# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 3 Methods and Results
### Improved Cooling via Saturation of Background Absorption in Yb:LiYF\({}_{4}\)
#### 3.1.2 Thermal Lens Spectroscopy (TLS)  
TLS was used to measure the thermal lensing signal by probing the refractive index variation of the crystal induced by a pump beam. The tunable Ti:Sapphire laser was used to pump the crystal and, by chopping its beam, synchronous detection was enabled. In general, two lensing effects are induced by the pump--a fast transient from photoinduced population transfer between the ground and excited states and a slower transient from thermal diffusion. Both of these lensing effects change the refractive index of the crystal and must be accurately modeled to account for their separate contributions [27; 28]. The TLS signal was obtained by measuring the intensity variation of a co-propagating He-Ne probe laser (Melles Griot 25-LHP-991-249), which sensed the crystal refractive index changes. Two normalized TLS signals and fits accounting for population and thermal effects [29] are shown in Figure 10. These best fits provided quantitative values for the thermal lens strength and the corresponding cooling efficiency \(\eta_{c}\) derived from it. The cooling efficiency determined at different intensities is plotted versus wavelength in Figures 11 and 12, based on the TLS and DLT data, respectively. The measured pump radii and intensities are shown in Table 1. The main result to note is a steepening of the cooling efficiency curve over most of the range as intensity is increased. This behavior arises from the different variations of the saturation intensity of the coolant and the impurity ions versus the wavelength, as discussed further below and in Ref. [30]. This effect, referred to here as differential absorption saturation, was observed in both the DLT and the TLS experiments and can be explained using the analysis of Section 2.1. Figure 11: Experimentally determined cooling efficiencies versus wavelength from TLS measurements in Yb:YLF at high and low intensities. Input powers in the range 50–530 mW focused to the tabulated spot sizes, and the resulting intensities are listed in Table 1. A host parameter value of \(\Theta=1.4\) W\({}^{-1}\) was used for analysis at all intensities. For low and high intensity data, the external quantum efficiency and the background absorption coefficient determined from fitting the data to Equation (10) were \(\eta_{ext}=1.01\pm 0.02\) and \(\alpha_{b}=(4\pm 2)\times 10^{-3}\) cm\({}^{-1}\). Figure 10: Normalized TLS signal versus time at 1015 nm and 930 nm, showing cooling and heating results in Yb:YLF, respectively. Black circles are the measured TLS signal, and red curves are best fits that account for population and thermal effects. The experimental cooling efficiency data coalesce at 1000 nm. There, all the curves pass through the same point, as predicted by the theory of Section 2. At such a crossing point, the cooling efficiency is clearly independent of intensity, indicating the saturation intensity of the background equals the saturation intensity of the coolant ions, which is \(\sim\)\(2.6\times 10^{4}\) W/cm\({}^{2}\). At longer wavelengths in the cooling range, a significantly higher cooling efficiency is observed at high pump intensity in both sets of data. The results are therefore consistent with a saturation intensity for background impurities that is less than that of the coolant ions, leading to reduced parasitic heating and higher cooling efficiency. At wavelengths shorter than 1000 nm, closer to the main absorption peak of Yb\({}^{3+}\), the cooling efficiency drops at high intensity, consistent with a drop in the Yb\({}^{3+}\) saturation intensity as resonance is approached. The separation of the experimental curves is somewhat larger than the theoretical efficiency curves of Figure 3. In the short wavelength range, this may be attributable to the easy saturation of Yb\({}^{3+}\), which would maintain the maximum population density in the excited state, thereby enhancing the well-known energy transfer from Ytterbium to other rare-earth ions such as Erbium [31, 32].