## 3 Results and Discussion
Furthermore, we calculated the Fourier transform limited duration of 515 nm pulse as 220 fs, as shown in Figure 4b. Considering an additional group delay dispersion (GDD) introduced by the MgO:PPLN crystal, a pulse stretching equation was employed to estimate the second harmonic pulse duration more accurately:  
\[\tau=\tau_{0}\sqrt{1+\left(4\ln 2\frac{\text{GDD}}{\tau_{0}{}^{2}}\right)}. \tag{4}\]  
where \(\tau\) and \(\tau_{0}\) are the stretched pulse duration and the initial pulse duration, respectively. The group velocity dispersion (GVD) parameter \(\beta_{2}\) of the extraordinary axis in MgO:PPLN crystal at 515 nm is 813.32 fs\({}^{2}\)/mm [30]. For a 3 mm length MgO:PPLN crystal, its GDD is calculated to be \(\sim\)2439.96 fs\({}^{2}\). Therefore, the final stretched pulse duration was calculated as 222.14 fs. The pulse duration of the 515 nm autocorrelated trace that we measured is \(\sim\)540 fs (Gaussian shape), as shown in Figure 4b. Compared to the fundamental light, the SHG pulse duration is longer, which may be cause by the limited pump acceptance bandwidth of the MgO:PPLN crystal. Additionally, it is essential to keep the SHG system operating bellow the LIDT of MgO:PPLN crystal, which is associated with the nonlinear refractive index and the characteristics of the fundamental light [31]. In our experiment, with a maximum incident fundamental average power of 2 W (0.92 GW/cm\({}^{2}\)), the laser induced damage did not occur. To find the LIDT of the MgO:PPLN crystal, another MOPA laser system with 100 MHz repetition rate, 150 fs pulse duration, and up to 5 W average power served as the fundamental laser. The maximum SHG conversion efficiency of the new SHG system kept below \(\sim\)20%, which was mainly caused by the decreased single pulse energy and rugged spectrum of the fundamental laser. The MgO:PPLN crystal also kept working properly by increasing the incident power to 2 W, corresponding to a pump intensity of 1.73 GW/cm\({}^{2}\). However, as the incident power increases, laser induced damage cracks were observed on the incident surface of the MgO:PPLN crystal under the pump intensity of 2.16 GW/cm\({}^{2}\). Figure 5a shows the magnified picture of damage point at the pump intensity of 2.16 GW/cm\({}^{2}\). Laser ablation might be caused by the thermal ion emission, for the repetition rate of the fundamental light is as high as 100 MHz and the pulse duration is as short as 150 fs [32]. The power stability of the 515 nm light was tested at 2 W incident fundamental light, as shown in Figure 5b. The RMS value of power stability was \(\sim\)0.68% at an average power of 820 mW for a four hour measurement. Compared to the power stability of incident fundamental  
Figure 4: Output characterization of the 515 nm light. (**a**) Optical spectra (black curve) and corresponding fitted curves (red curve) under incident power of 2 W. (**b**) Calculated Fourier transform limited pulse trace (green curve) and measured autocorrelation trace (black trace). light, the RMS value increased \(\sim\)0.24% attributed to the temperature fluctuation of the MgO:PPLN crystal. The pulse-to-pulse laser intensity was recorded by a fast InGaAs photodiode with 2-GHz bandwidth (Beijing Lightsensing Technologies Ltd., Beijing, China) and a 4-GHz digital oscilloscope (DSO9404A, Keysight Technology, Beijing, China). Nearly 40,000 pulses were recorded in 2 ms. As shown in Figure 5c, the maximum intensity of each pulse is presented as a normalized voltage. The distribution is close to a Gaussian curve with an R\({}^{2}\) of 0.997 and its standard deviation is about 0.007. The inset in Figure 5d is the outside view of the home-made aluminum alloy case. With a case, the SHG average power kept stable with an RMS of only 0.18% in an eight hour measurement.