3 ABSORPTION, EMISSION, AND DISPERSION OF LIGHT
3.12 ABSORPTION AND GAIN COEFFICIENTS
We now consider the propagation of narrowband radiation in a medium of atoms having a transition frequency equal, or nearly equal, to the frequency of the radiation. Our goal is to derive an equation describing the propagation of the intensityInof the field. A more rigorous treatment is given in Chapter 9. While the treatment here is simplified, it yields a correct and very important formula for theabsorption coefficientin the case of attenu- ation of light in an absorbing medium, and for thegain coefficientfor the amplification of light that occurs in a laser.
The intensityInis equal to the field energy densityuntimes the wave propagation velo- city. The rate at which electromagnetic energy passes through a plane cross-sectional area AatzisIn(z)A, and at an adjacent plane atzþDzthis rate isIn(zþDz)A; the difference is
[In(zþDz)In(z)]Affi In(z)þ@In
@z DzIn(z)
A¼ @
@z(InA)Dz (3:12:1) in the limit in whichDzis very small. This equation gives the rate at which electromagnetic energy leaves the volumeADz, that is,
@
@t(unADz)¼ @
@z(InA)Dz, (3:12:2)
whereunis the field energy density. SinceAandDzare constant, andun¼In/c, we may write this equation in the form
1 c
@In
@t þ@In
@z ¼0, (3:12:3)
the so-calledequation of continuity. Equation (3.12.3) is an example of Poynting’s theo- rem, in one space dimension, and is applicable to a plane wave propagating in vacuum.
If the wave propagates in a medium, however, we must replace the zero on the right- hand side of (3.12.3) by the rate per unit volume at which electromagnetic energy changes due to the medium. We can calculate this from the rate of change of the atomic level populations due to both absorption and stimulated emission. LettingN1
be the number of atoms per unit volume in the lower energy level of the resonant tran- sition, andN2the number per unit volume in the upper level, we can write the rate of change of the energy per unit volume in the medium due to stimulated emission and absorption as [recall Eq. (3.7.19)]
hndN2
dt ¼ s(n)In N2g2
g1N1
¼ hn
c InS(n)B21 N2g2
g1N1
¼ hnB21unS(n) N2g2 g1
N1
: (3:12:4)
Conservation of energy demands that the rate of change of field energy be minus the rate of change of the energy of the atoms of the medium. Thus, the change in intensity due to
stimulated emission and absorption is described in the plane-wave approximation by the equation
1 c
@
@tþ @
@z
In¼s(n) N2g2 g1
N1
In: (3:12:5) It is convenient to group the factors multiplyingInon the right-hand side into a single coefficientg(n) having units of (length)21:
g(n)¼s(n) N2g2
g1N1
¼l2A21
8p N2 g2
g1N1
S(n): (3:12:6) Depending on whetherN22(g2/g1)N1is positive or negative,g(n) is called thegain coefficient or the absorption coefficient, respectively. Thus, if N22(g2/g1)N1,0, we define the (positive) absorption coefficient as
a(n)¼l2A21 8p
g2 g1
N1N2
S(n) [absorption coefficient, (g2=g1)N1.N2]: (3:12:7) An important special case is that in which practicallyallthe atoms are in their ground states, so that N2 0 and N1 N, the total number of absorbing atoms per unit volume of the medium. In this case
a(n)l2A21 8p
g2 g1
NS(n): (3:12:8)
The termsabsorption coefficientandgain coefficientare easily understood by con- sidering the temporal steady state in whichIn is independent of time and varies only with the distancezof propagation in the plane-wave approximation we are assuming.
Then (3.12.5) simplifies to
dIn
dz ¼g(n)In: (3:12:9) If furthermore the numbers of atoms per unit volume in the two levels of the resonant transition are independent ofInandz, so thatg(n) is independent ofInandz, then
In(z)¼In(0)eg(n)z: (3:12:10) Thus, the intensity decreases or increases exponentially with distance of propagationzin the medium, depending on whether the medium is absorbing [a(n).0] or amplifying [g(n).0], respectively. The exponential attenuation formulaIn(z)¼In(0)e2a(n)zfor an absorber is often called Lambert’s law or Beer’s law, anda(n)21is called the Beer length (Problem 3.14).
The approximation of exponential attenuation of intensity in an absorber or exponen- tial growth in an amplifier is a useful and often very accurate one. However, we cannot in general assume, as we have done in obtaining (3.12.10), that the atomic level
3.12 ABSORPTION AND GAIN COEFFICIENTS 115
populations are independent of the field intensity. In general, the intensityIn and the populationsN1andN2are not independentbut are determined by thecoupleddifferen- tial equations (3.12.5) and (3.7.5). These equations account not only for the change in intensity as the field propagates in the medium but also for the change in the level popu- lations of the atoms due to absorption and emission induced by the field. That is, these equations determine the field intensity and the atomic level populationsself-consistently.
More often than not we must account for changes in intensity and atomic level popu- lations produced by effects other than absorption, stimulated and spontaneous emission.
In other words, it is generally necessary to add more terms to Eqs. (3.12.5) and (3.7.5).
