Text - Ed Maxfield
A brief introduction to a simple method that the amateur telescope maker uses to ensure that his or her mirror will produce a good image.
Terms
Purpose of the Foucault Test
Theory
Quantifying the Results
| spheroid | the shape of the surface of a spherical mirror. This surface is a circular segment of a large theoretical sphere of radius R |
| radius of sphere (R) | the radius of the large theoretical sphere |
| mirror axis | a line intercepting the center of the mirror and the large theoretical sphere |
A simple device, the Foucault tester, will make accurate measurements
of the surface of a mirror to a high degree of precision.
With the Foucault tester, a mirror maker can examine the overall
smoothness of the surface and measure variations in focus for
different areas of the mirror.
If a pinpoint of light is directed at the mirror from the center of the large theoretical sphere (at distance R) all points on the mirror will be equidistant from the light and will reflect light back to a point which will be superimposed on the source.
Having the returning beam land on the light source is not very helpful since the beam cannot be viewed. However, if the source is moved slightly to one side, the returning beam will move an equal distance in the opposite direction. This displacement of the source and returning beam will introduce a very small error in the setup (different points across the face of the mirror will be at slightly different distances from the source). However, if the separation between the source and the point that the returning beam lands is kept small, the error will be insignificant.
In practice, a narrow, vertical slit of light, rather than
a pinpoint of light, is used. A perfect pinpoint is difficult
to produce and a slit allows more light to be used.
A moveable knife edge, just in front of the location where your
eye will be placed to view the returning beam of light, is used
to block some of the returning light (the light does not focus
at one spot unless the surface of the mirror is perfectly spherical).
The shape of the mirror's surface can now be seen.
It is important to understand the appearance of the returning
light from the position of the eye placed near the radius, R.
If all points on the mirror are equal distance from R (the mirror
is spherical), the image will appear to be flat. Any deviations
from a perfect spheroid will appear as a bump or hollow on a flat
surface.
For the following discussion, assume that the source of light
is to the left (the beam returns to the right of the source) and
the knife edge is on the right of the returning beam.
Note that the knife edge can be moved into and out of the beam
at right angles to the mirror axis and can be moved back and forth,
parallel to the mirror's axis.
If the knife edge is positioned before the point of focus and
moved into the beam, it will cut off light from the right side
of the mirror producing a shadow moving from right to left. If
the knife edge is positioned after the point of focus and moved
into the beam, it will intercept light which has come from the
left side of the mirror and passed through the point of focus
and will now be on the right side. The shadow will now appear
to move from left to right.
Put another way, if the knife edge is moved from right to left
into the beam and the resulting shadow moves from right to left,
the knife edge is in front of the point of focus. If the shadow
moves from left to right, the knife edge is beyond the point of
focus. What happens if the knife edge is right at the point of
focus? The mirror will darken all over with the shadow appearing
to enter from neither the right nor the left. By moving the knife
edge backwards and forwards and observing the shadow as it is
moved into and out of the beam we can determine the point of focus.
If the surface is not 'perfect' but has, for example, a bump
on it, the edges of this bump will cause the reflected light to
come to a point of focus beyond the point of focus for the rest
of the mirror. Thus, if the knife edge is at the point of focus
for the rest of the mirror, the mirror will appear to darken evenly
but the knife edge will be ahead of the point of focus for the
edges of the bump and so the right half of the bump will be darkened
while the left side will still be brightly light. A hollow will
reflect through a point of focus short of the rest of the mirror
and the knife being beyond will show the left half of the hollow
in shadow and the right half will be lit. The bump or hollow will
stand out dramatically from the rest of the grayed mirror surface.
This test is so sensitive that the heat from a finger pressed
on the mirror face for a few seconds causing it to swell locally
will show at the tester as a hill with the left side lit and the
right side in darkness.
Seeing the overall shape of the mirror is useful, but the Foucault test can be used to measure the surface and plot the shape accurately.
The surface is isolated into circular segments with a cardboard mask made up of pairs of cutouts centered on the zone under test and the edges half way to the next zone. The height of each cutout is arbitrary and normally about equal to the width. In practice, the cutouts are staggered and the center of the zones are accurately measured and marked on the mask.
Each segment is tested to find the point of focus and the results plotted on a graph the yield the true shape. As he knife edge is moved in and out of the beam, the mirror maker notes the direction of the shadow movement. The carriage is adjusted until the pair of holes in the mask both gray equally which is the point of focus for that zone. The carriage position on a vernier caliper or dial indicator is recorded. This procedure is repeated for each zone.
The average of several measurements for each zone are used
and the average measurement for each zone is plotted on a graph
to show the slope of the surface. The graphed line is compared
to the desired shape.
Careful and consistent measuring techniques will yield results
in the order of a millionth of an inch or better than one tenth
the wave length of light.
Many methods are available to measure a mirror but the Foucault test is easy, accurate, and the equipment inexpensive to make.
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