## 2 Review of Cryogenic-Cooling Benefits
### Thermal Expansion Coefficient
Except where discussed, all data shown are for undoped host laser materials, and we have used published fits to the thermal expansion data where available to generate the plots shown here, Figure 20 shows the thermal expansion coefficient as a function of absolute temperature over the range 100-300 K for the oxide laser materials YAG, LuAG, and YALO [29]. YAG and LuAG are both isotropic crystals whereas YALO is biaxial and therefore has three principal thermal expansion coefficients. In Figure 21 we show thermal expansion as a function of temperature for the fluoride materials YLF and LuLF [29], both uniaxial crystals with two principal axes each. Figure 22 shows thermal expansion data for the biaxial laser materials BYF, KYW, and KGW, along their b-axes [29]. Figure 23 displays thermal expansion data for the biaxial laser material YSO b-axis as well as for the isotropic material GSAG [30]. In Figure 24 data are shown for the uniaxial laser materials YVO\({}_{4}\) and GdVO\({}_{4}\)[3; 30]. Thermal expansion coefficients as a function of absolute temperature for the undoped isotropic sesquioxide laser materials Sc\({}_{2}\)O\({}_{3}\)[30] and Y\({}_{2}\)O\({}_{3}\)[59], and the Yb-doped laser materials Yb:Y\({}_{2}\)O\({}_{3}\)[59], Yb:Sc\({}_{2}\)O\({}_{3}\)[59], and Yb:Lu\({}_{2}\)O\({}_{3}\)[59] are shown in Figure 25. Figure 26 shows thermal expansion data for the isotropic laser material CaF\({}_{2}\), for undoped and 10 at. % Yb doped crystals [59], while in Figure 27 we include some thermal expansion data for some other common laser and optical materials, uniaxial Al\({}_{2}\)O\({}_{3}\), isotropic MgO, and uniaxial ZnO [60]. Figure 18:0 Thermal expansion coefficient as a function of absolute temperature for the biaxial laser material YSO (blue) [30] and the isotropic laser material GSAG (red) [30]. YSO data is reported for the \(b\)-axis, and all data are for undoped crystals. Figure 18:1 Thermal expansion coefficient as a function of absolute temperature for the biaxial laser materials BYF (blue), KGW (green), and KYW (red) [29]. All data are reported for the \(b\)-axis and for undoped crystals. Figure 18:2 Thermal expansion coefficient as a function of absolute temperature for the uniaxial laser materials YVO4[30] along the \(a\) (pink) and \(c\) (green) axes, and GdVO4 along the \(a\) (red) and \(c\) (blue) axes [3]. All data are for undoped crystals. Figure 18:3 Thermal expansion coefficient as a function of absolute temperature for the undoped isotropic sesquioxide laser materials Sc2O3 (blue) [30] and Y2O3 (red) [59], and the 10 at. % Yb-doped laser material Yb:Y2O3 (green) [59], the 1 at. % Yb-doped material Yb:Sc2O3 (pink) [59], and 1 at. % Yb-doped material Yb:Lu2O3 (black) [59]. Most of the materials plotted in Figures 20-27 are well behaved in the sense that thermal expansion values are positive and converge towards 0 as temperature is lowered, in agreement with our previous discussion regarding Equation (66). In a few cases, however, the thermal expansion can become negative at lower temperatures, as may be seen in the YALO _a_-axis data in Figure 20, and the ZnO a and b axes data in Figure 18:4 In [61] the Gruneisen parameters for YAG and YALO are discussed, and the negative thermal expansion coefficient modeled as an additive negative Gruneisen parameter. Figure 18:5 Thermal expansion coefficient as a function of absolute temperature for the uniaxial laser materials Al\({}_{2}\)O\({}_{3}\) along the \(a\) (red) and \(c\) axes (blue) [60], ZnO along the \(a\) (black) and \(c\) (pink) axes [60], and for the isotropic laser material MgO (green) [60]. Figure 18:6 Thermal expansion coefficient as a function of absolute temperature for the isotropic laser material CaF\({}_{2}\) (blue) and 10 at. % Yb-doped Yb:CaF\({}_{2}\) (red) [59].