# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 4 Discussion
Associated with the forbidden transition is a long-lived excited state with a fluorescence lifetime of 170 \(\upmu\)s [42], making the transition easy to saturate. The resonance wavelength of this forbidden transition is 903 nm in YAG, with an absorption cross-section of \(\sim\)1.2 \(\times\) 10\({}^{-20}\) cm\({}^{2}\) at 1000 nm [19]. Unfortunately, the emission cross-section in YAG and both the absorption and emission cross-sections in LiYF\({}_{4}\) are currently unknown. However, based on the spectroscopy of Fe:YAG, it is reasonable to surmise that the background saturation intensity of trivalent iron in LiYF\({}_{4}\) could be extremely low in the same wavelength range. Using solely the value of the cross-section given above, the absorption saturation intensity of Fe\({}^{3+}\) is estimated to be 20% lower than that of Yb\({}^{3+}\) at 1000 nm, which is in quite good agreement with the experimental background saturation intensity reported here (albeit in a different material). For iron in other valence states, such as Fe\({}^{2+}\) or Fe\({}^{4+}\) ions, the transition near 1000 nm is allowed [43]. This would lead to a short-lived excited state and a much higher saturation intensity compared with Fe\({}^{3+}\). Thus, the divalent and tetravalent ions would not be expected to saturate at the low intensity measured in our experiments. However, these valence states may be less prevalent in YLF than the trivalent state. Hence, one could reasonably conclude that iron impurities may saturate easily and that the cooling efficiency should improve substantially if a sizeable fraction of background impurities become non-absorbing. The properties of trivalent iron seem consistent with the low-absorption saturation intensity determined in our experiments. Of course, a low value of background saturation intensity could arise from impurities other than iron. Several elements are significantly correlated with the background absorption in crystalline Yb:YLF [40], such as vanadium and chromium. Some of these elements could also lead to a low ionic saturation intensity near 1000 nm. For example, when the transition metal chromium is in the quadrivalent state in a YAG host, the saturation intensity of Cr\({}^{4+}\) is estimated to be only 10% of Yb\({}^{3+}\) at 1000 nm [44]. Consequently, the absorption of a variety of background ionic species could saturate similarly at relatively low intensities, contributing to the overall reduction in parasitic heating and enhanced cooling at elevated pump powers. Although we cannot state for certain what impurities contribute to heightened efficiency with increasing pump intensity in Yb:YLF, it seems evident that the dominant transition metal ion and others present in this crystal have properties that are consistent with the low saturation intensity we determined at \(\lambda_{cr}=1000\) nm. Our TLS experiments also indicate that laser cooling can take place on dipole-allowed transitions. The curves in Figures 10 and 14, which show heating and cooling at wavelengths shorter and longer than the mean fluorescence wavelengths of the corresponding ions, respectively, are qualitatively the same for Yb\({}^{3+}\):LiYF\({}_{4}\) (dipole-forbidden) and Ti\({}^{3+}\):Al\({}_{2}\)O\({}_{3}\) (dipole-allowed). The cooling behavior observed in Ti\({}^{3+}\):Al\({}_{2}\)O\({}_{3}\) reveals that configuration relaxation can be avoided by tuning to wavelengths longer than the mean fluorescence wavelength. This wavelength condition also determines the cooling range of rare-earth ions such as Yb\({}^{3+}\). However, the allowed transition of Ti\({}^{3+}\) has a much faster decay rate; so, faster cool-down times could be enabled for semiconductor sensor circuitry grown on Ti\({}^{3+}\):Al\({}_{2}\)O\({}_{3}\) substrates and cooled optically to cryogenic temperatures.