Laser Cooling of Solids
- Basic Physical Principles
- Historical Overview
- Experimental Measurements
- Cryocooler Applications
- Laser Applications
It is possible to cool a material by anti-Stokes fluorescence. This simply means that the material emits photons which have a higher mean energy than those it absorbs. The energy difference arises from thermal excitations in the sample. Effectively, heat is converted into light, which leaves the material and is dumped onto a heat sink elsewhere.
The cooling efficiency can be defined as the ratio of the cooling power in the active material to the input electric power to the pump light source. Experimental measurements and theoretical calculations indicate that the cooling efficiency of this anti-Stokes process is competitive with existing thermoelectric and Stirling-cycle cryocoolers. Thus a refrigerator based on this process could be manufactured which has no moving parts, is light-weight, and can in principle cool to any desired final temperature. A second application is to the construction of a portable laser which requires no external coolant system, using optical cooling via the pump beam directly. This could permit the scaling of solid-state lasers to much higher powers than is currently possible.
Basic Physical Principles
It is easiest to understand the process by considering a system of noninteracting impurities in a solid. Let us suppose that these impurities have a particularly simple energy-level structure: a ground state, which can be labeled "1", and a pair of excited levels, labeled "2" and "3". For the purposes of illustration, assume that the splitting between levels 2 and 3 is at most a few times kT, where k is the Boltzmann constant and T is the final temperature attained upon cooling. This ensures that these two levels will rapidly (i.e., on the timescale of at most a few nanoseconds) return to thermal equilibrium with each other whenever this equilibrium is disturbed. At the same time, we require that the energy spacing between the ground and excited states be at least an order of magnitude larger than this. This ensures, according to the well-known energy gap law , that nonradiative relaxation across this gap will be negligibly slow. Thus, an upper-level excitation can only relax to the lower state by emitting a photon. This is equivalently expressed by stating that the radiative (or fluorescence) quantum efficiency of the system is approximately unity: one photon is emitted for each one absorbed that is tuned to the gap energy.
This constitutes the simplest example of a system which can exhibit fluorescent cooling. For, if a narrow-band light source (a laser, say) is tuned to the 1 → 2 transition, level 2 will become overpopulated relative to thermal equilibrium with level 3. To restore equilibrium, some population will be almost instantly transferred to the highest level. The excited-state manifold will subsequently relax radiatively to the ground state: fluorescence from both the 2 → 1 and 3 → 1 transitions will occur. (Such fluorescent relaxation typically occurs on a millisecond timescale, much slower than the equilibration time within the excited manifold.) Evidently, the mean energy of the emitted photons is larger than that of the absorbed photons. The excess is due to the thermal absorption required to equilibrate the excited levels and is carried out of the solid on the fluorescent light beam, resulting in cooling.
Alternatively, we can suppose that the ground-state manifold is split into two levels, "1" and "2", and that the excited state consists of a single level "3". Cooling is now accomplished by tuning the laser to the 2 → 3 transition, with fluorescent emission from both the 3 → 2 and 3 → 1 relaxation. Again, the mean energy of the fluorescent photons is larger than that of the absorbed photons. More generally, we can imagine a situation like that depicted in the adjacent figure: both the upper and lower states consist of a manifold of many closely-spaced levels, with a large energy gap between the two manifolds. The pump laser is tuned as far to the red end of the absorption spectrum as possible, so that excitations are transferred from near the top of the lower manifold to near the bottom of the excited one. The subsequent fluorescence will typically occur at shorter wavelengths, or higher energies, resulting in cooling. Candidate materials which have an energy-level diagram similar to the one drawn here include semiconductors (excited across their band gaps), rare-earth or transition-metal doped crystals and glasses (excited between their 4f or 5d multiplets), and polyatomic molecules in any phase (excited between vibrational levels).
The cooling energy per photon, assuming unit radiative quantum efficiency, is equal to the difference in the mean energies of the fluorescent and absorbed photons. It proves convenient to write the incident pump wavelength as λ and the inverse of the mean fluorescent photon energy as λF/hc. Taking the time derivatives to give powers, it is clear that the cooling power, Pcool, can be written in terms of the absorbed pump power, Pabs, as
Pcool = Pabs(λ - λF) / λF.
