## 3 Experimental Results  
We measured the Ti:sapphire laser output power for various output coupler transmittances. To obtain the highest output power at the maximum incident pump power of 5.0 W with an output coupler transmittance of 10%, the laser cavity condition was optimized by adjusting the arm angles \(\theta_{1}\) and \(\theta_{2}\) and the distances from the crystal and mirrors \(L_{1}\) and \(L_{2}\), as shown in Figure 1. The laser output beams were measured to have an almost Gaussian beam profile. Figure 2 shows the laser output power as a function of the incident pump power. As shown in Figure 2, these functions representing the input-output characteristics of the laser are almost linear, and this linearity is evident at the output coupler transmittance of 10%. At the output coupler transmittance of 10%, a pump threshold of 1.19 W and maximum output power of 1.62 W were measured. The highest optical-to-optical conversion and slope efficiencies for the incident pump power were 32.4% and 42.5%, respectively, obtained at the maximum pump power. To increase the laser efficiencies, thermo-optic effects caused by the heat generated during the pumping process [38], which affect the beam propagation in the gain medium, should be considered. When considering the Taylor expansion of the thermo-optic distortion of the optical length as a function of distance from the optical axis, the thermal lensing effect is a second-order effect [39; 40]. In other words, thermal lensing is equivalent to an ideal quadratic distribution medium and does not degrade the beam quality of the Hermite-Gaussian mode, which is the eigenmode of the secondary distribution medium [27; 28; 29; 30; 31]. In our experiments, the laser beam radii inside the gain medium where the thermo-optic effect occurs are able to be considered to be less than several tens of micrometers (as mentioned below in Section 4). These radii are small enough that higher-order terms than the second-order term can be neglected. Furthermore, since the laser and pump beams can  
Figure 2: Laser output power as a function of the incident pump power for various output coupler transmittances (\(T=2\%\), \(5\%\) and \(10\%\)). The dashed lines represent the linear fit of the experimental data. be regarded as coaxial (if they were not coaxial, the input-output characteristics of the laser would not be almost linear as shown in Figure 2), the system is inversely symmetric with respect to this optical axis, so the odd-order terms in the distance can be ignored. Therefore, in our experiments, the thermo-optic distortion can be approximated by the lowest second order and can be considered as a thermal lensing effect. For reference, this approximation holds true whether the thermo-optic distortion is high or low. The effect of thermal lenses on the cavity mode is that of varying the beam diameter by changing or shifting the stability region of the cavity. Therefore, thermal lenses affect both the laser cavity and pump modes, but generally the effect on the cavity mode is higher. In general, for the above reasons, thermal lenses could affect the mode-matching efficiency between the laser and the pump beams. The slope efficiency is proportional to the mode-matching efficiency between the laser and the pump beams. Consequently, the thermal lenses could affect the slope efficiency. Therefore, in general, the slope efficiency is not constant when the thermal lens effect varies as the pump power increases. In our experiment, the maximum pump intensity was approximately 860 kW/cm\({}^{2}\), which corresponds to a thermal focusing length of a few millimeters when the gain medium is approximated by a thin lens and the intensity is extremely high in experiments with solid-state lasers, including Ti:sapphire lasers. In general, under these high pumping intensity conditions, the influence of the thermal lens effect on the laser cavity mode is considered to be often high.