Tunable Lasers Handbook

76 Charles Freed

where B is the principal rotational constant given in Eq. (2). At

tro t ----

400 K,

Jmax

= 19. This is the primary explanation of why the (0001)--[1000, 0200]i P(20)

transition dominates in a 12C160 2 laser. It also explains that in a long CO 2 laser

with a simple two-mirror cavity only the I-P(20) transition will oscillate. As an

example, a CO 2 laser with an optical cavity mirror spacing of L = 3 m, will have

longitudinal cavity modes [18,19] spaced every

c/2L

= (3 • 108)/6 = 50 MHz

apart. This mode spacing is less (as explained in the next section) than even the

Doppler-broadened gain profile of about 60 MHz, so that there always will be a

cavity mode under the gain profile no matter how far a cavity mirror is tuned.

Hence, a frequency-dispersive optical cavity element, such as a diffraction grating

for instance, should always be used when low-gain transitions are to be obtained.

6. LINESHAPE FUNCTIONS AND BROADENING DUE TO GAS

PRESSURE AND DOPPLER SHIFT IN CO 2 GAS

The phenomena of laser emission and saturable absorption are both the

result of an electromagnetic wave interacting with an atomic or molecular

medium. This interaction occurs over a finite frequency bandwidth.

Spontaneous emission occurs without the inducement of a radiation field

because there is a finite probability that an atom (molecule in the case of CO 2) in

level 2 of a system of energy levels

E i

will spontaneously undergo a transition to

level 1, emitting in the process a photon of energy hv =

E2-E 1.

It can be shown

[18,19] from basic quantum-mechanical considerations and verified experimen-

tally that both the emission and the absorption of radiation are described by the

same lineshape function g(v) that gives the distribution of emitted (or absorbed)

intensity as a function of frequency v. The lineshape function is usually normal-

ized so that

~_

+oo

_oo g(v)dv = 1. (6)

One of the possible causes for the frequency spread of spontaneous emission is

the finite lifetime

"1; i of

the emitting level. In the case of atomic or molecular tran-

sitions between an upper level (u) and a lower level (l), the coherent interaction

of an atom or molecule in either level (u or 1) with the electromagnetic field can

be interrupted by the finite lifetime of the level ('c u or a:t) or by an elastic colli-

sion, which erases any phase memory (a:cu or a:ct). In this case, a normalized line-

shape function with a Lorentzian profile is obtained:

12 /v 2