## 3 Results and Discussion  
In order to achieve a higher pulse energy MIR generation by DP-SRO configuration, the NLO crystal length needs to be carefully optimized to obtain a proper parametric gain. Suppressing the back conversion effect in NLO crystal, the pump energy can be transmitted to idler energy as much as possible. Firstly, a numerical simulation of the ns OPO process was performed to analyze the idler beam output characteristics, such as the output pulse energy, o-o conversion efficiency and temporal profile. In this simulation, we assume the pump beam with a Gaussian temporal and spatial profile and take into account the linear absorption, diffraction and beam spatial birefringent walk-off in BGSe crystal. For the collinear OPO process in ns regime, group velocity mismatch (GVM) can be ignored, so the mixing equations can be given by [24]:  
\[ \begin{array}{ll}\frac{\partial A_{j}(x,y,z,t)}{\partial z}&=\frac{i}{2K_{j} }[\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}]A_{j }(x,y,z,t)\\ &-\tan(\rho_{j})\frac{\partial A_{j}(x,y,z,t)}{\partial x}-\alpha_{j}A_{j}(x, y,z,t)+P_{j}(x,y,z,t)\end{array}  \tag{1}\]  
where \(j\) indexes the frequency (signal, idler and pump). \(\rho_{j}\) are the walk-off angles in the \(x\) direction, the linear absorption coefficients in the BGSe crystal are \(\alpha_{j}\) and \(A_{j}\) are the spatial and temporal complex amplitudes of three waves. \(P_{j}\) are nonlinear interaction terms, which are shown by the following:  
\[P_{s}(x,y,z,t)=i\frac{d_{eff}\omega_{s}}{cn_{s}}A_{p}(x,y,z,t)A_{i}^{*}(x,y,z,t) \exp(i\Delta kz) \tag{2}\]  
\[P_{i}(x,y,z,t)=i\frac{d_{eff}\omega_{i}}{cn_{i}}A_{p}(x,y,z,t)A_{s}^{*}(x,y,z,t) \exp(i\Delta kz) \tag{3}\]  
\[P_{p}(x,y,z,t)=i\frac{d_{eff}\omega_{p}}{cn_{p}}A_{i}(x,y,z,t)A_{s}(x,y,z,t) \exp(-i\Delta kz) \tag{4}\]  
where  
\[\Delta k=k_{p}-k_{s}-k_{i} \tag{5}\]  
is the wave vector mismatch, \(\omega_{j}\) is the angle frequency of the three waves, \(d_{eff}\) is the effective nonlinear coefficient of the BGSe crystal for type I phase-matching at 6.45 \(\upmu\)m pump by 1 \(\upmu\)m. \(n_{j}\) are the refractive indices of three waves in BGSe crystal and \(c\) is the speed of light in vacuum. Considering the optical characteristics of the BGSe crystal, the OPO numerical simulation in the DP-SRO is carried out based on the above three-wave mixing theory in SNLO free program by the Fourier transform and Cash-Karp Runge-Kutta method [25]. The relationship between idler output characteristics at 6.45 \(\upmu\)m and BGSe crystal length including output energy and o-o conversion efficiency is plotted under given input pulse energy of \(\sim\)100 mJ for the pump beam diameter of 6 mm, as shown in Figure 2. It can be seen that the idler output energy (red solid line) and o-o conversion efficiency (blue solid line) both increase gradually up to the highest values of \(\sim\)8.5 mJ and 8.5%, respectively, at the crystal length of \(\sim\)25 mm. Subsequently, the saturation effects on both output pulse energy and o-o conversion efficiency appear for longer crystal owing to the back conversion effect. It can be found that choosing the suitable length of the NLO crystal can improve the o-o conversion efficiency and output pulse energy. Therefore, we employed an optimal 25 mm long BGSe with an aperture of 8 \(\times\) 8 mm\({}^{2}\) as OPO NLO crystal in the following experiment, which was grown by the improved Bridgman-Stockbarger method, as illustrated in the upper right inset of Figure 1a. Figure 2: Calculated BGSe-OPO idler output pulse energy and o-o conversion efficiency at 6.45 \(\upmu\)m as a function of crystal length under given pump pulse energy of 100 mJ with a beam diameter of 6 mm. Based on the above experiments, the input-output characteristics of the generated MIR waves at 6.45 \(\upmu\)m were studied.