## 2 Pumping Intensity Distribution and Derivation of the Spatial Rate Equations  
### The Simulation of the Pumping Intensity Distribution in the Side-Pumped Laser Module  
The structural diagram of the five-dimensional laser diode (LD) side-pumped module is depicted in Figure 1. In the module's middle portion, the laser medium is enclosed by a transparent quartz flow tube. The deionized water serves as a cooling agent in the quartz flow tube, and the outer portion of the quartz flow tube is a highly reflecting material condenser [25; 26]. Five LD arrays are outside the condenser to pump the 808 nm laser into the laser medium. The different structural components and geometric distances of the LD side-pumped module result in various distributions of the pumping power across the gain medium. Hence, the structural parameters of the commercial LD side-pumped module from Table 1 are considered during the simulation. The research by Dr. [PERSON] presents a method to simulate the pump power intensity distribution of the side pump module, which is based on the fundamental principle of the side-pumping laser module and the ray tracing algorithm [27]. However, the crystal rod in that study is in saturation-pumping absorption when the side-pumped module begins pumping. In other words, the pumping power distribution simulated in that research is actually intended for the side-pumped laser module as an amplifier rather than an oscillation. Our paper focuses on analyzing the matching efficiency between the side-pumped laser's pumping beam and the resonator's oscillating laser. Hence, the crystal rod in the unsaturation-pumping absorption is considered, and the pumping power distribution is stimulated when the side pump module is utilized as the laser oscillation. The improved algorithm is utilized to simulate the distribution of pumping power, and the simulation program is executed accordingly. The distribution of pumping power intensity across the 5 mm-diameter Nd:YAG rod in the side-pumped module is shown in Figure 2. The finite element mesh used in the simulation consists of \(500\times 500\) elements. For the convenience of presentation, the axes are converted to millimeter units. In addition, it is normalized and plotted to represent the cross-sectional area of a Nd:YAG rod for the follow-up comparison and computation. The simulated distribution of pumping power intensity \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Numerical Values** \\ \hline Number of pumping directions & 5 \\ \hline Distance between LD bars and gain medium center & 6.1 mm \\ Manufacturer of LD bars & Coherent Inc. \\ LD bar model & 100 W 50 \(\mathrm{FF}\times 2\) mm TE \\ Fast axis divergence of LD bars (1/e\({}^{2}\)) & \(\sim\)50\({}^{\circ}\) \\ The refractive index of the gain medium & 1.82 \\ The refractive index of the cooling liquid & 1.33 \\ The refractive index of the flow tube & 1.45 \\ The inner diameter of the flow tube & 5.9 mm \\ The wall thickness of the flow tube & 1.67 mm \\ Effective pumping length & \(\sim\)50 mm \\ Rated centroid pumping wavelength & 808 nm \\ Laser medium & Nd:YAG \\ The doping concentration of the laser medium & 0.6 at.\% \\ Diameter of the medium gain rod & 3 mm, 4 mm, 5 mm, 6 mm \\ Gain medium-side surface diffusion (divergence) & \(\sim\)15\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of commercial side-pumping modules. Figure 1: The cross-section structure diagram of 5-dimensional laser diode (LD) side-pumped modules. across the five mm-diameter Nd:YAG rod findings suggest that the crystal rod's pumping power distribution is non-uniform and non-Gaussian, with an irregular intensity profile. A series of simulated pumping power distributions across the crystal rod is obtained by varying the diameter of the Nd:YAG rod while keeping all other parameters constant.