2.2 The two-mirror resonator

The linear optical resonator, also called a cavity is formed by two partially-transparent mirrors, arranged in parallel as shown in Figure 5View Image. This simple setup makes a very good example with which to illustrate how a mathematical model of an interferometer can be derived, using the equations introduced in Section 2.1.
View Image

Figure 5: Simplified schematic of a two mirror cavity. The two mirrors are defined by the amplitude coefficients for reflection and transmission. Further, the resulting cavity is characterised by its length D. Light field amplitudes are shown and identified by a variable name, where necessary to permit their mutual coupling to be computed.

The cavity is defined by a propagation length D (in vacuum), the amplitude reflectivities r1, r2 and the amplitude transmittances t1, t2. The amplitude at each point in the cavity can be computed simply as the superposition of fields. The entire set of equations can be written as

′ a1 = it1a0 + r1a3 a′1 = exp(− ikD ) a1 a2 = it2a′1 a = r a′ (4 ) 3′ 2 1 a3 = exp(− ikD )′ a3 a4 = r1a0 + it1a3
The circulating field impinging on the first mirror (surface) ′ a3 can now be computed as
′ ′ a3 = exp(− ikD ) a3 = exp (− ikD ) r2a1 = exp(− i2kD ) r2a1 (5 ) = exp(− i2kD ) r2 (it1a0 + r1a′3).
This then yields
′ ir2t1 exp(− i2kD ) a3 = a0-------------------- . (6 ) 1 − r1r2exp (− i2kD )
We can directly compute the reflected field to be
( 2 ) ( 2 2 ) --r2t1exp-(− i2kD-)- r1 −-r2(r1-+-t1)exp(−-i2kD-)- a4 = a0 r1 − 1 − r1r2exp (− i2kD ) = a0 1 − r1r2exp (− i2kD ) , (7 )
while the transmitted field becomes
− t t exp(− ikD ) a2 = a0-----12------------- . (8 ) 1 − r1r2exp (− i2kD )
The properties of two mirror cavities will be discussed in more detail in Section 5.1.
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