EMISSION AND ABSORPTION OF NARROWBAND LIGHT

bulb. When this compound comes in contact with the filament, however, the heat is sufficient to dissociate the compound into tungsten, which is redeposited on the filament, and the halogen, which is then available to continue the regenerative cycle.

The more efficient fluorescent tubes are, of course, not thermal light sources, but like thermal and all other nonlaser sources the light they generate derives from spontaneous emission. At each end of the tube is an electrode, one of which is a tungsten coil coated with a material that increases the efficiency with which electrons are ejected when the coil temperature exceeds about 1300K.

The low-pressure (0.008 Torr mercury and 1–3 Torr of other gases, 1 Torr¼1/760 atm of pressure) electric discharge along the axis of the tube causes the emission of 253.7 nm radiation from mercury atoms excited by collisions with electrons. The inner walls of the tube are coated with a “phosphor” that absorbs in the ultraviolet and emits in the visible. The operating lifetime in this case is determined primarily by the erosion of the electron emissive coating each time the lamp is turned on, so that the rated average life of fluorescent tubes is based on the number of starts, assuming 3 h of operation per start.

Lighting technology remains an active area of research and development, with particular focus in recent years on light-emitting diodes (LEDs). The reader wishing to pursue some of the many interesting aspects of the subject is referred not only to books and journals devoted to it but also to lighting and optical company catalogs or websites that sometimes contain tutorial information

about lighting products. †

both absorption and stimulated emission in a narrowband field is dN₂

dt ¼(absorption rate)N1(stimulated emission rate)N2

¼(absorption rate)(N₁N₂)

¼ 1 hn

l²A₂₁

8p ^{(N2N1)InS(n): (3:7:1)}

Similarly, we must modify Eq. (3.4.37) fordN1/dtto include the effect of stimulated emission:

dN1

dt ¼ 1 hn

l²A21

8p ^{(N2N1)InS(n): (3:7:2)} The absorption rate of narrowband light in these formulas was obtained assuming a linearly polarized, plane-wave, monochromatic field propagating in the z direction [recall Eq. (3.4.4)]. When absorption occurs, the field propagates in the same direction with the same polarization and frequency but with less intensity. In the case of stimu- lated emission the field also propagates in the same direction and with the same polar- ization and frequency as the incident field, but withgreaterintensity.

We have confined ourselves thus far to showing how absorption and stimulated emis- sion affect the atomic level populations, without regard for how these processes affect thefieldexcept insofar as they decrease or increase the field energy. In Section 3.12 we derive an equation describing how the field intensityIndepends on the level popu- lationsN₁andN₂.

The intensityIncan be expressed as the number of photons crossing a unit area per unit time, times the energyhnof a single photon. Thus,

Stimulated emission rate¼l²A₂₁ 8p ^S(n)

number of incident photons=(area-time): (3:7:3) The quantity

s(n)¼l²A₂₁

8p ^{S(n), (3:7:4)}

which has the dimensions of an area, is thecross sectionfor stimulated emission (and absorption). The relation (3.7.3) identifies the cross section as an effective area associ- ated with the atomic transition, such that every photon intercepted by this area would induce an atom to undergo stimulated emission (or absorption). Needless to say there is no actual geometric object associated with this area; the cross section is nothing more than a conventional measure of the absorption strength of a transition. Note that it depends not only on the transition wavelength andspontaneous emission rate, but also on the lineshape functionS(n).

Of course, the level populationsN1andN2can also change by spontaneous emission.

The rate equations describing all three emission and absorption processes are obtained by adding the spontaneous emission terms (3.3.8) and (3.3.9) to (3.7.1) and (3.7.2):

dN₂

dt ¼ s(n)

hn In(N2N1)A21N2, (3:7:5a) dN₁

dt ¼s(n)

hn I_n(N₂N₁)þA₂₁N₂: (3:7:5b) These are among the most important equations needed for a quantitative understanding of lasers. In general, it is necessary to modify them to include other processes, such as an external mechanism to supply and/or maintain atoms in the upper level, as well as col- lisions among atoms that can excite or deexcite the level populations. It is also necessary in general to account for the possibility that the two energy levels of the transition are degenerate and that the refractive index of the medium might differ significantly from unity. And in the case of semiconductor lasers, we must deal with a more complicated version of these rate equations because there the transitions are betweenbandsof levels rather than between two discrete states. Nevertheless Eqs. (3.7.5) are the foundation for a considerable part of the theory of lasers.

† A well-known result of the quantum theory of radiation is that the stimulated emission rate into a single mode of the field is equal to q times the spontaneous emission rate into that mode, whereqis the average number of photons initially occupying the mode. Thus, ifR_spon is the rate for spontaneous emission into a certain mode, then the rate for stimulated emission into that mode isR_stim¼qR_spon, or in other words the total emission rate into the mode is (qþ1)Rspon. We have already referred to this result following Eq. (3.6.21).

