# Thermo-Optic Numerical Research on Segmented Circular LD Arrays Side-Pumping a Nd:YAG Laser Rod
## 2 Materials and Methods
### Theory Models  
A ray-tracing method was used to build an inner heat-sources model of the laser rod. A linear LD array has an effective emitting area of about 1.0 \(\upmu\)m by 4 mm. In the direction of the LD bar's fast axis, the pump beam has an approximately Gaussian shape and a larger beam divergence angle of 40\({}^{\circ}\). The laser diodes are considered as point light sources because of the small emitting width of 1.0 \(\upmu\)m. In our simulation, the pump beam parallel to a profile of the laser rod was discretized to be  
Figure 2: (**a**) The annular liquid-cooling configuration; (**b**) compensated semicircular LD arrays side-pumping arrangement. rays carrying energy along the beam divergence angle. In contrast, the pump beam in the direction of the LD bar's slow axis had an approximate flat-top shape and a smaller beam divergence angle of 10\({}^{\circ}\). The pump power distribution along the direction of the laser rod axis (the LD's slow axis) was deemed to be uniform within a pump segment because of the longer-width (10 mm) LD bars. Therefore, the inner heat-sources distribution was determined by tracing all pump rays in the laser rod cross-section. Figure 3 illustrates the ray-tracing in a laser rod cross-section from any ray in the LD beam divergence angle. The \(x\),\(y\)-coordinate system is established with the origin at the center of the laser rod, and the angles between rays and \(x\)-axis are defined as positive angles in the condition of counterclockwise rotation from \(x\)-direction to the direction of ray propagation. The point \(O_{I}\) indicates a laser diode emitter. Any ray of \(O_{I}I\) is refracted into the laser rod at the incidence point \(I\), and the coordinates \((x_{I},y_{I})\) are obtained by:  
\[\left\{ \begin{array}{c}x^{2}+y^{2}=r^{2}\\ y=tg(\theta)[x+(r+d)]\end{array} \right., \tag{1}\]  
where \(\theta\) is the angle of the ray from a laser diode and the \(x\)-direction, \(r\) is the radius of the laser rod, and \(d\) is the distance between \(O_{I}\) and the surface of the laser rod. With some geometry and the law of refraction, the angle of \(\alpha\) between the ray \(IR_{I}\) and the \(x\)-direction is solved by:  
\[\left\{ \begin{array}{c}\frac{sin(\theta)}{r}=\frac{sin(\phi)}{d+r}\\ \sin(\varnothing)=nsin(\theta^{\prime})\\ \alpha=\theta-(\phi-\theta^{\prime})\end{array} \right., \tag{2}\]  
where \(\varnothing\) and \(\theta^{\prime}\) are the incidence and refraction angles at the incident point \(I\), respectively. With the coordinates \((x_{I},y_{I})\) and the angle \(\alpha\), the refraction ray of \(IR_{I}\) is expressed as:  
\[y_{0}(\theta)=tg(\alpha)(x-x_{I})+y_{I}. \tag{3}\]  
The refraction ray \(y_{0}(\theta)\) is reflected first at point \(R_{1}\), second at \(R_{2}\), and \(n\)th at \(R_{N}\). With the same method, the coordinates of \(R_{1}(x_{1},y_{1})\), \(R_{2}(x_{2},y_{2})\), \(R_{N}(x_{N},y_{N})\), and the angles between the corresponding  
Figure 3: Schematic of ray-tracing. reflection ray and the \(x\)-direction are determined. Using the method of induction, the general expression of the reflected rays is written as:  
\[y_{n}(\theta)=tg(\alpha+2n\theta^{\prime})(x-x_{N})+y_{N}, \tag{4}\]  
where \(n\) is the number of rays reflected in the inner surface of the laser rod. According to the Fresnel formulae, the reflectivity \(\mathcal{R}\) of a laser rod's inner surface is solved by:  
\[\left\{ \begin{array}{c}\mathcal{R}_{\mathrm{s}}=\frac{sin^{2}(\theta^{ \prime}-\phi)}{sin^{2}(\theta^{\prime}+\phi)}\\ \mathcal{R}_{p}=\frac{tg^{2}(\theta^{\prime}-\phi)}{tg^{2}(\theta^{\prime}+ \phi)}\\ R=\mathcal{R}_{\mathrm{s}}sin^{2}(\beta)+\mathcal{R}_{p}cos^{2}(\beta)\end{array}  \right., \tag{5}\]  
where \(\theta^{\prime}\) and \(\phi\) are the incidence and refraction angles in the plane of incidence, respectively; \(\mathcal{R}_{\mathrm{s}}\) and \(\mathcal{R}_{p}\) are the reflectivity of \(s\) and \(p\) waves, respectively; and \(\beta\) is the angle of the E vector of the incident wave with the plane of incidence.