[ATM] The zonal Foucault test is free of inherent correction bias -some supporting graphs

[Michael Peck <mpeck1@ix.netcom.com>]

I've added some supporting graphics describing my simulation of a zonal 
Foucault test of a large, fast mirror. For now I've put the graphs in the 
ATM list's contributors area, which I think are accessible to anyone. The 
base url is <http://www.atmlist.net/contrib/mpeck1-at-ix-dot-netcom-dot-com/>.

Trying to be as brief as possible, I performed a virtual 15 zone test on a 
500mm f/4 mirror using the test protocol described by Linfoot (1948). The 
numerical analysis used the same theory as in Linfoot, namely Fraunhofer 
diffraction, except of course that instead of laboriously working through a 
lot of mathematics I just plugged numbers into an FFT routine. If you want 
a very compact introduction to the math I have one at 
<http://home.netcom.com/~mpeck1/astro/foucault/ext_foucault.pdf>.

I'm going to skip the rest of the details of the computations. Suffice to 
say I used a dumb brute force approach that takes about 1 1/2 hours of CPU 
time on a reasonably fast laptop to perform a virtual test run. My 
criterion for a null at a given zone is that the average intensities in the 
interiors of the left and right mask openings be equal. I also computed "+- 
2 sigma" limits on readings as a contrast range of +-5%. That was a more or 
less arbitrary choice, but I think 10% contrast is supposed to be just 
detectable by eye.

The basic results are shown in 
<http://www.atmlist.net/contrib/mpeck1-at-ix-dot-netcom-dot-com/fvszone.png>. 
The top pane shows foucault readings against zone radius, with the solid 
curve the ideal readings for this mirror. The bottom pane shows the 
*differences* between readings and ideal. The error bars in both panes are 
the estimated +- 2 sigma limits.

I used the median zone radius as the effective radius for each zone. Notice 
my readings are slightly high except for the edge zone. That probably 
indicates that something like Nils' formula or the RMS zone radius is a 
better choice, but that will have no significant effect on the final outcome.

This plot 
<http://www.atmlist.net/contrib/mpeck1-at-ix-dot-netcom-dot-com/ipz1z10.png> 
shows intensity cross sections through zones 1 and 10 at their respective 
nulls. This confirms what I thought was one of the more interesting claims 
of Linfoot. The zone 1 profiles are mirror images, while those for zone 10 
are directly similar. I could see this being a source of "personal 
equation" bias, and it might be interesting to try to confirm it on a real 
mirror.

For completeness here are the intensity profiles for all 15 zones: 
<http://www.atmlist.net/contrib/mpeck1-at-ix-dot-netcom-dot-com/ipzall.png>.

Besides calculating the surface error profile from the readings I also 
performed a little Monte Carlo exercise. Notice the error bars are quite 
symmetrical around the nominal readings. I took the total length to be 4 
sigma, generated Foucault readings with Gaussian random errors around the 
central values, and computed the estimated surface error profile. That was 
repeated 1000 times. In the following plot 
<http://www.atmlist.net/contrib/mpeck1-at-ix-dot-netcom-dot-com/simsurf.png> 
the black line is the nominal surface error profile, and the light grey 
lines the results of the simulations. Notice the only significant 
systematic effect is the high edge. The center is subject to larger random 
errors than the rest of the mirror, but note there's no systematic trend to 
estimate it either high or low.

By the way it can be shown that Gaussian random measurement errors produce 
Gaussian random errors in surface profiles at any point. That applies to 
interferometry as well as Foucault.

Finally, this is not really relevant to the main results, but this plot 
shows a histogram of the surface RMS errors from the simulation: 
<http://www.atmlist.net/contrib/mpeck1-at-ix-dot-netcom-dot-com/simrmssurf.png>. 
The central value from the nominal readings is marked with a vertical line. 
The shape of this histogram is a generic feature of estimated RMS in the 
presence of random measurement errors -- on average the estimated RMS from 
a single test run will be biased upwards. This also is true of 
interferometry as well as Foucault, and holds for realistically defective 
mirrors as well as unrealistically perfect ones. Of course one expects the 
RMS from Foucault to be biased downwards in general. That's because 
systematic errors are likely to be dominant.

Mike Peck

------
Michael Peck
mpeck1@ix.netcom.com

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