## 2 Review of Cryogenic-Cooling Benefits
### Thermal Aberrations in Solid-State Lasers
Equation (2) results from the plane-strain approximation in which the aspect ratio of the rod \((L/2r_{0})>>1\). In that approximation, physical changes in the optical pathlength are ignored. In real amplifiers, bulging of the rod faces occurs, an effect that is directly attributable to the thermal expansion coefficient. A similar equation can be derived for the plane stress case (\((L/2r_{0})<<1\)), typical of active-mirror or thin disk amplifiers:  
\[ \begin{array}{ll}\Delta\phi_{r,\varphi}&=\frac{2\pi}{\lambda} \cdot\Delta n_{r,\phi}L\\ &=\left(\frac{2\pi}{\lambda}\right)\\ &\cdot\left(\frac{Q_{0}}{4\,k}\right)\left[\left[\beta+4n_{0}^{3} \alpha\cdot C_{0}\right]\cdot r_{0}^{2}+\left[\left(n_{0}-1\right)\alpha\left( \
u+1\right)-2n_{0}^{3}\,\alpha\,C_{r,\phi}\right]\cdot r^{2}\right]\cdot L \end{array}  \tag{3}\]  
For the plane-stress case of Equation (3), by inspection again we see that if \(\alpha=\beta=0\), the phases \(\Delta\phi_{r,\varphi}\) are again zero.In rare cases, most notably for certain Nd:Glass laser compositions with negative values of \(\beta\), and when stress-induced index changes can be ignored, one can achieve \(\Delta n_{T}=0\). We can write Equation (1), ignoring the stress term, as  
\[\Delta n_{T}=\Delta n_{\beta}+\Delta n_{\alpha}=\left[\beta+\left(n_{0}-1\right) \alpha\right]\Delta T \tag{4}\]  
Thus the index variation and resulting phase can be identically zero if  
\[S=\left(\beta+\left(n_{0}-1\right)\alpha\right)=0 \tag{5}\]  
This is the condition for athermal operation, and was first employed in connection with the development of Nd laser phosphate glasses for high-peak-power inertial confinement fusion laser systems to reduce thermal distortion [37]. For cryogenic-cooling, we employ a different strategy. While our goal is to achieve \(\Delta\varphi=0\), for most crystals finite values of \(\alpha\), \(\beta\), \(\
u\), and E at 77 K allow us to simply approach that condition. As we will see in the following paragraphs, the most successful strategy to date is to reduce the thermal aberrations to residual levels by exploiting the favorable scaling of critical parameters by reducing the operating temperature to the cryogenic regime, which we define here as \(<\)150 K. There are a few crystalline cases in which condition Equation (5) can be approached or satisfied over some temperature range. For Nd:YLF lasers for example, it is well known that one can exploit the negative value of \(\beta\) using the \(a\)-axis (\(\sigma\)) transition at 1047 nm. More recent measurements [29] of \(\beta\) for the \(c\) and \(a\) axes of the uniaxial crystal YLF show that negative values of \(\beta\) for both axes persist even down to 100 K, allowing the condition Equation (5) to be approached across the entire temperature range for both axes, with \(S_{\pi}=-1.88\)\(\times\)\(10^{-6}\)/K and \(S_{\sigma}=1.81\)\(\times\)\(10^{-6}\)/K at 300 K. At 100 K, one finds that \(S_{\pi}=-0.31\)\(\times\)\(10^{-6}\)/K and \(S_{\sigma}=0.56\)\(\times\)\(10^{-6}\)/K, so condition Equation (5) is even more closely approached at 100 K. Another rare example is the closely related material LuLF, a uniaxial crystal which also possesses the property that \(\beta\) is negative for the c and a axes [29], however as temperature is lowered both axes become increasingly negative down to about 170 K where \(\beta\) begins to increase, the opposite of YLF in which \(\beta\) increases as temperature is lowered. Calculations show that at a temperature near 250 K, Equation (5) can be satisfied identically. As we will see in the data we examine later in this article, for most crystals parameters \(k\), \(\beta\), and \(\alpha\) scale favorably as temperature is lowered such that thermal aberrations nearly vanish.