Tunable lasers handbook phần 6 ppsx

234 Norman P. Barnes

a Z b

C

e

d

Z

FIGURE 6 Orbits in octahedral symmetry. (a, u orbit. (b) 1' orbit. (cj x orbit. (d) y orbit. (e) :orbit.

6 Transition Metal Solid-state Lasers 35

S=-K(T)(Y2,(6.$) + &&$))/2': , ;5 j

Electron o'rbits described by these linear combinations of functions are graphed

in Fig. 6. As can be seen, the 3dT orbits are maximized along the .I, y, and I

axes. that is, the orbits are directed ton ard the positions of the nearest neighbors.

On the other hand, the 3d~ orbits are maximized at angles directed between the

nearest neighbors. Because the nearest neighbors usually have a net negative

charge, it is logical that the orbits directed toward the nearest neighbors uould

have a higher energy. In essence, the electrons are being forced to go where they

are being repulsed.

A calculation of the energies of the molecular bonding orbits must include

the effects of the mutual repulsion. Mutual repulsion energy contributions can be

expressed in terms of the Racah parameters, A. B, and C Racah parameters, in

turn. are expressed in term5 of Slater integrals: however, it is beyond the scope

of xhis chapter to delve into the details. Suffice it to say that the 4 term is an

additive term on all of the diagonal elements. When only energy differences are

to be calculated. this term drops out. The B and C energy terms occur on many

off-diagonal elements. However. Tanabe and Sugano observed that the ratio of

C/B is nearly constant and in the range of 4 to 5. A slight increase of this ratio is

noted as the nuclear charge increases while the number of electrons remains

constant. A. ratio of C/B of 3.97 was expected based on Slater integral formalism.

Thus. the mutual repulsion contribution to the energy levels can be approxi￾mated if only a single parameter is known. Usually this parameter is the Racah

parameter B. Hence, many of the Tanabe-Sugano calculations are normalized by

this parameter.

Crystal field contributions to the energy of the molecular orbits can be

described by the parameter Dq. Remember that lODq is the energy difference

between the 3dT and the 3~1e levels for a single 3d electron. Consider the case

where there are N electrons. These electrons can be split between the 3dT and

3d~ orbits. Suppose II of these electrons are in the 3de orbits. leaving N-n of

them in the 3dT orbits. Crystal field effect contributions to the energy can be

approximated as (6N - 1On)Dq. Crystal field energy contributions. in this simpli￾fied approach, occur only for diagonal energy matrix elements.

Energy differences between the various levels have been calculated for all

combinations of electrons in octahedral symmetry and are presented in Tanabe￾Sugano diagrams. Such diagrams often plot the energy difference between vari￾ous energy levels, normalized by the Racah B parameter. as a function of the

crystal field parameter, again normalized by the Racah B parameter. A

Tanabe-Sugano diagram for three electrons in the 3d subshell is presented in

236 Norman P. Barnes

Fig. 7. For this diagram, the ratio of C/B was assumed to be 4.5. Triply ionized

Cr is an example of an active atom with three electrons in the 3d subshell. Ener￾gies are calculated by diagonalizing the energy matrix. However, as the Dq term

becomes large, the energy differences asymptotically approach a constant or a

term that is linearly increasing with the parameter Dq. Such behavior would be

expected since, for large values of Dq, the diagonal terms dominate and the crys￾tal field energy contributions only appear on diagonal terms. Note that a Tanabe￾Sugano diagram is valid only for one particular active atom since other active

atoms may not have the same ratio of C/B.

Absorption and emission occur when an electron makes a transition

between levels. The energy difference between the initial and final levels of the

electron is related to the energy of the absorbed or emitted photon. In purely

electronic transitions, all of the energy between the two levels is taken up with

the emitted or absorbed photon. However, as will be explained in more detail,

some of the energy can appear as vibrations associated with the crystal lattice,

that is, phonons, in the vicinity of the active atom.

Selection rules indicate the strength of the transition between two levels of

different energy. Obviously, a transition that is allowed will have stronger

absorption and emission spectra than a transition that is not allowed. Two selec￾tion rules are particularly germane to the transition metals, the spin selection

1234

Dq/B

FIGURE 7 Tanabe-Sugano diagram for d3 electrons.

6 Transition Metal Solid-state lasers 37

rule and the Laporte selection rule. According to the spin selection rule, a transi￾tion can only occur between levels in which the number of unpaired electrons in

the initial and final levels is the same. In cases where a single electron undergoes

a transition, the spin must be the same for the initial and final levels. According

to one formulation of the Laporte selection rule, a transition is forbidden if it

involves only a redistribution of electrons having similar orbitals v, ithin a single

quantum shell. This formulation is particularly relevant to transition metals

because transitions tend to be between different 3d levels but within the same

quantum shell. For example, transitions involving only a rotary charge displace￾ment in one plane would be forbidden by this selection rule.

