# Laser Cooling and Trapping of Rare-Earth-Doped Particles
## 3 Optical Refrigeration of Rare-Earth-Doped Particles
### Laser Cooling of Rare-Earth-Doped Mie Resonant Samples
The cooling power generated in the sphere can be calculated as the difference between the pump power absorbed by the Yb\({}^{3+}\) ions in the sphere and the power radiated with anti-Stokes fluorescence at the mean fluorescence wavelength. The field intensity given by relation (19) can then be integrated over the nanosphere and then inserted into relation (6) as  
\[\overset{\rightarrow}{M}_{o1n}^{(1)}=cos\varphi\cdot\pi_{n}(cos\theta)\cdot j _{n}(\tilde{r})\delta_{\theta}-sin\varphi\cdot\pi_{n}(cos\theta)j_{n}(\tilde{ r})\delta_{\varphi} \tag{12}\]1  
which corresponds to the cooling power generated in the Mie-resonant nanosphere. Laser cooling of RE-doped low-phonon microparticles enhanced with Mie resonant modes was considered in [29]. By way of example, an Yb\({}^{3+}\):YAG nanosphere pumped with the plane electromagnetic wave propagating in vacuum at the pump wavelength \(\lambda_{P}=1030\) nm located in the long wavelength tail of the Yb\({}^{3+}\) absorption spectrum was considered (Figure 3). The mean fluorescence wavelength of the spontaneous radiation \(\lambda_{F}<\lambda_{P}\). A bulk Yb\({}^{3+}\):YAG sample was laser-cooled in [26]. A sample with an Yb\({}^{3+}\) ion concentration of \(\sim\)10% was investigated in [29]. This concentration is smaller than the critical concentration of Yb\({}^{3+}\):YAG (17%). Such samples are free from cooperative effects such as resonant-radiative transfer (reabsorption) and resonant nonradiative transfer, which deteriorate laser cooling [19]. Indeed, these cooperative effects result in the excited energy migration from the ion to ion inside the sample. This migrating energy can excite impurities in the sample and decay nonradiatively, causing heat generation in the sample. The pump power considered in [29] was \(I_{P}=0.5\cdot 10^{-4}\) W/\(\upmu\)m\({}^{2}\). The cooling power density generated in the nanosphere is equal to  
\[\overset{\rightarrow}{M}_{o1n}^{(1)}=cos\varphi\cdot\pi_{n}(cos\theta)\cdot j _{n}(\tilde{r})\delta_{\theta}-sin\varphi\cdot\pi_{n}(cos\theta)j_{n}(\tilde{ r})\delta_{\varphi} \tag{12}\]2where \(V=\frac{4}{3}\pi R^{3}\) is the volume of the sphere. \(R\) is the radius of the sphere. The cooling power density, \(\rho_{cool}\), for nanospheres with different radii in the range between 150 nm and 650 nm was estimated using relations (22) and (23). The results are plotted in \[\overset{\rightarrow}{M}_{o1n}^{(1)}=cos\varphi\cdot\pi_{n}(cos\theta)\cdot j _{n}(\tilde{r})\delta_{\theta}-sin\varphi\cdot\pi_{n}(cos\theta)j_{n}(\tilde{ r})\delta_{\varphi} \tag{12}\]3 It changes considerably on this range and has several peaks. The radii of the nanospheres corresponding to these peaks are very close to the resonant radii of the Mie resonant modes including the magnetic dipole (MD) modes, the magnetic quadrupole (MQ) modes, the electric dipole (ED) modes, and the electric quadrupole (EQ) modes. With the knowledge of the cooling power density, \(\rho_{cool}\), it is possible to estimate the temperature of the nanospheres placed on a low-contact and low-thermal-conductivity support in a vacuum chamber, using Equation (7). These results are presented in \[\overset{\rightarrow}{M}_{o1n}^{(1)}=cos\varphi\cdot\pi_{n}(cos\theta)\cdot j _{n}(\tilde{r})\delta_{\theta}-sin\varphi\cdot\pi_{n}(cos\theta)j_{n}(\tilde{ r})\delta_{\varphi} \tag{12}\]4 As one might expect, the radii of the nanospheres for which the temperature reaches minima almost perfectly coincide with those associated with the highest cooling power density (Figure 4). \[\overset{\rightarrow}{M}_{o1n}^{(1)}=cos\varphi\cdot\pi_{n}(cos\theta)\cdot j _{n}(\tilde{r})\delta_{\theta}-sin\varphi\cdot\pi_{n}(cos\theta)j_{n}(\tilde{ r})\delta_{\varphi} \tag{12}\]5 Cooling power density generated in nanospheres with different radii. Resonant radii of the nanospheres supporting MD, ED, MQ, and EQ modes are presented for simplicity of understanding [29].