Ultra-High Signal-to-Noise Ratio and Ultra-Narrow Linewidth Brillouin Fiber Laser for Linewidth Measurement at 2 $\mu$m

2 $\mu$m single frequency fiber lasers (SFFLs) have attracted wide attention thanks to their excellent laser performance and great application potentials in the operation waveband. This work presents an ultra-high optical signal-to-noise ratio (OSNR) and ultra-narrow linewidth Brillouin fiber laser (BFL) at 2 $\mu$m. A piece of ultrahigh numerical aperture fiber is employed as the Brillouin gain medium to construct a ring cavity, where a Sagnac loop interferometer is inserted to ensure the single-longitudinal-mode operation. The realized 2 $\mu$m BFL exhibits an OSNR up to 74 dB and a linewidth as narrow as 609$\pm$ 24 Hz. Applying such a BFL configuration to laser linewidth measurement, a measurement accuracy of 5 kHz at least is achieved. This BFL provides a convenient and simple measurement tool for linewidth characterization of 2 $\mu$m SSFLs without the long delay fiber and additional frequency shifter.