ATM Dave Rowe's Figure34 - a modern Foucault data reduction program

["Nils Olof Carlin" <nilsolof.carlin@telia.com>]

Before computers were common, the ATMer had to evaluate the Foucault test data using 
pencil-and-paper methods to estimate the figure, to decide how to go on figuring or when 
to stop.  The somewhat cumbersome method described in Texereau gave an estimate of the 
peak-to-valley deviations, using straight-line approximations for the zones.
With the Millies-Lacroix method, you made an estimate of the transverse aberrations of 
the different zones, and did a simpler evaluation by making a "trumpet" tolerance curve. 
Plotting the Foucault readings in a diagram, and sliding it up and down, you could see if 
the points could all fit inside the trumpet.

This has been such a long tradition that ATMers largely have come to think of transverse 
aberration and/or peak-to-valley aberration as entities of independent value, rather than 
as crude (but useful) substitutes for the real thing. 

So what is the real thing? We have alternatives - the RMS deviation (AFAIK long used by 
professionals) is one - it gives a weighted average of the deviation over the full 
surface, not just the worst case points as P-V. The quadratic weighting is no whim - as 
long as the deviation is small, the loss of phase coherence over the wavefront is 
proportional to the error squared*, and the total loss of intensity at the center of the 
Airy disk is thus proportional to the area-weighted mean square error (=the RMS 
squared!). 

Another measure is the Strehl ratio - it is the intensity at the center of the Airy disk 
at focus, relative to a perfect (no phase errors) aperture, and is a measure of the loss 
due to wavefront errors over the full aperture. As you would expect, at least if 
wavefronts are good (and otherwise why bother?), RMS error and Strehl ratio are closely 
related.
 
Yet another related measure is the Encircled Energy Ratio (EER) - for full details, see 
Suiter chapter 10.8**. This is somewhat more complex, but is a measure of the desired 
concentration of light at focus. It is also closely related to the Strehl ratio, but can 
accommodate the effects of a central obstruction in a way that the Strehl ratio can'
t.

The numeric integration necessary to compute these measures of quality can easily be done 
with a computer program, and any modern program should do it. 
In this sense, Jim Burrows' Foucault, and its successor Sixtests, are modern programs - I 
know of no other, until now that Dave Rowe offers his "Figure" Foucault evaluation 
program (the present version is 3.4).


Like Sixtests, Figure is a DOS program, unfortunately leaving out those who cannot run at 
least a DOS window. However, Figure has a more "graphic" interface, that lets you enter 
and save the test parameters from within the program. 

You enter the starting parameters (mirror ROC, units, fixed/moving etc), including mask 
openings (if you use "pinstick" readings, you can enter them twice!), and in the next 
window you enter the zonal readings, as offset from the given ROC.

Then you can choose one of several display windows: 
You can get the figure on glass, the transverse aberration, the Airy "profile" in a 
logarithmic plot, or a graphic plot of the point spread function.

The plot will open at focus defined by minimum RMS error, but you can "defocus" at will 
to decide your figuring strategy (e.g. by minimizing the amount of glass to remove) - or 
to see how the RMS, the P-V and transverse aberrations vary as you change focus. You can 
see that in general, neither P-V nor TA are minimized at best focus (min RMS, and max 
Strehl and EER will occur close to each other).

You may choose the deformation constant freely (-1 for paraboloid, 0 for sphere etc) - 
this may come in handy if you want to know how far you are from a sphere, even if you 
plan to eventually parabolize it.

The numeric accuracy for the few test cases I have tried, seems to be within few per 
cents of the known values, and well below the "noise" of actual mirror data - Sixtests 
may have an edge there, but I can't think the difference is significant. Still, this is a 
fairly new program and any unexpected results that may indicate bugs should be reported 
to Dave.

With sparse data, such as the 5th order aberrated test data of H R Suiter (see archive), 
Figure makes a conservative estimate, assuming no errors where no data are given - 
perhaps even more conservative than Sixtests, but this approach makes good sense to me.

While I may have briefly played the part of a co-advisor to Dave, the program is *his* 
work. Congratulations to it. It will likely be a favorite !***

Nils Olof

* for small values of x, cos(x)~1-x^2/2
** EER in Figure is EER(1) in Suiter, that is the energy inside the radius lambda/D 
relative to a perfect unobstructed aperture of the same size.
*** unless, of course, you want to use "Poor Man's Caustic" (where AFAIK Sixtests is the 
only available choice) or Gaviola Caustic.