## 2 Numerical Simulation and Device Design
### Numerical Simulation
As shown in Figure 2a, when I\({}_{\text{SOA}}\) and I\({}_{\text{DFB2}}\) were fixed at 2 and 55 mA, respectively, the magnitude of the PPR peak was raised from \(-\)2.4 to 5 dB, with I\({}_{\text{DFB1}}\) varying from 85 to 95 mA. This was attributed to the enhancement of the first-order pole frequency due to the increase of the photon number in the DFB1 cavity [11]. Then, I\({}_{\text{DFB2}}\) varied from 55 to 75 mA (Figure 2b), while I\({}_{\text{DFB1}}\) and I\({}_{\text{SOA}}\) were fixed at 95 and 2 mA, respectively. Since VPI does not consider the heating effect, the lasing wavelength of the solitary DFB2 was blue-shifted with an increased I\({}_{\text{DFB2}}\). Consequently, the detuning frequency \(\Delta\)f = f1-f2 decreased monotonically from positive to negative, as shown in the schematic inset of Figure 2b, wherein f1 and f2 are the lasing frequency of DFB1 and DFB2, respectively. This led to increased damping of the PPR peak and a decreased PPR frequency (Figure 2b), which coincided with a previous report's results [24]. When the currents of DFB1 and DFB2 were fixed at 95 and 55 mA, respectively, the SOA current varied from 1 to 5 mA. The change of coupling phase and change of gain in the SOA were estimated at about 0.51 \(\pi\) and 0.55 dB, respectively, as I\({}_{\text{SOA}}\) increased from 1 to 5 mA. As shown in Figure 2c, the magnitude of the PPR peak decreased primarily due to the coupling phase changes. However, the PPR frequency hardly changed, which may be because of the relatively small change in the coupling strength due to the small varying range of the SOA current. When the SOA section was biased above 5 mA, the IMC laser entered the unlocked state due to the phase condition change [33]. A maximal modulation bandwidth of approximately 40 GHz was obtained by adjusting the currents of the three sections. \begin{table}
\begin{tabular}{c c c} \hline
**Explanation** & **Values** & **Unit** \\ \hline Nominal Wavelength (\(\lambda\)) & 1.31 & \(\mu\)m \\ DFB Section Length (L\({}_{\text{DFB}}\)) & 220 & \(\mu\)m \\ SOA Section Length (L\({}_{\text{SOA}}\)) & 360 & \(\mu\)m \\ Internal Loss (\(\alpha_{\text{i}}\)) & 3000 & 1/m \\ Logarithmic Gain Coefficient & \(2.1\times 10^{5}\) & 1/m \\ Gain Compression Factor & \(1\times 10^{-23}\) & m\({}^{3}\) \\ Transparency Carrier Coefficient (N\({}_{\text{tr}}\)) & \(1.5\times 10^{24}\) & 1/m\({}^{3}\) \\ Linewidth Enhancement Factor (\(\alpha\)) & 3 & - \\ Linear Recombination Coefficient (A) & \(3\times 10^{8}\) & 1/s \\ Bimolecular Recombination Coefficient (B) & \(1\times 10^{-16}\) & m\({}^{3}\)/s \\ Auger Recombination Coefficient (C) & \(1.3\times 10^{-41}\) & m\({}^{6}\)/s \\ Index Grating Coupling Coefficient (\(\kappa_{\text{i}}\)) & 6000 & 1/m \\ Grating Phase Shift (\(\theta\)) & 90 & \(\circ\) \\ Reflectivity of Facets (R) & 0.1\% & - \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters used in the simulation. Figure 1: Conceptual diagram of the integrated mutually coupled (IMC) laser in the simulation. The cavity length is also a vital parameter in IMC lasers [24], but the relationship between the PPR frequency and cavity length has rarely been studied. To investigate its impact, we calculated the normalized small-signal response with different SOA lengths when the DFB sections were fixed at 220 \(\upmu\)m. The DFB1 current was fixed at 100 mA, while the DFB2 current was scanned to push the IMC laser into the MIL state. Meanwhile, the SOA current was adjusted to make the SMSR of lasing mode as similar as possible in IMC lasers with different SOA lengths. The modulation responses are depicted in Figure 3a. The PPR frequency increased as the SOA length decreased, and the dip deepened between the CPR and PPR frequencies.