# 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
\(R_{\text{p}}\) and \(W_{\text{s}}\) are given by:  
\[\frac{\mathrm{d}n_{\mathrm{up}}(\mathbf{r},t)}{\mathrm{d}t}=W_{\mathrm{up}}n_{4}^{2}( \mathbf{r},t)-\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}, \tag{1}\]1  
\[\frac{\mathrm{d}n_{\mathrm{up}}(\mathbf{r},t)}{\mathrm{d}t}=W_{\mathrm{up}}n_{4}^{2}( \mathbf{r},t)-\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}, \tag{1}\]2  
where \(c\) is the speed of light in vacuum, \(h\) is the Planck constant, \(\sigma_{\text{ap}}\) (\(\sigma_{\text{es}}\)) is the effective absorption (emission) cross section of pump light, \(\
u_{\text{p}}\) (\(\
u_{\text{s}}\)) is the frequency of pump light (laser) that is equal to \(c/\lambda_{\text{p}}\) (\(c/\lambda_{\text{s}}\)), \(I_{\text{p}}^{+}\) (\(I_{\text{p}}^{-}\)) is the intensity of forward (backward) propagating pump light, \(I_{\text{s}}^{+}\) (\(I_{\text{s}}^{-}\)) is the intensity of forward (backward) propagating laser, and \(n_{\text{c}}\) is refractive index of gain medium. For a-cut uniaxial crystal Nd:YVO\({}_{\text{4}}\) used in this work, the tiny difference of refractive indexes of ordinary light and extraordinary light can be neglected, thus we regard \(n_{\text{c}}=(n_{\text{o}}+n_{\text{e}})/2\) approximately. The spatial and temporal distributions of \(\sigma_{\text{ap}}\) and \(\sigma_{\text{es}}\) are nonuniform due to the influences of the temperature and heat generation distributions and the relationship has been detailedly studied in our previous work, the SDTI model [23]. The cavity, gain medium geometry and the corresponding coordinate frames are depicted in \[\frac{\mathrm{d}n_{\mathrm{up}}(\mathbf{r},t)}{\mathrm{d}t}=W_{\mathrm{up}}n_{4}^{2}( \mathbf{r},t)-\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}, \tag{1}\]3 Supposing that the propagation direction of the laser and pump beam is the \(z\)-axis direction, the conversion relations among intensity \(I_{\text{s}}^{\pm}\) (\(I_{\text{p}}^{\pm}\)), photon density \(\phi_{\text{s}}^{\pm}\) (\(\phi_{\text{p}}^{\pm}\)), and power \(P_{\text{s}}^{\pm}\) (\(P_{\text{P}}^{\pm}\)) along the propagation direction can be obtained as:  
\[\frac{\mathrm{d}n_{\mathrm{up}}(\mathbf{r},t)}{\mathrm{d}t}=W_{\mathrm{up}}n_{4}^{2}( \mathbf{r},t)-\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}, \tag{1}\]4  
\[\frac{\mathrm{d}n_{\mathrm{up}}(\mathbf{r},t)}{\mathrm{d}t}=W_{\mathrm{up}}n_{4}^{2}( \mathbf{r},t)-\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}, \tag{1}\]5  
where \(\omega_{\text{p}}^{+}\) (\(\omega_{\text{p}}^{-}\)) is the radius of forward (backward) pump beam, and \(\omega\) is the radius of laser beam. The distributions of pump and laser beams has been studied in previous work [14; 23]. In the gain medium, the coupled rate equations for laser and pump photon density propagation are given by:  
\[\frac{\mathrm{d}n_{\mathrm{up}}(\mathbf{r},t)}{\mathrm{d}t}=W_{\mathrm{up}}n_{4}^{2}( \mathbf{r},t)-\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}, \tag{1}\]6  
\[\frac{\mathrm{d}n_{\mathrm{up}}(\mathbf{r},t)}{\mathrm{d}t}=W_{\mathrm{up}}n_{4}^{2}( \mathbf{r},t)-\frac{n_{\mathrm{up}}(\mathbf{r},t)}{\tau_{\mathrm{up}}}, \tag{1}\]7  
where \(S\) is the rate at which spontaneous emission is added to the laser emission and population inversion \(\Delta n=n_{4}-n_{1}\). The value of \(S\) is a very tiny at about \(10^{-11}\) order in this system according to previous research [22]. Equations (12) and (13) are used for solving the power distributions in crystals. Actually, the speed of light is much larger than the speed of producing a Q-switched pulse. The intracavity photon density can be defined as \(\phi_{\text{s}}(t)=\int\!\!\int\phi_{\text{s}}^{+}(\mathbf{r},t)+\phi_{\text{s}}^{ -}(\mathbf{r},t)\text{d}V/V_{\text{m}}\), where volume differential \(\text{d}V=\text{d}x\text{d}y\text{d}z\) and the mode volume \(V_{\text{m}}=\int_{0}^{\dagger}\pi\omega^{2}(z)\text{d}z\).