## 1 Introduction  
With the development of technology, high precision measurement has become the mainstream. With low phase noise and long coherent length, narrow linewidth laser [1, 2, 3, 4] has been widely used in high precision detection fields, such as high precision spectrum, Lidar [5, 6], remote sensing, etc. It provides an excellent laser source for optical communication [7, 8], distributed sensing [9], and microwave generation [10]. The characteristics of the laser line width play a decisive role in the detection distance, precision [11], sensitivity, and noise characteristics of the systems. It becomes important to precisely measure the laser linewidth. Narrow linewidth fiber laser has a long service life, high conversion efficiency, comprehensive wavelength coverage, slight wavelength drift, high-quality light source, low noise, and long coherence length; thus, it is widely used in many fields. In optical communication systems, the linewidth directly affects phase information extraction. With the advancement of signal modulation formats, the line width has gradually reduced from 1 MHz to 10 kHz or even 100 Hz. In optical fiber sensing, the frequency modulated continuous wave technology and the principle of lightwave coherence can also be used to measure ultra-fine, ultra-long distance, and weak signals. In addition, narrow-linewidth lasers play important roles in oil and gas exploration systems, pipeline monitoring, gas leakage detection, cold atomic physics, and microwave photonics. In recent years, some groups had investigated KHz\(-\)order narrow linewidth. For example, [PERSON] et al. [12] established a kHz-order linewidth controllable 1550 nm single-frequency fiber laser using a low-pass filtered white Gaussian noise (WGN) signal applied on a fiber stretcher in an optical feedback loop. By using WGN signals with different signal amplitudes and different cutoff frequencies, we control the linewidth in the range of 0.8 to 353 kHz. The result indicates that optical signal-to-noise ratio is significantly higher than 72.0 dB. However, the structure they designed is expensive, which requires special FBG. On the other hand, [PERSON] et al. [13] proposed an efficient single-frequency fiber laser by using a newly developed Er3+/Yb3+ co-doped single-mode phosphate glass fiber. The laser power is 300 mW, and the linewidth is less than 2 kHz. Their laser requires two pump lasers, and the stretcher is strongly influenced by temperature. [PERSON] et al. [14] proposed that a widely tunable distributed Bragg reflector (DBR) laser with kilohertz linewidth output can be demonstrated by using a dual-loop self-injection scheme. The linewidth is less than 10 kHz. Their design also requires additional unique optical fiber. Moreover, [PERSON] et al. [15] proposed a Brillouin fiber laser with an ultra-narrow linewidth of 30 Hz; however, the structure is complex. It requires strong pump power and a long fiber length, which induces colossal loss. [PERSON] et al. [16] demonstrated a switchable single longitudinal mode (SLM) narrow linewidth fiber laser with cylindrical vector beam (CVB) output. At 1543.35 nm and 1544.38 nm, the OSNR is higher than 60 dB, and the laser linewidth is 5.7 KHz and 6.8 khZ, respectively. However, the design is complex including multiple cavities. which need to be controlled the grating and polarization state of the laser cavity. On the other hand, in this proposal, an extra-narrow laser linewidth can be obtained by using a simple structure. [PERSON] et al. [17] demonstrated a switchable dual-wavelength single-longitudinal-mode (SLM) narrow-linewidth Ytterbium-doped fiber laser (YDFL) based on a non-linear amplifying optical loop mirror (NALM). The OSNRs of both laser wavelengths are better than 50 dB. Their laser linewidth of them are 17.07 kHz and 18.64 kHz, respectively. However, the lasers linewidths are wide and the laser characteristic is unstable and susceptible to external influences.