In looking for possible sources of bias in performing the Foucault test, I have come across a characteristic of the shadows in the innermost zone of a typical Couder mask that makes it very tricky to read the zone correctly.
To motivate my ideas, I'm going to start with a challenge. In the first graphic, below, I have presented fifteen simulated images of the first zone in an 8-zone test of an 18" f/4.5 using a .1 mm (.004") slit source on a moving-source tester. The images are spaced at .002" increments of the stage. Your task is to look carefully at the image (without peaking at the answer, below) and to determine what you think is the image closest to the best null of the zone.
You might want to consider the following things while doing this task:
I have intentionally not placed the best null in the center image, so don't let image position influence your choice.
Before giving you the answer, I will point out the interesting shadow characteristic I mentioned. Take a look at the shadows in images A and O. In A, the gradient is from bright at the left to dim at the right for both openings. Image O is different. The left opening is dim in the center and bright at the edges. The right opening is the opposite. This difference doesn't happen in zones farther from the center of the mirror, and I believe this difference causes difficulty in reading zone one.
When doing this challenge myself, I noticed two effects of this characteristic of the shadows on my readings. First, the cup-shaped brightness gradient is still present at best null, and it makes it very difficult to compare the zone centers. In the left opening, the dark center is surrounded by light areas, which makes it look darker than it really is. Similarly, in the right opening the bright center looks brighter than it really is. This makes you want to read the null too far in.
However, even if you use the bracketing technique to try to avoid direct intensity comparisons of the centers of the openings, the cup-shaped gradient outside null causes problems. As mentioned above, the cup-shaped gradient makes the dark center look darker and the bright center look brighter. This gives the illusion that the brightness difference between the left and right openings increases faster as you move the stage outward from the null, than if you move the stage inward from the null. This also leads to reading the null too far in.
Now you know I think leads to zone reading problems in zone one. Try reading the null again. Does knowledge of this brightness gradient characteristic change your readings?
The following chart gives the stage positions relative to null, and a brightness ratio between the left and right zone openings. The best null is image J. I found that I usually selected image G (-.006") or H (-.004") as the best null, depending on what method I used.
image position (inches) brightness ratio A -.018 1.95 B -.016 1.81 C -.014 1.67 D -.012 1.55 E -.010 1.43 F -.008 1.33 G -.006 1.23 H -.004 1.14 I -.002 1.05 J null 0.97 K .002 0.90 L .004 0.83 M .006 0.77 N .008 0.71 O .010 0.66
The graph, below, shows the effect on the computed surface profile for various reading errors of zone one. In other words, assuming you were able to read zones 2 through 8 perfectly, but misread zone one, what would the analysis say. These surface profiles are in the direction that we have been discussing of Foucault possibly reporting more correct than actually exists.
For another view of this, consider what a camera would detect, using the flip and difference technique for finding the null. The camera finds the correct null to be image I or J.
Finally, here is a plot of brightness profiles across the openings. Blue is the left opening. Red is the reverse of the right opening. You can see how the hump shape of the brightness gradiant develops in the right opening (red) from top to bottom.
I would like to hear from anyone who reads the null perfectly the first time. In particular, what technique did you use to achieve this?
I suspect that some people will argue that this is just a simulation, and therefore proves nothing. Good point. However, the advantage of a simulation in this case is that we know a priori what the actual null position is. With an actual optic, you would have to use some other method (interferometry?) to be sure you knew what the shape of the optic was, then use some simulation technique to reverse-engineer what the actual null position is. Even with all this, you would have difficulty actually taking positioning the stage to the proper position, because you would have to measure the distance to the mirror to an accuracy of .001".
One valid complaint is that these are static images, and the actual visual Foucault test is not static. However, it's a bit difficult to present all the dynamics of the actual Foucault test in a way that would allow the challenge to proceed. I tried images that simulate different depths of knife cut, but they didn't seem to add any information that was not present in these images of an intermediate knife cut.