0198506961.pdf

142 The interaction of atoms with radiation

In a two-level atom the upper level can only decay to level 1 so Γ =A 21 ,

and for a transition of wavelengthλ 0 =2πc/ω 0 we find

σ(ω 0 )=3×

λ^20

2 π



λ^20

2

. (7.81)

This maximum cross-section is much larger than the size of the atom,

e.g. theλ 0 = 589 nm transition of sodium hasσ(ω 0 )=2× 10 −^13 m^2 ,

whereas in kinetic theory the atoms have a cross-section of onlyπd^2 =

3 × 10 −^18 m−^2 for an atomic diameter ofd=0.3 nm—‘collisions’ between

atoms and photons have a large resonant enhancement. The optical

cross-section decreases rapidly off resonance, e.g. light of wavelength

600 nm gives Γ/(ω−ω 0 )=10−^6 for the sodium transition above, so

thatσ(ω)=10−^12 ×σ(ω 0 )=2× 10 −^25 m^2. Clearly the absorption of

radiation has little relation to the size of the electronic orbitals.

7.6.2 The saturation intensity

In the previous section we calculated the absorption cross-section start-

ing from eqn 7.73 and we shall now use the same equation to determine

the population difference; we can write eqn 7.73 as (N 1 −N 2 )×r=N 2 ,

where the dimensionless ratior=σ(ω)I(ω)/(
ωA 21 ). This equation

andN 1 +N 2 =Ngive the difference in population densities as

N 1 −N 2 =

N

1+2r

=

N

1+I/Is(ω)

, (7.82)

where the saturation intensity is defined by

Is(ω)=

ωA 21

2 σ(ω)

. (7.83)

It is important to note that other definitions of saturation intensity are

also used, such as the above expression without 2 in the denominator.

From eqn 7.72 we find that the absorption coefficient depends on inten-

(^49) This equation is very similar to the sity as follows: 49
formula for the saturation of gain in a
homogeneously-broadened laser system
since gain is negative absorption.
κ(ω, I)=
Nσ(ω)
1+I/Is(ω)

. (7.84)

Theminimumvalue ofIsat(ω) occurs on resonance where the cross-

section is largest; this minimum value is often referred to asthesatura-

(^50) Here Isat =
ωA 21 /(2σ(ω 0 )) and tion intensity,Isat≡Is(ω 0 ), given by 50
σ(ω 0 ) is given by eqn 7.81.
Isat=
π
3
hc
λ^3 τ

, (7.85)

whereτ=Γ−^1 is the lifetime for radiative broadening. For example, the

resonance transition in sodium atλ= 589 nm has a lifetime ofτ=16ns

and for an appropriate polarization (as in Fig. 7.5) the atom cycles on

an effectively two-level transition. This leads to an intensityIsat =

60 W m−^2 ,or6mWcm−^2 , that can easily be produced by a tunable dye

laser.