## 2 Theoretical Analysis
For free-running lasers, the carrier number is clamped at the threshold level and the laser linewidth is in reverse proportion to the photon number. Optical injection increases coherent photon number in the laser cavity at a given current, thus improving coherence of the laser. Nevertheless, the increase of optical power is rather limited, and is estimated to result in a linewidth narrowing factor of only \(1.48\) according to our simulation. The second mechanism is mutual phase-locking between the coupled lasers. The behavior of mutually coupled lasers is determined by \((\kappa_{c}\varphi_{p}\),\(\delta)\), and the simulation results in Figure 2a depict the laser state under different detuning \(\delta\). Mutual injection locking occurs when \(\delta\) is around zero, with both lasers lasing at the same frequency \(f_{\mathrm{MIL}}\). It is important to notice that the variation of \(f_{\mathrm{MIL}}\) is small when \(\delta\) fluctuates. According to the third differential equation in Equation (1), the frequency noise of the lasers is mathematically equivalent to the fluctuation of \(\delta\). Hence the frequency noise of the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Value** & **Meaning** \\ \hline \(I\) & \(4\)\(I_{th}\) & Injection current (\(I_{th}\): threshold current) \\ \(\tau_{p}\) & \(7.15\) ps & Photon lifetime \\ \(\tau_{n}\) & \(0.33\) ns & Carrier lifetime \\ \(q\) & \(1.602\times 10^{-19}\) C & Electronic charge \\ \(g\) & \(1.13\times 10^{4}/\mathrm{s}\) & Differential gain \\ \(N_{0}\) & \(8.2\times 10^{6}\) & Transparent carrier number \\ \(\alpha\) & \(4\) & Linewidth enhancement factor \\ \(\overline{N}\) & \(2.05\times 10^{7}\) & Averaged carrier number at free running state \\ \(\beta\) & \(3.54\times 10^{-5}\) & Fraction of spontaneous emission coupled into lasing mode \\ \(\delta\) & - & Frequency detuning referred to free-running DFB\#1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Laser parameters. mutually injection locked lasers is reduced as compared with the free-running case. Figure 2b shows that \(df_{\text{MIL}}/d\delta<\)1 around zero detuning, indicating a suppression of frequency fluctuation. A negative feedback mechanism for frequency noise reduction exists in the mutually injection locked system, as depicted in Figure 2c. This results from the interplay between the photon and carrier, and their influence on the phase and frequency of the mutually coupled lasers. Supposing that the frequency of Laser#2\(f_{2}\) is increased slightly due to frequency noise, the resulting variation in laser phase leads to a decrease in photon number \(S_{1}\) and an increase in \(S_{2}\) through the cosine terms in Equation (1), which corresponds to the interference between the injected and the generated optical fields. The variation of photon number then induces an increase in \(N_{1}\) and a decrease in \(N_{2}\) through the stimulated emission process. As a result of the carrier plasma effect, the laser refractive index varies as described by the linewidth enhancement factor term, resulting in a decrease in \(f_{2}\) and an increase in \(f_{1}\). Hence, a negative feedback is formed and the variation of mutually injection locking frequency \(f_{\text{MIL}}\) is suppressed. To provide a quantitative demonstration of the above analysis, Figure 3a depicts the improvement of power spectrum density (PSD) of frequency modulation (FM) noise and optical spectrum under mutual injection locking. Compared with the free-running state, injection locked lasers exhibit reduced frequency noise and narrowed linewidth. A special feature of the mutual coupling configuration is the coupling delay. According to Figure 3b, the linewidth is narrowed from 2.25 MHz in the free-running state to 460 and 206 kHz under injection locking state with 5 and 30 ps delay, respectively, indicating that longer \(\tau_{d}\) helps enhance frequency noise suppression.