## 2 RSOA Direct Modulation Assisted by MRR-Based Notch Filtering
### Modeling
This fact confirms that the RSOA modulation response exhibits a low-pass characteristic, which is in agreement with the same trend noticed from extensive numerical calculations [49]. For the MRR, on the other hand, the modulation response is [24]  
\[ \begin{array}{ll}T_{MRR}&=A\times\left((1-j\alpha_{LEF})/2\right)\times \frac{A^{*}+\Omega_{m}^{2}+2\delta\Omega_{m}+j\mu_{e}^{2}\Omega_{m}}{\delta^{2 }\left((\delta+\Omega_{m})^{2}+1/\tau^{2}\right)}+\\ &+A^{*}\times\left((1+j\alpha_{LEF})/2\right)\times\frac{A+\Omega_{m}^{2}-2 \delta\Omega_{m}+j\mu_{e}^{2}\Omega_{m}}{\delta^{2}\left((\delta-\Omega_{m})^ {2}+1/\tau^{2}\right)}.\end{array}  \tag{7}\]  
In this expression, \(A=\delta^{2}-j\mu_{e}^{2}\delta\) (and \(A^{*}\) is the complex conjugate), \(\delta=2\pi c\frac{\lambda_{notch}-\lambda_{enc}}{\lambda_{notch}\lambda_{enc}}\), where \(\lambda_{notch}\) is the spectral position of the MRR transfer function notch (Figure 1) that obeys the condition of MRR resonance, \(2\pi Rn_{eff}=m\lambda_{notch}\), \(m\in Z^{*}\), and lies nearest to the encoded optical signal of wavelength, \(\lambda_{enc}\), in the vicinity of \(1550\) nm, and \(\mu_{e}\) is the coupling strength between the bus waveguide and the ring, \(\mu_{e}^{2}=\left[(1-r^{2})c/2\pi Rn_{eff}\right]=1/\tau\), where the last equality with the MRR \(1/e\) amplitude decay time, \(\tau\), holds under critical coupling [44]. The modulation response of the MRR-assisted RSOA is then obtained by taking the absolute value of (5) and multiplying it with the modulus of (7). Finally, an inevitable consequence of RSOA direct modulation is that the encoded pulses acquire an instantaneous frequency deviation, that is, chirp, across them [50]. The compensation of the chirp's irregular temporal variation via post-optical notch filter is converted by the latter to restored quality of encoded pulses and accordingly enhanced RSOA modulation capability. This means that quantifying this transient effect can also provide useful information for the operation and performance of the MRR-assisted directly modulated RSOA. For this purpose, the knowledge of the phase response of the RSOA and MRR is required since the chirp, \(\Delta v(t)\), is by definition linked to it through the derivative in the time domain [51]. Finding the chirp for the SOA case is straightforward by isolating the output phase term from (1), differentiating it in time and readily taking the differential of the time-delayed version of the RSOA-integrated gain response from the right-hand side of (2) using the transform \(t\to t-2 Ln_{g}/c\). For the MRR case, however, the phase of the transmitted light undergoes a steep variation around resonance [52], which compromises the direct numerical calculation of the corresponding response and accordingly of its temporal rate of change, which provides the chirp. Instead of taking the inverse tangent function of the ratio between the imaginary and real parts of the complex-valued electric field [53], we use the relevant information that is extracted when applying the signal phase-reconstruction technique in the optical domain [54,55]. This technique allows one to unambiguously recover the instantaneous frequency deviation profile of a random repetitive data signal which is inserted in a frequency discriminator whose spectral transfer function has a linear spectral amplitude variation around the detuning position. This element acts as a differentiator on the inserted signal, which is exactly what the MRR-based notch filter does in our case [52]. In this manner, the desired chirp can be directly extracted in analytical form from the knowledge of the time-domain intensity profiles of the signals that enter and exit the MRR.