Modern mirrors and beam splitters that make use of dielectric coatings are complex optical systems, see
Figure 8 whose reflectivity and transmission depend on the multiple interference inside the coating layers
and thus on microscopic parameters. The phase change upon transmission or reflection depends on the
details of the applied coating and is typically not known. In any case, the knowledge of an absolute value of
a phase change is typically not of interest in laser interferometers because the absolute positions of the
optical components are not known to sub-wavelength precision. Instead the relative phase between the
incoming and outgoing beams is of importance. In the following we demonstrate how constraints
on these relative phases, i.e., the phase relation between the beams, can be derived from the
fundamental principle of power conservation. To do this we consider a Michelson interferometer,
as shown in Figure 9
, with perfectly-reflecting mirrors. The beam splitter of the Michelson
interferometer is the object under test. We assume that the magnitude of the reflection
and
transmission
are known. The phase changes upon transmission and reflection are unknown. Due to
symmetry we can say that the phase change upon transmission
should be the same in both
directions. However, the phase change on reflection might be different for either direction, thus, we
write
for the reflection at the front and
for the reflection at the back of the beam
splitter.
Then the electric fields can be computed as
We do not know the length of the interferometer arms. Thus, we introduce two further unknown phases:It will be convenient to separate the phase factors into common and differential ones. We can write
with and similarly with For simplicity we now limit the discussion to a 50:50 beam splitter with We can test whether two known examples fulfill this condition. If the beam-splitting surface is the front
of a glass plate we know that ,
,
, which conforms with Equation (28
). A
second example is the two-mirror resonator, see Section 2.2. If we consider the cavity as an optical ‘black
box’, it also splits any incoming beam into a reflected and transmitted component, like a mirror or
beam splitter. Further we know that a symmetric resonator must give the same results for fields
injected from the left or from the right. Thus, the phase factors upon reflection must be equal
. The reflection and transmission coefficients are given by Equations (7
) and (8
) as
In most cases we neither know nor care about the exact phase factors. Instead we can pick any set which
fulfills Equation (28). For this document we have chosen to use phase factors equal to those of the cavity,
i.e.,
and
, which is why we write the reflection and transmission at a mirror or beam
splitter as
Please note that we only have the freedom to chose convenient phase factors when we do not know or do not care about the details of the optical system, which performs the beam splitting. If instead the details are important, for example when computing the properties of a thin coating layer, such as anti-reflex coatings, the proper phase factors for the respective interfaces must be computed and used.
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