It is not immediately obvious that Eqs. (3.7.5) are consistent with this “qþ1” rule, one reason being that the stimulated emission rateRstim¼s(n)I_n/hn¼(l²A21/8p)[I_n S(n)/hn] in these equations refers to a single field mode, that is, a field with a single frequency and polarization and propagating in a single direction, whereasA21is the total rate for spontaneous emission in all possible directions of propagation and polarization. Moreover the stimulated emission rate depends on the lineshape functionS(n) at the applied field frequencyn, whereasA21does not.

It is therefore instructive to show that Eqs. (3.7.5) are in fact consistent with theqþ1 rule and can indeed be inferred from it.

As the remarks above suggest,R_spon=A₂₁. Let us writeA₂₁[Eq. (3.3.7)] as

A21¼ 1 4pe₀

pe²f m

8p n²₀ c³

: (3:7:6)

We recall that (8pn²/c³)dnis the number of modes per unit volume in the frequency interval [n, nþdn], so that the factor in square brackets in (3.7.6) is the density, in frequency, of modes per unit volume at the frequencyn₀of the atomic transition. Thus, (3.7.6) expresses the spontaneous emission rateA21as a sum of rates into all possible modes having the frequencyn0. But for a single field mode in a volumeV—which for our purposes here might be imagined to be the volume of a laser cavity—the density in frequency of possible final states per unit volume is determined not by the field, which by assumption has only one frequency, n, but by the atomic lineshape functionS(n). Therefore, to obtain the rate of spontaneous emission into a single field modewe replace the factor in square brackets in (3.7.6) by the frequency – space

3.7 EMISSION AND ABSORPTION OF NARROWBAND LIGHT 95

density (1/V)S(n):

Rspon¼ 1 4pe₀

pe²f m

1

VS(n): (3:7:7)

Comparison with (3.3.7) shows that

Rspon¼l²A21

8p c

VS(n) (3:7:8)

and therefore, when we multiply this by the photon numberq, that the stimulated emission rate in Eqs. (3.7.5) is consistent with theqþ1 rule:

qRspon¼ 1 hn

l²A₂₁ 8p cqhn

V

S(n)¼ 1 hn

l²A₂₁

8p InS(n)¼s(n)

hn In¼Rstim, (3:7:9) where we have usedIn¼c(qhn=V), i.e., the intensity is the velocity of light times the field

energy density qhn/V. †

The relationA₂₁¼(8phn³₀=c³)B₂₁[cf. Eq. (3.6.17)] can be used to write the stimu- lated emission rate as¹⁰

Rstim¼s(n) hn In¼ 1

hn l²A₂₁

8p ^InS(n)¼ 1

cB21InS(n), (3:7:10) so that (3.7.5a), for instance, takes the form

dN2

dt ¼ 1

c(B21N2B12N1)InS(n)A21N2: (3:7:11) The reader is encouraged to compare this equation with the simple model laser equation (1.5.2).N2here corresponds tonthere, andN1is assumed zero there (complete inversion). What is the physical meaning of other differences? Can you identifyf?

In our treatment of emission and absorption thus far we have not dealt explicitly with the possibility of level degeneracy, that is, that there might be more than one quantum state associated with each of the energy levelsE2 and E1. The level populations N2

and N₁ in Eq. (3.7.11), for example, are the total populations of the two energy levels, regardless of any degeneracies. Let us now label different possible states of energyE2andE1bym2andm1, respectively. LetN2(m2) andN1(m1) denote the popu- lations in these specific states,A(m₂,m₁) the rate of spontaneous emission from the upper statem2to the lower statem1, andR(m1,m2) andR(m2,m1) the corresponding

10Recall that we are assumingnn₀, so that we can replacenbyn₀in these formulasexceptin the lineshape functionS(n).

rates for absorption and stimulated emission, respectively. Thus dN2(m₂)

dt ¼ X

m1

[R(m2,m1)N2(m2)R(m1,m2)N1(m1)]

X

m1

A(m₂,m₁)N2(m₂): (3:7:12)

Note that, for each of the three processes described by this rate equation, the total rate of change ofN2(m₂) is the sum of the rates for each possible channelm₂$m₁. We now sum both sides of (3.7.12) over the degenerate states m₂ and use the fact thatN2¼P

m2N2(m2):

dN2

dt ¼ X

m1,m2

[R(m2,m1)N2(m2)R(m1,m2)N1(m1)]

X

m1,m2

A(m₂,m₁)N2(m₂): (3:7:13)

This more general equation, and the one forN1that follows fromdN1/dt¼2dN2/dt, can sometimes be simplified. The most widely used simplification is to assume that all the states of a given level are equally populated, so thatN1(m1) andN2(m2) are independent ofm1andm2and equal toN1/g1andN2/g2, respectively, wheregjis the degeneracy, orstatistical weight, of levelj. In this case

dN2

dt ¼ X

m1,m2

N2

g₂R(m2,m1)N1

g₁R(m1,m2)

N2

g₂ X

m1,m2

A(m2,m1)

¼ N₂ g2

N₁ g1

X

m1,m2

R(m₁,m₂)N₂ g2

X

m1,m2

A(m₂,m₁): (3:7:14)

