This report collects in one place examples I have seen of interferometry not agreeing with Foucault test results on telescope mirrors made by amateurs or manufactured for the amateur market. My motivation for presenting this data is to contribute some concrete examples to the discussion about whether there is any sort of systematic bias in Foucault or interferometry that leads to consistent under- or over-correction of a finished optic.
Please note that all of the mirror examples shown are good to excellent mirrors. All, except perhaps one, meet the 1/4 wave P-V on the wavefront criterion for diffraction limited. However, if one is striving for perfection, even a small systematic correction error could lead to achieving results worse than the intended goal.
I am not going to draw any conclusions in this report about whether interferometry or Foucault has a systematic bias. My personal view is that if a systematic bias is present, it is likely to be in Foucault, but I realize that performing and interpreting interferometry is tricky enough that errors in this area cannot be discounted out of hand.
I have collected this data from various sources. I own one of the mirrors. Some of the examples come from professionally made mirrors tested by amateurs or other professionals. Some of the mirrors were certified by interferometry, and tested later by Foucault. Others were figured using Foucault, then tested later by interferometry.
Most of the mirrors in this report were professionally-made mirrors. I realize that identifying specific sources of the mirrors could prove embarrassing, depending on whether interferometry or Foucault is more correct. For that reason, I have not included any indication of the source of the mirrors.
I have also left out some possibly significant data, such as what sort of interferometer was used, how many zones were used in a Couder test, and whether or not a slit source was used. I did this, because I didn't have the same level of information on all the examples. However, I have indicated whether or not the test was performed by a professional: presumably, a professional's results will have higher reliability, over all, than the results of an amateur.
Various variants of the Foucault test were used:
Similarly, there were a variety of interferometer setups used. Some tests were done at the center of curvature. Others were done at focus with nulling optics (optical flat or Ross null lens). Some interferometers used a reference optical element, others did not.
In the graphs, I have made every attempt to put 2D interferometry and 1D Foucault on an even footing. To do this, I compare radial surface profiles.
Interferometric data was processed to produce a Zernike polynomial representation of the 2D surface. Where possible, I analyzed the interferograms myself, and checked that the results were similar to those reported by the interferometerist. I normalized the results to a wavelength of 550 nm, which was not always done in the source data. I then extracted the symmetric Zernike terms to derive the surface profile.
For zonal Foucault results, I analyzed most of the data myself using Mike Peck's algorithm for turning Foucault data directly into symmetric Zernike terms. The surface profile was derived from the Zernike terms. For a couple of the Foucault examples I only had a surface plot, and not the original readings. I presented those as-is.
I should note that because most of the surface profiles were derived from a Zernike representation, they have a characteristic "smooth lumpiness" typical of this method of data reduction. As with any method of deriving a surface profile from Foucault (or most interferometric) data, the result is an approximation. However, for the purpose of this comparison, I believe the Zernike representation is sufficient..
At this point in time, I have only a few observations to make about this data. First, nearly all cases shown here, the inside and outside edges of the surface profile are measured higher by interferometry and lower by Foucault. It is just the opposite for the middle region of the profile, where Foucault his higher and interferometry is lower. I have yet to see a case where the discrepancy is the opposite. In other words, a mirror measuring nearly perfect by Foucault, shows under-corrected by interferometry. Similarly, a mirror showing nearly perfect by interferometry shows over-corrected by Foucault.
Second, the two examples of hi-res Foucault follow interferometry much closer than they follow manual Foucault. For this reason, I have given it a distinct color in the graphs. It seems like maskless Foucault should perform the same as hi-res Foucault, since they both use a camera to measure the zone positions, but the example I have included does not.
This is only a start at collecting data on Foucault/Interferometry discrepancies. I will update this page when I get additional data.