[ATM] Re: Weird Foucault readings

[Vladimir Sacek <vladis.2@juno.com>]

Nils,

>> Yes, I did run into that article some time ago. Nils tried to
determine
>> at what point of zonal defocus the zonal null is likely to be most
>> "convincing" visually.

>Not quite. I determined the point on the axis where the diffraction peak
>falls from a given zone (of a perfect paraboloid), and from that,
determined
>the center of the zone in question as the point whose normal intersects
the
>optical axis at this point.

Well, and then you compared null efect at this point with those of two
different points, coming to a conclusion that your point's null effect is
more convincing that those of the other two. So, what is "not quite" so?

>>Both, "non-linearity" (uneven illumination across
>> the zone) and reversed illumination change for the two opposite zones
>> result from the zonal defocus, which causes different areas of the
zone
>> to be affected differently (illumination-wise) for any given position
of
>> the KE.

>Here, it seems you assume that you can treat the zone as having separate
>areas - this is simply not possible, for good wave-optical reasons. The
profile of the diffraction pattern is formed from the profile across the
whole zone aperture.

The illumination effects are obvious, and just as I described them. And
every zone does have separate areas, both in regard to the focus and -
consequently - illumination effect.
Profile of the diffraction pattern obviously doesn't make it impossible.

>>All three suggestions for the correct zonal radius are around the
>> best focus (also called diffraction focus) location. Couder is right
on
>> with (h^2+z^2)/2R, while Texereau and Carlin are slightly off.

>The diffraction peak is placed according to my formula (given the zone
is
>part of a parabola). When you say that Couder is "right on", what other
>criterion do you use?

If isolated from the rest of mirror's surface, every zone acts as a
section of an obstructed aperture. Axial shift from paraxial to
diffraction focus is given by 8Aw(1+c^2)F^2 where A is the total
(spherical) aberration (in the unit of wavelength), "w" is the
wavelength, "c" is the obstruction in units of the (effective) aperture
and F is the f#. The (1+c^2) factor can be directly related to the
longitudinal aberration for the entire mirror surface (8AwF^2) normalized
to 2. The expression shows that the diffraction focus location is always
in the middle between the focus of the upper zonal edge and that of the
lower zonal edge.
Since the focus location changes with the square of zonal hight, it won't
be the geometrical zonal center (with the appropriate radius) that
focuses at the diffraction focus, but a point slightly above it. This
hight's square is determined by the average of the squares of zone's top
and bottom hight, or h'^2=[(h^2+z^2)+(h^2-z^2)]/2, which reduces to
h'^2=h^2+z^2. Therefore, the focus shift for this point - which focuses
at the diffraction focus - is (h^2+z^2)/2R. It's easy to verify it does
focus at the midway of the zonal defocus section.

>>Much more substantial error
>> is already built in by averaging out the entire zone to its mean
radius.

>I don't quite see what you mean by "averaging out the entire zone" -
could
>you clarify?

Simply that the zone is assumed as perfect as its measured average
radius. Strictly talking, it only happens with perfect conics (assuming
perfect mesurements). And when we expand it to a pair of zones, the
built-in error increases.

>> Part of those real-life conditions is less than
>> perfect mirror, with less than symmetrical zonal surfaces which
changes
>> defocus structure and affects zonal illumination in an unpredictable
>> manner.

>By no means unpredictable - knowing the profile, the light distribution
can
>be calculated numerically to high accuracy...

The point is that we don't know the profile before the test; if we would
know it, what would be the testing for? To test the testing method?

 >>Consequently, while we could try to approximate error margin for a
>> perfect or near perfect mirror, it seems to be pretty much
unpredictable
>> for other - and rather common - scenarios.

>Conclusion - don't bother to measure your mirror unless it *is* perfect
-
>and if it is, why bother? <G>

I wouldn't go that far :) It serves the purpose to find out - or confirm
- that the mirror is 
very good or excellent. But the point that should be ephasized again is
that so-so and plain bad parabolizing methods - among other things - do
make it harder to get reliable results with the Foucault.

Regards, Vlad
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