To: atm{at}shore.net
From: Michael Peck 
Reply-To: Michael Peck 


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other two locations are slightly out of date.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{at}ix.netcom.com
Wildlife photography page http://home.netcom.com/~mpeck1/index.html Amateur
telescope making http://home.netcom.com/~mpeck1/astro/astro.html

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