Here, λF is determined experimentally by measuring the fluorescence spectrum and finding its mean energy. This enables one to easily calculate the expected cooling efficiency for any given material of interest. In practice, however, it must be remembered that the actual cooling power is limited both by the fraction of the incident pump beam which is actually absorbed in the sample and by nonradiative (heating) energy relaxation processes, particularly by energy transfer to or direct pump or fluorescence absorption by bulk and surface impurities.
The idea that anti-Stokes fluorescence might be used to cool a material is a surprisingly old one, proposed as early as 1929 by Pringsheim . This proposal led some 16 years later to a rather spirited debate between Pringsheim and Vavilov, with the latter claiming that its realization is impossible on thermodynamic grounds . Landau himself had to step into the controversy  and proved that the entropy lost by the sample upon cooling is more than compensated for by an increase in the entropy of the light, resulting from the loss of monochromaticity, phase coherence, and directionality of the beam.
A few years later, the French researcher Alfred Kastler  discussed two systems in which a "lumino-caloric" effect might be observed: Zeeman-split levels of sodium vapor illuminated by circularly polarized light, and vibrational sidebands of the electronic transitions of rare-earth ions in salt crystals. The next class of materials proposed for fluorescent cooling was semiconductors, in a 1957 paper  by a Czech theorist. In this and subsequent experimental  and theoretical  work until recently, the fluorescence resulted from current injection into an active junction (i.e., a LED) rather than from optical absorption using a laser. The essential concept is otherwise similar; however, none of these workers performed any thermal measurements. It is only in the past few years that reduced heating in a semiconductor has been reported . In this case, the excitations were generated using a laser tuned to the 1.4 eV bandgap of GaAs, a material chosen on the basis of an amazingly high quantum efficiency. But net cooling was not observed, owing to the large refractive index of GaAs which reduces the external quantum efficiency so that photons tend to be trapped inside the material and eventually captured by nonradiative sinks. If this problem could ever be solved, it might prove possible to cool electronic chips directly.
Other materials have also been discussed or investigated as possible fluorescent coolers. At a 1961 quantum electronics conference in Berkeley, Yatsiv  presented a talk in which trivalent gadolinium was considered. He chose this ion on the basis of its large (%7e300 nm) ground- to excited-state energy gap. However, at the time, the only available source would have been a filtered mercury-arc lamp, greatly reducing the efficiency of the process. Some years later, with the advent of lasers, the first experimental attempt to achieve real fluorescent cooling became possible. Kushida and Geusic  at Bell Labs used a Nd3+:YAG laser to pump a refrigerator sample of the same material. They reasoned that the stimulated laser emission should be of slightly lower mean energy than the spontaneous emission between the same pair of bands in the second crystal. Unfortunately, net cooling was not seen, due to parasitic heating resulting from absorption by and nonradiative relaxation of trace amounts of foreign rare-earth impurities in the sample. Another problem with Nd3+ is that a number of other 4f bands lie below and above the excited laser levels, thus promoting nonradiative relaxation. A few years later, a conference presentation on anti-Stokes fluorescence in organic dye solutions mentioned cooling as a possible application, which has only recently been successfully implemented . The first system in which actual cooling was observed involved vibrational transitions of carbon dioxide gas pumped by a CO2 laser . Specifically, the (100) → (001) transition was pumped at 10.6 µm, with a favorable branching ratio for anti-Stokes emission on the 4.3 µm (001) → (000) transition. The observed pressure changes were consistent with about one degree of cooling. However, given the low density of a gas compared to a solid (collisional de-excitation dominates at higher gas pressures), it is clear that this system is not a candidate for practical refrigeration.
On the theoretical side, a number of additional papers are worth mention. It was only a few years after the discovery of the maser that it was proposed that one could be run in reverse to effect cooling . Weinstein , Chukova , Kafri & Levine , and Landsberg & Tonge  have examined the thermodynamics of such processes in great detail. More recent theoretical analyses and reviews related to our experimental work below have been prepared by Lloyd , Lamouche et al. , and Mungan & Gosnell .
We have achieved net cooling of a bulk solid via laser-induced anti-Stokes fluorescence. This success was realized using a heavy-metal-fluoride glass doped with trivalent ytterbium. This rare-earth ion has an energy-level structure very similar to that depicted above. That is, there are only two manifolds of inhomogeneously broadened levels below the UV absorption edge of the host glass, namely a four-level 2F7/2 ground state and a three-level 2F5/2 excited state separated by an energy gap of approximately 1 µm. The glass we used was ZBLANP (ZrF4-BaF2-LaF3-AlF3-NaF-PbF2), a low-phonon material which has been intensively studied in the past decade as a possible telecommunications fiber medium. Ytterbium has, for our purposes, a favorably large nonradiative lifetime and average fluorescent photon energy in this host.
