## 2 Review of Cryogenic-Cooling Benefits
### Figures of Merit Comparison of Undoped Selected Laser Materials at 300 and 100 K
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \multirow{2}{*}{**Crystal**} & \multirow{2}{*}{
\begin{tabular}{c} \(k\) \\ **(W/cm** \(\cdot\)**K)** \\ \end{tabular} } & \(\begin{array}{c}\alpha\\ \mathbf{(10^{-6})}\\ \mathbf{(1/K)}\end{array}\) & \(\begin{array}{c}\beta\\ \mathbf{(10^{-6})}\\ \mathbf{(1/K)}\end{array}\) & \(\eta_{\rm R}^{\rm QD}\) & \(\
u\) & \(\begin{array}{c}\mathbf{\Gamma(10^{0})}\\ \mathbf{(g/cm^{2})}\end{array}\) & \(\Gamma_{\rm T}\) (\(\mathbf{10^{10})}\) & \(\Gamma_{\rm S}\) (\(\mathbf{10^{-5}}\)) & \(\Gamma\) & \(\mathbf{T}\) \\  & & & & & & & & & & & \\ \hline \multirow{2}{*}{YAG} & 0.112 & 6.14 & 7.80 & 0.086 & 0.24 & 3.080 & 2.72 & 5.23 & 6.71 & 300 \\  & 0.461 & 1.95 & 0.90 & 0.086 & 0.24 & 3.080 & 305.44 & 67.83 & 753.7 & 100 \\ \hline \multirow{2}{*}{LuAG} & 0.083 & 6.13 & 8.30 & 0.086 & 0.25 & 2.872 & 1.90 & 4.11 & 4.95 & 300 \\  & 0.254 & 2.46 & 0.70 & 0.086 & 0.25 & 2.872 & 171.5 & 31.35 & 447.86 & 100 \\ \hline \multirow{2}{*}{YLF-a} & 0.053 & 10.05 & \(-\)4.60 & 0.045 & 0.33 & 0.785 & 2.55 & 10.00 & 21.74 & 300 \\  & 0.242 & 3.18 & \(-\)0.50 & 0.045 & 0.33 & 0.785 & 338.23 & 144.34 & 453.93 & 100 \\ \hline \multirow{2}{*}{YLF-c} & 0.072 & 14.31 & \(-\)6.60 & 0.045 & 0.33 & 0.785 & 1.69 & 9.54 & 14.45 & 300 \\  & 0.337 & 2.36 & \(-\)1.80 & 0.045 & 0.33 & 0.785 & 176.29 & 270.81 & 1504.50 & 100 \\ \hline \multirow{2}{*}{YALO-a} & 0.117 & 2.32 & 7.70 & 0.096 & 0.23 & 3.220 & 6.82 & 12.56 & 16.31 & 300 \\  & 0.649 & \(-\)1.16 & 1.00 & 0.096 & 0.23 & 3.220 & 582.79 & 139.36 & 1393.60 & 100 \\ \hline \multirow{2}{*}{YALO-b} & 0.100 & 8.08 & 11.70 & 0.096 & 0.23 & 3.220 & 1.10 & 3.08 & 2.63 & 300 \\  & 0.544 & 3.24 & 4.50 & 0.096 & 0.23 & 3.220 & 388.66 & 41.82 & 92.93 & 100 \\ \hline \multirow{2}{*}{YALO-c} & 0.133 & 8.70 & 8.30 & 0.096 & 0.23 & 3.220 & 1.92 & 3.81 & 4.59 & 300 \\  & 0.776 & 3.00 & 1.20 & 0.096 & 0.23 & 3.220 & 224.54 & 64.43 & 536.92 & 100 \\ \hline \multirow{2}{*}{Lu\({}_{2}\)O\({}_{3}\)} & 0.114 & 6.10 & 7.1 * & 0.080 & 0.29 & 2.019 & 3.29 & 8.22 & 11.58 & 300 \\  & 0.340 & 2.90 & 2.3 * & 0.080 & 0.29 & 2.019 & 63.72 & 51.54 & 224.09 & 100 \\ \hline \multirow{2}{*}{Sc\({}_{2}\)O\({}_{3}\)} & 0.147 & 6.40 & 8.12 & 0.088 & 0.24 & 2.254 & 3.21 & 8.80 & 10.84 & 300 \\  & 0.455 & 0.75 & 2.20 & 0.088 & 0.24 & 2.254 & 313.36 & 265.66 & 1207.55 & 100 \\ \hline \multirow{2}{*}{Y\({}_{2}\)O\({}_{3}\)} & 0.130 & 6.30 & 6.08 & 0.074 & 0.30 & 1.723 & 4.59 & 11.33 & 18.64 & 300 \\  & 0.520 & 0.90 & 2.40 & 0.074 & 0.30 & 1.723 & 325.33 & 156.86 & 653.58 & 100 \\ \hline \multirow{2}{*}{CaF\({}_{2}\)} & 0.080 & 19.20 & \(-\)12.70 & 0.091 & 0.30 & 1.122 & 0.36 & 2.86 & 2.25 & 300 \\  & 0.390 & 10.60 & \(-\)7.5 * & 0.091 & 0.30 & 1.122 & 5.32 & 25.22 & 23.79 & 100 \\ \hline \multirow{2}{*}{Al\({}_{2}\)O\({}_{3}\)-c} & 0.330 & 5.15 & 9.80 & 0.335 & 0.23 & 3.844 & 1.95 & 3.83 & 3.91 & 300 \\  & 5.150 & 0.71 & 4.05 & 0.335 & 0.23 & 3.844 & 534.62 & 433.72 & 1070.91 & 100 \\ \hline \multirow{2}{*}{Al\({}_{2}\)O\({}_{3}\)-a} & 0.360 & 5.93 & 12.80 & 0.335 & 0.23 & 3.844 & 1.42 & 3.63 & 2.84 & 300 \\  & 3.440 & 0.90 & 1.90 & 0.335 & 0.23 & 3.844 & 600.51 & 228.55 & 1202.89 & 100 \\ \hline \end{tabular}
\end{table} 
Table 3: Thermal Conductivity \(k\), thermal expansion coefficient \(\alpha\), thermo-optic coefficient \(\beta\), quantum defect heat fraction \(\eta_{\rm R}^{QD}\), Poisson’s ratio \(\
u\), Young’s Modulus \(E\), and calculated Figures of Merit \(\Gamma_{T}\), \(\Gamma_{S}\), and \(\Gamma\) for selected laser crystals at 100 and 300 K. Absolute values of negative parameter values were used to calculate figures-of-merit. -fracture stress; in some cases such cracks and other irregularities can be removed by etching processes. As can be seen from Equations (72) and (73), the maximum extractable average power is achieved when the surface stress is equal to the fracture stress.