Foucault Test

Background

If you take a spherical mirror and shine a light onto it from a pinhole source located at the center of the sphere, the light will reflect back to that source:
If you offset the pinhole to one side, the reflection offsets the same amount to the other side:
A parabola is flatter on the outer edges, so reflections from there will focus farther away from the mirror.

Geometry tells us that the center zone on the mirror will focus a distance R away from the surface of mirror. And a zone on the mirror of radius r from the center will focus a distance R+ r²/R away from the surface of the mirror.

What we will do is find those focus points by introducing a knife edge from the side. If the knife edge hits the focus point of a zone on the mirror, then the image on the mirror in that zone will turn grey all at once. So we first find the point where the knife edge makes the center zone of the mirror turn grey, then draw the knife edge back away from the mirror along the optical axis. As you do this, you will grey out zones farther and farther out along a parabolic mirror. Note down how far back you had to draw the knife edge back to grey out a particular zone -- this should correspond to r²/R.

Note that r²/R is a fairly small number. For a 4.25" f/10 mirror, at the very edge of the mirror, where r²/R is greatest, it is only 0.053" (1.34mm). For zones closer in to the center, it is even smaller. You'll also note that it gets bigger for shorter focal lengths, since R gets smaller.

How to do the Test

First, you need to prop up the mirror somewhere where it won't move. Then let it come to room temperature. This test is sensitive enough to detect the minute shape changes when the mirror changes temperature.

Then set up the Foucault tester away from the mirror, fairly close to one radius of curvature R away. You'll need to set it up along the optical axis of the mirror -- I reflect a small flashlight off the mirror and look for the reflection to help me do this.

Turn on the lamp and look past the knife edge at the mirror, looking for the image of the slit. Once you find it, move the tester toward or away from the mirror until the slit fills the entire mirror -- you'll be near the focus then.

Now bring the knife edge in from the side.

If you cut in at the center zone's focus, the center zone will go grey. The right side of the mirror (middle and edge zones) will go dark, and the left side (middle and edge zones) will be bright. The simulated image is via the shareware program DIFFRACT.EXE by Jim Burrows.
If you cut in at the middle zone's focus, interesting things happen. The middle zone, of course, goes grey. But the outer zone right half goes dark, as does the center left half. The center right half is bright along with the outer zone left half. The simulated image is for a 4.25" f/10 mirror with the knife edge 0.647mm back from the center focus.
Here we cut in at the edge zone's focus. The edge zone, of course, goes grey on both sides. Now the center and middle left are dark. The center and middle right are bright. The simulated image is for a 4.25" f/10 mirror with the knife edge 1 mm back from the center focus.
Here's what it looks like as you draw the knife edge back along the optical axis away from the mirror from the center focus to the edge focus.

And here is what it looks like as you cut the knife edge in from the side at the middle zone focus.

The shadows above are fairly faint. This is because the mirror has a large f- ratio. Smaller f-ratio mirrors have sharper shadows. Here are the shadows for an f/6 mirror:

Ok, I can see the shadows. But how can I tell what radius the zones are on the mirror?
Some people use a ruler, others hang across the front of the mirror a stick with pins accurately stuck in it. The old-fashioned way is to make a mask, called a Couder mask, that delineates the zones. This mask has openings on the right and the left. The idea is to look at a pair of holes and try to judge when the shadows are evenly grey.

Center Zone
Middle Zone
Edge Zone

More detail to come...

If you cut in at the center zone's focus, the center zone will go grey. The right side of the mirror (middle and edge zones) will go dark, and the left side (middle and edge zones) will be bright. The simulated image is via the shareware program DIFFRACT.EXE by Jim Burrows.
If you cut in at the middle zone's focus, interesting things happen. The middle zone, of course, goes grey. But the outer zone right half goes dark, as does the center left half. The center right half is bright along with the outer zone left half. The simulated image is for a 4.25" f/10 mirror with the knife edge 0.647mm back from the center focus.
Here we cut in at the edge zone's focus. The edge zone, of course, goes grey on both sides. Now the center and middle left are dark. The center and middle right are bright. The simulated image is for a 4.25" f/10 mirror with the knife edge 1 mm back from the center focus.