The microscopic cooling properties of a sample of ZBLANP doped with 1 wt % Yb3+ were investigated using a collinear photothermal deflection technique, as depicted in the above figure. A counterpropagating HeNe laser beam was focused into the same sample volume that was exposed to a chopped infrared pump beam; the resulting thermally-induced angular deflections were synchronous with the 0.5 Hz chopped pump beam and measured with a digitizing oscilloscope. The amplitudes of the observed deflection signals as a function of pump wavelength were normalized by the absorptivities of the sample and by the infrared pump powers, to give a measure of the cooling efficiency. Noteworthy in this experiment was an unmistakable 180 degree phase shift in the photothermal deflection signal for pump wavelengths long and short of the mean fluorescent-photon energy, λF.
Next, a steady-state bulk cooling experiment was performed by suspending a matchstick-shaped sample in a vacuum chamber and illuminating the sample along its long axis with 800 mW of pump power at 1008 nm, as shown in the adjacent figure. Blackbody emission from a small spot on the sample was measured using an InSb focal-plane array in order to determine the sample temperature (relative to a reference sample outside of the beam path). We observed net cooling of 0.3 K below ambient temperature. A detailed analysis was published as a Letter to Nature .
Subsequently , we achieved more dramatic cooling of a solid, by using a sample geometry favorable for such purposes, namely an ytterbium-doped ZBLANP optical fiber. A short piece was cooled from 298 to 282 K by pumping it with 770 mW of laser light at 1015 nm. The temperature was measured directly from the shape of the fluorescence spectrum, since this is affected by the Boltzmann populations of the ground and excited state levels. In addition, it was confirmed that the dominant heat load on the sample is background blackbody radiation, by measuring the exponential thermal relaxation of the sample after suddenly exposing it to the pump laser beam. More recently , the final temperature of the sample has been reduced to 236 K, as limited by optical saturation of the absorption in the core of the fiber. In the past year, these cooling results have been extended to other systems, including Tm3+ in ZBLANP  and Yb3+ in YAG .
The experimental cooling efficiency measured at room temperature in the above experiments was approximately 2% at a pump wavelength of 1015 nm. This can of course be increased by pumping at longer wavelengths. However, since the absorption coefficient falls off rapidly in the red tail of the spectrum, this requires that the path length of the laser beam in the sample be correspondingly increased. There are two ways in which this can be accomplished. First, the active material can take the form of a long fiber, and the fluorescence allowed to escape laterally through the transparent cladding. The fiber would be wound around a cold finger, which is gold coated to reflect the fluorescent radiation. However, not all materials can be drawn into fibers of the requisite bend radius and robustness. A more general method of ensuring a long path length is to place a sample having polished faces inside of a confocal Fabry-Pérot cavity into which the laser light would be injected through a small hole in one of the mirrors. The light would then be trapped for many bounces until it is absorbed in the sample. Theoretical calculations of the estimated absorption path length, combined with measured absorption and fluorescence spectra at low temperatures, allow us to estimate the expected cooling efficiency at cryogenic temperatures .
The above table compares different spaceflight-qualified cryocooling systems. The laser cooler in the last column combines the key advantages of the other systems: low operating temperature, light weight, long mean-time-to-failure, high efficiency, lack of vibrations, and relatively low cost. For these reasons, we expect that such a device would be useful in space, military, and electronic applications, if the current engineering and scientific challenges can be successfully addressed.
Solid-state lasers could be constructed in which the cooling of anti-Stokes fluorescence would offset heat generated by stimulated emission . This mode of laser operation is referred to as radiation-balanced (RB) lasing. Unlike conventional exothermic laser systems, RB lasers would exhibit little or no internal heat generation. In principle, this technique would allow them to be scaled up to much higher average powers than conventional solid-state laser systems. The key idea is to balance the upward (pump absorption) and downward (stimulated lasing and spontaneous fluorescence) rates of transfer of both population and energy. For this purpose, the spectral parameters of eighteen different ytterbium-doped hosts were analyzed and figures-of-merit for RB lasing developed. The optimal materials were found to be potassium yttrium tungstate (KYW) and potassium gadolinium tungstate (KGW) crystals . Photothermal deflection spectroscopy was used to verify that these systems do in fact laser cool, so the next phase of work will be to demonstrate simultaneous optically-pumped cooling and lasing. The efficiency of such a laser is limited by both the first and second laws of thermodynamics .
