EXAMPLE: SODIUM VAPOR

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 I_n¼(c/n)u_n by I_n¼v^gu_n. This leads again to the cross sections(n)¼(l²A₂₁/8pn²)S(n) and therefore to Eq. (3.12.11) for

the gain coefficient. †

there is an intrinsic (spin) angular momentumIof the nucleus in addition to the orbital and spin angular momenta of the electrons. The angular momentumFobtained by the addition ofIandJhas allowed quantum numbers found by applying again the quantum- mechanical rule for the addition of two angular momenta:F¼jJ2Ij,jJ2Ij þ1,. . ., JþI. Since I¼³₂ for the sodium nucleus, we can have either F¼³₂¹₂

or F¼³₂þ¹₂

for 3S1/2, and these states have an energy difference corresponding to about 1772 MHz. For 3P3/2, similarly, Fcan have the values 0, 1, 2, or 3 obtained fromJ¼I¼³₂, while for 3P1/2 Fcan be 1 or 2. The hyperfine splittings of the 3P levels are considerably smaller than the 1772-MHz hyperfine splitting of 3S1/2

(Fig. 3.19.)

In the absence of any magnetic field there is associated with eachFlevel a set of 2Fþ1 degenerate substates, MF¼2F, 2Fþ1,. . ., F21, F. The D2 line, for instance, actually involves a total of 24 states (Fig. 3.19).

We will consider specifically the absorption of narrowband radiation with frequency nnear the D2resonance. ForT200K and vapor pressures.0:1 Torr, the D2line is

D₂ D₁

3P_1/2

3S_1/2 3P_3/2

Figure 3.18 The sodium D lines associated with fine structure splitting.

–2 –1 0 1

0 1

2 0

–1

–1 0 1

–2 –1 0 1 2

–3 –2 –1 0 1 2 3

F = 3 F = 2

59.8 MHz 35.5 MHz 16.5 MHz

1772 MHz F = 2

F = 1 F = 1 F = 0

3S_1/2 3P_3/2

Figure 3.19 Hyperfine splittings of the sodium 3S1/2and 3P3/2levels. The energy differences indi- cated are not to scale.

3.13 EXAMPLE: SODIUM VAPOR 119

Doppler broadened, with Doppler width [Eq. (3.9.11)]

dnD¼2:1510¹¹ 1 589

200 23 ₁₌₂

" #

ffi1 GHz: (3:13:1)

This is less than the 1.772-GHz hyperfine separation of the 3S1/2(F¼1) and 3S1/2(F¼ 2) levels, but large compared to the separation of the different 3P3/2levels. Therefore we will ignore the hyperfine splittings of the 3P3/2level, giving a degeneracy 7þ5þ3þ 1¼16 for it. (See the black-dot section below.)

If the population of 3P3/2is negligible, the rate of change due to absorption of the population of 3S1/2(F¼1) is simply [cf. Eq. (3.7.13)]

dN₁⁽¹⁾ dt

" #

abs

¼ X¹

m1¼1

X

m2

R(m₁,m₂)N⁽¹⁾₁ (m₁), (3:13:2)

where the superscript (1) is used to designate 3S1/2(F¼1) states. Thus,N₁⁽¹⁾(m₁), with m1¼21, 0, 1, are the populations of atoms in the three 3S1/2(F¼1) states. In thermal equilibrium atT200K the 3S1/2(F¼1)23S1/2(F¼2) splitting is small compared to kBT, so that each of the eight (¼g1) 3S1/2 states has practically the same popu- lation, namelyN₁⁽¹⁾(m₁)¼N₁=g₁, whereN₁is the total 3S_1/2population. Therefore, from (3.7.16),

dN₁⁽¹⁾ dt

" #

abs

¼ N₁ g1

X¹

m1¼1

X

m2

R(m1,m2)

¼ N1

g₁g₂1

cI_nS⁽¹⁾(n)B₂₁

¼ l²A⁽¹⁾ 8phn

g2

g₁InS⁽¹⁾(n)N₁, (3:13:3) where

S⁽¹⁾(n)¼ 1 dn_D

4 ln 2 p 1=2

e^{4(nn(1)0 )2ln 2=dn2D}, (3:13:4)

n⁽¹⁾₀ is the 3S1/2(F¼1)!3P3/2transition frequency, andA⁽¹⁾is the rate of spontaneous emission from 3P3/2 due to 3P3/2!3S1/2(F¼1) transitions. Similarly, using the superscript (2) to designate 3S1/2(F¼2) states, we have

dN₁⁽²⁾ dt

" #

abs

¼ I_n hn

l²A⁽²⁾ 8p

g₂

g₁N₁S⁽²⁾(n), (3:13:5)

where A⁽²⁾ is the spontaneous emission rate of 3P3/2 due to 3P3/2!3S1/2(F¼2) transitions and

S⁽²⁾(n)¼ 1 dn_D

4 ln 2 p ₁₌₂

e^{4(nn0(2))2ln 2=dn2D}, (3:13:6) with n⁽²⁾₀ the 3S_1/2(F¼2) ! 3P_3/2 transition frequency, n⁽²⁾₀ ¼n⁽¹⁾₀ 1772 MHz.

