# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 2 Theory
### Improved Cooling Efficiency Using Optical Saturation
\(\lambda\) and \(I\) are the wavelength and intensity of the pump laser. The polarization-averaged mean fluorescence wavelength \(\lambda_{f}\) is determined from emission spectra. The absorption coefficients of the coolant ions, \(\alpha_{r}\), and background impurities, \(\alpha_{b}\), were assumed to be intensity-dependent. The form of this dependence [17] was taken to be \(\alpha(I)=\alpha(0)/(1+I/I_{sat})\), where the saturation intensity \(I_{sat}\) was assigned the values \(I_{r}\) or \(I_{b}\) for the coolant or background impurity ions, respectively. The saturation intensity of the coolant ions varies with wavelength through its dependence on absorption and emission cross-sections. It is defined by the expression  
\[\frac{dN_{2}^{\prime}}{dt}=\frac{\sigma^{\prime}I_{p}\lambda_{p}}{hc}.\big{[} N_{1}^{\prime}-N_{2}^{\prime}\big{]}-\frac{N_{2}^{\prime}}{\tau^{\prime}}. \tag{2}\]1  
where \(\tau_{f}\) is the fluorescence lifetime. The effective absorption cross-section \(\sigma_{a}(\lambda)\) was measured directly with a spectrophotometer (SHIMADZU UV-3600) and corrected for instrument response. The effective emission cross-section \(\sigma_{e}(\lambda)\) was then calculated using the McCumber relation [18]. This procedure ensured that the multiple Stark levels of Yb\({}^{3+}\) and the [PERSON] population distribution among them were taken into account. Figure 1 displays the effective absorption and emission cross-sections of the 10% Yb:YLF crystal studied in our experiments and shows the position of the polarization-averaged mean fluorescence wavelength at 997.6 nm. Based on these cross-sections and a literature value for the Yb\({}^{3+}\) fluorescence lifetime of 2.2 ms [19], the theoretical saturation intensity of the Yb\({}^{3+}\) coolant ions was determined using Equation (11) and plotted as the continuous curve in Figure 2 versus wavelength. Surmising that an impurity such as Fe\({}^{3+}\) is the main source of background absorption [9] and that it has a broadband spectrum centered around 1 micron [20], the saturation intensity \(I_{b}\) of the background may be presumed to be only weakly dependent on wavelength over the range of our experiments. With this one assumption, the experimental value of \(I_{b}\) can be estimated using the empirical approach outlined next and illustrated as a dashed line in \[\frac{dN_{2}^{\prime}}{dt}=\frac{\sigma^{\prime}I_{p}\lambda_{p}}{hc}.\big{[} N_{1}^{\prime}-N_{2}^{\prime}\big{]}-\frac{N_{2}^{\prime}}{\tau^{\prime}}. \tag{2}\]2  
\[\frac{dN_{2}^{\prime}}{dt}=\frac{\sigma^{\prime}I_{p}\lambda_{p}}{hc}.\big{[} N_{1}^{\prime}-N_{2}^{\prime}\big{]}-\frac{N_{2}^{\prime}}{\tau^{\prime}}. \tag{2}\]3 Absorption and emission cross-sections plot for E\(||\)c in 10% Yb\({}^{3+}\):LiYF\({}_{4}\). The blue curve is the effective absorption cross-section measured by a spectrophotometer, and the red is the effective emission cross-section calculated using the McCumber theory. The dashed vertical line indicates the polarization averaged mean fluorescence wavelength, 997.6 nm. \[\frac{dN_{2}^{\prime}}{dt}=\frac{\sigma^{\prime}I_{p}\lambda_{p}}{hc}.\big{[} N_{1}^{\prime}-N_{2}^{\prime}\big{]}-\frac{N_{2}^{\prime}}{\tau^{\prime}}. \tag{2}\]4 Theoretical saturation intensities versus wavelength. The blue curve shows the wavelength-dependent saturation intensity \(I_{r}\) of Yb\({}^{3+}\) (from Equation (11)). The dashed line shows an arbitrary, constant saturation intensity level \(I_{b}\) of the background impurities. The plot illustrates an intersection point at \(\mathbf{\lambda}_{cr}=1010\) nm on the low energy side of the Yb\({}^{3+}\) absorption band, where \(I_{b}=I_{r}(\mathbf{\lambda}_{cr})=3\times 10^{4}\) W/cm\({}^{2}\). The experimental determination of \(I_{b}\), the intersection at \(\mathbf{\lambda}_{cr}\) is determined by the wavelength where cooling efficiency curves recorded at different intensities intersect. At \(\mathbf{\lambda}_{cr}\), the wavelength-dependent saturation intensities are equal.