Page 94 - Tunable Lasers Handbook

4 CO, Isotope Lasers and Their Applications 75

_-
Ev,J
30 NJ = (2J + 1)e kT

19 29 39 49 59
ROTATIONAL QUANTUM NUMBER
FIGURE 5 Boltzmann distribution of population densities as a function of the rotational quan-
tum nnmber J. for Tro, = 100 and 1000 K, respectively. (After C. K. N. Patel.)

the (0001) upper laser level, the requirement to maintain the Boltzmann distribu-
tion will result in a transfer of excited molecules from all other rotational levels
into the rotational level that directly contributes to the lasing transition. This
cross-relaxation among all the rotational levels in the inverted population of the
(0001) upper vibrational level results in very strong competition among the pos-
sible laser transitions, and once a transition (usually the one with the highest
gain) starts oscillating it will draw on all available inverted population in the
upper laser level so that other transitions will not have sufficient gain to oscillate.
This phenomenon also explains the high saturation intensity of CO, lasers. and
the fact that the entire available power may be extracted in a single regular band
transition of a well-designed cw CO, laser.
The gain itself varies approximately in accordance with the Boltzmann dis-
tribution of the upper laser level population, as given by

The theoretical derivation and experimental verification of the gain in regular
band CO, - laser transitions was first accomplished by Patel [1,2,1,5]. By com-
puter matching the theory to experimental data, Patel found [ 1.21 a good match
at a rotational temperature Trot of about 300 K. By differentiating Eq. (4) and
setting d NoOol(J)/dJ = 0, we get