## 2 Review of Cryogenic-Cooling Benefits
### Thermal Parameters
#### 2.2.7 Other Reported Thermal Conductivity Measurements Including Commonly Used Ancillary Materials  
Since our 2005 paper [2], the measurement of thermal conductivity values has accelerated, with most common laser materials being tabulated. In this section, we show plots of most of the data reported in the literature for single-crystal growth. Measurements for ceramic laser materials have not been included due to their well-known dependence on pore size. We showed previously (Figure 4) the thermal conductivity of YAG and Yb-doped YAG as a function of temperature and doping density (Figure 4), and for a number of undoped and Yb-doped sesquioxide laser materials (Figures 5 and 6). Here we show similar plots for many other laser materials. We start with the biaxial laser material YALO (YAlO\({}_{3}\)), where Figure 10 shows thermal conductivity as a function of absolute temperature for undoped and 5 at. % Yb doped cases. Measurements of k were performed for the \(a\), \(b\), and \(c\) crystal axes [29]. Due to the moderate mass mismatch between Yb and Y (Table 1), the \(k\) values for 5 at. % Yb doping are significantly less than the undoped values. Figure 11 shows k values for undoped YAG and LuAG [3, 29, 45], whilst Figure 12 plots the thermal conductivities of YAG and LuAG as a function of Yb doping density [50] and for undoped and Tm-doped Lu\({}_{2}\)O\({}_{3}\)[51]. These data show that for less than about 4 at. % Yb Yb:YAG is superior while for greater than 4 at. % Yb:LuAG is the better choice. The small change in \(k\) for Yb:LuAG as doping density increases is due to the very small mass mismatch between Yb and Lu. Similarly, for Tm:Lu\({}_{2}\)O\({}_{3}\), the mass mismatch between Tm and Lu is very small, and the thermal conductivity decreases very slowly with Tm concentration. Figure 13 plots \(k\) as a function of absolute temperature for the uniaxial laser material YLF, for undoped and 5 at. % and 25 at. % doped Yb cases [29, 30]. The large drop in \(k\) for 5 and 25 at. % Yb dopings is a consequence of the moderate mass mismatch between Yb and Y. Figure 14 shows k for the recently developed undoped biaxial laser materials BYW, KGW, and KYW [29]. In Figure 15 we show the thermal conductivity of the important uniaxial ultrafast laser material CaF\({}_{2}\), for undoped, 1 at. %, 3 at. % and 5 at. %, obtained from a number of sources [44, 52, 53, 54]. It is important to point out that the very rapid drop in k for increased Yb doping density is partially due to the very large mass mismatch between Ca and Yb (see Table 1). According to [54], between about 1-3 at. % Yb doping, where \(k\) is almost flat from 100-300 K, larger dopings drive the crystal structure towards clustering and increasingly glasslike behavior, resulting in \(k\) values that decrease with decreasing temperature, rather than the increase in \(k\) which we have seen is almost a universal phenomenon with pure crystalline materials. This type of behavior is also displayed by laser glasses, and other important optical materials often used in ultrafast laser systems. One reviewer of this paper has pointed out that mass-mismatch may play the lesser role in this case. Figure 16 for example, shows the thermal conductivity of fused silica (SiO\({}_{2}\)) [55], an anaxial optical material, as a function of temperature. It displays a monotonic decrease in \(k\) as temperature is lowered, as expected due to a locally disordered crystal structure that severely limits the phonon mean free path. In Figure 17 the thermal conductivity values for the GdVO\({}_{4}\)\(c\) and \(a\) axes, the YSO \(b\)-axis, and for GSAG are shown as a function of absolute temperature [3, 30]. Figure 11: Thermal conductivity as a function of absolute temperature of the undoped anaxial laser materials LuAG (green) [45], YAG (blue) [3; 29] and LuAG (red) [29]. Figure 10: Thermal conductivity as a function of absolute temperature for the biaxial laser material YALO for undoped \(a\) (blue), \(b\) (red), and \(c\) (green) axes and 5 at.