# Accurate Characterization of the Properties of the Rare-Earth-Doped Crystal for Laser Cooling
## 2 LITMoS Test Theory  
The cooling efficiency \(\eta_{c}(\lambda,T)\) is used to deduce the cooling grade of the material and expresses as [36]:  
\[\eta_{c}(\lambda,T)=1-\eta_{ext}\frac{\alpha_{r}(\lambda,T)}{\alpha_{r}( \lambda,T)+\alpha_{b}}\frac{\lambda}{\lambda_{f}(T)} \tag{1}\]  
where \(\eta_{ext}\) is the external quantum efficiency, and \(\alpha_{r}(\lambda,T)\) and \(\alpha_{b}\) are the resonance absorption coefficient and the background absorption coefficient, respectively. The four parameters \(\eta_{ext},\alpha_{b},\lambda_{f}(T)\) and \(\alpha_{r}(\lambda,T)\) in Equation (1) describe the optical cooling grade of the sample. It is necessary to precisely measure the four parameters above to calculate the wavelength- and temperature-dependent cooling efficiency contour map (cooling window). The effect of various thermal loads on the sample can be expressed as:  
\[C_{v}\frac{dT}{dt}=-\eta_{c}P_{abs}+\frac{\varepsilon_{s}A_{s}\sigma}{1+ \chi}\Big{(}T_{a}^{4}-T_{s}^{4}\Big{)}+A_{S}k_{h}(T_{a}-T_{s})+\frac{Nk_{l}A_{ l}}{d_{l}}(T_{a}-T_{s}) \tag{2}\]  
In the right side of the equation, the first term represents the heat extracted from the anti-Stokes cooling. The second term represents the radiative heat load from the ambience. The third and fourth terms represent the convective and conductive heat loads on the sample, respectively. \(\sigma=5.67\times 10^{-8}\) W/mm\({}^{2}\)K\({}^{4}\) is the Stefan-Boltzmann constant and \(\chi=(1-\varepsilon_{a})\varepsilon_{s}A_{S}/\varepsilon_{a}A_{a}\). Here \(T_{a,s}\), \(A_{a,s}\) and \(\varepsilon_{a,s}\) are the temperature, the surface area and the thermal emissivity, respectively. Subscripts \(a\) and \(s\) denote the ambience and the sample. \(N\) is the number of contacting points with the length \(d_{l}\) of support, the area \(A_{l}\) of the contact points and the conductivity \(k_{l}\) of support. \(k_{h}\) is the convective heat transfer coefficient of the surrounding. \(C_{v}=c_{v}\rho V\) is the heat capacity, where \(\rho\), \(c_{v}\) and \(V\) are specifically the heat, density and volume of the sample, respectively. The last two terms in Equation (2) represent the convective heat load and the conductive heat load on the samples, respectively. Their contributions can be neglected for a sample supported by two optical fibers of 100 um diameters inside the high vacuum cavity. Since the cavity surface area \(A_{a}\) is much larger than the sample surface area \(A_{s}\), the influence of \(\chi\) can be neglected. For the case of the small temperature change \(\Delta T(T_{a}-T_{s}\leq 5\) K), Equation (2) is simplified as [46]. \[C_{\
u}\frac{dT}{dt}\approx-\eta_{c}P_{abs}-4\epsilon_{s}A_{s}\sigma T_{a}{}^{3 }(T_{s}-T_{a}) \tag{3}\]  
The experimentally measured cooling efficiency \(\eta_{c}(\lambda)^{exp}\) under steady-state (\(dT/dt=0\)) condition can be expressed as Equation (4). \[\eta_{c}(\lambda)^{exp}=K_{rad}\cdot\Delta T/P_{abs}(\lambda) \tag{4}\]  
where \(K_{rad}=4\epsilon_{s}A_{s}\sigma T_{a}{}^{3}\) and \(P_{abs}(\lambda)\propto k\int S(\lambda)d\lambda\). Here, \(k\) and \(S(\lambda)\) are the scaling factor and the fluorescence spectrum, respectively. The cooling efficiency of the sample \(\eta_{c}(\lambda)^{exp}\) can be obtained by precisely measuring the sample temperature change \(\Delta T\) as a function of the pump wavelength \(\lambda\) and the absorption power. To accurately measure the laser-induced temperature change \(\Delta T\), a special experimental set-up has been designed to minimize the sample's external thermal load and a calibrated thermal imaging camera is used to precisely monitor the sample temperature in real time. With the measured values of the average fluorescence wavelength of the sample \(\lambda_{f}(T)\) and the resonance absorption coefficient \(\alpha_{r}(\lambda,T)\), the external quantum efficiency \(\eta_{ext}\) and the background absorption coefficient \(\alpha_{b}\) can be obtained by fitting the cooling efficiency \(\eta_{c}(\lambda)^{exp}\) measured in the experiment utilizing Equation (1).