## 2 Materials and Methods
### Theory
The absorption coefficient was obtained from the experiment. As for the laser distribution function \(\phi_{\mathrm{o}}\), it was assumed to be a Gaussian beam profile with the same focusing position as the pump beam and beam qualities \(M^{2}\) of 1.0. Since gain saturation must occur when the laser oscillates, the dependence of the laser distribution function on the direction of the beam propagation was assumed to be constant. The laser output power \(P_{\mathrm{o}}\) and optical-to-optical conversion efficiency \(\eta_{\mathrm{oo}}\) as functions of the incident pump power \(P_{\mathrm{P}}\) were obtained by using these equations. Therefore, the pump and laser distribution dependence of the laser output power \(P_{\mathrm{o}}\) and optical-to-optical conversion efficiency \(\eta_{\mathrm{oo}}\) can be obtained from the theory. This means that, more specifically, we can obtain the pump and laser spot radii dependence of the laser output power \(P_{\mathrm{o}}\) and optical-to-optical conversion efficiency \(\eta_{\mathrm{oo}}\). While the normalized intracavity laser power \(S\) is zero at the pump threshold \(P_{\mathrm{th}}\), the normalized pump threshold \(F_{\mathrm{th}}\) is obtained from Equation (8) as  
Figure 1:0  
where \(V_{\mathrm{eff}}\) is the effective mode volume between the pump and laser beams in the gain medium and is defined by  
Figure 1:1  
From Equation (9), the pump threshold \(P_{\mathrm{th}}\) is obtained as  
Figure 1:2  
Equation (12) shows the pump threshold \(P_{\mathrm{th}}\) is proportional to the total loss \(L\) and the effective mode volume \(V_{\mathrm{eff}}\), as is well known. The physical parameters of the gain medium, e.g., the emission cross-section \(\sigma_{\mathrm{em}}\) of 28 \(\times\) 10\({}^{-20}\) cm\({}^{2}\) and the lifetime \(\tau\) of 3.2 \(\upmu\)s, rely on the material data from [35]. The crystal length, Ti\({}^{3+}\)-ion doping concentration and FOM were identical to those in the experiment (measured by the manufacturer). The intrinsic residual loss of the laser cavity \(L_{\mathrm{i}}\) used in the analysis was 4.0%, which was obtained from the Caird analysis [36]. The reasons for using the result of the Caird analysis are as follows: In general, there are two known methods for obtaining the intrinsic residual loss of the laser cavity, one by Findlay and Clay [37] and the other by Caird [36]. The Findlay and Clay analysis uses the assumption that the effective mode volume of the pump and laser beams remains unchanged when the pump threshold is changed by changing the cavity loss. The Caird analysis, on the other hand, uses the assumption that changing the coupling efficiency changes the slope efficiency, which in turn changes the laser intensity inside the cavity, but does not change the mode-matching efficiency. As explained below in Section 3, in general, the effect of thermal lensing on beam propagation is often non-negligible under high-intensity pumping. Therefore, the effective mode volume may change significantly depending on the pump power. Hence, changing the output coupler transmittance to change the cavity loss and the pump threshold could also change the effective mode volume. This means that the conditions that satisfy the reliability of the Findlay and Clay analysis may not be maintained. On the other hand, the slope efficiency is almost constant in our experiment, as shown below in Figure 2, even though the beam propagation could vary due to thermal lensing effects. This means that even if the laser intensity in the gain medium could vary due to variation in beam propagation, the mode-matching efficiency remained almost constant. In this case, the laser intensity in the gain medium is changed also by changing the coupling efficiency by changing the output coupler transmittance; however, the mode-matching efficiency is expected to remain almost constant. Namely, in our experiments, the conditions that satisfy the reliability of the Caird analysis are considered to hold, so we used the results of the analysis.