## 2 Theoretical Analysis
For convenience of comparison, the simulated optical spectra in Figure 3b are shifted to the same central frequency and the same central power of 0 dB. Figure 2: (**a**) Variation of the photon number, carrier number and lasing frequency of mutually coupled lasers as the detuning varies from \(-\)15 to 15 GHz. (\(\tau_{d}\) = 5 ps, \(\kappa_{c}\) = 2 \(\times\) 10\({}^{10}\)/s, \(\varphi_{p}\) = 0); (**b**) relation between injection locked lasing frequency \(f_{MIL}\) and detuning \(\delta\); (**c**) negative feedback diagram. Next, we consider the situation that delay time is increased to \(\tau_{d}\) = 5 ns, an order of magnitude greater than \(1/f_{RO}\). The frequency noise suppression is also valid for long delayed mutual coupling, although locking is found to be attainable only for weak coupling. The frequency noise in the time domain is calculated in Figure 4. For free-running lasers with 0.3 GHz detuning, the lasing frequency of either laser fluctuates with a standard deviation of 1.92 \(\times\) 10\({}^{6}\) Hz. When injection locking occurs, the standard deviation reduces to 1.36 \(\times\) 10\({}^{5}\) Hz, corresponding to a frequency noise suppression by a factor of 14.1. Unlike the case of unidirectional injection locking, where the frequency noise of the slave laser can be improved by adopting a low frequency noise master laser [9; 10], frequency noise reduction via mutual injection locking does not require the adoption of a low frequency noise laser, making it attractive for practical applications. The third mechanism is the enhanced photon lifetime due to expanded cavity length. In the mutual coupling structure, a fraction of the photons are stored in the coupling path, which should be taken into consideration. When stable state is obtained at strong coupling, the entire system can be regarded as a compound resonator. Here, \(\lambda/4\) shifted DFB lasers are employed to evaluate the change of photon lifetime. We extend the methods for photon lifetime determination in standard DFB lasers [33; 34] to this compound cavity. Firstly, field distribution is derived by transfer matrix method. Then photon lifetime is derived by calculating the energy stored in the system and energy dissipated due to absorption and facet output. Typical field distribution of system with \(8L\) coupling length is  
Figure 4: Frequency fluctuation in the time domain of free-running and mutually injection locked lasers with long delay. (\(\tau_{d}\) = 5 ns, \(\kappa_{c}\) = 5 \(\times\) 10\({}^{8}\)/s, \(\varphi_{p}\) = 0, \(\delta\) = 0.3 GHz). Figure 3: (**a**) Calculated power spectrum density (PSD) of frequency modulation (FM) noise of mutually injection locked lasers with different delay time; (**b**) calculated optical spectrum of mutually injection locked lasers with different delay time. (\(\kappa_{c}\) = 2 \(\times\) 10\({}^{10}\)/s, \(\delta\) = 0). shown in Figure 5b. The equivalent mirror loss is \(\alpha_{m}=44.6\) cm\({}^{-1}\) for a \(\lambda/4\)-shifted DFB with a typical coupling coefficient of \(\kappa L=1.5\) (\(L=500\)\(\mu\)m), while the typical absorption loss is \(\alpha_{i}=10\) cm\({}^{-1}\)[35]. As the coupling path is lengthened, equivalent mirror loss is reduced due to the expanded cavity length. Consequently, photon lifetime is increased as shown in Figure 5a. However, this trend slows down when \(\alpha_{m}\) is comparable to \(\alpha_{i}\). The limit of photon lifetime is 10.8 ps determined by \(\alpha_{i}\), five times longer than that of a solitary laser. Employing the low loss coupling region may reduce the laser linewidth further.