4 CO 2 Isotope Lasers and Their Applications 79
range of
CO 2
lasers becomes increasingly dominated by the Doppler-limited
linewidth for laser gas fill pressures much below 10 Torr.
As a corollary, one can also deduce that gas lasers are generally tunable over
a frequency range that is at least as wide as the Doppler-broadened lineshape.
Since the invention of the laser, various techniques have been sought to defeat
the limits imposed by Doppler broadening so that the inherently great spectral
purity of lasers may be more fully utilized (e.g., in high-resolution spectroscopy).
The various techniques of Doppler-free spectroscopy utilize the laser's inherently
high intensity, spectral purity, and low divergence to produce some nonlinear
effect that can discriminate against Doppler broadening. Saturation spectroscopy,
two-photon spectroscopy, and laser-induced line-narrowing are the best known
methods developed so far for overcoming Doppler broadening. The line-center
stabilization of CO 2 lasers to be discussed in Sec. 8 of this chapter is based on
the nonlinear saturation resonance that can be observed in low-pressure room-
temperature CO 2 gas when the cell containing it is subjected to a strong standing-
wave field of CO 2 laser radiation. However, prior to a more thorough discussion
of the standing-wave saturation resonance [48], it is appropriate to briefly review
the spectral purity and short-term stability of CO 2 lasers [55,56].
7. SPECTRAL PURITY AND SHORT-TERM STABILITY
The output waveform of a stable, single-frequency
CO 2
laser far above the
threshold of oscillation may be approximated by an almost perfect sine wave
with nearly constant amplitude and frequency. For a laser operating in an ideal
environment, the spectral purity is measured by a linewidth that is determined by
frequency fluctuations caused by a random walk of the oscillation phase under
the influence of spontaneous emission (quantum) noise. In their fundamental
1958 paper, Schawlow and Townes predicted [57] that the quantum-phase-noise-
limited line profile will be a Lorentzian with a full width between the half-power
points (FWHM), which may be approximated by:
AVFwHM N
anhvo
e0
Vo / 2
~-)-), (14)
where a,
h,
v 0, P0, and
Qc
denote the population inversion parameter, Planck's
constant, the center frequency, power output, and "cold" cavity Q of the laser,
respectively. In a well-designed small CO 2 laser the "cold" cavity Q is given by:
2rtLv 0
Q c ctr '
(15)
where
L, c,
and
t r
denote the cavity length, velocity of light, and mirror transmis-
sion, respectively (diffraction losses are usually negligible compared to output

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