## 2 Pumping Intensity Distribution and Derivation of the Spatial Rate Equations
### Computation of Spatial Overlap Efficiency  
The cross-sectional area of output laser \(A_{\rm e}\) and spatial overlap efficiency \(S\) are provided by [16; 17; 23; 24]. \(R(r,z)\) represents the spatial distribution of the pumping power throughout the gain rod. \(\phi(r,z)\) is the intensity distribution across the section of the laser-oscillated mode in the resonator.  
\[A_{\rm e}=[\iint R(r,z)\phi(r,z)\mathrm{d}x\mathrm{d}y]^{-1}, \tag{1}\]  
\[S=\frac{[\int R(r,z)\phi(r,z)\mathrm{d}x\mathrm{d}y]^{2}}{\int R(r,z)\phi^{2}( r,z)\mathrm{d}x\mathrm{d}y}. \tag{2}\]  
The previous section has already computed and simulated the distribution of the pumping power \(R(r,\,z)\). Therefore, the intensity distribution across the section of the oscillating laser mode needs to be determined utilizing Equations (3) and (4).  
\[\phi_{1}(r,z)=\frac{2}{\pi\omega_{1}^{2}(z)}\exp[-2\frac{r^{2}}{\omega_{1}^{2} (z)}], \tag{3}\]  
\[\omega_{1}(z)=\omega_{0}\sqrt{1+\left(z\lambda/\pi\omega_{0}^{2}\right)^{2}} \approx\omega_{0}. \tag{4}\]  
From Equations (3) and (4), the intensity distribution of the fundamental transverse mode is directly related to the beam waist size, which is determined by the parameters of the gain medium and the optical element positioned within the resonator. Thus, based on the primary parameters of the Q-switched Nd:YAG side-pump laser utilized in the experiment, the optical software can calculate the beam waist size of 2.09 mm. Figure 4 depicts the intensity distribution of the fundamental transverse mode \(\phi_{1}(r,z)\).  
Because side-pumping lasers typically exhibit high-order mode oscillation, it is necessary to calculate the intensity distribution of the high-order laser mode. The expression for high-order mode can be represented by Equation (5), where \(J_{0}\) is the Bessel function of the first order. The actual oscillated laser mode \(\phi(r,z)\) in a resonator of the side-pumped laser is the superposed intensity of the fundamental transverse mode \(\phi_{1}(r,z)\) and the higher-order mode \(\phi_{2}(r,z)\), as expressed by Equation (6).  
\[\phi_{2}(r,z)=A_{0}\exp[-(\frac{r}{\omega_{0}(z)})^{n}]J_{0}(\alpha r), \tag{5}\]  
\[\phi(r,z)=\phi_{1}(r,z)+\phi_{2}(r,z). \tag{6}\]  
Figure 4: The intensity distribution of fundamental transverse mode in the resonator.  
The intensity distribution after superposition is depicted in Figure 5. Compared to the distribution of the fundamental transverse mode \(\phi_{1}(r,z)\), the superposed intensity \(\phi(r,z)\) shifts to a particular level. A Gaussian distribution still characterizes its central region, while its periphery has an explicit intensity distribution.  
In previous calculations of spatial overlap efficiency, the distribution of pumping power in a side-pumped laser is approximated as a Gaussian distribution, and the high-order mode oscillation in a resonator is ignored. To guarantee the accuracy of computed results, our simulated outcomes that closely resemble the actual situation are wholly integrated into the subsequent calculation process, including the pumping power distribution \(R\) and oscillated laser mode \(\phi\).  
When \(R(r,z)\) and \(\phi(r,z)\) are plugged into Equations (1) and (2), the spatial overlap efficiency \(S\) and the output laser cross-sectional area \(A_{\text{e}}\) can be computed by mathematical software. Figure 6 shows the computed results of the cross-sectional area of the output laser and spatial overlap efficiency in a Nd:YAG side-pumped laser with various rod diameters. As the diameter of the Nd:YAG rod increases, so does the cross-sectional area of the output laser \(A_{\text{e}}\), but the spatial overlap efficiency \(S\) declines. Several of these computed results are applied to calculating the following spatial rate equations.  
Figure 5: The intensity distribution of oscillated mode in the resonator of the side-pumped laser.  
Figure 6: The spatial overlap efficiency \(S\) and the cross-sectional area of output laser \(A_{\text{e}}\) with the different Nd:YAG rod diameters.