## 2 Theoretical Analysis  
Figure 1 depicts a generalized mutual injection locking configuration between two semiconductor lasers. The optical path can be realized on chip, in free-space or through fiber link. The light emitted from one laser is injected into the other with a coupling strength \(\kappa_{c}\) and a coupling phase \(\varphi_{p}\) as well as a delay \(\tau_{d}\). Hence, the optical field of each laser is distorted not only by its internal noise sources, but by the noise of the other lasers via optical coupling as well. To investigate the linewidth characteristics of mutually coupled lasers, coupled single mode rate equations are adopted to analyze their frequency noise. Different from commonly used Lang-Kobayashi equations [29; 30; 31], laser field \(E(t)\) is replaced by \(E(t)=S^{1/2}\text{exp}(\text{i}\varphi)\) to facilitate the introduction of Langevin noise. The modified equations are given in Equation (1), where \(S\), \(N\) and \(\varphi\) represent photon number, carrier number and optical phase, respectively. Both lasers are assumed to share the same parameters listed in Table 1, together with the definition of each symbol. \(F_{n}\), \(F_{s}\) and \(F_{p}\) are Langevin forces driving the equations of carrier, photon and phase. Due to carrier-photon interaction in a semiconductor laser, \(F_{n}\) and \(F_{s}\) are correlated and  
Figure 1: Generalized configuration of mutually coupled semiconductor lasers. cannot be described by white Gaussian noise directly. A noise generation technique as described in [32] is employed and the laser-parameter related noise sources are given by Equation (2). \[ \begin{array}{l}\frac{\mathrm{d}N_{12}(t)}{\mathrm{d}t}=\frac{\mathrm{I}_{12 }}{q}-\frac{\mathrm{N}_{12}(t)}{\mathrm{t}_{n}}-g(\mathrm{N}_{12}(t)-\mathrm{ N}_{0})\mathrm{S}_{12}(t)+F_{n12}(t)\\ \frac{\mathrm{dS}_{12}(t)}{\mathrm{d}t}=g(\mathrm{N}_{12}(t)-\mathrm{N}_{0}) \mathrm{S}_{12}(t)-\frac{\mathrm{S}_{12}(t)}{\mathrm{t}_{p}}+\frac{\mathrm{ \delta}N_{12}(t)}{\mathrm{t}_{n}}+\ldots\\ 2\kappa_{c}\sqrt{\mathrm{S}_{2,1}(t-\mathrm{t}_{4})\mathrm{S}_{12}(t)}\cos( \varphi_{2,1}(t-\tau_{d})-\varphi_{p}-\varphi_{1,2}(t))+F_{n12}(t)\\ \frac{\mathrm{d}\varphi_{12}(t)}{\mathrm{d}t}=\frac{\mathrm{e}}{2}g(\mathrm{N} _{1,2}(t)-\overline{\mathrm{N}_{1,2}})+\kappa_{c}\sqrt{\frac{\mathrm{S}_{12 }(t-\tau_{d})}{\mathrm{S}_{12}(t)}}\sin(\varphi_{2,1}(t-\tau_{d})-\varphi_{p}- \varphi_{1,2}(t))+\delta_{1,2}(t)+F_{p1,2}(t),\\ \end{array}  \tag{1}\]  
\[ \begin{array}{l}F_{n}(t)=\sqrt{\frac{2N(t)}{\mathrm{t}_{n}\mathrm{d}t}}x_{1 }(t)-\sqrt{\frac{2\mathrm{\delta}N(t)S(t)}{\mathrm{t}_{n}\mathrm{d}t}}x_{2}(t) \\ F_{s}(t)=\sqrt{\frac{2\mathrm{\delta}N(t)S(t)}{\mathrm{t}_{n}\mathrm{d}t}}x_{2 }(t)\\ F_{p}(t)=\sqrt{\frac{\mathrm{\delta}N(t)}{2\mathrm{t}_{n}\mathrm{d}t}}x_{3}(t).\\ \end{array}  \tag{2}\]  
Here \(x_{1}(t)\), \(x_{2}(t)\) and \(x_{3}(t)\) are independent random sequences taken from standard normalized distribution and \(\Delta t\) is the simulation step. Laser frequency noise and spectrum characteristics under mutual injection locking is investigated numerically, and three mechanisms are found for linewidth narrowing. The first one is photon number increase due to optical injection. Figure 2a shows the variation of photon number, carrier number and lasing frequency of both lasers when the detuning is swept from \(-15\) to \(15\) GHz. In the range of \(-8\)-\(7\) GHz, carrier and photon number in both lasers are stable and their lasing frequency are the same, indicating stable mutual injection locking. In our simulations, the two coupled lasers are assumed to be identical. The strong bidirectional optical injection leads to an increase in photon number within the laser cavity by \(1.7\) dB at zero detuning. As a result, the carrier density is reduced by \(17.2\%\) via enhanced stimulated emission. The refractive index of active region decreases due to the carrier plasma effect and lasing frequency is redshifted by \(10.9\) GHz. Laser frequency noise results from spontaneous emission and carrier fluctuation.