# Thermo-Optic Numerical Research on Segmented Circular LD Arrays Side-Pumping a Nd:YAG Laser Rod
## 3 Results and Discussion  
### Temperature Distributions  
The temperature distributions and values indicate the change in the refractive index, through which the thermo-optic effects are further investigated, and the gain distribution in the laser rod, which influences the laser beam quality [1]. Figure 7 shows the temperature distribution of the laser rod in the segmented scheme. The distribution was axisymmetric about the center of the laser rod, whose maximum temperature was 30.11 \({}^{\circ}\)C and the minimum temperature was 25 \({}^{\circ}\)C at the cooled cylindrical surfaces of the laser rod. The radial temperature gradient as a function of the length of the rod is illustrated in Figure 8.  
There were six peaks at the joints of the pump and holder segment as a result of the heat flow propagating along the length of the rod. Notably, the thermal resistance between the holders and the surface of the rod was not considered. As a result, some peaks are steep. The averaging temperature profile of the laser rod, obtained by averaging the temperatures of points along the length of the rod, showed a rather small radial temperature gradient of 0.35 \({}^{\circ}\)C/2.5 mm as shown in Figure 9. Figure 9 shows that the gain had a good Gaussian distribution due to the symmetrical pump arrangement. The segmented symmetrical side-pumping configuration induces heat flows to propagate along the \(z\)-axis of the laser rod rather than directly in the radial direction to the surface of the laser rod in the pumped segments because the boundary conditions on the pumped cylindrical surfaces of the laser rod are in an adiabatic condition (Neumann condition). The heat flows into the cooled segments and is removed by the segmented holders. So, the structure setup avoids higher temperature gradients in the radial direction. Figure 10 illustrates the averaging temperature profile of the laser rod in the annular liquid-cooling scheme. The distribution is symmetrically axial, with a maximum temperature of 25.314 \({}^{\circ}\)C, and the radial temperature gradient of 0.314 \({}^{\circ}\)C/2.5 mm is rather small. The radial temperature distribution revealed that the gain has a good Gaussian shape. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Boundary Conditions/Parameters** & **Segmented Model** & **Liquid-Cooling Model** & **Compensated Model** \\ \hline Number of semicircular LD & & & \\ modules & 6 & 6 & 6 \\ Total peak power of pump (W) & 6000 & 6000 & 6000 \\ Length of circular cool & \(3\times 8\) mm & 65 mm & — \\ Length of semicircular cool & — & — & \(2\times 32\) mm \\ Length of circular pump & \(3\times 10\) mm & \(3\times 10\) mm & — \\ Length of semicircular pump & — & — & \(2\times 30\) mm \\ Initial temperature conditions (\({}^{\circ}\)C) & 25 & 25 & 25 \\ Boundary conditions on cooled (\({}^{\circ}\)C) & 25 & 25 & 25 \\ surfaces of crystal & (Dirichlet condition) & (Dirichlet condition) & (Dirichlet condition) \\ Boundary conditions on pumped & Adiabatic condition & Adiabatic condition & Adiabatic condition \\ surfaces of crystal & (Neumann condition) & (Neumann condition) & (Neumann condition) \\ Thermal conductivity K(\(T\)) of crystal & 14 & 14 & 14 \\ (Wm\({}^{-1}\)K\({}^{-1}\)) & 1 & 1 & 1 \\ Number \(n\) of reflections in crystal & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The boundary conditions and parameters in the thermal simulation. Figure 8: The radial temperature gradient as a function of the length of the rod. Figure 7: The temperature distribution of the laser rod with the segmented scheme. Figure 9: The averaging temperature profile of the laser rod in the segmented side-pumping scheme. The compensated semicircular LD array side-pumping scheme has two identical pumps and heat-removal structures. The temperature profile of the laser rod in each pump and heat-removal structure has a circular asymmetrical distribution, as shown in Figure 11, but is symmetrical about the \(y\)-axis.