# Four-Dimensional Thermal Analysis of 888 nm Pumped Nd:YVO4 Dual-Rod Acousto-Optic Q-Switched Laser
## 2 Theoretical Model  
### AO Q-Switched Laser Dynamics  
The energy levels and manifolds for trivalent neodymium in Nd:YVO\({}_{4}\) noted by corresponding wave numbers are depicted in Figure 1. In the Nd:YVO\({}_{4}\) 888 nm in-band pumping scheme, the ions in \(Z_{2}\) sublevel of ground state manifold \({}^{4}I_{9/2}\) are directly pumped to the laser upper level that \(R_{1}\) sublevel of manifold \({}^{4}F_{3/2}\) and lasing at 1064 nm, which is usually analyzed as a quasi-three-level system. According to our previous work [23], the rate equations including GSD, ETU, CR and fluorescence branching ratio are established as:  
\[\frac{\mathrm{d}n_{\mathrm{up}}(\mathbf{r},t)}{\mathrm{d}t}=W_{\mathrm{up}}n_{4}^{2}( \mathbf{r},t)-\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}, \tag{1}\]  
\[\frac{\mathrm{d}n_{4}(\mathbf{r},t)}{\mathrm{d}t}= R_{\mathrm{P}}(\mathbf{r},t)n_{0}(\mathbf{r},t)-\frac{n_{4}(\mathbf{r},t)}{\tau_{ 4}}\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}-2W_{\mathrm{up}}n_{4} ^{2}(\mathbf{r},t)-W_{\mathrm{cr}}n_{0}(\mathbf{r},t)n_{4}(\mathbf{r},t) \tag{2}\] \[-W_{41}(\mathbf{r},t)n_{4}(\mathbf{r},t)+W_{14}(\mathbf{r},t)n_{1}(\mathbf{r},t),\] \[\frac{\mathrm{d}n_{3}(\mathbf{r},t)}{\mathrm{d}t}=\beta_{43}\frac{n_{4 }(\mathbf{r},t)}{\tau_{4}}-\frac{n_{3}(\mathbf{r},t)}{\tau_{3}}+2W_{\mathrm{cr}}n_{0}( \mathbf{r},t)n_{4}(\mathbf{r},t), \tag{3}\]  
\[\frac{\mathrm{d}n_{2}(\mathbf{r},t)}{\mathrm{d}t}=\beta_{42}\frac{n_{4}(\mathbf{r},t) }{\tau_{4}}+\frac{n_{3}(\mathbf{r},t)}{\tau_{3}}-\frac{n_{2}(\mathbf{r},t)}{\tau_{2}}, \tag{4}\]  
\[\frac{\mathrm{d}n_{1}(\mathbf{r},t)}{\mathrm{d}t}= \beta_{41}\frac{n_{4}(\mathbf{r},t)}{\tau_{4}}+\frac{n_{2}(\mathbf{r},t)}{ \tau_{2}}-\frac{n_{1}(\mathbf{r},t)}{\tau_{1}}+W_{\mathrm{up}}n_{4}^{2}(\mathbf{r},t) \tag{5}\] \[+W_{41}(\mathbf{r},t)n_{4}(\mathbf{r},t)-W_{14}(\mathbf{r},t)n_{1}(\mathbf{r},t),\] \[\frac{\mathrm{d}n_{0}(\mathbf{r},t)}{\mathrm{d}t}=\beta_{40}\frac{n_{4 }(\mathbf{r},t)}{\tau_{4}}+\frac{n_{1}(\mathbf{r},t)}{\tau_{1}}-R_{\mathrm{P}}(\mathbf{r},t)n_{0}(\mathbf{r},t)-W_{\mathrm{cr}}n_{0}(\mathbf{r},t)n_{4}(\mathbf{r},t), \tag{6}\]  
where \(\mathbf{r}=(x,y,z)\) is the position vector in laser crystal, \(t\) is time, \(n_{i}\) and \(\tau_{i}(i=0,1,2,3,4)\) are the population density and the lifetime of the lowest five manifolds, respectively. \(n_{\mathrm{up}}\) and \(\tau_{\mathrm{up}}\) are the population density and lifetime of ETU manifolds, respectively. \(\beta_{4i}(i=0,1,2,3)\) is the fluorescent branching ratios from the fourth manifold to the \(i\)-th manifold, \(R_{\mathrm{P}}\) is the pumping rate, \(W_{\mathrm{up}}\) is the upconversion coefficient, \(W_{\mathrm{cr}}\) is the cross relaxation coefficient, and \(W_{41}\) (\(W_{14}\)) are stimulated radiation (absorption) rate. The total density of Nd\({}^{3+}\) ions \(n_{\mathrm{tot}}=n_{0}+n_{1}+n_{2}+n_{3}+n_{4}+n_{\mathrm{up}}\). For each energy manifold lifetime \(\tau_{i}\), we have [24],  
Figure 1: Energy sublevels and main transitions of Nd\({}^{3+}\) ions doped in YVO\({}_{4}\) crystal. \[\frac{\mathrm{d}n_{\mathrm{up}}(\mathbf{r},t)}{\mathrm{d}t}=W_{\mathrm{up}}n_{4}^{2}( \mathbf{r},t)-\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}, \tag{1}\]0  
where \(\tau_{i,\text{sp}}\) (\(\tau_{i,\text{nr}}\)) is the spontaneous radiation (nonradiative) transition lifetime of the \(i\)-th manifold, and the total transition rate is equal to the sum of spontaneous radiation rate and nonradiative transition rate. According to [PERSON]'s theory of stimulated emission, \(W_{41}=W_{14}=W_{\text{s}}\).