# Accurate Characterization of the Properties of the Rare-Earth-Doped Crystal for Laser Cooling
## 1 Introduction  
Optical cooling of solids is based on the anti-Stokes fluorescence principle, in which the average wavelength of fluorescence (\(\lambda_{f}\)) emitted by a transparent solid is shorter than the wavelength of the coherent light (\(\lambda\)) for excitation [1, 2]. The extra energy of the fluorescence over the excitation light comes from the kinetic energy of the lattice vibrations in the substrate [3]. The kinetic energy of the lattice vibrations is continuously pumped away, causing a temperature drop of the substrate until the cooling effect is balanced by the heating effect [4, 5, 6, 7]. Optical cooling of solids is effective for developing both the cryogenic optical refrigerator of free vibration with high reliability and radiation-balanced lasers (RBLs) of no gain medium heating from quantum defects and non-radiative attenuation [8, 9, 10, 11, 12, 13, 14]. [PERSON] and co-workers first demonstrated the anti-Stokes fluorescence cooling concept on the 1 wt.% Yb\({}^{3+}\)-doped ZBLANP fluoride glass in 1995 [1]. Since then, researchers have focused on exploring new promising potential materials for cryogenic optical cooling. So far, net cooling by laser has been demonstrated not only in a variety of rare-earth-doped (Yb\({}^{3+}\)-, Tm\({}^{3+}\)-, Er\({}^{3+}\)- and Ho\({}^{3+}\)-) bulk and nano crystals [2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], but also in glassy materials [32, 33, 34] and semiconductors of nano structure [35, 36, 37]. Among all the crystals tested, fluoride crystals have exhibited the best cooling performances [38, 39, 40]. The Yb\({}^{3+}\)/Tm\({}^{3+}\) co-doped LiYF\({}_{4}\) crystal has been cooled to a record temperature of 87 K [41] and the Yb\({}^{3+}\) doped LuLiF\({}_{4}\) crystal has the potential to be cooled to 89 K [29]. Recently, the Yb\({}^{3+}\) doped fibers (such as ZBLAN and silica fibers) have been attracting researchers as the platform for RBLs. Especially, the fibers can be further improved in purity and doped with modifiers to increase the Yb ion solubility [42, 43]. Demonstration of RBLs and radiation-balanced amplifiers based on the fiber platform has also been reported [14, 44, 45]. [PERSON] and [PERSON] developed a four-level model in theory to describe the laser-cooling processes of rare-earth-doped hosts with four parameters: the external quantum efficiency \(\eta_{ext}\), the resonant absorption coefficient \(\alpha_{r}(\lambda,T)\), the background absorption coefficient \(\alpha_{b}\) and the mean fluorescence wavelengths \(\lambda_{f}(T)\) being derived for characterizing their cooling performances [7]. Accurate characterization of the lasercooling properties in both theory and experiment is very important for determining high-cooling-grade materials [46; 47; 48]. The external quantum efficiency \(\eta_{ext}\) and the background absorption coefficient \(\alpha_{b}\) are two crucial parameters for assessing the laser cooling grade of a material. The laser-induced temperature modulation spectrum (LITMoS) method has been developed to accurately measure these two parameters [48]. The laser-cooling limits of both rare-earth doped fluoride glass of ZBLAN(P) and fluoride crystal of LiYF\({}_{4}\) were accurately predicted from their corresponding \"cooling windows\" obtained with the help of the LITMoS method [49; 41; 48; 42; 41]. For instance, the minimum achievable temperature (MAT) of the Yb\({}^{3+}\)-doped LiYF\({}_{4}\) crystal was predicted to be as low as 60 K, which is lower than the temperature of liquid nitrogen [41; 51; 52]. However, the MAT of the Yb\({}^{3+}\)-doped ZBLAN glass was estimated to be only about 200 K [33]. Recently, a Yb\({}^{3+}\)/Tm\({}^{3+}\) co-doped LiYF\({}_{4}\) crystal picked up by LITMoS test was used to cool a Fourier Transform Infrared detector (HgCdTe) down to 135 K [53; 54].