# Thermo-Optic Numerical Research on Segmented Circular LD Arrays Side-Pumping a Nd:YAG Laser Rod
## 2 Materials and Methods
### Simulation Methods and Boundary Conditions  
The details of the simulation methods and processes are described in the following. Firstly, we used MATLAB software (MathWorks, Natick, MA, USA) to calculate the inner heat source density \(Q_{G_{mn}}\) of the laser rods in three different side-pumping schemes according to the same boundary conditions with the help of the derived Equation (13). Secondly, the temperature distributions of \(T(x,y,z)\) in the laser rods were analyzed and discussed by deriving Equation (14) considering heat conduction using FEA in ANSYS software (ANSYS, Canonsburg, PA, USA), because the major contribution to the thermal lens was from the temperature-dependent variation of the refractive index. Thirdly, MATLAB software (MathWorks, Natick, MA, USA) was used again to obtain the change \(\Delta n(x,y,z)_{T}\) in the refractive index in the laser rod using Equation (17) and the complex amplitude equivalent transfer function \(t(x,y)\) of laser rod by deriving Equation (20). Next, a planar wave was employed to measure the thermal lensing in the laser rod. The wave-front distortion \(U(x^{\prime},y^{\prime})\) after a planar wave passing through the laser rods was determined using Equation (21) in MATLAB (MathWorks, Natick, MA, USA). Finally, the thermo-optic effects in the laser rods were further evaluated in three different side-pumping schemes. In our simulations, the variations in the refractive index caused by thermal stress were not considered due to the minimal influence in low- and medium-power applications. In addition, the end effects were ignored because their influence is rather limited in the side-pumping configuration. To facilitate comparison, it was necessary to set the same boundary conditions in the three different side-pumping configurations. In the side-pumping configurations models, the same Nd:YAG rods with a diameter of 5 mm and length of 65 mm were employed. The properties of the Nd:YAG used in the simulation are listed in Table 1.  
The same semicircular LD array modules were used in the three different side-pumping schemes. A three-dimensional (3D) geometrical model of semicircular LD arrays is shown in Figure 6.  
Ten bars 10 mm in length and 1.4 mm in width were fixed on the surface with a 5 mm radius of curvature. A bar of peak power of 100 W contained 10-20 independent laser emitters and the laser output produced by an emitter was emitted from a \(200\times 1.0~{}\upmu\)m area with a \(40^{\circ}\) fast axis \(\times 10^{\circ}\) slow axis beam divergence. The peak power output of a semicircular LD array is about 1000 W and six semicircular LD arrays were employed in the three different side-pumping schemes, corresponding to a total peak power of 6000 W. The LD arrays with a center wavelength of 808 nm worked at a pumping rate of 10 Hz with a pump pulse duration of 230 \(\upmu\)s, and the wavelength changed with the temperature \begin{table}
\begin{tabular}{c c} \hline \hline
**Properties/Parameters** &
\begin{tabular}{c} **Neodymium Doped Yttrium** \\ **Aluminum Garnet (Nd:YAG)** \\ \end{tabular} \\ \hline Diameter (mm) & 5 \\ Length (mm) & 65 \\ Dopant concentration (atm \%) & 1.0 \\ Absorption coefficient \(\alpha\) (m\({}^{-1}\)) & 400 \\ Density \(\rho\) (gcm\({}^{-1}\)) & 4.56 \\ Specific heat c (Wsg\({}^{-1}\)K\({}^{-1}\)) & 0.59 \\ Refractive index n\({}_{0}\) & 1.86 \\ Change in refractive index with temperature \(\frac{d\mu}{dT}\) (K\({}^{-1}\)) & \(7.3\times 10^{-6}\) \\ Heat conductivity K\({}_{0}\) (Wm\({}^{-1}\)K\({}^{-1}\)) & 14 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The properties of the Nd:YAG used in the simulation. at 0.3 nm/\({}^{\circ}\)C. The copper base on which the semicircular LD arrays were mounted was a cuboid with a length of 20 mm, a width of 10 mm, and a height of 15 mm, with screws on the heat sink.