Comparing with (3.7.11), we make the identifications A21¼ 1

g2

X

m1,m2

A(m2,m1), (3:7:15)

1

cInS(n)B₂₁¼ 1 g₂

X

m1,m2

R(m1,m2), (3:7:16)

1

cInS(n)B₁₂¼ 1 g₁

X

m1,m2

R(m1,m2): (3:7:17)

Note therefore that

g₂B₂₁ ¼g₁B₁₂, (3:7:18)

3.7 EMISSION AND ABSORPTION OF NARROWBAND LIGHT 97

which generalizes (3.6.13). Using (3.7.15) – (3.7.17) in (3.7.14), we have dN₂

dt ¼ dN₁ dt ¼ 1

cInS(n)[B₂₁N2B12N1]A21N2

¼ 1

cI_nS(n)B₂₁ N₂g2

g₁N₁

A₂₁N₂

¼ 1 hn

l²A21

8p ^N2 g2

g₁N₁

I_nS(n)A₂₁N₂, (3:7:19) Situations where we cannot assume that degenerate states are equally populated can arise under excitation by polarized light, as discussed in Section 14.3.

We can use these relations to write theAandBcoefficients in terms of the oscillator strength and level degeneracies. Equations (3.6.17) and (3.7.15) – (3.7.17) give

A₂₁¼8phn³₀

c³ B₂₁¼g₁ g₂

8phn³₀

c³ B₁₂¼g₁ g₂

2pe²f

e₀mcl²₀^{, (3:7:20)} where we have used Eq. (3.6.19) forB12. In particular, if the transition wavelengthl₀is expressed in meters,

A₂₁¼(6:6710⁵)g1

g₂ f l²₀ ^s

1: (3:7:21) The appearance of the dimensionless factor (g1/g2)fin this formula explains why the spontaneous emission rate is not simply a universal constant timesl²₀ , as predicted by classical theory and discussed following Eq. (3.3.6).

† Denoting the nondegenerate states 1 and 2 in Eq. (3.A.26) bym₁andm₂, we can write A(m2,m1)¼ e²v³₀

3pe0hc³jxm1m2j², (3:7:22) wherexm1m2is the matrix element between statesm₁andm₂of the electron coordinatex. (For a multielectron atom,xis the sum of the position vectors of all the electrons.) In the case ofg2

degenerate statesm₂ associated with an atomic energy level E₂, each statem₂ has the same total radiative decay rate, namelyA21¼P

m1A(m2,m1), the sum of the spontaneous emission rates fromm₂ to all possible lower states m₁ of the transition. This must be so because the states m2 correspond simply to different “z components” of angular momentum, but the zdirection is chosen arbitrarily; in other words, spherical symmetry requires that

A21¼X

m1

e²v³₀

3pe0hc³jxm1m2j²¼ 1 g2

X

m1,m2

e²v³₀

3pe0hc³jxm1,m2j²: (3:7:23) Thus, the last equality in (3.7.20) implies that

f ¼ 1 g1

2mv₀ 3h

X

m1,m2

jxm1m2j², (3:7:24) which generalizes (3.A.25) to the case of degenerate levels.

Let us write Eq. (3.7.24) as

f₁₂¼ 1 g1

2mv₂₁ 3h

X

m1m2

jxm1m2j² (3:7:25)

for the transition between levels 1 and 2, wherev21¼(E2E1)=h (.0). Interchanging 1 and 2, we define

f21¼ 1 g2

2mv12

3h X

m1m2

jxm1m2j²¼ g1

g2

f12, (3:7:26)

which is negative;f12andf21are called oscillator strengths for absorption and emission, respect- ively. One motivation for introducing a separate oscillator strength for emission, which is not really necessary for our purposes, may be found at the end of Section 3.14. † The degenerate states belonging to a given atomic energy level are those correspond- ing to different magnetic quantum numbersm. The application of a relatively weak mag- netic fieldBestablishes a preferredzdirection and removes the degeneracy: Each of the states is shifted in energy by an amount proportional tomjBj. This is theZeeman effect discussed in many textbooks on quantum mechanics. The different values of m are defined with respect to some “z” direction, such as the direction of the magnetic field in the Zeeman effect. If the atom is exposed to isotropic (e.g., thermal) radiation, then by spherical symmetry the different magnetic substates must have equal populations, so that (3.7.19) is an exact consequence of the more general (3.7.12). In the case of atoms in unidirectional narrowband light, simplified rate equations such as (3.7.19) are often a good approximation if collisions between atoms or between atoms and con- tainer walls are effective in maintaining a nearly equal population distribution among degenerate magnetic substates, or if the intensity of the light is not large enough to pro- duce a significant change in an initial thermal distribution of level populations. In the latter case the generalization of Eq. (3.6.7),

N₂

N₁¼g₂N2(m₂) g₁N1(m₁)¼g₂

g₁e^(E2E1)=kBT, (3:7:27) ensures the validity of (3.7.19). As discussed in Section 14.3, however, atoms can be

“optically pumped” or “aligned” preferentially in certain magnetic substates.

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