# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 3 Methods and Results
### Laser Cooling on an Electric-Dipole-Allowed Transition in Ti:Al\({}_{2}\)O\({}_{3}\)  
A Ti\({}^{3+}\):Al\({}_{2}\)O\({}_{3}\) sample (GT Advanced Technologies) was grown using the heat exchange method (HEM) known to produce high-quality crystals with exceptionally high figures of merit. In this material, the figure of merit (FOM) is defined as the ratio of the absorption coefficients at two specific wavelengths (\(\alpha_{532\mathrm{mm}}/\alpha_{800\mathrm{nm}}\)) and is used as a measure of crystal \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Beam Radius (\(\mu\)m)** & **Intensity (W/cm\({}^{2}\))** \\ \hline DLT High & 32.4–39.3 & (1.6–2.4) \(\times\) 10\({}^{4}\) \\ DLT Low & 284–318 & (2.0–3.3) \(\times\) 10\({}^{2}\) \\ TLS High & 316–364 & (1.1–1.3) \(\times\) 10\({}^{3}\) \\ TLS Low & 316–364 & (1.1–1.3) \(\times\) 10\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Pump beam radius (half-width at 1/e\({}^{2}\) intensity) and intensity values used for the measurements presented in Figures 11 and 12. Entries indicate the small variations of intensity at multiple wavelengths for each intensity category. Figure 12: Experimentally determined cooling efficiencies versus wavelength from DLT measurements in Yb:YLF at high and low intensities. The pump power was 0.8 W for all the measurements, and beam size was varied to reach the intensities shown in Table 1.  
quality and laser performance. The GTAT sample had a quoted FOM of 844 and was Brewster-cut with the dimensions of \(4\times 5\times 20\) mm\({}^{3}\). The normalized absorption and emission spectra of this sample are shown in Figure 13. Two other laser-grade samples with lower FOMs were included in the cooling tests. Thermal lens spectroscopy (TLS) in a mode-mismatched configuration [28] was used to investigate the thermal characteristics of the three sapphire samples for pump light tuned close to the absorption peak of Ti\({}^{3+}\) and in the absorption tail. An experimental approach that corrects for intensity variations of the pump laser [33] was applied in experiments on Yb:YLF but not in the GTAT sapphire sample because weak fluorescence prevented any reliable correction. A weak helium-neon laser (\(\lambda_{p}=633\) nm) was used as the probe laser. A test was performed with a thermal camera to ensure that the probe power was low enough to avoid any detectable heating. The GTAT sample was first pumped with 532 nm light (Coherent Verdi V6) to show heating. The TLS transient signal for this excitation wavelength (Figure 14a) shows the positive slope expected for a material with \(ds/dT>0\) with increasing temperature. To pump at red-detuned wavelengths in the absorption tail, the tunable Ti:Sapphire laser was used. For some wavelengths longer than the mean fluorescence wavelength, the slope of the TLS signal then became negative, as shown in Figure 14b, which was indicative of sample cooling within the pumping region. Using the known material parameters and the measured experimental values, in both cases, the data fit very well with TLS theory. The TLS signals in the other samples showed only positive sloping curves, such as that in Figure 14a, at all wavelengths. Interestingly, the cooling behavior at wavelengths longer than the mean fluorescence wavelength was not uniform. Instead, sharp resonances were observed at which heating abruptly switched to cooling. This behavior is shown as TLS signals with negative polarity at discrete wavelengths in Figure 15, consistent with the discussion of Section 2.2 and Figure 5. In the case of well-defined local or host optical phonons, the cooling transitions should be discrete. The spectrum of infrared-active modes is well-known for sapphire [21], but the frequencies of local modes exhibiting site-dependent perturbations with respect to host modes are not. Nevertheless, the cooling resonances observed in Figure 15a,b are well accounted for by assignments that assume four mildly perturbed sites for Ti\({}^{3+}\) ions in sapphire, as listed in Tables 2 and 3.