## 3 Results and Discussion  
Subsequently, the incident fundamental laser was focused on a MgO:PPLN crystal to achieve type-0 (e + e\(\rightarrow\)e) QPM. By using an open-source Gaussian-beam propagation calculator (GaussianBeam), the beam diameter was calculated as 99.16 \(\upmu\)m, since the waist of the incident fundamental laser beam was 660 \(\upmu\)m (full width at 1/e\({}^{2}\) maximum). Figure 3a  
Figure 2: Output characters of the home-made MOPA laser system. **(a)** The optical spectrum of the oscillator pulse (pink curve) and stretched pulse (blue curve). **(b)** The 1030 nm amplified power (red curve) and the compressed pulse power (blue curve) as a function of the diode pump power. **(c)** The optical spectrum of main amplifier with 3.36 W output power. Inset: Measured autocorrelation trace of the compressed pulse (solid line) and a Gaussian fitting curve (dashed line). **(d)** Power stability of the compressed pulse. Inset: Magnified scale. shows the average power (black curve) and conversion efficiency (red curve) of SHG laser versus the incident fundamental average power. By optimizing the temperature of the MgO:PPLN crystal and the position of the focus, the SHG average power at 515 nm could be optimized to 820 mW with an incident power of 2 W (0.92 GW/cm\({}^{2}\)), corresponding to a SHG conversion efficiency of \(\sim\)40%. The curves in Figure 3b represent the optimal distance (black curve) between the lens and the MgO:PPLN crystal and the optimal temperature (yellow curve) versus the incident fundamental average power, respectively. The optimal position for the incident power of 0.6 W was set as the zero point, and the SHG conversion efficiency was \(\sim\)26.4%, shown in Figure 3a. Increasing the incident power from 0.6 to 1.9 W, the optimal distance decreased 2.2 mm. During this process, more thermal effect and nonlinear refractive index variation were expected. Moreover, the optimal temperature decreased from 45 to 39.5 \({}^{\circ}\)C as the incident power increased from 0.6 to 1.2 W, and kept around 39 \({}^{\circ}\)C at 1.6 W.  
Figure 4a shows the optical spectrum of the 515 nm light with an 820 mW average power, recorded by a Fourier transform optical spectrum analyzer (OSA201C, Thorlabs, Shanghai, China) with a resolution of \(\sim\)6.4 pm. The spectrum roughly matches \(sinc^{2}\) fitting curves. Assuming there is no depletion of the fundamental light, the normalized SHG conversion efficiency function can be simplified as:  
\[\eta=\sin c^{2}(\frac{\Delta k_{Q}L}{2}), \tag{1}\]  
where \(L\) is the length of the MgO:PPLN crystal. \(\Delta k_{Q}\) is the QPM wavevector mismatching expressed as:  
\[\Delta k_{Q}=k_{H}-2k_{F}-\frac{2\pi}{\Lambda}. \tag{2}\]  
where \(k_{H}\) and \(k_{F}\) are the wavevectors of the SHG and fundamental light, respectively, and \(\Lambda\) is the poling period of the MgO:PPLN crystal. Meanwhile, the equation of pump acceptance bandwidth (\(\Delta\lambda_{F}\)) in theory is given as [28]:  
\[\Delta\lambda_{F}=\frac{0.4429\lambda_{F}}{L}\bigg{|}\frac{n_{H}-n_{F}}{ \lambda_{F}}+\frac{\partial n_{F}}{\partial\lambda_{F}}-\frac{1}{2}\frac{ \partial n_{H}}{\partial\lambda_{F}}\bigg{|}^{-1}. \tag{3}\]  
Figure 3: Output power and optimal parameters of the SHG system. (**a**) SHG power (black curve) and SHG conversion efficiency (red curve) with different incident fundamental average power. (**b**) The optimal position (black curve) and the optimal temperature (yellow curve) corresponding to the highest conversion efficiency at different incident fundamental average power. The \(n_{F}\) (2.15@ 1030 nm) and \(n_{H}\) (2.23@ 515 nm) can be calculated from the latest Sellmeier equation [29]. For the crystal length of 3 mm and the fundamental light at 1030 nm, the pump acceptance bandwidth was calculated as 0.6 nm (0.3 nm at 515 nm). It can be observed that the spectral bandwidth is much boarder than the theorical value, which may be caused by the high pump intensity.