## 2 Review of Cryogenic-Cooling Benefits
### Figures of Merit Comparison of Undoped Selected Laser Materials at 300 and 100 K  
Table 3 shows a comparison between the same material at 300 and 100 K, and between different materials, by use of the figures of merit \(\Gamma_{T}\), \(\Gamma_{S}\), and \(\Gamma\), for laser materials for which most or all of the needed parameters are available in the literature. Most values reported are within a few degrees of 100 or 300 K, and are listed for those values to simplify the presentation. For the heat fraction, here we assume for simplicity that the only contribution to the heat fraction is the quantum defect between the pump and lasing phonons, denoted as \(\eta_{h}^{QD}\). Data used in Table 3 were taken from the references associated to Figures 10-32 as well as Table 2 and its references. It is important to note that the results shown are for undoped materials only. Concentrating first on the thermal aberrations figure-of-merit \(\Gamma_{T}\), we see that for all materials considered, the figure of merit value is significantly larger at 100 K than at 300 K. This is a consequence of the increased values of \(k\) as temperature is lowered, and reduced values of \(\alpha\) and \(\beta\). The materials with the largest increase in \(\Gamma_{T}\) between 300 and 100 K are YAG, YLF, YALO, Sc\({}_{2}\)O\({}_{3}\), Y\({}_{2}\)O\({}_{3}\), and Al\({}_{2}\)O\({}_{3}\). These are the materials that will display the largest reductions in thermal aberrations. The material with the smallest increase in the figure-of-merit is CaF\({}_{2}\). If this simple comparison, we ignore the fact that additional decreases in thermal aberrations may be achievable by balancing a negative value for \(\beta\) with the thermal expansion coefficient to achieve an athermal result as described in connection with Equation (5). YLF and CaF\({}_{2}\) for example from benefit from that approach. It should also be appreciated that adding Yb doping density as an additional parameter, for example, will degrade the expected increases in \(\Gamma_{T}\), very significantly in some cases. As we have seen previously in this paper, Lu\({}_{2}\)O\({}_{3}\) and LuAG have minor decreases in k as doping density increases, whereas for YAG and Sc\({}_{2}\)O\({}_{3}\) the drop in thermal conductivity as doping density increases is severe. Perhaps the worst case is CaF\({}_{2}\) where for 3 or 5 at. % Yb doping the thermal conductivity does not increase at any temperature (see Figure 15). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Elastic** & \(\mathbf{E}_{0}\left(0\text{ K}\right)\) & \(\mathbf{a}\) & \(\mathbf{E}\left(\text{300 K}\right)\) & \(\mathbf{B}_{0}\left(\text{0 K}\right)\) & \(\mathbf{b}\left(\text{$\times$10^{-4}$/K}\right)\) & \(\mathbf{B}\left(\text{300 K}\right)\) & \\
**Parameters** & **(Gpa)** & **(10\({}^{-4}\)/K)** & **(Gpa)** & **(Gpa)** & **b** (\(\text{$\times$10^{-4}$/K}\)) & **(Gpa)** & \(\
u\) \\ \hline Al\({}_{2}\)O\({}_{3}\) & 393 & 1.33 & 377 & 241 & 0.84 & 235 & 0.23 \\ CaF & - & - & 110 & - & - & - & 0.30 \\ Lu\({}_{2}\)O\({}_{3}\) & 204 & 1.03 & 198 & 161 & 0.24 & 160 & 0.29 \\ MgAl\({}_{2}\)O\({}_{4}\) & 278 & 1.98 & 262 & 187 & 1.97 & 176 & 0.25 \\ MgO & 310 & 1.63 & 295 & 164 & 1.23 & 158 & 0.19 \\ Sc\({}_{2}\)O\({}_{3}\) & 229 & 1.22 & 221 & 148 & 0.98 & 144 & 0.24 \\ Y\({}_{2}\)O\({}_{3}\) & 176 & 1.37 & 169 & 147 & 1.93 & 139 & 0.30 \\ YAG & - & - & 302 & - & - & - & 0.24 \\ YbAG & - & - & 257 & - & - & - & 0.25 \\ LiYF\({}_{4}\) & - & - & 77 & - & - & - & 0.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Young’s modulus, bulk modulus, and Poisson’s ratio for selected laser materials. For the stress figure-of-merit \(\Gamma_{\rm S}\), the laser material with the largest increase in figure-of-merit between 300 and 100 K are YAG, YLF, YALO, and Al\({}_{2}\)O\({}_{3}\). Recall that a large figure-of-merit here means that the stress values achieved are low. Materials with the lowest increase in \(\Gamma_{S}\) are LuAG and CaF\({}_{2}\).