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

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