# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 4 Discussion
Sapphire is an excellent substrate for GaN, Al\({}_{x}\)Ga\({}_{1\times}\)N or In\({}_{x}\)Ga\({}_{1\times}\)N, and ZnO epitaxy [21; 45; 46]; so, if cooling were to be demonstrated with laser diodes, the improved performance of imaging arrays in outer space could be achieved in the near future. In view of the general agreement between calculated and observed wavelengths in Tables 2 and 3, it can be concluded that the discrete resonances in cooling response in Ti:Al\({}_{2}\)O\({}_{3}\) (Figure 15) arise from absorptions on all-electronic transitions or optical phonon sidebands. The discrepancies between the calculated and observed wavenumbers were less than \(\sim\)400 cm\({}^{-1}\) in all cases (\(<\)3%) and better than 1.2 cm\({}^{-1}\) (\(<\)0.01%) on average for fifteen \(\pi\) resonances and 60 cm\({}^{-1}\) (\(<\)5.6%) on average for ten \(\sigma\) resonances. Experimentally, more resonances were observed for \(\pi\)- than \(\sigma\)-polarization, even though \(\sigma\)-polarization excites more optical modes in sapphire [21]. Whether or not a particular absorption transition gives rise to observable cooling in Ti\({}^{3+}\):Al\({}_{2}\)O\({}_{3}\) seems to be dictated by the level of background impurity absorption. The background absorption coefficient for \(\sigma\)-polarization has a peak in the 750-790 nm range, with a value nearly twice that for \(\pi\)-polarization [34]. So, more resonances may be observed in this range for \(\pi\)- than for \(\sigma\)-polarization despite the greater number of infrared active \(E_{u}\) modes [21]. The relative efficiencies of cooling transitions also depend on the occupation of the various levels in question, determined by the Boltzmann distribution, and the nature of the transitions, which in general is of mixed electronic and vibrational character. Due to these complexities, initial and final states were assigned to each resonance, but no attempt was made to analyze the relative intensities of the cooling resonances. Because the dipole-allowed transition of Ti\({}^{3+}\) has proven useful in this work for the laser cooling of sapphire, the analysis of quantum interference in the 3-level system of Ce\({}^{3+}\), which has two such transitions, acquires heightened interest. Trivalent Cerium can undergo quantum interference effects of several kinds. This was evident from the calculations in Figure 16a,b, where gain was induced on the two dipole-allowed absorption transitions by a single incident pump wave. Both figures indicate that lasing without inversion can take place on these transitions as the two lower states are always more occupied than the upper level (given that state 3 is long-lived in Ce\({}^{3+}\)). Gain on the 13 transition in Figure 16a is due to Fano resonance as it has no threshold and requires the mixing of the excited states (\(V\
eq 0\)). Gain on the 12 transition in Figure 16b is induced only at high pump powers and does not require excited state mixing (\(V=0\)). Hence, quantum interference that originates in two distinct ways is possible in this system. At the same time, laser cooling can be sustained theoretically on the red side of the transition at \(\omega_{13}\). Therefore, the intriguing possibility emerges of accomplishing self-cooled lasing without inversion in materials with allowed transitions such as Ce\({}^{3+}\):LiCaAlF\({}_{6}\).