ATM Back to Robo meets the interferometer

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

James Lerch sent some of us a complete set of Zernike coefficients for the 
mirror he had interferometrically tested yesterday. The complete data set 
confirmed a suspicion I had about the interferometry, so I'm going to make 
one final longish post on Robo vs. the interferometer.

Here's the conclusion first. Discussion of my reasoning with links to 
pictures and graphs follows. Recall that the major zonal defect found on 
James' test mirror by Robo-Foucault was a hump starting at about the 80% 
zone, peaking at ~90% with the outer 10% strongly rolled off.

My conclusion is that in fact the interferometry, properly analyzed, does 
show those features. Furthermore the Zernike coefficients for the higher 
order spherical aberration terms are in reasonable quantitative agreement 
with Robo.

This may not be much consolation to James Lerch, since it seems he got the 
fine details right but completely missed the big picture. The 
interferometry indicates that this mirror is grossly undercorrected, which 
is strongly at odds with what the Foucault data indicated. I still have no 
theory for why there would be such a large discrepancy in the estimated 
correction between Robo and other methods. Contrary to one constructive 
criticism that was offered recently I think qualitative testing - a star 
test and Ronchi at focus if possible - would be useful at this point. It 
would be nice to know, and easy to decide, if this mirror is strongly 
overcorrected (as the report indicates), strongly undercorrected (my 
analysis of the interferometry), or neither (Robo).

OK, on to gory details. I've uploaded a number of pictures and graphs to my 
web pages under the directory 
<http://home.netcom.com/~mpeck1/astro/interferometry/>. I also joined 
atm_free for at least long enough to put some stuff in the files area. 
That's in a folder named "lerch_interferogram". James also downloaded those 
to his web space at 
<http://lerch.no-ip.com/atm/2ndTry/Lerch_interferogram/>. With the complete 
data set from the interferometry report available I modified a couple 
graphs and added a few new ones to the directory on my web pages, so the 
other two locations are slightly out of date.

I spent much of last week figuring out how Quickfringe does the final 
stages of data reduction and learning how to do it myself. That was 
actually pretty easy since (a) they tell you how they do it, and (b) I 
already knew how anyway and had all the tools I needed to do my own fringe 
analysis. I just had to write about 10 new lines of code to collect the 
results from manual fringe tracing.

Here's what I think happened. The person who prepared the interometry 
report got the fringe ordering reversed. To be more precise, he initially 
had it right and changed his mind for some reason. That's what the line 
that says Scale -1.00 means in the report that was provided to James.

I calculate the 4th order spherical Zernike coefficient (that's Z8 in the 
Quickfringe output) for a paraboloid  of the size and focal ratio of this 
test mirror relative to a spherical wavefront with source at center of 
curvature to be -1.782. A paraboloid is *overcorrected* relative to a 
sphere (correct?), which implies a negative coefficient for the Zernike 
term representing spherical aberration. The interferometry on this mirror 
was done at center of curvature in a single pass setup with no nulling 
optics. I initially assumed that the raw interferogram had a SA coefficient 
of -1.782-0.239 = -2.021 (-0.239 being the reported corrected coefficient). 
After learning how to do the analysis myself I realized that hypothesis 
must have been wrong. In fact the raw value of the coefficient was 
estimated to be +1.543. The person who did the interferometry then 
subtracted +1.782 (right value, wrong sign) to get -0.239, indicating a 
strongly overcorrected mirror.

But, that's not quite right. What he really got for Z8 was -1.543. He 
should have subtracted -1.782 from that to obtain a coefficient of +0.239, 
indicating a strongly undercorrected mirror.

Does it matter? In one sense, no. The estimated RMS wavefront error is the 
same, and the mirror's just as crappy. But the wavefront itself is inverted 
from what was shown in the report.

As an outside observer the only reason that matters to me is it clears up a 
mystery, namely why there was no apparent trace in the interferometry of 
the rolled edge that both the person who prepared the report stated was 
there and that the Foucault data showed.

The data, as reported, indicate (once spherical aberration is removed) an 
edge that is turning *up* not down. That is largely masked by the overall 
overcorrection, but you can see signs of the edge curling up over parts of 
the mirror in the wavefront map (shown here as reconstructed by me from the 
latest data 
<http://home.netcom.com/~mpeck1/astro/interferometry/wf_royce.png> ). The 
upturning edge will be more evident in this map, where I've removed all the 
4th order coefficients from the wavefront 
<http://home.netcom.com/~mpeck1/astro/interferometry/wf_royce_no4.png>.

So, how did I make the inferences that I claimed 4 paragraphs up? As I said 
a few paragraphs before that, I learned how to do the analysis myself (the 
only thing I lack at the moment is a way to automate the fringe tracing, 
but doing it manually seems to be just about as accurate). Here is an 
example of the raw interferogram provided to James, with one of my traces 
of fringe centers overlaid: 
<http://home.netcom.com/~mpeck1/astro/interferometry/intfit.jpg>. I assumed 
the lowest order fringe to be at the top. There's no way to tell from a 
static interferogram which way the fringes go (AFAIK), but in this case I 
think it should be obvious assuming this mirror isn't actually oblate.

