## 1 Introduction  
Semiconductor lasers feature small foot-prints, low power consumption, high flexibility and compatibility with photonic integration circuits, hence they are widely used in various optical systems. Lasers suffer from frequency noise, which results from spontaneous emission and carrier noise, and the spectrum purity always casts a performance limit on different applications. In coherent optical communications with high-order modulation format, laser phase noise disperses the constellation and causes bit error [1; 2; 3]. In atomic sensing, laser frequency noise converts to amplitude noise after interaction with atom gas, limiting the stability or sensitivity of atomic sensors [4; 5]. In addition, the spectrum width also determines the resolution of spectroscopy detection [6], sensitivity of fiber sensing [7] and the dynamic range of lidar [8], to name just a few. Therefore, it is quite worthwhile to devote efforts to narrowing laser linewidth for further improvement of system performance. The linewidth of semiconductor lasers is typically in the range of a few MHz. Optical injection provides a new degree of freedom for improving the performance of semiconductor lasers. In the case of unidirectional injection locking, the linewidth of the slave laser follows that of the master [9], so that a narrow linewidth master laser is required to improve the spectral purity of the slave laser [10]. As for self-injection or external cavity, a fraction of light emitted from the gain material is fed back through a low loss external cavity, hence the laser linewidth is reduced by increased photon lifetime. Recent research centers on the realization techniques of self-injection, such as low loss Si [11; 12] or Si\({}_{3}\)N\({}_{4}\)[13; 14] ring resonator, fiber components based feedback circuit [15; 16], assembled confocal Fabry-Perot (FP) cavity [17] and CaF\({}_{2}\) cavity [18]. These structures rely on high Q cavity or multiple mode selection components. In this work, we propose and demonstrate a simple but effective approach to linewidth narrowing based on mutual injection locking (MIL) of semiconductor lasers. Previous research on mutual coupling mainly focuses on the nonlinear dynamics [19; 20], which can be exploited to realize random sequences generation [20], chaos synchronization [21; 22], modulation characteristics enhancement [23] and photonic microwave generation [24]. In addition, it is also used for coherent combining [25; 26] and lasing threshold reduction [27]. However, to the best of our knowledge, there have been relatively few reports on laser linewidth or frequency noise characteristics upon mutual injection locking. A systemic study on the linewidth narrowing mechanism, locking condition and frequency noise reduction is carried out in the investigation. Modified mutual coupling rate equations incorporating Langevin noise sources are presented to analyze the mechanisms for frequency noise reduction of mutually injection locked lasers. Previous experiments indicate that laser behaves differently for short delay (\(\tau_{d}<1/f_{\text{RO}}\)) and long delay (\(\tau_{d}>1/f_{\text{RO}}\)) [19; 20; 28], where \(\tau_{d}\) is the coupling delay time and \(f_{\text{RO}}\) represents the relaxation oscillation frequency. Therefore, experiments are carried out in both regimes: monolithically integrated mutually coupled Distributed Feedback(DFB) lasers are employed to investigate frequency noise performance in short delay regimes, while a fiber link based system is adopted for the study of long delay regimes. Locking condition and the dependence of laser frequency noise on system parameters are investigated in both regimes, confirming mutual injection locking as a practical method for linewidth narrowing. This paper is organized as follows. In Section 2, a theoretical model is presented to clarify the frequency noise suppression in mutually injection locked lasers.