[ATM] Foucault and diffraction

["vladimir sacek" <vlad@copper.net>]

Recent thread that once again raised the question of
astigmatism detection/measurement with the Foucault
test made me re-think its underlying principles. While most 
everyone learns these principles based on geometrical optics, 
this is not what the reality of the test is.

Due to diffraction, there is no focal "point", not even a disc,
rather a cylinder of light extending in the focal zone and widening 
into defocused pattern on both sides. Therefor,
there is a certain length along which an intercepting edge
would cause whole of the reflecting surface darken nearly
instantly. I don't know what exactly determines length of this section, 
but will arbitrarily take 1 wave (~1/3.3 RMS) of defocus error as a 
rough limit at which the darkening begins to be gradual.

In such case, the length of travel along which an intercepting edge would 
darken the surface nearly instantly would be about 0.7mm at f/9 
(typical f/4.5 mirror at the r.o.c.) for a perfect unobstructed focus. However, 
parabola at the r.o.c. has as much of spherical aberration in the wavefront as 
its spherical counterpart at the infinity focus, only of opposite sign. The effect 
of s.a. is to extend the cylinder of light, also throwing more energy out of it. 
Where the cylinder starts disintegrating into defocused pattern in the presence of s.a.? 
My guess is roughly beyond marginal and paraxial foci. So its length would be roughly 
approximated by the longitudinal aberration, which is at the r.o.c. four times that at the 
infinity focus of the corresponding sphere, or D/8F for D in mm. For a 300mm f/4.5 mirror 
that would give 8.3mm.

Assuming the outermost mask opening being 0.1 the width of radius, 
its longitudinal spherical would be 1-0.9^2 or about 1/5 of the total longitudinal, 
which for the above mirror comes to ~1.7mm. But for a pair of openings, 
the three-dimensional diffraction form of the focal zone is altered by two other factors, 
shape/size of the openings and their separation. While the former probably can be safely 
neglected, the latter likely further expands the "blank" section within which the zone 
darkening is nearly simulatneous. This would be due to the factor of an effective 
90% c.obstruction, which diminishes defocus error by a factor (1-0.9^2) for a circular 
annular aperture. Diffraction pattern formed by a pair of opposing segments of such 
aperture would also be radially asymmetrical, being considerably extended in the 
direction perpendicular to the line connecting the openings.

Main implication is that the accuracy of zonal radii measurement is limited by the 
diffractive form of the focal zone. Most so for the outermost zones, usually regarded as 
those most reliable measurement-wise. For the above mirror, the "blank" section of the 
KE travel within which there is little or no change in the uniformity of zonal darkening 
might be up to a few mm for the outermost zone. If so, it would likely be more precise 
to determine points at which each of the two zones noticeably darkens first, and take 
the mid-value as that of the zonal r.o.c. than to try to pinpoint it directly. A 1mm difference 
here can change a lot in the extracted wavefront profile. 

The two-dimensional pattern elongation could  make it relevant whether the KE moves along 
the line connecting the openings, or perpendicularly to it. But I don't have a clear idea of what 
the three-dimensional diffraction form of the focal zone looks like, needed to make any specific 
conclusions. As for the astigmatism, looking at both, side cross-section of the diffraction form 
in the focal zone and intra- and extra-focal patterns, the astigmatic foci seem to be too distorted 
by the energy thrown around by both spherical aberration and aperture profiled by the openings
to allow reliable detection. 

Well, that would be my take at the actual, diffraction aspect of the Foucault test. I'm sure it's been 
addressed much better somewhere else, but my impression is that it didn't
make it into the mainstream of the Foucault know-how. It is reasonable to expect that 
better insight into the actuality of it would result in its more successful application. 
Hope someone with more information and knowledge would follow up on this.

Vlad




 
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