## 3 Experimental Results and Discussion
### Single-Pass Sum-Frequency Generation (SFG) to 852.3 nm  
Then, in a single-pass SFG experiment, we find the optimized phase matching by adjusting the temperature of the PPMgO:LN crystal. When the power of fundamental-wave 1560.5 nm and 1878.0 nm lasers are both 1.0 W, the best phase matching of the PPMgO:LN-crystal is achieved at a temperature of 80.6 \({}^{\circ}\)C. The results are shown in Figure 6.  
The results of SFG output power and efficiency are shown in Figure 7. The power of the 1878.0 nm and 1560.5 nm lasers are 1.77 W and 4.94 W, respectively; 276 mW of the 852.3 nm laser can be obtained, and the corresponding optical-optical conversion efficiency is 4.1%. Fitting to the data of the linear  
Figure 5: Doppler-broadened absorption spectra of 55\({}_{1/2}\)–5P\({}_{3/2}\) transition (D\({}_{2}\) line) of \({}^{87}\)Rb and \({}^{85}\)Rb atoms by using SHG laser at 780.2 nm, while the 1560.5 nm fundamental-wave laser is frequency-scanned linearly. Clearly the continuously tunable range of the 780.2 nm laser frequency is more than 6.8 GHz. Figure 6: Temperature dependence of 852.3 nm laser power. Circles represent the experimental data. The optimal quasi-phase-matched temperature is 80.6 \({}^{\circ}\)C.  
region yields a nonlinear conversion efficiency of 0.62% (W cm) \({}^{-1}\). The SFG nonlinear conversion efficiency can be written as [23]  
\[\eta=\frac{p_{3}}{p_{1}p_{2}l}=\frac{8\omega_{0}^{3}d_{eff}^{2}}{\pi\varepsilon_{ 0}c^{4}n_{0}n_{3}}\Bigg{[}\frac{\big{(}1-\delta^{2}\big{)}\big{(}1-\gamma^{2} \big{)}}{(1+\delta\gamma)}\Bigg{]}h(\mu,\zeta)\sin^{2}(\pi\Delta)\]  
where \(p_{i}\)(i = 1, 2, 3) represent the laser power of 1560.5, 1878.0, and 852.3 nm, respectively. \(n_{o}\) = (\(n_{1}\)+\(n_{2}\))/2, \(\omega_{0}\) = (\(\omega_{1}\)+\(\omega_{2}\))/2,\(\omega_{i}\) are the frequency of the corresponding laser, and \(d_{eff}\) is the effective nonlinear coefficient. \(\delta=1-2\omega_{1}/(\omega_{1}+\omega_{2})\) and \(\gamma=1-2n_{1}/(n_{1}+n_{2})h(\mu,\zeta)\) indicates the B-K focusing factor, which is related to the focusing parameter \(\zeta\). The last term is the correct for the crystal's non-ideal grating duty cycle. In addition, the spatial mode matching of the two fundamental beams within the crystal is the most important factor. The optimal SFG can be obtained when the confocal parameters of these two fundamental beams are equal and the focusing parameter \(\xi\) = 2.84. We also use an f = 100 mm lens to match the crystal, and the corresponding optical-optical conversion efficiency is 3.7%. From the experimental results, we can see that when the lens is 75 mm, it can yield a high-power 852.3 nm laser. According to the optimal B-K focus factor \(\xi\) = 2.84, the optimal waist spot radii of 1560.5 nm and 1878.0 nm lasers are 45.3 \(\upmu\)m and 49.7 \(\upmu\)m, respectively. When using a matching lens with f = 75 mm, the waist spot radii of 1560.5 nm and 1878.0 nm lasers are 41.6 \(\upmu\)m and 46.5 \(\upmu\)m, respectively. The radii for f = 75 mm are closer to the optimal waist spot radius than those for f = 100 mm. In a single-pass sum-frequency experiment, the matching lens is not the only factor that influences efficiency, but the overlapping of the two fundamental-wave lasers in the crystal is also important. Although an achromatic doublet can make 1560.5 nm and 1878.0 nm lasers at one point in the crystal, but it is not easy to customize. Therefore, we will also study the method of separate focusing to make the fundamental waves match well, including size and position. The transverse beam quality of 852.3 nm laser is evaluated by using the beam quality factor parameter (\(M^{2}\)) in the two orthogonal transverse directions \(X\) and \(Y\). Figure 8 shows the 1/\(e^{2}\) beam radius versus the axial position \(Z\) using a plano-convex lens with a focal length of 50 mm. Fitting of the experimental data gives \(M_{X}^{2}\) = 1.07 and \(M_{Y}^{2}\) = 1.13. Figure 7: Experimental data for single-pass SFG using a PPMgO:LN crystal with matching lens f = 75 mm.