# Recent Development of High-Energy Short-Pulse Lasers with Cryogenically Cooled Yb:YAG
## 2 Changes in Crystal Properties with Temperature
### Thermodynamic Properties  
The thermodynamic properties of Yb:YAG crystals related to temperature change are mainly thermal conductivity, thermo-optic coefficient, and thermal expansion coefficient. Figure 1 shows the thermal conductivity, thermo-optic coefficient, and thermal expansion coefficient of undoped YAG crystal as a function of temperature [10]. The thermal conductivity of undoped YAG crystal at room temperature is about 11 W/(m\(\cdot\)K), while the thermal conductivity of Ti:sapphire at room temperature can reach about 33 W/(m\(\cdot\)K). Therefore, the thermal conductivity of the YAG crystal is not superior to that in other matrix materials at room temperature, but when the YAG crystal is cooled to 77 K, the thermal conductivity increases by about 9 times. When doped with a moderate concentration of Yb\({}^{3+}\), the advantage of increasing thermal conductivity is reduced, but there is still a 3-5 times improvement, compared with that at room temperature [11]. In addition, as shown in the figure, the thermo-optic coefficient and thermal expansion coefficient of YAG crystal decrease with a decrease in temperature, and the change in thermo-optic coefficient and thermal expansion coefficient of Yb:YAG crystal with temperature is similar to that of YAG crystal. Compared with normal temperature, the thermo-optic coefficient and the thermal expansion coefficients are reduced to 1/7 and 1/4 of the original, respectively [12]. Figure 1: Thermal conductivity, thermo-optic coefficient, and thermal expansion coefficient of YAG crystal as a function of temperature [10]. Reprinted/adapted with permission from Ref. [IEEE]. 2007, [PERSON]. Although compared with flashlamp pumping, the efficiency of laser diode pumping is greatly improved, and the thermal load of the crystal is greatly reduced, but in high-power solid-state lasers, the influence of the thermal load of the crystal cannot be ignored. The uneven temperature distribution leads to uneven distribution of stress and refractive index, which causes thermal effects such as thermally induced birefringence and thermal lensing effect, resulting in beam distortion, beam quality degradation, and even crystal damage [13]. When isotropic material is subjected to mechanical stress, it becomes anisotropic, resulting in a change in refractive index, which is called stress-optic effects. Since the pump laser will generate heat and temperature differences in the medium, due to the uneven expansion inside and outside the laser crystal, a temperature gradient is formed in the crystal, resulting in mechanical stress. The stress-optic effects caused by the uneven heat distribution are also called thermally induced birefringence. For nonlinear optical research and engineering applications such as laser frequency doubling and electro-optical Q-switching, linearly polarized beams are required. Especially for high-power solid-state lasers, the thermal depolarization effect caused by thermally induced birefringence is very significant, which will greatly affect the performance and beam quality of solid-state lasers. A figure of merit of birefringence (\(FOM_{B}\)), which can measure the stress-optic effects can be defined by  
\[FOM_{B}=\kappa/\left(\chi_{QL}|\alpha|\right), \tag{1}\]  
the smaller the \(FOM_{B}\), the greater the stress-optic effects, which means the greater the thermally induced birefringence effect [10]. In the formula above, \(\kappa\) is the thermal conductivity, \(\alpha\) is the thermal expansion coefficient, and \(\chi_{QL}\) is the heat production rate, which represents the ratio of the quantum loss heat production to the laser energy. \(\chi_{QL}\) can be expressed as  
\[\chi_{QL}=\chi_{QD}/\left(1-\chi_{QD}\right), \tag{2}\]  
where \(\chi_{QD}\) is quantum loss, which can be expressed as  
\[\chi_{QD}=1-\lambda_{p}/\lambda_{l}, \tag{3}\]  
where \(\lambda_{p}\) is pump wavelength, and \(\lambda_{l}\) is laser wavelength.