# Four-Dimensional Thermal Analysis of 888 nm Pumped Nd:YVO4 Dual-Rod Acousto-Optic Q-Switched Laser
## 2 Theoretical Model
### Thermal Analysis  
The heat generation in laser crystal is caused by the total nonradiative transition processes, in which the power of photon is transformed into heat by multi-phonon relaxations [22; 24]. For precisely calculating spatial and transient heat generation, the coupled rate equations and heat conduction equation must be solved. The total thermal power density is the sum of products of energy gap between each pair of manifolds multiplying the corresponding nonradiative transition rate, that:  
\[Q(\mathbf{r},t)=\sum_{i}\sum_{j<i}\Delta E_{ij}\beta_{ij}\frac{n_{i}(\mathbf{r},t)}{ \tau_{i,{\rm nr}}}=\sum_{i}\sum_{j<i}hc\left(\widetilde{\lambda}_{i}- \widetilde{\lambda}_{j}\right)\beta_{ij}\frac{n_{i}(\mathbf{r},t)}{\tau_{i,{\rm nr }}}, \tag{17}\]  
where \(\Delta E_{ij}\) is the energy gap between manifolds \(i\) and \(j\). The energy of each manifold here is in the form of wavenumber, \(\widetilde{\lambda}=1/\lambda\). In the Nd:YVO\({}_{4}\) 888 nm in-band pumped scheme, the heat generation is mainly due to the decays from \({}^{4}I_{11/2}\) and ETU manifolds, nearly all of which are nonradiative transitions [6; 22]. The ETU induced downward transition is mainly to the \({}^{4}I_{11/2}\) manifold and the CR effect is much weaker than ETU so that the CR can be neglected here [20; 22; 26]. Thus, the thermal power density distribution can be simplified as:  
\[Q(\mathbf{r},t)=hc\left(\widetilde{\lambda}_{1}-\widetilde{\lambda}_{0}\right) \frac{n_{1}(\mathbf{r},t)}{\tau_{1}}+hc\left(\widetilde{\lambda}_{4}-\widetilde{ \lambda}_{1}\right)\frac{n_{\rm up}(\mathbf{r},t)}{\tau_{\rm up}}+hc\left( \widetilde{\lambda}_{4}-\widetilde{\lambda}_{f}\right)\frac{n_{4}(\mathbf{r},t)}{ \tau_{4}}, \tag{18}\]  
where \(\lambda_{f}\) is the average fluorescence wavelength. For anisotropy medium, the thermal conductivity is a tensor and we also considered temperature-dependent thermal conductivity; thus, the heat conduction equation is given by:  
\[\rho C_{\rm P}\frac{\partial T(\mathbf{r},t)}{\partial t}-\
abla\cdot\left( \begin{bmatrix} K_{x}(\mathbf{r},t)&0&0\\ 0&K_{y}(\mathbf{r},t)&0\\ 0&0&K_{z}(\mathbf{r},t)\end{bmatrix} \cdot\
abla T(\mathbf{r},t)\right)=Q(\mathbf{r},t), \tag{19}\]  
where \(\rho\) is the density of the medium, \(C_{\rm P}\) is the specific heat capacity of the medium, and \(K_{x}\), \(K_{y}\), \(K_{z}\) are the thermal conductivity along the three orthogonal principle axes. As for a-cut uniaxial crystal Nd:YVO\({}_{4}\) used in this work, the optical axis is set coincident with the \(x\)-axis, i.e., \(K_{x}=K_{c}\), \(K_{y}=K_{z}=K_{a}\), where \(K_{c}\) (\(K_{a}\)) is the thermal conductivity parallel (orthogonal) to the optical axis. The temperature distribution in the laser crystal is obtained by solving the coupled initial and boundary problems of parabolic partial differential equations (PDE) of heat conduction Equation (19). The initial temperature distribution in crystal is equal to environmental temperature \(T_{0}\); therefore, the initial condition is:  
\[T(\mathbf{r},0)=T_{0}. \tag{20}\]  
Because the convection heat transfer coefficient with air is much smaller than with heat sink [27], we can suppose that the crystals are approximately adiabatic with air at the end faces and have convection heat transfer coefficient \(H\) with heat sink at side faces. The boundary conditions are  
\[K_{x}\frac{\partial T}{\partial x}\Big{|}_{x=\frac{h}{2}}=H\left[T_{0}-T\left( \frac{h}{2},y,z\right)\right], \tag{21}\]  
\[-K_{x}\frac{\partial T}{\partial x}\Big{|}_{x=-\frac{h}{2}}=H\left[T_{0}-T \left(-\frac{h}{2},y,z\right)\right], \tag{22}\]  
\[K_{y}\frac{\partial T}{\partial y}\Big{|}_{y=\frac{w}{2}}=H\left[T_{0}-T\left( x,\frac{w}{2},z\right)\right], \tag{23}\]  
\[-K_{y}\frac{\partial T}{\partial y}\Big{|}_{y=-\frac{w}{2}}=H\left[T_{0}-T \left(x,-\frac{w}{2},z\right)\right], \tag{24}\]  
\[\frac{\partial T}{\partial z}\Big{|}_{z=0}=\frac{\partial T}{\partial z}\Big{|} _{z=l_{c}}=0, \tag{25}\]  
where \(w\) and \(h\) are width and height of crystal.