## 2 Review of Cryogenic-Cooling Benefits
### Thermal Aberrations in Solid-State Lasers
For the case where the prime motivation is to eliminate or reduce aberrations to residual levels, we can define two figures-of-merit \(\Gamma\) as follows:  
\[\Gamma_{T}=\frac{k}{\alpha\ \beta\ \eta_{\mathrm{\mu}}{}^{T}} \tag{6}\]  
\[\Gamma_{S}=\frac{k\left(1-\
u\right)}{\alpha\ E\ \eta_{\mathrm{\mu}}{}^{T}} \tag{7}\]  
Figure of merit \(\Gamma_{T}\) addresses changes in the index of refraction due to temperature gradients, and \(\Gamma_{S}\) the change in the index due to stress (or strain) effects. To reduce thermal aberrations, we wish to maximize the value of \(\Gamma_{T}\). This is achieved by maximizing the value of \(k\) to minimize the thermal gradients, minimizing the values of \(\alpha\) and \(\beta\), and by minimizing the amount of heat generated by minimizing the value of the total heat fraction \(\eta_{\mathrm{\mu}}{}^{T}\). The heat fraction is formally defined as the fraction of the absorbed light converted to heat. It includes contributions from the quantum defect, fluorescence, and stimulated-emission. Yb based materials are particularly attractive for producing low \(\eta_{\mathrm{\mu}}{}^{T}\) values, due to the lack of concentration-quenching, excited-state absorption, and upconversion, processes that plague other laser materials like Nd:YAG or Nd:YVO\({}_{4}\), and Nd:YLF for example. For Yb based laser materials such as Yb:YAG, we have published a detailed kinetics model [38] that allows the simple calculation of \(\eta_{\mathrm{\mu}}{}^{T}\). The model recognizes two distinct heating mechanisms, the first due to  
Boltzmann heating (or cooling), and the second due to the photon related processes of fluorescence and stimulated-emission. In general, the heat fraction for Yb:YAG is defined by the equation  
\[\eta_{h}^{T}=\frac{Q_{h}^{B}+Q_{h}^{P}}{Q_{A}}=A+B\eta_{ex}+C\left(1-\eta_{ex}\right) \tag{8}\]  
where \(Q_{A}\) is the total absorbed power density, \(Q_{h}^{B}\) the Boltzmann heat density, and \(Q_{h}^{P}\) the photon heat density. The quantities \(A\), \(B\), and \(C\) are constants and \(\eta_{ex}\) is the laser extraction efficiency. The constants \(A\), \(B\), and \(C\) are associated with Boltzmann heating or cooling, stimulated-emission, and fluorescence respectively. For complete laser extraction only the first and second terms contribute, since the fluorescence is zero, whereas for no laser extraction the first and third terms contribute. In most cases the extraction efficiency is between those two limits, and Equation (8) allows the simple calculation of the total heat load when both stimulated-emission and fluorescence are operative. Equation (7) is the figure of merit for considering stress effects only. It arises naturally from the consideration of the stresses generated in laser crystals due to thermal gradients. For a rod amplifier, using the plane-strain approximation [21], the stresses can be calculated from the well-known equations  
\[\sigma_{rr} =\frac{Q_{0}}{16M_{s}}\left(r^{2}-r_{0}^{2}\right) \tag{9}\] \[\sigma_{\varphi\varphi} =\frac{Q_{0}}{16M_{s}}\left(3r^{2}-r_{0}^{2}\right)\] (10) \[\sigma_{zz} =\frac{Q_{0}}{8M_{s}}\left(2r^{2}-r_{0}^{2}\right) \tag{11}\]  
and for a slab amplifier in the plane-strain approximation, from [22]  
\[\sigma_{xx}=\sigma_{zz}=\frac{Q_{0}}{2M_{s}}\left(2y^{2}-\frac{t^{2}}{12}\right) \tag{12}\]  
Equations (9)-(12) show that the obtained stress is proportional to the heat density and inversely proportional to \(M_{s}\), a quantity known as the materials parameter, and expressed as  
\[M_{s}=\frac{k\left(1-\
u\right)}{\alpha E} \tag{13}\]  
The materials parameter is then a mixture of thermal and elastic constants. To reduce the stress one wants to maximize the value of \(M_{s}\), and that is the motivation for the use of the figure of merit \(\Gamma_{S}\). To minimize stress contributions to the change in the index of refraction, the appropriate strategy is to maximize \(k\), minimize \(\alpha\), \(E\), and \(\eta_{h}^{T}\), and minimize the value of \(\
u\).