Laser Cooling in Rare Earth–Doped Glasses and Crystals

For the first time the second law of thermodynamics was applied to luminescence processes by Landau in 1946 [8]. In 1960, Weinstein

estimated the limiting efficiency of electroluminescent processes [20]. In 1969, Chukova considered luminescence in general way and obtained the relation for the upper limit of efficiency [21, 22]. Chukova has shown that the limiting efficiency of photoluminescence cannot be larger than the thermodynamic limit on the efficiency of electroluminescence [23, 24]. Let us consider a system consisting of the pump radiation, a luminescent body, which is in thermal equilibrium with an ambient, and the fluorescence radiation. In the steady-state condition, the energy conservation requires that  E Q Ep f+ - = 0 (2.1)where Ep is the pump laser radiation rate, Q is the heat flow power delivered to the body, and Ef is the fluorescence radiation rate. The points over the symbols denote the derivative d/dt, where t is time. The second law of thermodynamics requires that the entropy of an isolated system does not decrease. It gives rise to the inequality   S Q

T Sf p-- ≥ 0 (2.2)where Sp is the entropy of the pump source, Sf is the entropy of

fluorescence radiation, and T is ambient temperature. The entropy of radiation (“photon gas”) is determined by the well-known formulae of Bose statistic. S l

c n n n n d d= + + -ÚÚk n nn n n nB

3 1 1[( )ln( ) ln ] W (2.3)where nν is the mean photon occupation number per unit volume of the k-space, where k is the wave vector. In the case of polarized radiation l = 1, l = 2 specifies unpolarized radiation. The entropy flux density of the radiation in a frequency interval (ν, ν + dν), which passes the unit area of a surface per unit time into solid angle dΩ in a direction under an angle θ with the normal to the surface (Fig. 2.1) is S l

c n n n n d d= + + -ÚÚk n q nn n n nB cos

2 1 1[( )ln( ) ln ] W (2.4)

Figure 2.1 Entropy flux density of the radiation emitted from the surface of a sample into a solid angle dΩ. The polar angle θ is measured relative to the surface normal direction n. The factor cos θ projects the unit area of the surface to be normal to radiation. The energy flux density of the same radiation can be described asE l c

h n d d= ÚÚ n n q nn 2

2 cos W (2.5) The effective flux temperature, which is the mathematical substitution, can be introduced as the relation between the energy

flux density and entropy flow densityT E S

eff =  

(2.6) 2.2.1 Entropy for Near-Monochromatic Radiation Flow

Suppose that the radiation flow is spread in the frequency interval ∆ν around a central frequency ν0 (∆ν << ν0) and propagates into the angle ∆Ω. In this case Eqs. 2.4 and 2.5 can be written asS l

c n n n n= + + -k

2 1 1[( )ln( ) ln ] DWD (2.7)

E l c

h n= n

n q nn 2

2 cos DWD (2.8)

respectively. Let us suppose that ether ∆ν or ∆Ω go to zero. Since the energy flow density is nonzero the relation nν∆Ω∆ν in Eq. 2.8 must be finite, that is, nν tends to infinity (nν Æ ∞). Consequently,

considering Eq. 2.7 for the entropy flow density one can conclude that for monochromatic or unidirectional radiation fluxes the entropy flow density vanishes. For the first time that was proved by Landau in 1945. Using Eq. 2.6 one can conclude that the effective flux temperature for monochromatic or unidirectional radiation fluxes approaches infinity (Teff Æ ∞). 2.2.2 Conversion Efficiency

The maximum conversion efficiency, η, of the pump energy flux density, Ep , into fluorescence energy flux density, Ef , can be estimated with Eqs. 2.1, 2.2, and 2.6. h = £

-( ) - Ê

Ë ÁÁ

ˆ

¯ ˜˜

  E

E

T

T T

T

1 (2.9) where Tfeff and Tpeff are the effective flux temperature of the fluo-rescence and the pump radiation, respectively. In the case of nearmonochromatic pump radiation the effective flux temperature

ficiency depends only on the fluorescence effective flux temperature Tf

eff . This result for the first time was obtained by Weinstein in 1960 for electroluminescence [20]. 2.3 Fundamentals of Laser Cooling in Rare

2.3.1 The Two-Level Model of Laser Cooling in Rare Earth-Doped HostsRE ions (lanthanides) have a long history of optical applications. REs have unique characteristics that distinguish them from other optical ions. RE ions absorb and emit over narrow frequency ranges. The frequencies of absorption and emission transitions are almost insensitive to host materials. The lifetimes of metastable levels are

long (μs to ms). These optical properties of RE ions are dictated by their unique electronic structure. All the RE ions have the same

outer electronic structure 5s25p66s2, which are filled shells. The optical properties of the RE ions are connected with the electrons occupying the inner 4f shielded quite effectively orbital. As a result, the RE ions maintain much of their “free ion” properties when are doped into hosts than other ions. For example, the absorption and emission spectra of the REs are less dependent on external electric field than the absorption and emission spectra of the transition elements, which do not experience electronic shielding. While the crystal field interactions are weak in RE ions, they are suitable for laser cooling. The idea to use the RE ions for cooling was proposed by Kastler in 1950 [9]. Let us consider the laser cooling process with ytterbium ions, Yb3+, in the two-level system. Laser cooling in the four-level system has been considered in Refs. [25, 26]. Ionization of the REs usually results in trivalent states. As an example the energy-level diagram of Yb3+ ions doped in the YAG host is presented in Fig. 2.2. The energy gap between the 2F7/2 and 2F5/2 manifolds is ~10,000 cm-1.