## 2 Theoretical Calculation of the Laser Beam Divergence  
Figure 1 shows a schematic cross-sectional view of a typical unstable resonator. The convex output mirror has a diameter \(D_{1}\) and radius of curvature \(R_{1}\); the concave reflecting mirror has a diameter \(D_{2}\) and radius of curvature \(R_{2}\). The radii of curvature are defined as positive for concave mirrors and negative for convex ones. The mirror separation \(L\) is the resonator length. There are two types of unstable resonators: positive-branch and negative-branch. For high-power DF lasers, negative-branch unstable resonators should be avoided because of their solid focus in the cavity. In this paper, we only considered a confocal positive-branch unstable resonator, which produces a collimated output beam. The structures of the confocal positive-branch unstable resonator used in this paper were designed according to Ref. [28]. Table 1 shows the detailed parameters of our resonator. The magnification \(M\) is from 1.65 to 2.25, the resonator length \(L\) and reflecting mirror diameter  
\(D_{2}\) are fixed with \(L\) = 2.113 m and \(D_{2}\) = 5 cm, and \(C\) is the output coupling coefficient calculated by \(1-1/M^{2}\). As we can see from Figure 1, for a strictly adjusted confocal positive branch resonator, the output beam can be seen as an ideal plane beam. The far-field distribution of the output laser can be given according to the Fraunhofer diffraction field distribution formed by the plane wave passing through the annular aperture [29]. The angular distribution of far-field intensity can be written as follows:  
\[I(\theta)=\left\{\frac{M^{2}}{(M^{2}-1)}\left[\frac{2J_{1}(ka\theta_{s})}{ka \theta_{s}}-\frac{2J_{1}(ka\theta_{s}/M)}{Mka\theta_{s}}\right]\right\}^{2}I_ {0} \tag{1}\]  
where \(J_{1}\) is the Bessel function of the first kind, \(\theta_{s}\) is the semi-divergence angle, \(k=2\pi/\lambda\), \(a\) is the radius of reflecting mirror, and \(I_{0}\) is the beam intensity at the center of the beam in far-field distribution. Let \(Z=ka\theta\), and then expression (1) can be rewritten as follows:  
\[I(Z)=\left\{\frac{M^{2}}{(M^{2}-1)}\left[\frac{2J_{1}(Z)}{Z}-\frac{2J_{1}(Z/M )}{MZ}\right]\right\}^{2}I_{0} \tag{2}\]  
The far-field relative energy distribution is as follows:  
\[E(Z)=\int\limits^{Z}_{0}\left\{\frac{M^{2}}{(M^{2}-1)}\left[\frac{2J_{1}(Z)}{ Z}-\frac{2J_{1}(Z/M)}{MZ}\right]\right\}^{2}ZdZ \tag{3}\]  
According to expressions (2) and (3), the far-field relative intensity and energy for different magnifications were plotted based on different \(Z\) values as shown in Figures 2 and 3. Given the total energy of 100%, the energy proportion of the central main lobe for unstable resonator from \(M\) = 1.65 to 2.25 is 36.6%, 43.5%, 49.2%, and 53.8%, respectively. The larger the magnification, the higher the energy contained by the main lobe, meaning a better beam quality of the laser beam. As we can see from Figure 3, within the same divergence angle, the energy content of the far-field energy distribution of the unstable cavity with larger magnification \(M\) was bigger than that of the resonator with a small \(M\). Therefore, with the gradual increase of \(M\), the beam quality of the unstable resonator became better and better. Because circular laser beams usually have a Gaussian-shaped energy density distribution from the center of the beam to the edges, we used 86.5% encircled energy to definite the far-field spot size. Therefore, the far-field divergence angle can be calculated based on \(\theta=2Z/(ka)\). The calculated results for different magnifications with wavelength \(\lambda=3.8\) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(M\) & \(C\) & \(R_{1}\)/mm & \(R_{2}\)/mm & \(D_{1}\)/mm & \(D_{2}\)/mm & \(L\)/mm \\ \hline
1.65 & 0.633 & \(-\)6501.5 & 10,727.5 & 30.2 & 50 & 2113 \\
1.85 & 0.708 & \(-\)4971.9 & 9197.8 & 26.9 & 50 & 2113 \\
2.05 & 0.762 & \(-\)4024.8 & 8250.8 & 24.2 & 50 & 2113 \\
2.25 & 0.802 & \(-\)3380.8 & 7606.8 & 22.2 & 50 & 2113 \\ \hline \hline \end{tabular}
\end{table}