## 2 Review of Cryogenic-Cooling Benefits
### Thermal Parameters  
#### 2.2.1 Thermal Conductivity  
Thermal conductivity \(k\) is a very important laser design parameter, and understanding its' dependence on temperature and dopant density is crucial to implementing cryogenic laser designs. We first examine the standard quadratic equation used for determining the CW temperature distribution in a rod or slab laser [21; 22]. For a slab laser the temperature \(T(y)\) as a function of slab thickness is given by  
\[T\left(y\right)=T_{0}-\frac{Q_{0}y^{2}}{2k} \tag{15}\]  
The slab center temperature is \(T_{0}\), \(Q_{0}\) is the (assumed uniform) heat power density, \(k\) the thermal conductivity, and \(y\) the distance from the slab center, where \(0\leq y\leq(t/2)\) and t is the slab thickness. Newton's law of cooling is used as he boundary condition at \(y=\pm t/2\).To reduce the temperature differential between the center and the edge of the slab, important in reducing the thermal aberrations, we want to reduce or eliminate the thermal gradient given by  
\[\frac{dT}{dy}=-\frac{Q_{0}}{k}y \tag{16}\]  
From Equation (16) we see that reducing the thermal gradient is equivalent to reducing the ratio \((Q_{0}/k)\). Reducing the heat fraction in a laser material reduces \(Q_{0}\); the other way to reduce the gradient is to increase \(k\). Both of these strategies are used in modern diode-pumped lasers; for diode-pumped Yb lasers in particular, heat fractions are much reduced when compared to older materials such as Nd:YAG, Nd:YLF, or Nd:YVO\({}_{4}\). Here we recount how cryogenic-cooling has been used to obtain much larger k values. We can also see from Equation (16) that for flashlamp-pumped lasers where \(Q_{0}\) values can be an order of magnitude or two larger than the modest values obtained with diode-pumped lasers, thermal gradients are very much larger. Thermal aberrations can be substantially reduced by using diode-pumping to reduce the heat fraction, and cryogenic-cooling to increase \(k\). In Equation (15), the thermal conductivity is treated as a constant, independent of temperature or doping density. One can include the temperature dependence using a number of methods. We may write the CW heat equation for an infinite slab element as  
\[-\frac{d}{dy}\left[k\left(T\right)\cdot\frac{dT}{dy}\right]=Q_{0} \tag{17}\]  
As we will explore later in this article, theoretically the thermal conductivity varies approximately as \((1/T)\) in the region of interest from room temperature to 77 K. We can write a simple relationship for the thermal conductivity dependence on \(T\) as  
\[k\left(T\right)=k_{0}\cdot\frac{T_{A}}{T} \tag{18}\]  
where \(k_{0}\) is the thermal conductivity at ambient temperature \(T_{A}\) = 300 K. Substituting in Equation (16) and integrating results in the analytical solution  
\[T\left(y\right)=T_{0}\cdot\text{Exp}\left(-\frac{Q_{0}y^{2}}{2k_{0}T_{A}}\right) \tag{19}\]  
This equation allows the computation of the temperature in any slab in which the thermal conductivity varies as \((1/T)\). It should be noted that, as first reported in [20], the equation is thenno longer quadratic. The derived equation satisfies the boundary condition that \(T(y=0)=T_{0}\). For \(y=\pm t/2\), we find the edge temperature \(T_{e}\):  
\[T_{e}=T_{0}\cdot\text{Exp}\left(-\frac{Q_{0}t^{2}}{8k_{0}T_{A}}\right) \tag{20}\]  
Equation (20) allows the calculation of the edge temperature, knowing \(T_{0}\). Often \(T_{e}\) is fixed at room temperature by a temperature controlled water chiller, or by liquid nitrogen (LN\({}_{2}\)), and in that case Equation (20) may be used to calculate the slab center temperature \(T_{0}\). It should be noted that as expected, for small \(Q_{0}\), Equation (20) can be expanded in a Taylor series. Keeping only the first term, we find  
\[T\left(y\right)=T_{0}\left(1-\frac{Q_{0}y^{2}}{2k_{0}T_{A}}\right)=\lim_{\left( T_{0}/T_{A}\right)\to 1}T_{0}-\frac{Q_{0}y^{2}}{2k_{0}}\cdot\left[\frac{T_{0} }{T_{A}}\right]=T_{0}-\frac{Q_{0}y^{2}}{2k_{0}} \tag{21}\]  
or the equation reverts to the \(k=\) constant case.