Laser cooling of rare-earth doped solids has been demonstrated across a wide range of material platforms, inspiring the development of simple phenomenological models such as the four-level model to elucidate the universal properties of laser cooling under various operating conditions. However, these models usually require the input of full absorption spectra that must be provided experimentally or by additional complicated atomic modeling. In this Letter, we propose that a four-level model, when extended to admit effective energy levels adaptive to the pumping photon energy, can accurately predict the cooling efficiency as a function of temperature and pumping frequency using only few inputs such as the absorption coefficient measured at a single frequency and temperature. Our model exploits the quasi-equilibrium properties of the excitation of rare-earth ions for the determination of the effective four energy levels. The model is validated against published experimental results for a span of materials including ytterbium/thulium-doped glass and crystals. With the verified model, we derive explicit expressions for the optimal frequency and the operating bandwidth of pumping lasers. Our model significantly simplifies the modeling process of laser cooling and is expected to stimulate further development of optical refrigeration.
Laser cooling of rare-earth doped solids has been demonstrated across a wide range of material platforms, inspiring the development of simple phenomenological models such as the four-level model to elucidate the universal properties of laser cooling under various operating conditions. However, these models usually require the input of full absorption spectra that must be provided experimentally or by additional complicated atomic modeling. In this Letter, we propose that a four-level model, when extended to admit effective energy levels adaptive to the pumping photon energy, can accurately predict the cooling efficiency as a function of temperature and pumping frequency using only few inputs such as the absorption coefficient measured at a single frequency and temperature. Our model exploits the quasi-equilibrium properties of the excitation of rare-earth ions for the determination of the effective four energy levels. The model is validated against published experimental results for a span of materials including ytterbium/thulium-doped glass and crystals. With the verified model, we derive explicit expressions for the optimal frequency and the operating bandwidth of pumping lasers. Our model significantly simplifies the modeling process of laser cooling and is expected to stimulate further development of optical refrigeration.
A mechanism for laser cooling of crystals doped with rare-earth ions is proposed. The mechanism involves two-photon Raman scattering through the dipole-allowed 5d ion level, electron–phonon transitions between the Stark-split sublevels, and fluorescence from the excited ion level. It is shown that, under the conditions of coherent pumping, the total population of the ground level is transferred to the excited level of the rare-earth ion, thus increasing the cooling power and accelerating the cooling process. The involvement of dipole-allowed transitions yields an increase in the cooling efficiency by 14% compared to the efficiency of direct pumping at transitions between the 4f levels. Estimations show that the temperature of the sample cooled by the proposed mechanism can reach 82.9 K for current materials with a background absorption coefficient of 4.0×10−4 cm−1.
Expectedly there are challenging issues to be tackled. For example, the purity level of the (undoped) fiber needs to be high to avoid excessive parasitic absorption of photons by impurities. When the fiber cooler and the object to be cooled are macroscopic (centimeter scale or larger), managing the (conductive and convective) heat and photon flows is also a pertinent issue. These and related issues will be discussed below. In Section 3.2, we describe our theoretical developments in this investigation. We discuss the calculation of the cooling powers of the Tm atoms and of a model isolated fiber and the simulation of heat flow in the cooling of the heat load. In Section 3.3, we discuss experimental progress in fabricating high-purity fiber and temperature measurements under optical pumping conditions. 3.2 Theoretical Developments 3.2.1 Modeling the Cooling of Thulium AtomsThe basic cooling agents in our setup are the thulium (Tm) atoms doping the fiber. We describe in this subsection a photoexcitation model of a Tm atom in a glass host, which we use to estimate its laser cooling efficiency. More detail can be found in [7]. The model, given in Ref. [8], is shown in Fig. 3.2. It consists of four energy levels, each with a degree of degeneracy gi, as designated in the figure. The lower two levels represent the manifold of states created by the splitting of the 13-fold degenerate ground state (3H6) of Tm by interactions with the glass host, and the upper two levels represent the split ninefold excited-state (3F4) manifold. The pump laser energy hn is tuned to the energy between |1> and |2>, which is the lowest excitation energy between the two manifolds. The pumped Tm atoms de-excite (incoherently) to a level in the ground-state manifold via luminescence, at rate Wrad, and nonradiative transitions, at a rate Wnr. If the de-excitation happens sufficiently slowly, the excited atoms have time to thermalize in the excited-state manifold. The luminescent photons are thus at higher energies than that of the pump photons. In each cycle of operation in which a Tm atom absorbs a pump photon and emits a luminescent photon, which then escapes, heat is removed from the system. The population dynamics of the four-level model obeys the following rate equations [8]: ( ) g B/inc1 1 112 1 2 2 3 1 1 02 02 E k TIdN g gRN N N N w N N edt g h g ds n -Ê ˆÊ ˆ= - - + + - -Á ˜Á ˜Ë ¯ Ë ¯ (3.1) u B/inc 32 1 dN dt RN w N g 2 2= - - - Ê ËÁ ˆ ¯˜ -d u B/ (3.3) N N N N Ntotal = + + +0 1 32 (3.4)where Ni and gi are the population density and degree of degeneracy, respectively, of level |i>, and Ntotal is the total density of Tm atoms. The pump laser has intensity Iinc and frequency n. R = 2Wrad + 2Wnr, and s12 denotes the photoabsorption cross section of the Tm atoms. dEg and w1 are the energy difference between the two levels and the intramanifold relaxation rate in the ground-state manifold, and dEu and w2 are the corresponding quantities for the excited-state manifold. T is the sample temperature, and kB is the Boltzmann constant. In Eq. 3.1, the first term on the right side represents stimulated absorption from and emission into the pump laser mode. The second term models the decay of the excited manifold population and the third term the thermalization within a manifold. The terms in Eqs. 3.2 and 3.3 have the same interpretation.
