Under appropriate conditions, absorption of light by a solid can initiate a process by which it is cooled. In particular, energy is extracted from a material when its absorption of a photon is followed by emission of a photon of higher energy. This up-conversion requires some of the solid's electrons to garner energy from atomic vibrations. Here, two schemes for laser cooling via localized electronic states are addressed. The first scheme utilizes the ground state and an excited state of a localized center. In this two-level scheme, the cooling process is initiated with photon absorption in the extreme low-energy tail of a localized state's vibrationally broadened absorption spectrum. The subsequent atomic relaxation transfers energy of especially large vibratory atomic strains into electrical energy that is then extracted via photon emission. The second scheme involves the ground state and two excited states of a localized center. Cooling is facilitated when (i) the photoexcitation of an electron from its ground state to the lower excited level is followed by (ii) electron-phonon-induced promotion to the uppermost level and the subsequent (iii) return of the electron to its ground state with emission of a photon of higher energy than that of the absorbed photon. However, competing relaxation processes contribute to heating. The net cooling power per unit volume is maximized for both schemes, thereby determining characteristics of localized electronic systems that foster optical cooling. The cooling power per unit volume is greatest at high temperatures and falls rapidly as the thermal energy is reduced below each system's luminescence Stokes shift. Moreover, cooling via the three-level scheme is most effective when (i) the energy separation between excited states is smaller than the thermal energy and (ii) the degeneracy of the highest-lying excited state is much larger than that of the center's middle level. These restrictive conditions appear to be satisfied for the three-level systems that to date exhibit laser cooling at or somewhat below room temperature: severely localized crystal-field-split $f$-electron states of rare-earth ions.
Under appropriate conditions, absorption of light by a solid can initiate a process by which it is cooled. In particular, energy is extracted from a material when its absorption of a photon is followed by emission of a photon of higher energy. This up-conversion requires some of the solid's electrons to garner energy from atomic vibrations. Here, two schemes for laser cooling via localized electronic states are addressed. The first scheme utilizes the ground state and an excited state of a localized center. In this two-level scheme, the cooling process is initiated with photon absorption in the extreme low-energy tail of a localized state's vibrationally broadened absorption spectrum. The subsequent atomic relaxation transfers energy of especially large vibratory atomic strains into electrical energy that is then extracted via photon emission. The second scheme involves the ground state and two excited states of a localized center. Cooling is facilitated when (i) the photoexcitation of an electron from its ground state to the lower excited level is followed by (ii) electron-phonon-induced promotion to the uppermost level and the subsequent (iii) return of the electron to its ground state with emission of a photon of higher energy than that of the absorbed photon. However, competing relaxation processes contribute to heating. The net cooling power per unit volume is maximized for both schemes, thereby determining characteristics of localized electronic systems that foster optical cooling. The cooling power per unit volume is greatest at high temperatures and falls rapidly as the thermal energy is reduced below each system's luminescence Stokes shift. Moreover, cooling via the three-level scheme is most effective when (i) the energy separation between excited states is smaller than the thermal energy and (ii) the degeneracy of the highest-lying excited state is much larger than that of the center's middle level. These restrictive conditions appear to be satisfied for the three-level systems that to date exhibit laser cooling at or somewhat below room temperature: severely localized crystal-field-split f-electron states of rare-earth ions.
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.
Light emission occurs as a result of interaction between light and matter. Let us assume a two-level atom with levels 1 (ground state) and 2 (excited state). The energies of the ground and excited states are E 1 and E 2 , respectively, and the energy difference is therefore given by the difference ΔE 12 =E 2 −E 1 . When light with photons of energy equal to this difference is absorbed by the atomic or molecular system, an electron will be excited from level 1 to level 2. The energy of the photons is given by ΔE 12 =E 12 =hν 12 (h is Planck's constant and 1→2 is the frequency of light resulting from the level 1→2 transition). This energy is absorbed by an atom or a molecule that has energy levels, separated by ΔE, where these energy levels are the ground and the excited states. This kind of “quantum jump” of an electron between two states occurs in atomic systems between electronic levels; it can be extended to molecular systems, where vibrational and translational energy levels participate in the quantum jump. The interaction between light and matter involves the transition of electrons between different states. This interaction results in the absorption of photons (stimulated absorption) as well as spontaneous and stimulated emission. These processes can be described using Einstein's A and B coefficients, as will be described in the next paragraphs.
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.
Publisher Summary This chapter focuses on the concept of laser cooling of a solid, which may occur when the average energy of the photons emitted by the solid is larger than the energy of the ones it absorbs. More formally, the anti-Stokes emission, which occurs at frequencies larger than that of the pump laser, must dominate the Stokes emission that occurs at smaller frequencies. A crucial additional requirement is that the nonradiative decay rates of the laser-pumped states be negligible in comparison to their radiative decay rates. For the sake of specificity, these states will be taken to be those of a set of isolated ions embedded within an insulating host, although in general they could equally well be those of gas-, liquid-, or solid-phase neutral atoms or molecules, or even those of the energy bands of intrinsic semiconductors. Outside the realm of impurity-doped insulators, recent experiments with semiconductors make net anti-Stokes fluorescence cooling of these materials appear achievable. A long-term idea for improving laser cooling of solids takes advantage of photonic band gap materials. The idea is to fabricate a refractive-index structure within the cooling element that would establish a three-dimensional photonic gap immediately below the pump frequency. In this way, emission at Stokes frequencies falling within the gap would be inhibited, thereby enhancing the cooling power derived from the anti-Stokes emission.
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.
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.