## 2 Pumping Intensity Distribution and Derivation of the Spatial Rate Equations
### Solution of Spatially Coupled Rate Equations  
The Q-switching process in the resonator is divided into two processes. When the pump source begins pumping (the Q-switch is off), laser oscillation cannot be generated, and the population inversions in the upper level of the gain medium can accumulate in large quantities. The time that energy can be stored is determined by the lifetime of the upper energy level of the laser medium. When the Q-switch turns on, the stimulated emission quickly intensifies. In a brief period, population inversions are consumed, converted into laser energy, and a laser pulse is emitted through the coupled mirror.  
The gain medium utilized in the laser is the Nd:YAG crystal, which is a four-level atomic system. The four-level Q-switched rate equations are employed to depict the process of generating laser output pulses. The spatial rate equations for a four-level AQS operation are provided by  
\[\frac{dN}{dt}=-B\Phi N, \tag{7}\]  
\[\frac{d\Phi}{dt}=[BA_{\mathrm{e}}lN-\frac{1}{\tau_{\mathrm{c}}}]\Phi, \tag{8}\]  
where \(N\  
In previous models [25; 28], the output pulse energy \(E_{\text{out}}\), peak power \(P_{\text{out}}\), and pulse width \(\Delta\tau_{\text{p}}\) are provided by  
\[E_{\text{out}}=\frac{SA_{\text{e}}h\upsilon}{2\sigma\gamma}[N_{\text{i}}-N_{ \text{f}}], \tag{12}\]  
\[P_{\text{out}}=\frac{\gamma_{2}SA_{\text{e}}h\upsilon}{2\sigma\tau_{\text{c}}} \bigg{[}\frac{N_{\text{i}}}{N_{\text{p}}}-\ln(\frac{N_{\text{i}}}{N_{\text{p}} })-1\bigg{]}, \tag{13}\]  
\[\Delta\tau_{\text{p}}=\tau_{\text{c}}\frac{(N_{\text{i}}-N_{\text{f}}/N_{\text {p}})}{[(N_{\text{i}}/N_{\text{p}})-\ln(N_{\text{i}}/N_{\text{p}})-1]}. \tag{14}\]  
The side-pumped laser's gain medium is Nd:YAG rod, with the stimulated emission cross section \(\sigma\) = 2.8 \(\times\) 10\({}^{-23}\) m\({}^{2}\) (1064 nm) and photon energy \(h\upsilon\) = 1.87 \(\times\) 10\({}^{-19}\) J (1064 nm). The effective cavity length is approximately \(L_{\text{e}}\) = 350 mm, and the Nd:YAG rod length is \(l\) = 105 mm. The transmittance of the output coupling mirror is T = 30%, and the output coupling mirror loss \(\gamma_{2}=-\ln(1-T_{2})\) = 0.35. The loss factor of the resonator is \(\gamma=0.1+\gamma_{2}/2=0.28\) when the Q switch is on. In addition, the spatial overlap efficiency and output spot area at varying Nd:YAG rod diameters have previously been computed. Consequently, the functional relationship between single pulse energy, pulse width, peak power, and average pump power (808 nm laser) is shown in Figure 7 by plugging the values of the related parameter.  
The linear relationship between the single pulse energy of the Q-switched pulse and the average pump power is observed. The pulse width is approximately inversely proportional to the average pump power. Peak power initially increases logarithmically  
Figure 7: Computed output laser performance based on spatial rate equations versus average optical pumping power for various Nd:YAG rod diameters. (**a**) Single pulse energy versus average pump power; (**b**) pulse width versus average pump power; (**c**) peak power versus average pump power.  
and then approaches a linear relationship with the average pump power. As the diameter of the Nd:YAG rod increases, the threshold pump power of the laser drops, and the slope of the single pulse energy increase as the pump power rises. Despite this, installing the large Nd: YAG rod in the side-pumped laser reduces the peak power and widens the pulse width of the output laser. The observed phenomenon can be attributed to the non-uniform distribution of pumping power intensity and the consequent reduction in spatial overlap efficiency. Detailed experiments will be conducted to verify the accuracy of the computed results.