[ATM] Re: [atm_free] The zonal Foucault test is free of inherentcorrection bias - some supporting graphs

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

Steve Koehler wrote about his simulations:

> I have been doing similar simulations, although I'm not trying to follow
> Linfoot.  I have been  using an 10" f/5 tested with six zones.

> 1. For a perfect optic, Couder always shows a bit of undercorrection.  The
> choice of zone center makes a little difference, but not a lot.  Flip and
diff
> does much better, in general.

I think that the actual kind of test should be better defined in the
discussion. "Foucault" has been sometimes used to mean just about any test
with KE on axis (not by you!). "Couder" can mean masked testing with
apertures of unspecified width - with the apertures not under test open or
closed.
>
> 2. The Couder results are often highly dependent on the depth of knife
cut.
> Results are worse for a deeper cut, and go toward perfect for a very, very
light
> knife cut.  Robo does best with the knife cut in the middle.  I'm
experimenting
> with a variety of slit widths (.1 mm to .7 mm) and knife cut depths.

Perhaps it would be worth the trouble investigating the actual diffraction
patterns from one aperture - in amplitude and phase - will other apertures
affect the pattern? Linfoot suggests not by much, but could this explain the
discrepancies? For instance, it should be easy to check what happens to the
zonal image when both of the pairs are open and when the opposite one is
closed (or at the opposite side of the main diameter, see below). As I
understand Linfoot, this could explain the puzzling mirror symmetry of the
inner zone. Or in other words, is the null affected by alternating between
apertures instead of having both open? If the position of the null is
displaced from the theoretical even with only one aperture, the explanation
should have interesting implications if found.

Given the pattern is sufficiently symmetric (as I believe it would be with
narrow zones, but not necessarily with the widths commonly used), I have
predicted the maximum will occur on-axis for an average zonal radius between
linear and quadratic averages (assuming a paraboloisd mirror). Is this
correct? Perhaps some better mathematician than I ought to check. However,
there must be some asymmetry, and naively one would believe that shallow
cutting would affect the most asymmetric parts.

Also, the light distribution (Linfoot's "halo") *outside* the aperture
limits is usually ignored or even denied, but they may contain a significant
fraction of the total light. What effect could this have?


>
> 3. The shape of the Couder mask makes a significant difference.  When
using the
> mask like in Nils Olof's page where the zones alternate above and below
the
> horizontal center line, the results change when you reverse the mask L-R.

As the KE test (as Linfoot points out) is strictly one-dimensional (p.429,
first paragraph: "...the appearance seen along any line running
perpendicular to the knife-edge in the plane of the zonal screen depend only
on the figure of the exposed parts of the mirror immediately behind that
line"). If a single zone with two apertures are used, the apertures can
interact if both are the same side of the diameter, but not if they are not.

The
> results are also different for a mask that is symmetrical around the
center
> line.  It appears that a primary contributor to these smallish errors in
> (simulated) Foucault is the portions of the mask openings away from the
center
> line.  Currently, I'm running the test with individual mask pairs open.
(For
> some reason I haven't figured out, my simulation runs off in one direction
when
> I use the full mask in place.)  I'm curious to find out how using the full
mask
> changes the results.

So am I ;-)
>
> 4. The most common shape of surface error in the Couder cases is a dip at
about
> 50%, and a rise toward the middle and the edge.  Most often, the P-V error
is
> about 10 nm on the surface, which is larger than you found.  With some
> combinations, this can be coaxed to go a factor of 2 or 3 higher.

Something like this is what I found with Diffract's Couder mask but not with
the separate Ritchey masks. Still, the effect was quite small for that
rather small, slow mirror.

>
> 5. As you found, too, these results are exactly opposite to what has been
> showing up in the Foucault/interferometry comparisons.  It entirely
possible
> that we are running in to the limits of the simulation.
>
> In summary, my experiments so far show that maskless Foucault with the
knife at
> 50% cut is much more free of bias than a Foucault test with a Couder mask,
and
> that changing the orientation of the Couder mask can change the bias.  I'm
> guessing that the taller the Couder mask openings, the worse the bias.

For simulations, you could use the main diameter, or one line above/below
it. For an ideal rotational symmetric mirror, all points of equal radius
will have their normals intersect the axis at the same distance - but with
the tall zones commonly used near the center, the zonal line defined by a
line parallel to but distant from the main diameter and limited by circular
edges may not have the same effective ROC as on the main diameter.
>
> Another thing I have considered, but not proven, is that actual or test
stand
> astigmatism may be a contributing factor to errors in Couder readings.  If
> there's enough astigmatism, the flip-and-diff circle can go up or down
with
> respect to the horizontal center line.  This could affect couder mask
openings
> differentially.

Considering the difficulty in detecting astigmatism in COC tests, it seems
less likely to me.

Linfoot, it seems to me, gives little consideration to the optimum width of
the zone (WRT the dimensions of a near-paraboloid mirror). with a zonal
illumination like fig.8, it should be relatively easy to compare
illuminations - fig. 9 shows a very much less favorable light distribution,
due to a too wide zone (this is the situation I try to avoid in my zone
calculator).

regards,

Nils Olof
PS Thanks Mike for pointing out the Linfoot article - even if most of the
maths are far beyond me, I am convinced it is a great piece of work. I have
a copy of the chapter "The Foucault test" from "recent advances in Optics"
by Linfoot 1954, covering much of what is in the 1948 paper but also a lot
more interesting reading - admittedly it was a long time since I tried to
read it but I'll take another stab when I can....




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