## 2 Numerical Simulation and Device Design  
### Numerical Simulation  
Several simulation studies on IMC lasers have been conducted based on the modified rate equations model [20; 22; 24], which provides a physical intuition of the dynamics of IMC lasers with the coupling strength and detuning frequency. However, this is less practical for integrated device designs, considering the complicated interactions of each section and difficulty extracting the coupling parameters under different bias conditions. Thus, a method based on the traveling-wave (TW) model is preferable for analyzing the static and dynamic performances in a multi-section laser, which comprises the spatial-temporal effects along the cavity [30; 31]. The slowly varying envelopes of the propagating electric field obey the following equations:  
\[ \begin{bmatrix}\frac{1}{v_{g}}\frac{\partial}{dt}+\frac{\partial}{\partial x }\end{bmatrix} E_{f}\ =\ -\bar{j}\overline{\delta}E_{f}-j\kappa_{i}E_{b}+F_{f}\\ \frac{1}{v_{g}}\frac{\partial}{dt}-\frac{\partial}{\partial x}\end{bmatrix}E_{b} \ =\ -\bar{j}\overline{\delta}E_{b}-j\kappa_{-l}E_{f}+F_{b} \tag{1}\]  
\[\overline{\delta}\ =\ \delta+j\frac{(\Gamma g-\alpha_{i})}{2} \tag{2}\]  
where \(E_{f}\) and \(E_{b}\) are the complex-valued envelopes of the forward and backward traveling optical fields, respectively, \(v_{g}\) is the group velocity of the optical mode, \(\kappa_{\pm l}\) is the coupling coefficient in the DFB section, \(\delta\) is the deviation of the propagation constants, \(\Gamma\) is the confinement factor, \(g\) is the gain coefficient, \(\alpha\) is the internal loss, and \(F_{f,b}\) is the stochastic noise donated by the spontaneous emission. Here, to simulate the IMC laser, we used a commercial simulation package, VPItransmissionMakerTM (VPI), with a transmission line model based on the TW model. Figure 1 shows the structure of the IMC laser in our simulation, which is composed of two DFB sections on both sides and an SOA section in the middle that serves as a tuning section to adjust the coupling strength and coupling phase. The IMC laser works under the MIL state by adjusting the current of three sections, which leads to an appropriate detuning frequency, coupling strength, and coupling phase. The DFB section under modulation was named DFB1, while the DFB section under only DC bias was named DFB2. The length of both DFB sections was fixed at 220 \(\upmu\)m to ensure an adequate output power and CPR frequency. The grating phase shift and reflectivity of the facets were based on the device's manufacturing structure. The other parameters used in the simulation were chosen after referring to [24; 31; 32] and are listed in Table 2. In both the simulation and experiments, the laser output was coupled from the facet of DFB1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference** & **Type of Laser** & **\(\mathbf{f_{R}}\)****1** **or \(\mathbf{f_{3 dB}}\)****2** & **Bit Rate** \\ \hline
[19] & Monolithic injection-locked DFB lasers & \(\mathbf{f_{R}}\): 23 GHz & - \\
[21] & Monolithically integrated coupled-cavity DBR lasers & \(\mathbf{f_{R}}\): 30 GHz & - \\
[23] & Integrated two-section discrete mode laser & \(\mathbf{f_{3 dB}}\): 10 GHz & - \\
[24] & Monolithically integrated DFB lasers & \(\mathbf{f_{R}}\): 34.3 GHz & - \\
[28] & Monolithic integrated two-section DFB laser & \(\mathbf{f_{R}}\): 26.8 GHz & - \\
[29] & Integrated coupled three-section DFB lasers & \(\mathbf{f_{3 dB}}\): 18.7 GHz & - \\
[25] & Coherently coupled 2 \(\times\) 1 VCSEL arrays & \(\mathbf{f_{3 dB}}\): 37 GHz & 36 Gb/s \\ This work & Integrated coupled DFB lasers & \(\mathbf{f_{3 dB}}\): 38.7 GHz & 40 Gb/s \\ \hline \hline \end{tabular} \({}^{1}\) Enhanced resonance frequency. \({}^{2}\) 3-dB bandwidth.  
\end{table} 
Table 1: Representative modulation performances of different integrated mutually coupled (IMC) lasers. Initially, the small-signal modulation response of the IMC laser under various DC biasing was investigated via simulation.