&\Delta I(\Omega_{m})=\mathcal{F}[\Delta I(t)]=\\ &=2I_{m}\Bigg{\{}\frac{1}{\Omega_{m}}\big{[}1-\exp(-j\Omega_{m} T)\big{]}+\big{[}\exp(-j\Omega_{m}T)-1\big{]}\bigg{[}0.5\sqrt{\pi}t_{r}\exp \big{(}-\Omega_{m}^{2}t_{r}^{2}/4\big{)}-it_{r}\mathbb{F}(\Omega_{m}t_{r}/2) \bigg{]}\Bigg{\}},\end{split}  \tag{6}\]where function \(\mathbb{F}(x)=\exp(-x^{2})\int\limits_{0}^{x}\exp(t^{2})dt\) represents [PERSON]'s integral [PERSON] (1958), which is available in Matlab software. Taking the squared modulus and after proper algebraic manipulations, (6) can be brought into a form that allows one to approximate its values for the boundary conditions of the modulation frequency [46]. The obtained result is analogous to the square of the product of the electrical modulation pulses' peak amplitudes and period when \(\Omega_{m}\to 0\), while it tends to null when \(\Omega_{m}\rightarrow\infty\). In other words, the level of the specific function in these two limits is in the opposite direction to that of the modulation frequency variation.