# Thermo-Optic Numerical Research on Segmented Circular LD Arrays Side-Pumping a Nd:YAG Laser Rod
## 2 Materials and Methods
### Theory Models
Then, the power carried by the \(n\)th reflective wave at the reflective point of \(R_{N}\) is expressed as:  
\[P[y_{n}(\theta)]=\rho^{n}\mathbb{P}[y_{n-1}(\theta)], \tag{6}\]  
where \(P[y_{n-1}(\theta)]\) indicates the power of the incident wave at the reflective point \(R_{N}\). Line \(O_{1}I\) in Figure 3 indicates any incident ray from a laser diode whose light power \(P(\theta)\) is:  
\[P(\theta)=A\exp(-\frac{\theta^{2}}{\theta_{1/2}^{2}}), \tag{7}\]  
where \(\theta\) is the angle of the ray and the \(x\)-axis, \(\theta_{1/2}\) is half of the beam divergence angle in the direction of the fast axis, and \(A\) is a normalized coefficient:  
\[A=\frac{P_{0}}{\int\exp(-\frac{\theta^{2}}{\theta_{1/2}^{2}})}, \tag{8}\]  
where \(P_{0}\) is the power produced by a laser diode. Based on gridding the profiles of the laser rod, the absorbed power of the grids was calculated by tracing all rays emitted from around laser diodes within the beam divergence angle of \(-\theta_{1/2}\) to \(\theta_{1/2}\). Figure 4a shows a gridding profile of the laser rod and Figure 4b depicts the absorbed power of a grid from any ray of \(y_{n}(\theta)\). In Figure 4a, the laser rod cross-section is discretized as the grids of \(2\mathrm{N}\times 2\mathrm{N}\), and the length \(\delta\) of a grid is:  
\[\delta=\frac{r}{N}, \tag{9}\]  
In Figure 4b, the length \(d_{mn}(\theta)\) of the ray \(y_{n}(\theta)\) through any grid \(G_{mn}\) is calculated using the geometry method. The absorbed power of grid \(G_{mn}\) from the ray of \(y_{n}(\theta)\) is written in the form:  
\[P_{G_{mn}}[y_{n}(\theta)]=P_{IG_{mn}}[y_{n}(\theta)][1-\exp(-\alpha d_{mn}( \theta))], \tag{10}\]  
where \(P_{IG_{mn}}[y_{n}(\theta)]\) is the power carried by the ray of \(y_{n}(\theta)\) before inputting the grid \(G_{mn}\), and \(\alpha\) is the absorption coefficient of the laser rod. Then, the absorbed power of grid \(G_{mn}\) from a laser diode is obtained using the integral in the beam divergence angle:  
\[P_{G_{mn}}^{(i)}=\int_{-\theta_{1/2}}^{\theta_{1/2}}P_{G_{mn}}[y_{n}(\theta)]d\theta, \tag{11}\]where \(i\) is the mark number of a laser diode around the profile of the laser rod. The \(x\),\(y\)-coordinate system rotation around origin \(O\) is used to trace the rays from laser diodes not on the \(x\)-axis, and the rotation equations are written as:  
\[\left\{ \begin{array}{c}x_{\varphi}=-ysin(\varphi)+xcos(\varphi)\\ y_{\varphi}=ycos(\varphi)+xsin(\varphi)\end{array} \right., \tag{12}\]  
where \(\varphi\) is the rotation angle of the \(x\),\(y\)-coordinate system. There are \(j\) laser diodes around the profile of the laser rod. Thus, the inner heat source density profile \(Z_{0}\) along the length (\(z\)-direction) of the laser rod is expressed as:  
\[Q_{G_{mn}}\jmath_{z=Z_{0}}=(1-\eta)\tau\sum_{i=1}^{j}P_{G_{mn}}^{(i)}, \tag{13}\]  
where \(\eta\) is the quantum efficiency and \(\tau\) is the pump time. The temperature distributions of \(T(x,y,z)\) in an isotropic Nd:YAG rod is solved using the finite element analysis (FEA) method [13] from the heat equation:  
\[\rho c\frac{\partial T}{\partial t}=\
abla\cdot(K(T)\
abla T)+Q_{G_{mn}}, \tag{14}\]  
where \(\rho\) is the density, \(c\) is the specific heat, \(K(T)\) is the thermal conductivity, \(Q_{G_{mn}}\) is the inner-sources density, and \(\
abla\) denotes the gradient operator. In steady-state conditions, Equation (14) can be rewritten as:  
\[\
abla\cdot(K(T)\
abla T)+Q_{G_{mn}}=0. \tag{15}\]  
For Nd:YAG crystal, \(K(T)\) is normally assumed to be constant with moderate temperature increases. Under higher heat load conditions, \(K(T)\) is given in the first approximation [14] by:  
\[K(T)=K_{0}\frac{T_{s}}{T_{s}+\Delta T}, \tag{16}\]  
where \(T_{s}\) is the reference temperature of the cooling heat sinks, \(K_{0}\) is the heat conductivity at \(T_{s}\), and \(\Delta T\) is the difference in \(T_{s}\). Figure 4: (**a**) Gridding profile of laser rod; (**b**) absorbed power of a grid from any ray of \(y_{n}(\theta)\).