Appendix D: The line

shape in saturated

absorption spectroscopy

D

The description of saturated absorption spectroscopy in Section 8.3 ex-
plained qualitatively how this technique gives a Doppler-free signal.
This appendix gives a more quantitative treatment based on modify-
ing eqn 8.11 to account for the change in populations produced by the
light. We shall useN 1 (v) to denote the number density of atoms in
level 1 with velocitiesvtov+dv(along the direction of light) and
N 2 (v) to denote those in level 2 within the same velocity class. At low
intensities most atoms remain in the ground state, soN 1 (v)Nf(v)
andN 2 (v) 0 .Higher intensity radiation excites atoms close to the
resonant velocity (given in eqn 8.12) into the upper level, as shown in
Fig. 8.4. Within each narrow range of velocitiesvtov+dvthe ra-
diation affects the atoms in thesameway, so we can use eqn 7.82 for
homogeneous broadening to write the difference in population densities
(for atoms in a given velocity class) as

N 1 (v)−N 2 (v)=Nf(v)×

1

1+(I/Isat)L(ω−ω 0 +kv)

. (D.1)

This includes the Doppler shift +kvfor a laser beam propagating in the
opposite direction to atoms with positive velocities, e.g. the pump beam
in Fig. 8.4. The Lorentzian functionL(ω−ω 0 +kv) is defined so that
L(0) = 1, namely

L(x)=

Γ^2 / 4

x^2 +Γ^2 / 4

. (D.2)

For low intensitiesIIsat, we can make the approximation

N 1 (v)−N 2 (v)Nf(v)

{

1 −

I

Isat

L(ω−ω 0 +kv)

}

. (D.3)

The expression inside the curly brackets equals unity except nearv=
−(ω−ω 0 )/kand gives a mathematical representation of the ‘hole burnt’
in the Maxwellian velocity distributionNf(v) by the pump beam (as
illustrated in Figs 8.3 and 8.4). In this low-intensity approximation the
hole has a width of ∆v=Γ/k. Atoms in each velocity class absorb light
with a cross-section given by eqn 7.76, with a frequency detuning that
takes into account the Doppler shift. The absorption of a weak probe