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Chapter 2.2.2 - Required Accuracy in a Beam Splitter Plate

2.2.2.   Required Accuracy in a Beam Splitter Plate

When constructing a Twyman–Green interferometer with a beam splitter plate, it is
very important to determine the required surface quality in each one of the two faces
of the beam splitter, for a desired interferometer accuracy. These may be found
considering the following model of a beam splitter with three small localized defects
with thicknesses δ t1, δ t2, and δ t3 as shown in Figure 2.8.

In this model we use the fact that in the external reflection in defect 1, if the
reflection is displaced a small distance t from the plate, then the optical path is
reduced by an amount

 

FIGURE 2.8. Effect of errors in the faces of the beam splitter of a Twyman-Green interferometer.

For beams reflected in the two faces of glass plate with plane and parallel faces, there
is an optical path difference given by

For beams reflected in the two faces of glass plate with plane and parallel faces, there
is an optical path difference given by

 

Thus, in the internal reflection at the localized defect 1, the optical path is increased.
Upon transmission through a glass plate with thickness t, the additional optical path
introduced by the plate is

 

Thus, the optical path introduced by the plate defects for the beam reflected from
mirror A is

 

and the optical path introduced by the plate defects for the beam reflected from mirror
B is

 

Thus, the difference between these two quantities is the optical path difference OPD
introduced by the plate errors:

 

As expected, this optical path difference does not depend on the error δ3, since it is
common for both beams. Of course, any error δ2 is also an error δ3 for some
other ray.

If OPD is the desired interferometer accuracy, the required accuracy δ1 on the
plates face 1 is given by

 

and the required accuracy δ t2 on the plate face 2 is

 

These results mean that, roughly, the reflecting face must be polished with about
twice the interferometer required accuracy, while the other face must have a quality
of only about half the interferometer accuracy.

 

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