0198506961.pdf

156 Doppler-free laser spectroscopy

gH(ω−kv)≡δ(ω−ω 0 −kv)thatpicksoutatomsmovingwithvelocity

v=

ω−ω 0

k

. (8.12)

Integration overvtransformsf(v) into the Gaussian line shape function

(^16) This is a convolution of the solution in eqn 8.4: 16
for a stationary atom with the velocity
distribution (cf. Exercise 7.9). gD(ω)=

∫

f(v)gH(ω−kv)dv. (8.13)

Thus sinceκ(ω)=Nσ(ω) (from eqn 7.70) we find from eqn 8.11 that

the cross-section for Doppler-broadened absorption is

σ(ω)=

g 2

g 1

π^2 c^2

ω^20

A 21 gD(ω). (8.14)

Integration ofgD(ω) over frequency gives unity, as in eqn 7.78 for ho-

mogeneous broadening. Thus both types of broadening have the same

(^17) The cross-section only has a signifi- integrated cross-section, namely 17
cant value nearω 0 ,sotakingthelower
limit of the integration to be 0 (which is
realistic) or−∞(which is easy to eval-
uate) makes little difference.

∫∞

0

σ(ω)dω=

g 2

g 1

λ^20

4

A 21. (8.15)

The line broadening mechanisms spread this integrated cross-section out

over a range of frequencies so that the peak absorption decreases as

the frequency spread increases. The ratio of the peak cross-sections

approximately equals the ratio of the line widths:

[σ(ω 0 )]Doppler

[σ(ω 0 )]Homog

=

gD(ω 0 )

gH(ω 0 )

=

√

πln 2

Γ

∆ωD

. (8.16)

The numerical factor

√

πln 2 = 1.5 arises in the comparison of a Gaus-

sian to a Lorentzian. For the 3s–3p resonance line of sodium Γ/ 2 π=

10 MHz and at room temperature ∆ωD/ 2 π= 1600 MHz, so the ratio of

the cross-sections in eqn 8.16 is 1 /100. The Doppler-broadened gas

gives less absorption, for the sameN, because only 1% of the atoms

interact with the radiation at the line centre—these are the atoms in

the velocity class withv=0andwidth∆vΓ/k. For homogeneous

broadening all atoms interact with the light in the same way, by defini-

tion.

8.3.1 Principle of saturated absorption spectroscopy

This method of laser spectroscopy exploits the saturation of absorption

to give a Doppler-free signal. At high intensities the population differ-

ence between two levels is reduced as atoms are excited to the upper

level, and we account for this by modifying eqn 8.11 to read

κ(ω)=

∫∞

−∞

{N 1 (v)−N 2 (v)}σabs(ω−kv)dv. (8.17)