# Compact 50 W All-Solid-State Picosecond Laser System at 1 kHz
## 2 Experiment and Results
### Side-Pumped Amplifier  
Since the power of the side-pumped module is extremely high, the thermal effect of the gain medium is relatively obvious. The non-uniform temperature distribution could cause thermally induced birefringence inside the crystal. For a Nd:YAG rod in which the z axis is in the [1 1 1] direction, the principle of axes of thermally induced birefringence are radial and tangential within the rod cross-section. Therefore, when the linear polarized beam traverses the gain medium, the tangential and radial components could experience different phase retardation, resulting in significant depolarization [15]. The compensation of birefringence is a significant issue for the Nd:YAG rod side-pumped amplifier [16; 17; 18; 19]. The Nd:YAG crystal is an isotropic medium. The refractive index ellipsoid is a regular sphere without thermally induced birefringence. The refractive index ellipsoid is  
\[B_{1}x_{1}^{2}+B_{2}x_{2}^{2}+B_{3}x_{3}^{2}=1, \tag{1}\]  
where the \(B_{i}=1/n_{i}^{2}=1/n_{o}^{2}\), \(i=1,2,3\) are principal refractive indexes. The refractive index ellipsoid with thermally induced birefringence is  
\[B_{11}x_{1}^{2}+B_{22}x_{2}^{2}+B_{33}x_{3}^{2}+2B_{23}x_{2}x_{3}+2B_{13}x_{1} x_{3}+2B_{12}x_{1}x_{2}=1, \tag{2}\]  
\[B_{ij}=\Delta B_{ij}+B_{0 ij}\,i,j=1,2,3. \tag{3}\]  
In the Equation (3), \(\Delta B_{ij}\) implies the change of index with thermally induced birefringence, and \(B_{011}=B_{022}=B_{033}=1/n_{o}^{2}\), \(B_{012}=B_{013}=B_{023}=0\). According to previous studies,  
\[\Delta B_{ij}=\pi_{ijkl}o_{kl}, \tag{4}\]  
where the elastomeric coefficient \(\pi_{ijkl}\) is a fourth-order tensor and the \(o_{kl}\) is deformation tensor. In order to simplify the calculation, the form converted into a matrix is  
\[\Delta B_{m}=\pi_{mm}o_{n},m,n=1,2,3,4,5,6. \tag{5}\]  
Figure 4: Power curve and beam image of end-pumped single-pass amplifier. The beam propagates along the z-axis direction, so there is no polarization component in the z-direction. Equation (2) can become  
\[\left( \begin{array}{cc}x_{1}&x_{2}\end{array} \right)\left( \begin{array}{cc}1/n _{0}^{2}+\Delta B_{1}&\Delta B_{6}\\ \Delta B_{6}&1/n_{0}^{2}+\Delta B_{2}\end{array} \right)\left( \begin{array}{c} x_{1}\\ x_{2}\end{array} \right)=1. \tag{6}\]  
The new principle refractive indexes can be given by the following formula:  
\[n_{\pm}=\frac{1}{\sqrt{\frac{1}{n_{0}^{2}}+\frac{1}{2}(\Delta B_{1}+\Delta B _{2})\pm\frac{1}{2}\sqrt{\left(\Delta B_{1}-\Delta B_{2}\right)^{2}+4\Delta B _{6}^{2}}}}. \tag{7}\]  
The angle \(\alpha\) between the axis with thermal birefringence and the original axis can be given by the formula:  
\[\tan(2\alpha)=\frac{2\Delta B_{6}}{\Delta B_{1}-\Delta B_{2}}. \tag{8}\]  
The difference of phase retardation between tangential and radial components is  
\[\delta=\frac{2\pi l}{\lambda}(n_{+}-n_{-}). \tag{9}\]  
By swapping the phases of the tangential and radial components of polarization and traversing the thermally-induced birefringent crystal again, an equal phase retardation can be achieved at each point within the crystal cross section. When the beam passes through the depolarized crystal for the second time, if the beam distribution at each point in the z direction is the same as the first, the depolarization can be perfectly compensated. Figure 5 shows the simulation patterns of thermal depolarization and perfect compensation for the side-pumped amplifier. The gain medium is equivalent to a thick lens in the side-pumped amplifier, and the collimated beam will become a convergent beam after passing through the crystal. A convergent beam can usually be collimated by a concave lens, however, the convergence angle is different at different pump powers in the side-pumped amplifier. To solve this problem, the side-pumped amplifier can apply the structure shown in Figure 6. The matrix after the beam passes through the gain crystal can be defined as  
Figure 5: Simulation patterns of thermal depolarization and compensation (**a**) without compensation; (**b**) perfect compensation.