Optical Shop Testing

Chapter 2.2.1 - Optical Path Difference Introduced by a Beam Splitter Plate

2.2.1.   Optical Path Difference Introduced by a Beam Splitter Plate

We can show with a few algebraic steps and the law of refraction that a beam splitter
or compensating plate shifts the optical axis laterally and parallel to itself by the
following amount:

 

where θ is the incidence angle, t is the plate thickness, and n is the refractive index.

The light going to the observer from mirror M1 has traversed the beam splitter
only once, whereas the light from mirror M2 has gone through it three times. An
interferometer that has more glass in one arm than in the other, as in this case, is said
to be uncompensated. The interferometer can be compensated by inserting another
piece of glass in front of mirror M1 as shown in Figures 2.1 and 2.2.

The importance of compensating an interferometer is clearly seen in the following
section. Adjustable compensators for Williams configurations (Steel, 1963) and
Twyman–Green configurations (Connes, 1956; Mertz, 1959; Steel, 1962) have
been described in the literature.

As pointed out before, an interferometer is said to be uncompensated when it
has more glass in one of its arms than in the other, because (a) an optical
component (lens or prism) is present in one arm in order to test it or (b) the light
travels once through the beam splitter in one path and three times in the other
path, and the compensating plate is absent. Both of these situations can be
included in a general case in which an inclined plane glass plate is placed in
one of the arms. The unfolded optical paths for both arms of the interferometer

FIGURE 2.5. Light paths for both interfering beams in an uncompensated interferometer.

are shown in Figure 2.5. Here we may see that the complete effect is equivalent to
going through a system of two plates. The optical path difference (OPD) introduced
by one passage through a glass plate is a function of the angle of incidence
of the light, as shown in Figure 2.6, yielding

 

and then

 

If the plates are inclined at an angle φ0 with respect to the optical axis and the ray
direction is defined by the angles θ and ψ as shown in Figure 2.7, the OPD introduced
by both passages may be computed by

 

FIGURE 2.6. Optical path difference introduced by a plane parallel plate.


FIGURE 2.7. Light passing through an inclined plane parallel plate.

where subscripts 1 and 2 designate the first and second passages, respectively,
through the plate. The last term corresponds to an additional OPD introduced by a
shift t0 of one of the mirrors along the optical axis. Angles φ1 and φ2 are obtained
from angles φ1, c, and φ0 by means of the relations

 

If the glass plate is normal to the optical axis, φ0 = 0 and φ1 = φ2 = θ thus
φ1' = φ2' = θ0'. In this case Eq. (2.5) reduces to

 

 

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