## 1 Introduction  
The laser is the core component in inter-satellite laser communication systems, primarily employed for the generation and receiving of optical signals. Inter-satellite laser communication imposes stringent requirements on system performance metrics such as data capacity, phase noise, and sensitivity. Moreover, due to relative motion, Doppler frequency shift may occur, necessitating laser attributes including narrow linewidth, low phase noise, high frequency stability, and tunability [1, 2, 3]. Narrow linewidth semiconductor lasers, with advantages including compact size, extended operational lifespan, ease of integration, and direct modulation capability, have emerged as ideal light source devices in fields like inter-satellite laser communication [4, 5, 6]. In satellite laser communication systems, the choice of communication wavelengths has primarily focused on 1064 nm and 1550 nm in recent years. This is intended to achieve larger antenna gains, minimize signal pointing attenuation, and reduce the impact of solar background and sunlight scattering. As technology advances, compatibility with both 1064 nm and 1550 nm communication wavelengths is gradually becoming the development trend for future satellite laser communication systems [7]. Considering the widerapplication of the 1550 nm laser communication wavelength and the mature preparation of industrial-grade optical components at this wavelength, this paper selects a narrow linewidth external-cavity laser operating in the 1550 nm band for investigation. Existing research indicates that the bit error rate (BER) of space coherent optical communication systems is highly sensitive to changes in laser linewidth. As the laser linewidth increases, the phase noise also increases, leading to a corresponding decrease in the system's signal-to-noise ratio (SNR) and an increase in the BER. Moreover, in cases where the laser beam experiences significant jitter and boresight errors, a laser with a linewidth less than 8 kHz is required to meet the BER requirements for high-quality communication [8]. Additionally, [PERSON], [PERSON] et al. investigated an 8-QAM coherent free-space optical communication system employing amplitude compensation and phase recovery, where the combined linewidth tolerance of the transmitter and local oscillator is 10 kHz [9]. Therefore, in terms of the linewidth of lasers, it is essential to study lasers with a linewidth less than 10 kHz to better meet the requirements of inter-satellite laser communication in terms of communication distance and quality. The concept of linewidth was first introduced in 1958 by the inventors of lasers, [PERSON], along with the classical Schawlow-Townes linewidth calculation formula [10]. However, subsequent experimental studies revealed a significant discrepancy between the theoretically calculated linewidth based on the Schawlow-Townes formula and the actual measurements of lasers. In 1982, [PERSON] and colleagues proposed a modification to the theory of semiconductor laser linewidth, emphasizing that adjusting the laser cavity length through appropriate means, such as introducing optical feedback, can narrow the linewidth and enhance output performance [11]. Subsequently, the team led by [PERSON] conducted practical measurements and data analysis on lasers with different cavity lengths, leading to the crucial conclusion that linewidth is inversely proportional to output power and cavity length. This laid the foundation for the research and development of external cavity semiconductor lasers and their linewidth and noise suppression [12]. In 2006, the Institute of Semiconductors at the Chinese Academy of Sciences utilized external cavity feedback principles to create an external cavity grating feedback semiconductor laser, reducing the laser linewidth by one order of magnitude and achieving a leap from THz to MHz levels [13]. Since then, various types of external cavity strong feedback structures for narrow linewidth semiconductor lasers have emerged.