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Re: [ATM] Foucault images
Mike,
Here's my best effort at deciphering the astigmatism in your hall of
shame #5 (http://bi-staff.beckman.uiuc.edu/~melockwo/mirror_making/foc_hall/foc_hall.html).
First, I set up a test where I assumed a perfect 32" f/4 with only
primary astigmatism at a 45 degree angle (where it shows up strongest
in Foucault), and compared it to the worst of your 12 images. The
following image (on my area of atmsite) shows how vaying amounts of
primary astigmatism look. The annotations are RMS wavefront error. I
chose 2.5 as the closest value. The rest of my images explain and
confirm this value. 2.5 RMS waves on the wavefront equates to 12.25
waves P-V on the wavefront.
http://tinyurl.com/kafnc
As a first confirmation, here's a 133 lpi Ronchi simulation taken 2mm
outside paraxial focus. I also put a link to your Ronchi image for
comparison.
http://tinyurl.com/hp4cm
http://bi-staff.beckman.uiuc.edu/~melockwo/mirror_making/foc_hall/ronchiS.jpg
There are some remarkable similarities between the two in the subtle
features. Note how the outer bands get clearer in both. Note how the
second band (from the middle) gets less distict as you go down. Note
how the second, third, and fourth bands start to braid togther into an
indistict mass.
One interesting note for the technically inclined is that I needed to
use a pupil size of 1000 pixels wide. Combined with a padding of 1.2,
I was doing 1200x1200 FFTs for the simulation. This was an
unfortunate consequence of the mirror's large diameter and short focal
ratio. These are the largest FFTs that I have had to use, to date.
I don't know the thickness of this mirror, but I suspect that it is
thinnish, and that it shows a fairly large amount of thest stand
astigmatism, oriented vertically. This is overlayed on the mirror's
astigmatism in your composite Foucault in 12 orientations. When two
astigmatisms are combined, the angle and strength of the composite
varies. The following graph shows how 1 wave RMS of test stand
astigmatism combines with 2.5 waves RMS of mirror astigmatism. The
lines show the orientation and strength of the combination for 100
increments.
http://tinyurl.com/o4p5m
What I did was to experiment with the starting angle of the mirror
astigmatism (-70 degrees was best) and the strength of test stand
astigmatism (somewhere between .5 and 1 wave RMS) that made the
combination that most closely matched the 12-frame composite Foucault
images. Here's the result, along with the original:
http://tinyurl.com/pj5s3
http://bi-staff.beckman.uiuc.edu/~melockwo/mirror_making/foc_hall/rotate_foc.jpg
The match is by no means perfect. My simulation is annotated with the
degree or rotation and the size of the combined astigmatism for that
frame. Note how focus shifts back and forth from frame to frame.
There's a lot of similarity with the actual Foucault images, but also
a lot that is not the same. The focus shift could be part of this. I
took all the simulations at the same stage position, but I suspect
that Mike was not able to control this as well. Also, test stand
deformation often changes from one settiing of the mirror to another,
which throws off the combined astigmatism.
One thing I noted from the simulations is that the astigmatism should
disappear at four evenly-spaced positions. Between each adjacent pair
of these four, the astigmatism should switch back and forth (left
lean, versus, right lean). If you look carefully at the original
composite, there is no way to pick the four frames one above the other
that lack astigmatism. The best compromise I found was to place the
"zero point" just outside of the 60 degree frame, at about 70 degrees.
There are only two or three frames that match very poorly in overall
structure with this choice.
Thanks for the interesting puzzle. I'm wondering if anyone else has
an estimate, and whether different means were used to achieve it.
--
Steve Koehler
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