## 4 Discussion
### Theoretical Analysis for Higher Efficiency  
As explained above, our theory was in good agreement with the experimental results, and the discrepancy between the theoretical and experimental results was explained by the difference in the laser spot radii. In other words, we succeeded in confirming the reliability of our theory. From these results, we are able to provide a reliable theoretical analysis of how to further improve the efficiency of SHG laser-pumped CW Ti:sapphire lasers through optimization by high-intensity pumping. This analysis was conducted under the same crystal conditions and beam qualities as above. The pump spot radii, which were chosen to minimize the effective mode volume for high-intensity pumping, were 12.2 m for both the vertical and horizontal axes. The pump absorption efficiencies were selected to be 90.5% for the single-pass pumping, the same as in the experiment, and 99.1% for the double-pass pumping. The intrinsic residual losses were selected to be 4.0%, the same as that obtained in the experiment, and 8.0%, twice as high as that obtained in the experiment. The reason for assuming twice as much loss as in the experiment is to include loss due to the insertion of wavelength-selective elements and mode-locking. In general, these losses can be kept within the range of assumed loss values with sufficient adjustments. The maximum pump powers were selected to be 10.0 W, the maximum output power of Spectra-Physics Millennia Pro, and 25.0 W, the maximum output power of Spectra-Physics Millennia eV [41]. Figure 6 compares the input-output characteristics for single-pass pumping and double-pass pumping under these conditions, and Table 3 compares the corresponding highest efficiencies. In Figure 6, the intrinsic residual loss of the laser cavity is 4.0%, and we optimized the output coupler transmittance and the laser spot radii to improve the optical-to-optical conversion efficiency at the maximum pump power of 10.0 W. An output coupler transmittance of 31.0% was selected for the single-pass pumping, and 32.6% was selected for the double-pass pumping. The laser spot radii were selected to be 14 m for both the vertical and horizontal axes for both pumping conditions. For the single-pass pumping, the highest optical-to-optical conversion and slope efficiencies were obtained at the maximum pump power and were 46.5% and 51.1%, respectively. For the double-pass pumping, the highest optical-to-optical conversion and slope efficiencies were increased over those for single-pass pumping, at 51.2% and 57.4%, respectively. Figure 6: _Cont._Figure 6: Theoretical laser output powers and optical-to-optical conversion efficiencies as functions of the incident pump power. The thin blue dotted lines and thin red solid lines show the optical-to-optical conversion efficiency and output power for single-pass pumping, respectively. The bold gray dotted lines and bold orange solid lines show the optical-to-optical conversion efficiency and output power for double-pass pumping. The pump spot radii were optimized to obtain the minimum mode volume of the pump beam, which was 12.2 μm in both the vertical and horizontal directions. The laser spot radii were also optimized to obtain the highest optical-to-optical conversion efficiency at the maximum pump powers, which were 14 μm in both the vertical and horizontal directions. The optimization conditions for (**a**) were 10 W maximum pump power and 4.0% intrinsic residual loss, for (**b**) were 25 W maximum pump power and 4.0% intrinsic residual loss and for (**c**) were 25 W maximum pump power and 8.0% intrinsic residual loss. In Figure 6b, the intrinsic residual loss is 4.0%, the same in Figure 6a, but we optimized at the higher maximum pump power of 25.0 W. We selected an output coupler transmittance of 51.3% for the single-pass pumping and 53.9% for the double-pass pumping. The highest optical-to-optical conversion and slope efficiencies were increased over those optimized at the maximum pump power of 10.0 W in Figure 6a.