 S. Vavilov, Some remarks on the Stokes law, J. Phys. (Moscow) 9, 68 (1945); P. Pringsheim, Some remarks concerning the difference between luminescence and temperature radiation: Anti-Stokes fluorescence, J. Phys. (Moscow) 10, 495 (1946); S. Vavilov, Photoluminescence and thermodynamics, J. Phys. (Moscow) 10, 499 (1946).
 A. Kastler, Quelques suggestions concernant la production optique et la détection optique d'une inégalité de population des niveaux de quantification spatiale des atomes: Application à l'expérience de Stern et Gerlach et à la résonance magnétique, J. Phys. Radium 11, 255 (1950).
 R.J. Keyes and T.M. Quist, Recombination radiation emitted by gallium arsenide, Proc. I.R.E. 50, 1822 (1962); G.C. Dousmanis, C.W. Mueller, H. Nelson, and K.G. Petzinger, Evidence of refrigerating action by means of photon emission in semiconductor diodes, Phys. Rev. 133, A316 (1964).
 P. Gerthsen and E. Kauer, The luminescence diode acting as a heat pump, Phys. Lett. 17, 255 (1965); P.T. Landsberg and D.A. Evans, Thermodynamic limits for some light-producing devices, Phys. Rev. 166, 242 (1968); J.I. Pankove, Optical refrigeration, in Optical Processes in Semiconductors (Dover, NY, 1975), pp. 193-197; P. Berdahl, Radiant refrigeration by semiconductor diodes, J. Appl. Phys. 58, 1369 (1985).
 M.S. Chang, S.S. Elliott, T.K. Gustafson, C. Hu, and R.K. Jain, Observation of anti-Stokes fluorescence in organic dye solutions, IEEE J. Quantum Electron. 8, 527 (1972); C. Zander and K.H. Drexhage, Cooling of a dye solution by anti-Stokes fluorescence, in Advances in Photochemistry, edited by D.C. Neckers, D.H. Volman, and G. von Bülau (Wiley, NY, 1995) Vol. 20, p. 59; J.L. Clark and G. Rumbles, Laser cooling in the condensed phase by frequency up-conversion, Phys. Rev. Lett. 76, 2037 (1996); C.E. Mungan and T.R. Gosnell, Comment, Phys. Rev. Lett. 77, 2840 (1996); J.L. Clark, P.F. Miller, and G. Rumbles, Red edge photophysics of ethanolic rhodamine 101 and the observation of laser cooling in the condensed phase, J. Phys. Chem. A 102, 4428.
 Yu. P. Chukova, Maximum light yield of luminescent light sources, Opt. Spektrosk. 26, 251 (1969); Thermodynamic limit of the luminescent efficiency, JETP Lett. 10, 294 (1969); Thermodynamic limit to the efficiency of broad-band photoluminescence, Bull. Acad. Sci. USSR - Phys. Ser. 35, 1349 (1971); Influence of excitation-line characteristics on efficiency of spectral conversion of energy by ions of trivalent neodymium in yttrium-aluminum garnet, Bull. Acad. Sci. USSR - Phys. Ser. 38, 57 (1974); The region of thermodynamic admissibility of light efficiencies larger than unity, Sov. Phys. JETP 41, 613 (1976).
 C.E. Mungan and T.R. Gosnell, Laser cooling of solids, in Advances in Atomic, Molecular, and Optical Physics, edited by B. Bederson and H. Walther (Academic Press, San Diego, 1999) Vol. 40, p. 161.
 C.E. Mungan, M.I. Buchwald, B.C. Edwards, R.I. Epstein, and T.R. Gosnell, Spectroscopic determination of the expected optical cooling of ytterbium-doped glass, Mat. Sci. Forum 239-241, 501 (1997); G. Lei, J.E. Anderson, M.I. Buchwald, B.C. Edwards, R.I. Epstein, M.T. Murtagh, and G.H. Sigel, Jr., Spectroscopic evaluation of Yb3+-doped glasses for optical refrigeration, IEEE J. Quantum Electron. 34, 1839 (1998).