# Modeling and Analysis of High-Power Ti:sapphire Laser Amplifiers-A Review
## 3 Phase Distortion Analysis
### Thermal Birefringence  
Leading research groups are focused on improving the repetition rate and pulse quality instead of successfully generating PW-class pulses [3; 4; 5; 6; 9; 10; 11]. Applications of ultra-high-power lasers require extraordinarily high quality, and it is necessary to increase the repetition rate to use them practically [8; 12]. The thermal distortion that occurs in this process is one of the major issues to overcome. Ultra-high-power laser amplifiers use high-energy lasers as the source of the pump energy, imposing high thermal loads on the medium. This thermal distortion effect is one of the major causes of the degradation of the spatio-temporal quality of ultra-high-power laser pulses [21]. Because the non-involvement of the energy in the amplification process is a reason of the thermal effect, it becomes serious with the repetition rate. If a compact design for improved usability is applied, then the thermal problem becomes even more severe. To be active in an actual industrial field at the laboratory level, a solution to the thermal distortion problem should be developed first. In this section, we present the results of a study on the thermal effects of ultra-high-power laser systems that our group has conducted [21]. We will introduce the photo-elastic (PE) effect, which is relatively disregarded among the causes of the thermal distortion occurring in a CPA system using Ti:sapphire as the medium. The PE effect is neglected in most cases because the degree of distortion is not significant compared to other thermal distortions. However, the distortion due to the PE effect cannot be ignored under the severe thermal conditions under which ultra-high-power laser pulses reaching PW at a repetition rate of several tens of Hz or more are generated. Thermal birefringence is caused by thermal stress [1]. Thermal stress and the relative impermeability (B) are related to the \(\pi\)-tensor (piezo-optical tensor, Figure 13) [60; 61]. The square root of \(B\) is the refractive index. The refractive index change induced by the heat can be expressed by Equations (24)-(26),  
\[\Delta B_{m}=\pi_{mn}\sigma_{n}, \tag{24}\]  
\[\Delta B_{p}=2\sqrt{\left(\frac{\Delta B_{xx}+\Delta B_{yy}}{2}\right)^{2}+ \Delta B_{xy}^{2}}, \tag{25}\]  
\[\beta=\frac{1}{2}\tan^{-1}\frac{2\Delta B_{xy}}{\Delta B_{yy}-\Delta B_{xx}}, \tag{26}\]  
\[\Delta k_{t}=\frac{2\pi}{\lambda}\left(\frac{n^{3}\Delta B_{p}}{2}\right), \tag{27}\]  
where \(\pi\)-tensor is a piezo-optical tensor, \(\sigma\) is the thermally induced stress, \(\beta\) is the angle between the principal axis of the thermal stress and Cartesian axis. Figure 12: (**a**) The output spectrum of the amplifier chain and the accumulated atomic phase shift, (**b**) the compressed pulse distortion due to the atomic phase shifts. However, because the deformation of a beam experiences not only thermal birefringence but also intrinsic birefringence and the amplification effect of the medium, they must be considered simultaneously [21]. To model the complete PE effect, the simultaneous consideration of the amplification of the medium and intrinsic birefringence is required. Therefore, we use the Jones matrix (Equations (28)-(30)) to express the instantaneous change so that the three effects can be considered simultaneously in the medium. \[A_{gain}= \begin{pmatrix}e^{\frac{ik_{c}}{2}z}&0\\ 0&e^{\frac{ky_{c}}{2}z}\end{pmatrix}  \tag{28}\]  
\[A_{c}= \begin{pmatrix}e^{i\frac{k_{c}}{2}z}&0\\ 0&e^{-i\frac{k_{c}}{2}z}\end{pmatrix}  \tag{29}\]  
\[A_{t}= \begin{pmatrix}\cos\beta&\sin\beta\\ -\sin\beta&\cos\beta\end{pmatrix}\begin{pmatrix}e^{i\frac{k_{t}}{2}z}&0\\ 0&e^{-i\frac{k_{t}}{2}z}\end{pmatrix}\begin{pmatrix}\cos\beta&-\sin\beta\\ \sin\beta&\cos\beta\end{pmatrix}  \tag{30}\]  
Using the above matrices, a first-order differential equation can be obtained, as expressed in Equation (32). By solving Equation (32), we were able to obtain the shape of the phase distortion due to the thermal birefringence.