## 4 Discussion
### Theoretical Analysis for Higher Efficiency
For single-pass pumping, the highest optical-to-optical conversion and slope efficiencies were 50.7% and 54.2%, respectively. For the double-pass pumping, the highest optical-to-optical conversion and slope efficiencies were increased at 55.9% and 59.6%, respectively. In Figure 6c, the intrinsic residual loss is 8.0%, twice as high as that in Figure 6a,b, and we optimized at the maximum pump power of 25.0 W, the same in Figure 6b. We selected an output coupler transmittance of 70.3% for the single-pass pumping and 73.9% for the double-pass pumping. In Figure 6c, the highest efficiencies were of course reduced compared to those in Figure 6b because the intrinsic residual loss is twice as high as that in Figure 6b, but the reductions were limited to approximately 3%. The reason for the limitation of the reduction in the efficiencies is the high pump intensity. For the single-pass pumping, the highest optical-to-optical conversion and slope efficiencies were 47.5% and 52.1%, respectively. For the double-pass pumping, the highest optical-to-optical conversion and slope efficiencies were increased over those for single-pass pumping, at 52.5% and 57.4%, respectively. These efficiencies are higher than those optimized at the maximum pump power of 10.0 W with the intrinsic residual loss of 4.0% in Figure 6a. The reasons why the highest efficiencies at the intrinsic residual loss of 8.0% in Figure 6c are higher than the highest efficiencies at the intrinsic residual loss of 4.0% shown in Figure 6a are as follows: First, the intrinsic residual loss of 8.0% in Figure 6c is twice the intrinsic residual loss of 4.0% in Figure 6a. In contrast, the maximum pump power of 25.0 W for the optimization condition in Figure 6c is 2.5 times the maximum pump power of 10.0 W for the optimization condition in Figure 6a. As indicated previously, in the analysis in Figure 6, the unsaturated gain is proportional to the pump power because the pump beam propagation is fixed. Thus, the ratio of the unsaturated gain of the optimization condition of 2.5 is higher than the ratio of the intrinsic residual loss of 2.0, so the highest efficiencies in Figure 6c are higher than the highest efficiencies in Figure 6a. As shown in Table 3, high-intensity pumping resulted in optimum coupling efficiencies of approximately 90% or higher, even at high loss conditions of 4.0% to 8.0%, and these high coupling efficiencies are one of the reasons for very high efficiencies. These high optical-to-optical conversion and slope efficiencies are obtained from the theory that has been confirmed reliable as explained above. Optimum pump and laser spot radii are easily obtained by adjusting the focusing optics and controlling the stability region of the cavity. Furthermore, as explained in Section 4.1, in high-intensity pumping, it is easy to increase the laser intensity inside the resonator at the optimum coupling efficiency, so that even if the spot radii deviate from the optimum values, the efficiencies decrease only in a range of a few percent, as shown in Figures 3 and 5. In other words, these efficiencies are sufficiently feasible experimentally because they are values obtained from our reliable theory, including the reduction in efficiency due to deviations from optimal conditions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Maximum** & **Intrinsic Residual** & **Pumping Pass** & **\(\eta_{\infty}\) (\%)** & **\(\eta_{s}\) (\%)** & **\(\eta_{c}\) (\%)** \\ \hline
10 W & 4.0\% & Single & 46.5 & 51.1 & 88.6 \\  & & Double & 51.2 & 57.4 & 89.1 \\  & & Single & 50.7 & 54.2 & 92.8 \\  & & Double & 55.9 & 59.6 & 93.1 \\
25 W & 8.0\% & Single & 47.5 & 52.1 & 89.8 \\  & & Double & 52.5 & 57.4 & 90.2 \\ \hline \hline \end{tabular}
\end{table}