I repeated this exercise twice, getting a total of 813 points which I 
combined for my analysis. I also did a 3rd trace of the bright fringes 
(similar results but not used here), and finally a 4th where I made an 
effort to trace the fringes closer to the edge (more on that later). I ran 
a least squares fit of the Zernikes to the fringe centers (this is exactly 
the analysis Quickfringe does) and obtained a value of the coefficient of 
Z8 of -1.554, with an estimated standard error of 0.005. Here is a plot of 
the estimated fringe centers from the least squares fit with my measured 
points overlaid 
<http://home.netcom.com/~mpeck1/astro/interferometry/wf_ifitc.png>. Notice 
the Zernike polynomial fit agrees pretty well with my manual fringe trace. 
There's one (probably real) localized defect that's smoothed out by the 
fit, and I apparently didn't do a very good job of tracing the oval fringe 
at the bottom (the fit fringe is closer to the real center, I think). The 
fit predicts the presence of the "order 0" fringe that's partially visible 
at the top of the image, and which I made no attempt to trace initially. 
The standard deviation of the residuals was 0.04 waves.

 From that analysis I easily inferred that the uncorrected coefficient in 
the report could not have been around -2. Errors considerably larger than 
the statistical errors wouldn't surprise me, but a 100 sigma error seems 
unlikely. Therefore I concluded that the person who did the interferometry 
made two offsetting sign errors. The corrected SA coefficient from my 
fringe tracing is -1.554-(-1.782) = +0.228. The uncorrected value in the 
report must have been +1.543, from which +1.782 was subtracted to get 
-0.239, the reported value. What clinched the case in my mind was getting 
the remaining high order coefficients from James yesterday, which filled in 
the blanks in the original report. Here is a summary of the spherical 
aberration terms as reported, and as I measured (about a week ago, FWIW):

Coefficient     Order   Reported        Measured by MLP
Z8              4       +1.543 (inferred)       -1.554
Z15             6       +0.094                  -0.120
Z24             8       +0.044                  -0.071
Z35             10      +0.056                  -0.059
Z36             12      +0.005                  -0.004

All of these are reasonably close in magnitude, with opposite signs. All of 
the coefficients in my analysis are estimated to be statistically 
significantly different from 0 except for the 12th order.

Here is the wavefront map from my fringe tracing 
<http://home.netcom.com/~mpeck1/astro/interferometry/wf_ifitc_adj.png>. 
This may be rotated and/or mirror imaged from the one reconstructed from 
the interferometry report, but it's obviously similar except for the 
inversion of the wavefront.

And here's a version of my estimated wavefront with all 4th order 
aberrations removed 
<http://home.netcom.com/~mpeck1/astro/interferometry/wf_ifitc_no4.png>. 
This clearly shows an irregularly shaped hump at about the zonal location 
measured by the Foucault data with a rolled edge of varying width and 
depth. And for a final comparison here is the wavefront I had estimated 
from the Foucault data and uploaded some time ago 
<http://home.netcom.com/~mpeck1/astro/interferometry/wf_mike.png>. The 
Foucault data were taken on just 3 diameters so features get smeared out in 
azimuth, but these are pretty clearly similar.

For a final quantitative comparison, here are estimates of the RMS 
contributions to surface errors of spherical aberration terms. First is 
what I got from the Foucault data, then my fringe analysis, followed by 
that in the interferometric report. Reported standard errors from least 
squares analysis are shown in parentheses. Values are in nanometers on the 
surface.

Order           Foucault                Measured by MLP Report
4               -9.4 (1.3)              +32.3 (0.6)             -33.8
6               -14.5 (0.9)             -14.4 (0.6)             11.3
8               -12.3 (0.7)             -7.5 (0.6)              4.7
10              -8.8 (0.6)              -5.7 (0.6)              5.4
12              -2.4 (0.5)              -0.4 (0.6)              0.4

I threatened to say more about a 4th round of fringe tracing, but this post 
is too long. The upshot is, when I traced closer to the edge the estimated 
high order coefficients got larger in magnitude (more negative), slightly 
improving the overall correspondence between Robo and interferometry. 
Mostly that tells me that the algorithm used to trace fringes matters and 
that statistical errors are probably on the optimistic side. No surprise to me.

To conclude, finally, I'm not sure how this helps James. I think the 
interferometry does partially validate Robo, but unfortunately not the most 
important part.

Here's one possible long term benefit of this exercise. ATM's keep looking 
for "the poor man's interferometer." I'm starting to think the best poor 
man's interferometer might be an interferometer, specifically a Shack cube 
interferometer. Ric Rokosz made some posts about them some years ago, but I 
don't know if ATM's have made them. They have the advantage of needing only 
one precision part, which is a plano-convex lens that has to be good only 
on the convex side. I'm not 100% sure the reference element isn't causing 
systematic error here, but if not it seems to be a reasonably buildable 
piece of equipment.

Software need no longer be an obstacle. I now know how to do the analysis, 
and I work cheap. And there are plenty of other ATMs who already know how 
to do it or who can learn easily enough.

Mike Peck

_________________

Michael Peck
email mpeck1@ix.netcom.com
Wildlife photography page http://home.netcom.com/~mpeck1/index.html
Amateur telescope making http://home.netcom.com/~mpeck1/astro/astro.html