## 2 Materials and Methods
### Theory  
In the theoretical analysis, we used 4-level laser rate equations with CW pumping, which incorporate the spatial distribution of the pump and laser beams [32; 33; 34]. First, for simplicity, the incident pump power to the gain medium \(P_{\text{P}}\) is nondimensionalized as  
\[F\equiv\frac{2f\sigma_{\text{em}}\ \tau\ l\ \eta_{\text{P}}\ \eta_{\text{a}}\ P_{ \text{P}}}{h\
u_{\text{P}}\ L\ V_{\text{L}}}, \tag{1}\]  
Figure 1: Schematic diagram of the CW Ti:sapphire laser. The arm angles of concave mirrors \(M_{1}\) and \(M_{2}\) are \(\theta_{1}\) and \(\theta_{2}\), respectively. The distances from the end faces of the Ti:sapphire crystal and the concave mirrors \(M_{1}\) and \(M_{2}\) are \(L_{1}\) and \(L_{2}\), respectively. and the laser output power \(P_{\rm o}\) is also nondimensionalized as  
\[S\equiv\frac{\sigma_{\rm em}\ \tau\ l\ P_{\rm o}}{h\
u_{\rm L}V_{\rm L}\ T}, \tag{2}\]  
where the gain medium length is \(l\), pump quantum efficiency is \(\eta_{\rm p}\), pump photon energy is \(h\
u_{\rm p}\), laser photon energy is \(h\
u_{\rm L}\), output coupler transmission is \(T\), total loss of the laser cavity is \(L\), laser mode volume in the gain medium is \(V_{\rm L}\) and sum of the localized Boltzmann distribution of the upper and lower laser levels is \(f\). The optical-to-optical conversion efficiency for the incident pump power \(\eta_{\rm oo}\) is defined by the ratio between the laser output power \(P_{\rm o}\) and incident pump power \(P_{\rm p}\) and is obtained from the ratio between the normalized intracavity laser power \(S\) and normalized incident pump power \(F\) derived from Equations (1) and (2) as  
\[\eta_{\rm oo}\equiv\frac{P_{\rm o}}{P_{\rm p}}=\eta_{\rm c}\eta_{\rm p}\eta_{ \rm q}\eta_{\rm a}\frac{S}{F}, \tag{3}\]  
where the coupling efficiency between the laser output and intracavity laser powers \(\eta_{\rm c}\) is defined by  
\[\eta_{\rm c}\equiv\frac{T}{L}, \tag{4}\]  
and the atomic quantum efficiency \(\eta_{\rm q}\) is defined by  
\[\eta_{\rm q}\equiv\frac{h\
u_{\rm L}}{h\
u_{\rm p}}. \tag{5}\]  
The slope efficiency for the incident pump power \(\eta_{\rm s}\) is also defined by the derivative of the laser output power \(P_{\rm o}\) with respect to the incident pump power \(P_{\rm p}\) and is obtained from Equations (1) and (2) by  
\[\eta_{\rm s}\equiv\frac{dP_{\rm o}}{dP_{\rm p}}=\eta_{\rm c}\eta_{\rm p}\eta_{ \rm q}\eta_{\rm a}\eta_{\rm m}, \tag{6}\]  
where the mode-matching efficiency between the laser and pump beams \(\eta_{\rm m}\) is defined by the derivative of the normalized intracavity laser power \(S\) with respect to the normalized pump power \(F\) by  
\[\eta_{\rm m}\equiv\frac{dS}{dF}. \tag{7}\]  
From Equation (6), it can be seen that the slope efficiency \(\eta_{\rm s}\) is proportional to the coupling efficiency \(\eta_{\rm c}\) and mode-matching efficiency \(\eta_{\rm m}\), as is well known. During steady-state laser oscillation, from the condition that the power amplified inside the cavity is equal to the power lost, the relationship between the normalized pump power \(F\) and the normalized intracavity laser power \(S\) is obtained by  
\[F=\frac{1}{V_{\rm L}\int\frac{r_{\rm p}\ \dot{\phi}_{\rm o}}{1+fS\ \dot{\phi}_{\rm o}} dV}, \tag{8}\]  
where the pump distribution function \(r_{\rm p}\) and laser distribution function \(\phi_{\rm o}\) are normalized by the volume integral inside the gain medium as  
\[\int r_{\rm p}dV=\int\phi_{\rm o}dV=1. \tag{9}\]  
These distribution functions \(r_{\rm p}\) and \(\phi_{\rm o}\) are defined by the spot positions, radii and beam qualities of the pump and laser beams. In the analysis, the pump distribution function \(r_{\rm p}\) was assumed to be a Gaussian beam profile with the same spot position and beam qualities\(M^{2}\) as the experiment. Since the saturation of the pump absorption should disappear when the laser oscillates, the dependence of the pump distribution function on the direction of the beam propagation was assumed to decay exponentially.