The total rate of change of 3S1/2 population due to absorption is [dN₁=dt]_abs¼ [dN₁⁽¹⁾=dt]absþ[dN₁⁽²⁾=dt]abs, that is,

dN₁ dt

abs

¼ I_n hn

l² 8p

g₂

g₁N₁A⁽¹⁾S⁽¹⁾(n)þA⁽²⁾S⁽²⁾(n)

: (3:13:7)

Equivalently, the absorption cross section for narrowband radiation with frequency near the D2line is

s(n)¼ l² 8p

g₂ g1

A⁽¹⁾S⁽¹⁾(n)þA⁽²⁾S⁽²⁾(n)

, (3:13:8)

and the absorption coefficient isa(n)¼N1s(n).

The sodium 3P3/2 radiative lifetime is known experimentally to be about 16 ns, corresponding to a spontaneous emission rateA21¼A⁽¹⁾þA⁽²⁾¼1/(16 ns)¼6.2 10⁷s²¹. It might be expected, since there are five possible lower states in 3P3/2! 3S1/2(F¼2) spontaneous emission, and three possible lower states for 3P3/2! 3S1/2(F¼1), that A⁽²⁾¼ ⁵₃ A⁽¹⁾ and therefore that A21¼ ⁸₃ A⁽¹⁾, A⁽¹⁾¼ ³₈ A21, and A⁽²⁾¼ ⁵₈ A₂₁. These relations are in fact correct. They imply that

s(n)¼l²A21

8p g2

g₁ 3

8S⁽¹⁾(n)þ5 8S⁽²⁾(n)

: (3:13:9)

ForT¼200K andn¼n⁽²⁾₀ ,

s(n)ffil²A₂₁ 8p

g₂ g1

5 8

1 dn_D

4 ln 2 p ₁₌₂

¼(589010¹⁰m)²

8p ^(6:210

7s¹) 16 8

5 8 1

1:0810⁹s¹ 4 ln 2

p ₁₌₂

¼9:310¹⁶ m²: (3:13:10) The cross section (3.13.9) is plotted in Fig. 3.20 for several different values of the temp- eratureT, each of which implies a different Doppler width. As discussed in Section 14.1,

3.13 EXAMPLE: SODIUM VAPOR 121

the dependence of the sodium absorption cross section on temperature has been used to measure the temperature variation in the mesosphere.

The transmission coefficient for narrowband radiation of frequencynnear the D2

resonance is

In(z)

I_n(0)¼e^a(n)z¼e^N1s(n)z, (3:13:11) where z is the distance of propagation through the sodium vapor. The exponential dependence on N1s(n)z, together with the curves for s(n) shown in Fig. 3.20, indicates how strongly dependent on frequency the transmission coefficient can be (Problem 3.17).

These results are based on the assumption that the 3P3/2population is very small, so that (3.13.2), (3.13.3), and (3.13.5) are valid. In other words, we have assumed that the effect of stimulated emission can be neglected in writing the rate equations forN₁⁽¹⁾andN₁⁽²⁾. For the population of 3P3/2to remain small, its loss rateA21due to spontaneous emission—which is assumed here to be larger than any collisional deexci- tation rate—must be large compared to its growth rates(n)In=hndue to absorption:

A21s(n)In=hn, or

InhnA₂₁

s(n) 20 kW=m²¼2 W=cm² (3:13:12) if we assumen¼n⁽²⁾₀ , in which cases(n) is given by (3.13.10). In the following chapter we discuss in more detail what it means for a field intensity to be large or small in its effect on level populations.

–2.0 0.0 2.0 4.0 Absorption cross section (cm2)

6.0 8.0 10.0 12.0 14.0

*10–12

–1.0 0.0 1.0 n – n₀

2.0 (GHz)

3.0 T = 400K T = 300K

T = 200K T = 100K

4.0

(2)

Figure 3.20 Absorption cross section of the sodium D2line [Eq. (3.13.9)] for different values of the temperatureT. The hyperfine splitting shown is obviously unresolved atT¼5800K and so inaccess- ible to Fraunhofer when he first resolved and named D2as distinct from D1.

† Note that, if the 3S_1/2(F¼1)23S_1/2(F¼2) splitting were very small compared to the Doppler width, we would haveS⁽¹⁾(n)ffiS⁽²⁾(n);S(n) and

s(n)ffil²A21

8p g2

g1

3 8þ5

8

S(n)¼l²A21

8p g2

g1

S(n): (3:13:13) We would in effect be justified in ignoring the energy difference between theF¼1 andF¼2 levels.

The same sort of arguments as used in Section 3.5 for broadband radiation can be made to show that, if theradiation has a spectral width dn large compared to the 3S_1/2(F¼1)2 3S_1/2(F¼2) separation, then the absorption is also given approximately by (3.13.13). In other words, if either the spectral width of the transition or the spectral width of the radiation is large com- pared to the 3S_1/2hyperfine splitting, we can treat 3S_1/2as a single “unresolved” level. In particular, this approximation can be made if the radiation is in the form of a pulse of durationt_p(1=dn) that is short compared to 1/(2p1772 MHz)¼90 ps.

Such considerations can also be applied, of course, to excited states and can be used to justify our treatment of 3P_3/2as a single unresolved level in the calculation ofs(n). †

In document Laser Physics by Milonni P.W., Eberly J.H. .pdf (Page 137-142)