## 1 Introduction  
Green ultrafast lasers have attracted increasing attention in scientific and industrial fields for various applications, such as dual-comb spectroscopy [1], optical parametric oscillators [2], direct laser writing [3, 4], and biomedical imaging [5]. In particular, green lasers with ultrashort pulse duration (\(-\)fs) and high repetition rate (\(\sim\)MHz) are ideal light sources for two-photon photopolymerization (TPP), an essential technique for flexible and high-resolution fabrication of three-dimensional nanoscale structures. During the TPP process, femtosecond pulses with appropriate peak power are required to achieve the TPP threshold and a high repetition rate makes it possible to fabricate rapidly over a large area [6]. Moreover, in comparison to lasers at 780 nm generated from Ti: Sapphire lasers or frequency-doubling Er-doped fiber lasers, femtosecond lasers at 515 nm capacitate a smaller focal spot and a higher single photon energy, which could approach to an improved fabrication resolution [7, 8, 9]. Indeed, there have been various wavelength-conversion methods, such as third-harmonic generation in nematic liquid crystals [10], supercontinuum generation in fused silica [11], self-frequency doubling lasers [12] and optical sum-frequency from dual-wavelength lasers [13]. For reliable and efficient green femtosecond laser generation, the conventional route applies frequency doubling from femtosecond lasers around 1.0 \(\upmu\)m. Among those, Yb-doped fiber laser possesses many intrinsic properties, such as robustness, compactness, and low cost. The way to obtain mode-locked laser in all-fiber based Yb-doped oscillators is diverse. Nonlinear polarization rotation (NPR) based fiber oscillators are sensitive to environment perturbations due to non-polarization-maintaining (PM) fibers, and the non-PM fiber increases the pulse-to-pulse power fluctuation and even degrades the mode-locking stability [14]. The all-PM fiber architecture can be applied in semiconductor saturable absorber mirror (SESAM) or nonlinear amplify loop mirror (NALM) mode-locking mechanism. However, SESAMs suffer from a low optical damage threshold, which is unreliable for long-term operation. In 2016, [PERSON] et al. reported an integrated all-PM NALM oscillator using a non-reciprocal phase shifter, producing pulses with 538 fs duration after de-chirping [15]. Traditionally, large-mode-area photonic-crystal fiber amplifiers with free-space bulky components are utilized to achieve higher average power, which would inevitably degrade the integrability and stability of laser systems [16]. Therefore, a monolithic PM Yb-doped femtosecond fiber laser was developed instead. In 2017, [PERSON] et al. obtained a diffraction-limited laser beam with high polarization extinction radio at Watt-level average power, which is crucial for efficient second harmonic generation (SHG), from a direct fiber-spliced laser system [17; 18]. To date, considerable studies on femtosecond SHG have been conducted in all kinds of nonlinear crystals. The birefringence phase match (BPM) technique was widely applied in these experiments. In 2014, [PERSON] et al. demonstrated a 176 fs, 28.8 nJ, 78 MHz green laser with a BiB\({}_{3}\)O\({}_{6}\) (BIBO) crystal, and the corresponding conversion efficiency was 46.5% [19]. In 2016, [PERSON] et al. realized a 150 fs, 2.3 W, 78 MHz green laser using a 4-mm-length LiB\({}_{3}\)O\({}_{5}\) (LBO) crystal with an incident fundamental average power of 4.8 W [20]. In 2017, [PERSON] et al. proposed a monolithic laser system and a \(\beta\)-BaB\({}_{2}\)O\({}_{4}\)(BBO)-based SHG module, from which \(\sim\)350 fs, 1.25 W, 17 MHz green laser was produced, and the corresponding conversion efficiency was 26.6% [21]. However, the BPM-based SHG process are inevitably along with the spatio-temporal beam distortion caused by the walk-off effect, which notably degrades the SHG laser beam quality [22].