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Optical Sources, Detectors, and Systems by Robert H. Kingston

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2.4 The Fabry-Perot Resonator 41
2.4 The Fabry-Perot Resonator
Stimulated emission was first observed at microwave frequencies by
Gordon, Zeiger, and Townes in 1955 (Gordon et al., 1955) They used a
beam of ammonia (NH3) molecules which entered a microwave cavity
tuned to approximately 20 GHz. This resonant frequency corresponds
to a transition of the molecule from an upper to a lower vibrational state.
The higher energy state has an energy that increases with applied electric
field, while the lower state energy decreases with field. By sending the
beam of particles along the axis of a "sorter", consisting of a quadrupole
electric field in which the field increases with radius, the low-energy
particles are deflected and only particles in the upper state enter the
cavity. As a result, there is an inverted distribution, the initial molecules
emit spontaneously, the radiated field is amplified and the result is an
oscillator of exceptional frequency purity. Later microwave masers
(microwave amplification by stimulated amission of radiation) used
solids as well as gases and are excellent frequency references as well as
low-noise microwave amplifiers. In 1958 Schawlow and Townes
published a paper discussing the possibilities of "maser" action at optical
and infrared frequencies. (Schawlow and Townes, 1958) Although there
were possible methods of obtaining population inversion, the short
spontaneous emission time, inversely proportional to the cube of the
frequency, was one of the difficulties to be surmounted. In addition, a
"cavity" at optical frequency would be miniscule and alternative
structures were needed. Schawlow and Townes proposed the use of a
Fabry-Perot etalon, a filter consisting of two highly reflecting, partially
silvered mirrors, which has narrow transmission peaks at wavelengths
where the spacing is an integral number of half-wavelengths. It was
with this structure that laser action was first demonstrated in ruby by
Maiman (Maiman, 1960)and most laser oscillators use the Fabry-Perot
structure or a modified form.
Figure 2.5 shows two partially transmitting mirrors with amplitude
reflectivity, r, and transmissivity, f. The incident complex field E^ and
the transmitted field E^ obey the general relation, E(t) = Re [Ee'J^]. A
wave crossing the cavity in either direction experiences a change in
amplitude and phase given by e ^, where S
=
-jk + yjl - ajl. Here k is
the wave vector,
InjX,
7 is the power gain coefficient [Eq.
(2.5)],
and a
is the power attenuation coefficient in the laser material, independent of
42
Chapter 2 Interaction of Radiation with Matter
the gain. The factor of one-half in the 5 expression is required since we
are considering a field amplitude that is proportional to the square root
of the power or intensity. Using the multiple reflections as indicated in
the figure we may write the transmitted electric fields at the successive
outputs 2, 6,10, etc. as
E2 =tit2e^Ei
Ee=he%e%e%
Ejo =he^(r2e%e^)^t2 etc.
(2.8)
This leads to an infinite series and a final expression given by
E^
= tit2e^ll +
r-^r2e^^
+
(r-^r2e^^)'^
+...
__tM
6d
l-rjr2e
t
2dd
E,
1
1
t
(2.9)
'2
^
^
'2
f^.
Figure 2.5 Fabry-Perot schematic. Numerals indicate successive passages of a multiply
reflected wave.

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