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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