# 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\({}_{4}\)
#### 3.1.1 Differential Luminescence Thermometry (DLT)  
DLT was employed to deduce the crystal temperature from variations of the fluorescence line shape [3]. The infrared fluorescence was collected with a multimode optical fiber (Ocean Optics QP600-2-VIS-NIR; NA = 0.4) connected to a 0.25 m grating spectrometer (Oriel 74100) equipped with a CCD detector (Andor DU491A-1.7). The difference between the measurement and the reference was spectrally integrated to calculate the DLT signals, which were pre-calibrated by the direct heating or cooling of the sample using a temperature controller (Quantum Northwest Flash 300) with an accuracy of \(\pm 0.01\) K. A thermal camera (FLIR A655 sc) was used to monitor the crystal surface temperature and cross-check the temperature measured by DLT [24]. Figures 8 and 9 plot the crystal temperature change versus the time measured by DLT at wavelengths of 1015 nm and 920 nm, showing the representative signals for the cooling and heating ranges. Figure 8: Temperature vs time after the cooling beam is introduced into the Yb:YLF sample. Data were obtained by the DLT method with pump power of 0.8 W at \(\lambda=1015\) nm. At this wavelength, the sample cools. A linear fit to the temperature change during the first minute had a slope of \(-2.11\times 10^{-3}\) K/s (inset). Figure 9: Temperature change versus time based on DLT measurements in Yb:YLF with pump power of 0.8 W at \(\lambda=920\) nm. At this wavelength, the sample heats. A linear fit to the temperature change for the initial excursion in temperature of \(\Delta T=1\) K had a slope of 3.48 \(\times\) 10\({}^{-2}\) K/s (inset). The cooling efficiency can be computed directly from the temperature evolution measurements. Assuming a crystal of thermal emissivity \(\varepsilon\) and a surface area \(A_{surf}\) enclosed by a blackbody, the thermal balance equation relating the various contributions to heating power at a crystal temperature of \(T\)[25; 26] is:  
\[cM\frac{dT}{dt}=-\eta_{c}P_{abs}+A_{surf}\epsilon\sigma_{B}\left(T_{0}^{4}-T^{ 4}\right)+\left(h_{c}A_{surf}+k\frac{A}{L}\right)(T_{0}-T), \tag{13}\]  
where \(\sigma_{B},\ T_{0},\ h_{c},\ k,\ A\), and \(L\) are the Stefan-Boltzmann constant, environmental temperature, convective heat transfer coefficient, thermal conductivity of the link, cross-sectional area, and sample length, respectively. The specific heat \(c\) of 10% Yb:YLF is 0.79 J/(g K) at room temperature [27]. The crystal mass (\(M=0.408\) g) was determined using a precision balance (QUINTIX213-1S) with an accuracy of 0.001 g.  
The first term on the right side of Equation (13) is the laser-induced heat density. The second term is the power exchange between the crystal and its surroundings through blackbody radiation. The last term is the conductive heat load, which takes into account the contact with the air and sample support. The blackbody radiation and conductive heat load depend on the temperature difference between the crystal and the environment. As indicated by the data of Figures 8 and 9, the temperature change is linear at very short times, reflecting the negligible difference between \(T\) and \(T_{0}\) in the second and third terms on the right of Equation (13). During the first minute after the pump beam enters the sample, only the first term on the right of Equation (13) is responsible for the temperature changes of the crystal. This term is numerically equal to the cooling power; so, the cooling power can be determined directly from the initial slope of the sample temperature versus time. At long times, the difference between \(T\) and \(T_{0}\) increases. Consequently, the last two terms in Equation (13) become significant, and the temperature dependence on time becomes nonlinear. By restricting the data analysis to the early times (or to temperature excursions \(dT<1\) K) and the measuring power absorbed by the sample, the blackbody radiation and conduction terms can be ignored, permitting the cooling efficiency to be determined directly from Equation (13) in a simple way.