For example, inelastic collisions of atoms will causeN2andN1to change in ways that are not accounted for by Eqs. (3.7.5), and these equations certainly do not account for the physical mechanisms responsible for creating “gain” [g(n).0] in a laser. And there may be scattering, diffraction, and other “loss” processes that cause the field intensity to change but are not included in Eq. (3.12.10). We shall deal with such effects in the following chapters.
Expression (3.12.6) for the gain coefficient may be generalized to include the refrac- tive index of the host medium. This generalization, which is derived below, is11
g(n)¼l2A21
8pn2 N2g2
g1N1
S(n), (3:12:11)
wherenis the refractive index at the frequencyn. This modification is significant in solid-state lasers, wherenmay differ appreciably from unity.
† To derive this result we first return to the thermal radiation energy density (3.6.1) and see how it is modified when the radiation is in a medium with refractive indexn(n). The denominator in (3.6.1) is unaffected by the refractive index, but the following argument shows that the numer- ator, the number of modes per unit volume in the frequency interval [n,nþdn], must depend uponn(n).
Let the medium of refractive indexn(n) be a box with sides of lengthLx,Ly, andLz, such that radiation of frequencynconsists of standing waves inside the box. Along each of thex,y, andz directions there is one node of the field of frequency n for each integral multiple of the wavelength. The number of nodes along thex direction, for instance, is Nx¼kxLx/2p, where kx is the x component of the wave vector k and jkj¼k¼2pn(n)n/c is the wave number (see Sec. 8.2). A small changeDkx in kx, therefore, implies a change DNx¼Lx Dkx/2p in the number of nodes along thex direction. (We assume Nx1.) Equating the change in the number of nodes with the change in the number of modes, we see that the number of modes of frequencynin a volumeDkxDkyDkz ! d3k of “kspace” must be DNx DNy DNz¼[LxLyLz/(2p)3]d3k¼Vd3k/(2p)3, where V is the volume of the box.
Multiplying by 2 in order to account for the two independent (orthogonal) polarizations for each frequency and direction of propagation, we obtain the number of field modes of frequencynper unit volume,
dNn¼ 2
(2p)3 d3k¼ 2
(2p)34pk2dk, (3:12:12) in the volume elementd3k¼4pk2dkofkspace.
11lin this expression, and throughout this book, is the wavelength invacuum.
Ifn(n)¼1,k¼2pn/cand
dNn¼8p n2
c3 dn: (3:12:13)
This result for the number of field modes per unit volume in the frequency interval [n,nþdn]
has already been used (Section 3.6), and we have now derived it.
Ifn(n)=1,
dNn¼ 2
8p34p 2pnn c 22p
c d
dn[nn]dn, (3:12:14) where we have used the fact thatdk¼d[2pnn/c]¼(2p/c)d[nn].
Suppose thatd[nn]/dn¼ndn/dnþnnat the frequencyn. Then dNn8pn3n3
c3 dn: (3:12:15)
In this approximation the spectral energy density of thermal radiation is [cf. (3.6.20)]
r(n)¼hn(dNn=dn)
ehn=kBT1 ¼8phn3n3=c3
ehn=kBT1: (3:12:16) According to Eq. (3.6.15), therefore,
A21
B21¼8phn3(n0)n30
c3 , (3:12:17)
whereA21andB21are now the EinsteinAandBcoefficients for the case where the atom is inside a host medium of refractive indexn(n).
Equation (3.12.4) implies that the stimulated emission cross section may be expressed as s(n)¼hnB21un
In
S(n): (3:12:18)
If we relate the intensity to the energy density by the formulaIn¼(c/n)un, then (3.12.18) gives s(n)¼hnn
cS(n) c3A21
8pn3n3¼l2A21
8pn2 S(n), (3:12:19) and, using this result for the cross section in (3.12.6), we obtain (3.12.11).
The termsA21,B21,In, andunall depend on the refractive index. In the formula (3.12.11) for the gain coefficient,A21is the spontaneous emission rate in the host medium; it may be shown that, aside from a possible Lorenz – Lorenz local field correction factor, the spontaneous emission rate in the medium isn(n0) times the rate in free space in the case of electric dipole transitions. In this case the gain (or absorption) coefficient is actually 1/ntimes its value when there is no “host”
medium of refractive indexn.
The generalization of (3.12.15) whend[nn]/dnis not well approximated bynis dNn¼8pn2n3
c3 d
dn[nn]dn¼8pn2n3 c2vg
dn, (3:12:20)
3.12 ABSORPTION AND GAIN COEFFICIENTS 117
where
vg; c
d[nn]=dn (3:12:21)
is thegroup velocityat frequencyn(Section 8.3). Note that (3.12.15) differs from (3.12.20) by the replacement of the group velocity by the phase velocity,c/n. Equation (3.12.11) is valid even if the group velocity is not well approximated by the phase velocity. In this case (3.12.17) is replaced by A21=B21¼8phn2(n0)n30=vgc2, and the formula In¼(c/n)un by In¼vgun. This leads again to the cross sections(n)¼(l2A21/8pn2)S(n) and therefore to Eq. (3.12.11) for
the gain coefficient. †