The enhanced laser cooling performance of rare-earth-ion-doped nanocrystalline powders is predicted, compared to the bulk material, using Yb3+:Y2O3 as the model material. This is achieved by enhancing the off-resonance, phonon-assisted absorption, which is proportional to the three factors considered in this paper: the dopant concentration, the pumping field energy, and the excitation coefficient. Using the energy transfer theory for concentration quenching, the optimum concentration corresponding to the maximum cooling power is found to be considerably larger than the currently used value, suggesting noticeable enhancement effects for laser cooling. The pumping field energy is enhanced in random nanopowders compared with bulk crystals under the same irradiation, due to the multiple scattering of photons. Photons are thus localized in the medium and do not propagate through, increasing the photon absorption of the pumping beam (and it is shown that the reabsorption of the fluorescence is negligible). Using molecular dynamics simulations, the phonon density of states (DOS) of the nanopowder is calculated, and found to have broadened modes, and extended small tails at low and high frequencies. The second-order electronic transition rate for the anti-Stokes luminescence is calculated using the Fermi golden rule, which includes the influence of this phonon DOS, and is shown to have enhancement effects on the laser cooling efficiency using nanopowders. It is finally concluded that these three enhancement mechanisms are essentially to increase the population of the three participating carriers (electron, photon, and phonon) in the interacting volume, and this also points out directions for enhancing laser cooling performance in bulk materials.
The recent achievements devoted to cooling of solids with a laser are presented in this paper. We discuss the latest results of traditional laser cooling of solids based on rare earth ions and new techniques based on colloidal lead-salt quantum dots doped in a glass host, laser cooling in Tm3+-doped oxy-fluoride glass ceramic. Relatively short (microsecond) lifetime of the excited level of the PbSe QDs compared to the millisecond lifetime of the excited level of RE ions allows an acceleration of the cooling process and provides an opportunity to use new materials with higher phonon energy as hosts, which are normally considered unsuitable for cooling with RE ions. Another new approach to the laser cooling problem based on super-radiance has been considered in this paper. The advantages of optical refrigeration with rare earth doped semiconductors, in which not only optically active electrons of the 4f shell but the valence and conduction bands of the host material are involved in cooling cycle is discussed. It is shown that involving the valence and conduction bands of the host in the cooling cycle allows the pump wavelength to be shorter than mean fluorescence wavelength. Raman laser cooling of solids as well as observation of spontaneous Brillouin cooling have been presented.
This review discusses the progress and ongoing efforts in optical refrigeration. Optical refrigeration is a process in which phonons are removed from a solid by anti-Stokes fluorescence. The review first summarizes the history of optical refrigeration, noting the success in cooling rare-earth-doped solids to cryogenic temperatures. It then examines in detail a four-level model of rare-earth-based optical refrigeration. This model elucidates the essential roles that the various material parameters, such as the spacing of the energy levels and the radiative quantum efficiency, play in the process of optical refrigeration. The review then describes the experimental techniques for cryogenic optical refrigeration of rare-earth-doped solids employing non-resonant and resonant optical cavities. It then examines the work on laser cooling of semiconductors, emphasizing the differences between optical refrigeration of semiconductors and rare-earth-doped solids and the new challenges and advantages of semiconductors. It then describes the significant experimental results including the observed optical refrigeration of CdS nanostructures. The review concludes by discussing the engineering challenges to the development of practical optical refrigerators, and the potential advantages and uses of these refrigerators.
The material metrics for optimal laser cooling of ion-doped solids are derived using atomic and molecular dynamics properties of the constituents. The anti-Stokes process is modeled as an optical phonon coupling of the bound electron, followed by a photon absorption. The transition dipole moment is estimated using a simplified charge-displacement model and the Judd--Ofelt theory of rare-earth ions both suggesting that transitions with high-energy gaps and similar angular momentum states should be used. The electron-phonon coupling is interpreted as a derivative of electronic energy with respect to displacement of the nearest neighboring ligands whose stretching mode frequency is approximated using the molecular data. The Debye-Gaussian model is used for the phonon density of states of diatomic crystal. Then, the Fermi golden rule is used for photon-induced, phonon-assisted electronic transition probability and applied to the cooling rate equation by defining a phonon-assisted transition dipole moment. Based on the material metrics, an example blend is investigated for its cooling performance and a general guide is proposed for selection of better performing laser cooling hosts. Furthermore, the cooling rate limits are discussed and three distinct characteristic times are identified with the photon-induced, phonon-assisted transition time controlling the rate. The metrics guide the selection of host materials for optimal cooling, and predict a noticeable increase in the absorption rate when using a blend of cation atoms.