Selection rules were also considered by Tanabe and Sugano. Usually the

strong interaction that allows a transition between levels with the emission of a

photon is the electric dipole interaction. However, for the 3d electrons, all transi￾tions between the various levels are forbidden since all levels have the same par￾ity. Consequently, three other transition interactions were considered: the electric

dipole interaction coupled with a vibration, the electric quadrupole interaction,

and the magnetic dipole interaction. The strengths of these various interactions

LA ere estimated. From these estimations. it was concluded that the electric dipole

transition coupled with vibration, that is, a vibronic transition, u as the strongest

interaction. Vibronic transitions involve emission or absorption of a photon and a

quantized 3mount of lattice vibrations referred to as a phonon. Vibronic interac￾tions were estimated to be about 2 orders of magnitude stronger than the nexl

strongest interaction, the magnetic dipole interaction.

McCumber [ 101 investigated the absorption and emission that results from

vibromc interactions. Terminology used in the original paper refers to phonon￾terminated absorption and emission rather than vibronic transitions. McCumber

analyzed the absorption, emission. and gain of the transition metal Ni in the ini￾tial paper. Emission spectra from Ni:MgF, were characterized by sharp emission

lines and a broad emission spectra on &e long-wavelength side of the sharp

emission lines. Sharp lines were associated nith electronic transitions, whereas

the long-wavelength emission was associated with vibronic emission. Since

then. this general analysis has been extended to many of the transition metals.

Through the use of an analysis similar to the McCumber analysis, the gain

characteristics of an active atom can be related to the absorption and emission

spectra. Relating the gain to the absorption and emission spectra is of consider￾able practical importance since the gain as a function of wavelength is a more

difficult measurement than the absorption and emission. Emission and absorp￾tion spectra often display relatively sharp electronic, or no phonon. transitions

accompanied by adjacent broad vibronic transitions associated with the emission

and absorption of phonons. General absorption and emission processes appear in

Fig. 8. At reduced temperatures only phonon emission is observed since the

average phonon population is low. In this case. the vibronic emission spectra

238 Norman P. Barnes

extends to the long-wavelength side of the electronic transitions. On the other

hand, the vibronic absorption spectra extends to the short-wavelength side of the

electronic transition. In some cases, the absorption spectra and emission spectra

are mirror images of each other. Although in general this is not true. at any

wavelength the absorption, emission, and gain are related by the principle of

detailed balance.

Several assumptions must be met in order for the McCumber analysis to be

valid. Consider a system consisting of an upper manifold and a lower manifold.

As before, the term manifold will be used to describe a set of closely spaced

levels. To first order approximation, levels within the manifold can be associ￾ated with a simple harmonic motion of the active atom and its surrounding

atoms. While the simple harmonic oscillator energy level spacings of the upper

and lower manifolds may be the same. in general they do not have to be. Fur￾thermore, the position of the minimum of the simple harmonic potential wells

may be spatially offset from each other due to the difference in size of the

active atom in the ground level and the excited level. Population densities of

these manifolds are denoted by N, and N,. One of the assumptions used by the

theory is that a single lattice temperature-can describe the population densities

of these manifolds. For example, suppose the upper manifold consists of a

series of levels commencing with the lowest energy le\7el which is an energy

hvZp above the ground level. Levels within the manifold are separated by an

energy hvv where this energy represents a quantum of vibrational energy asso￾ciated with the simple harmonic motion of the upper level. According to this

assumption, the active atoms in the upper manifold will be distributed among

the various vibrational levels associated with the upper manifold according to a

simple Boltzmann distribution. In turn. the Boltzmann distribution can be char￾acterized by a single temperature T. Thus, with all of the vibrational levels

equally degenerate, the population of any particular vibrational level will be

given by N,exp (-JhvJkT) (1 - exp (-kv, /W)) where J is the integer denoting

the energy ievel, k is Boltzmann's constant, and T is the lattice temperature. The

last factor simply normalizes the distribution since it represents the summation

over all levels within the manifold. Furthermore, the same temperature can

describe the relative population of the levels comprising the lower manifold.

Another assumption is that the time interval required for thermal equilibrium

for the various population densities is very short compared with the lifetime of

the upper level. For example. suppose all of the population of the upper mani￾fold may be put initially in a single level by utilizing laser pumping. The sec￾ond assumption says, in essence, that the closely spaced levels achieve thermal

equilibrium in a time interval short with respect to the lifetime of the upper

manifold. A third assumption is that nonradiative transitions are negligible

compared with the transitions that produce the absorption or emission of a pho￾ton. Although this is not always true. the lifetime of the upper level may be