0198506961.pdf

308 Appendix D: The line shape in saturated absorption spectroscopy

beam travelling along thez-direction through a gas with a strong pump

beam in the opposite direction is given by the integral in eqn 8.17:

κ(ω, I)=

∫

{N 1 (v)−N 2 (v)}σ(ω−kv)dv

=

∫

Nf(v)

1+IsatI Γ

(^2) / 4
(ω−ω 0 +kv)^2 +Γ^2 / 4

×

σ 0 Γ^2 / 4

(ω−ω 0 −kv)^2 +Γ^2 / 4

dv.

(D.4)

Note the opposite sign of the Doppler shift for the probe (−kv) and pump

(+kv) beams. (Both beams have angular frequencyωin the laboratory

frame.) For low intensities,I/Isat1, the same approximation as in

going from eqn D.1 to D.3 gives

κ(ω, I)=Nσ 0

∫

f(v)L(ω−ω 0 −kv)

{

1 −

I

Isat

L(ω−ω 0 +kv)

}

dv.

(D.5)

AsI→ 0 ,this reduces to eqn 8.11 for Doppler broadening without satu-

ration, i.e. the convolution off(v)andL(ω−ω 0 −kv). The intensity-

dependent part contains the integral

∫∞

−∞

f(v)L(ω−ω 0 −kv)L(ω−ω 0 +kv)dv

=f(v=0)

∫∞

−∞

Γ^2 / 4

x^2 +Γ^2 / 4

×

Γ^2 / 4

{2(ω−ω 0 )−x}^2 +Γ^2 / 4

dx

k

.

(D.6)

The product of the two Lorentzian functions is small, except where both

ω−ω 0 +kv=0andω−ω 0 −kv=0.Solving these two equations we

find that the integrand only has a significant value whenkv=0and

ω−ω 0 = 0. The Gaussian function does not vary significantly from

f(v= 0) over this region so it has been taken outside the integral. The

change of variables tox=ω−ω 0 +kvshows clearly that the integral

is the convolution of two Lorentzian functions (that represent the hole

burnt in the population density and the line shape for absorption of the

probe beam). The convolution of two Lorentzian functions of widths Γ

and Γ′gives another of width Γ + Γ′(Exercise 8.8). The convolution of

two Lorentzian functions with the same width Γ = Γ′in eqn D.6 gives a

Lorentzian function of width Γ + Γ′= 2Γ with the variable 2 (ω−ω 0 );

this is proportional togH(ω), as defined in eqn 7.77 (see Exercise 8.8).

Thus a pump beam of intensityIcauses the probe beam to have an

absorption coefficient of

κ(ω)=N× 3

π^2 c^2

ω^20

A 21 gD(ω)

{

1 −

I

Isat

πΓ

4

gH(ω)

}

. (D.7)

The function in the curly brackets represents the reduced absorption at

the centre of the Doppler-broadened line—it gives the peak in the probe

beam intensity transmitted through the gas whenω=ω 0 ,asshownin