## 2 Review of Cryogenic-Cooling Benefits
### Thermal Aberrations in Solid-State Lasers  
Crystalline solid-state lasers are optically-pumped devices. The first solid-state laser, demonstrated by Maiman [36], was a flashlamp-pumped ruby laser. Xe and Kr flashlamps and other similar sources, still in use today for some applications, were the pump source of choice for many decades of the twentieth century, and beginning in the 1980's their dominance has been slowly eroded as narrow band diode pump sources of increasing power have become available. Diode laser technology has developed rapidly and today is the pumping source of choice for most applications, with only the specific cost (Monetary Unit/Watt) preventing its adoption in the highest power applications. Typical Xe flashlamps used in inertial confinement fusion (ICF) applications produced output irradiances with black-body temperatures near 9000 K or more, and with spectral outputs spanning the range of 150 to 1700 nm. Due to the finite number of Nd absorption bands and the continuous nature of the black-body radiation, the absorption efficiency was poor, with most of the Xe pump radiation being deposited in the pump chamber housing the lasing element, even for typical Nd:Glass compositions where the absorption bands were significantly wider than say Nd:YAG. For the absorbed light, ratios of inversion to heat density were typically only 1.5, hence for every joule of inversion energy produced, 0.67 joules of heat were generated as well. Nd lasers dominated the development landscape, and Yb based lasers were largely ignored due to their poor absorption efficiency. The low repetition rate of ICF laser systems may largely be attributed to the use of flashlamp pump sources. As of this writing, the conversion from flashlamp-pumping to diode-pumping is well under way and accelerating, and modern diode-pumped lasers, many based upon Yb crystalline materials, produce only about 80 mJ of heat energy per joule absorbed, a decrease when compared to Nd:Glass laser systems of a factor of over 8. The reduction in the amount of heat generated is greater than two orders of magnitude when the poor absorption of Xe flashlamp radiation is taken into account. Thermal aberrations are produced in laser crystals by the production of heat, primarily from the optical-pumping process, by stimulated-emission, and by fluorescence. In some crystals, upconversion, two-photon absorption, non-unity quantum efficiencies, and other effects may also provide additional heating. The total index change \(\Delta n_{T}\) in an isotropic (anaxial) laser crystal due to temperature changes may be written as the sum of three terms:  
\[\Delta n_{T}=\Delta n_{\beta}+\Delta n_{\alpha}+\Delta n_{S} \tag{1}\]  
where \(\Delta n_{\beta}\) is due to the thermo-optic coefficient \(\beta\), \(\Delta n_{\alpha}\) due to the thermal expansion coefficient, and \(\Delta n_{S}\) is produced by the stresses (or equivalently the strains) in the crystal. For a laser rod whose barrel is maintained at a constant temperature, the proportional optical phases \(\Delta\phi_{r,\varphi}\) in the radial (\(r\)) and azimuthal (\(\varphi\)) directions due to thermal heating producing a temperature change \(\Delta T\) can be written [21]  
\[ \begin{array}{ll}\Delta\phi_{r,\phi}&=\frac{2\pi}{\lambda} \cdot\Delta n_{r,\phi}L\\ &=\left(\frac{2\pi}{\lambda}\right)\cdot\left(\frac{Q_{0}}{4 \,k}\right)\left[\left[\beta+4n_{0}^{3}\alpha\cdot C_{0}\right]\cdot r_{0}^{2 }-\left[\beta+2n_{0}^{3}\,\alpha\,C_{r,\varphi}\right]\cdot r^{2}\right]\cdot L \end{array}  \tag{2}\]  
In Equation (2), \(L\) is the crystal length, \(n_{0}\) the index of refraction, \(Q_{0}\) the uniform heat density, \(k\) the thermal conductivity, \(r_{0}\) the rod radius, and \(C_{0}\), \(C_{r}\), and \(C_{\varphi}\) are photoelastic constants. It can be seen that the phase varies quadratically with the radius r. It is also clear that if \(\alpha=\beta=0\), the phases \(\Delta\varphi_{r,\varphi}\) would be zero for all \(r\), regardless of the value of \(Q_{0}\).