We demonstrate a femtosecond fiber master oscillator power amplifier system of high-stability and high-quality pulse compression, which is applied to corneal refractive surgery. The nonlinear-polarization-evolution mode-locking in a hybrid-cavity Yb-fiber oscillator consisting of both polarization-maintaining (PM) and non-PM fibers is demonstrated. In this paper, the PM-fibers introduced into the mode-locked fiber oscillator partially replace non-PM fibers. This alternative approach is conducive to reducing the adverse effect of uncontrolled fiber birefringence, which originates from the on-PM fiber suffering environmental temperature fluctuation and mechanical vibration. Once the length of non-PM fiber is comparable to the fiber beat length, the uncontrolled fiber birefringence caused by environment starts to abate the laser robustness, repeatability, and reliability of mode-locking. The stability becomes notoriously worse for long-cavity Yb-fiber oscillators. The PM-fibers adopted in the mode-locked fiber oscillator could improve the mode-locked stability of nonlinear-polarization-evolution self-started with long-cavity Yb-fiber oscillators. We study the dependence of the compressed pulse quality on the parameters of input pulse pre-chirp injected into the Yb-doped fiber amplifier. Due to the nonlinear-chirp and third order dispersion, the mode-locked pulse shape in time-domain will produce distortion during power being amplified in the Yb-doped fiber amplifier. A diffraction grating to adjust the pre-chirp of the input pulse from positive value to negative value launched into the Yb-doped fiber amplifier is placed between mode-locked Yb-fiber oscillator and Yb-doped fiber amplifier. We vary the pre-chirp by changing the distance between the diffraction grating and triangular prism and then adjust the second grating pair to compress the amplified pulses into its shortest pulse duration of full-width at half maximum measured by an autocorrelator. The experimental results show that the best compression quality of mode-locked pulse occurs at the negative pre-chirp with a measured pulse width of 183 fs. Deviation from this optimum pre-chirp degrades the compressed-pulse quality and features an increased temporal pedestal. The fiber laser produces self-started mode-locking at a repetition rate of 19.4 MHz, an average power of 1.2 W, a pulse width of 183 fs. The homemade femtosecond fiber laser is used to perform surgery on ex vivo pig corneas. The surgey shows that the light burst of femtosecond laser in corneal tissues could cut pig corneas, which proves that the femtosecond fiber laser satisfies the surgical operation requirements of animal corneas.