Re: Foucault test limitations was [ATM](no subject)

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

Geoff Nelson wrote:

>If we remove 
can remove the human interpretation would the accuracy be in the 
region always of 1/40th wave or 1/100th wave.<

Not likely; accurate readings require certain level of symmetry within
zonal defocus. If the defocus symmetry is seriously compromised, there is
nothing that can determine accurate radius, simply because there is none.
Besides, I wouldn't take those numbers too seriously. Foucault test
procedure creates a fictitious surface - or, rather, wavefront - that
serves its purpose in indicating where the deviations of the actual
wavefront are relative to a perfect reference wavefront. But the nominal
deviations it expresses are also fictitious, and smaller than those of
the real wavefront.

For example, if we apply the procedure to a 300mm  f/4.5 mirror which is
a perfect ellipse with -0.96 conic, using 6 zones, with 0.707 reduction
zone, perfect measurements will give that the largest transverse
aberration is for the outermost zone - 0.63 - while wavefront deviation
from the central to the end zone are 1/34.6 wave, 
1/21.2 wave, 1/19.2 wave, 1.22.6 wave, -1/42.3 wave and -1/98.2 wave,
respectively. That gives direct wavefront p-v error as  1/13.2 wave
(which is likely to be reduced when calculated relative to a perfect
wavefront that have the "best fit" to the wavefront data). "Quickie" RMS
comes from the p-v error divided by 3.4, giving 1/44.9 wave RMS error,
and resulting 0.98 Strehl.

By automatically applying "wavefront doubles surface error" principle, we
come to characterize this surface as 1/26.4 wave.

The anatomy of the real surface/wavefront of this mirror is somewhat
different. Surface deviation from a perfect parabola of this
diameter/radius is given by (1+K)h^4/8R^3 with K being the conic, and R
the r.o.c. So, surface deviation of this -0.96 ellipse from a perfect
parabola reaches the maximum at the edge, and it is 0.0001286mm, or
1/4.277 waves for the 0.00055mm wavelength. The wavefront deviation is
doubled at both marginal and paraxial infinity focus, 1/2.14 wave, but at
the best focus in the middle it is four times smaller (or only a half of
the surface error) - 1/8.55 wave of spherical aberration. The appropriate
RMS is 1.28.7 wave, giving 0.953 Strehl. Real transverse aberration at
the best focus is 1.53.

Quite a bit of discrepancy. It seems that the main reason for the
significantly smaller test numbers is the treatment of transverse
aberration. It is assumed to be half of  that given by the longitudinal
aberration at c.o.c., while the geometry clearly suggests it is twice
larger at infinity focus than at c.o.c. I don't see the reason for this
"minimization" of the transverse aberration. Can someone enlighten me?

Vlad


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