# Modeling and Analysis of High-Power Ti:sapphire Laser Amplifiers-A Review
## 3 Phase Distortion Analysis
### Thermal Lens Model  
Methods such as optical parametric amplification (OPA) are being attempted to eliminate the cause of the thermal problems. However, the limit of the power is insufficient compared to that in CPA [65; 66]. Therefore, even if OPA is used, a hybrid-CPA method using CPA together for generating PW-level pulses is showing its potential. Most CPA laser systems, the medium for generating ultra-high-power pulses is Ti:sapphire, which has excellent mechanical and thermal durability, and so, there is not much demand for research on the thermal distortion effect. In comparison, thermal distortion was a significant issue in the early solid-state laser systems such as Nd:YAG. Related studies have been conducted from the beginning of laser development to the present. In the case of the Ti:sapphire, there was little need to study the thermal effects because of its excellent thermal durability and its thin-disk shape, which is advantageous for reducing thermal-induced distortions [5; 17]. Figure 14: (**a**) The linear polarized E-field amplitude gain map of Ti:sapphire amplifier, (**b**) The depolarized component ratio of Ti:sapphire amplifier, (**c**) The wavefront distortion by the phase retardant of amplified beam, (**d**) The temporal shape of amplified pulse, the red line is amplified beam and the blue one is depolarized component of the beam (Simulation condition: \(E_{p}\) is 120 )@ 1 Hz, the radius is 3.5 cm and the thickness is 2 cm). The thermal distortion effects that consider both the induced thermal birefringence described in the previous section and the phase delay effect due to the thermal expansion are discussed in this section. The thermal distortion of the medium is generally analyzed by dividing its effect into the temperature dependence of the refractive index, thermal expansion, and thermal birefringence, as shown in Figure 15. Even though this is a general and traditional method, it is suitable for multi PW and tens of hertz high-power laser systems to apply. Also, although not included in our model, a distortion due to the acousto-optical(AO) effect from a vibration caused by Stoke's conversion should be considered to improve the beam quality [5; 17]. The temperature dependence of the refractive index depends on the temperature distribution in the medium. Generally, because ultra-high-power lasers have a pump beam with a uniform distribution, it can be assumed that the heat generation due to the quantum defects in the medium is uniform. In this case, the temperature distribution of the medium appears parabolic in the center. By such a temperature distribution, a lens shape similar to that of a GRIN (Gradient-index) lens is derived and can be expressed by Equation (33). Attempts to homogenize the temperature of the entire medium by controlling the doping rate or junction of undoped YAG, which is often used in high-power solid-state lasers, are intended to prevent changes in the refractive index with temperature. \[f_{T}^{-1}=\frac{dn}{dT}\frac{\partial^{2}T\left(r,\theta\right)}{\partial r^{ 2}} \tag{33}\]  
Temperature dependence of refractive index, \(dn/dT\), in Equation (33) refers to the amount of change in the refractive index with temperature, and \(\Delta T\) is the temperature difference between the center of the medium and each position. Another effect of temperature rise is thermal expansion. As the temperature of a medium rises, it expands, and this leads to a change in the light path. A rod-shaped medium can be approximated by \(\Delta L=\alpha TD\). However, this formula is not suitable for application to the disc-shaped media employed in high-power laser systems. Related studies are being conducted on a wide variety of materials. Considering both the anisotropy of the material and a disc shape, it is more appropriate that the Ti:sapphire medium used in ultra-high-power lasers is expressed by Equations (34)-(36).