We report on the high power amplification of narrow-linewidth polarization-maintained (PM) laser in all-fiber PM MOPA configuration, which can operate with linewidth around 2GHz at above 1kW power level. Pump-limited maximum narrow-linewidth output power is 1.4kW without SBS effect, and the linearly-polarization single-mode output power was limited to ~550W by mode instabilities (MI). The beam quality of the fiber amplifier (M2 ) was measured under different power, which degrades gradually from ~1.4 to ~2.2 after the onset of MI. The polarization extinction rate is measured to be about 90% before the onset of MI and reduce dramatically after MI sets in. The experimental results are analyzed based on a novel semi-analytical model, which has taken the effects of gain saturation into account. The theoretical results agree well with the experimental results. Mitigating MI by coiling the gain fiber has been analyzed and demonstrated numerically. It shows that, by tight coiling of the gain fiber to a radius of 4cm, the MI threshold can increase to 2.5 times higher than that without coiling or loose coiling, which means that the amplifier have the potential to achieve MI-free 1.4kW output power.