Page 97 - Tunable Lasers Handbook

78 Charles Freed

The spectral distribution of the spontaneous emission observed in such situations
is due to the spread in the individual transition frequencies, and the resulting
broadening is called inhomogeneous [ 18,191.
In the CO, system, inhomogeneous broadening occurs when the transition
frequency v of the gaseous CO, is Doppler-shifted due to the finite thermal
velocity of the molecules according to:

1’
v=v,+*v,, (9)

where vo is the frequency of the stationary molecule and v, is the component of
the velocity (vector) along the direction connecting the observer with the mov-
ing molecule. The Maxwellian velocity distribution function of a gas with
atomic mass M at equilibrium at temperature Tis given by

where k is the Boltzmann constant andf(vy, I,,, v-j dvl dv, dv- corresponds to the
fraction of all the molecules whose s. y, and 2 components of velocity are con-
tained in the velocity intervals v, to i~, + dvL, v, to i-, + ch,. and v- to y-+ h--,
respectively. One can show that this physical situation will give rise to the so-
called normalized Doppler-broadened lineshape:

The functional dependence of the line profile in Eq. (1 1 j is referred to as Gauss-
ian, with a FWHM Doppler linewidth given by:

(12)

Equation (1 1) can be written in terms of Av, as:

For the most commonly used (00~1)-[1000, 02001, band *2C1602 P(20) transi-
tion (at a 10.6-ym wavelength) the Doppler linewidths calculated for T= 300 and
400 K result in Avo= 53 and 61 MHz. respectively. Thus, the frequency-tuning