From: Mark Holm 
To: atm{at}shore.net
Reply-To: Mark Holm 


OK, I'm not fully up on the arguments here, but I think I know enough to
say a few things for sure.

1. Wavefront errors are additive, period.  It doesn't matter how close or
far away an optical surface is.  (Assumes intervening medium: air, etc., is
homogeneous.)  This also means that they are subtractive: A high on one
surface might by chance, or design, partly cancel a low on another surface.
(Texerau has
been quoted (in translation) as saying that tolerances could be relaxed on
the diagonal.  I haven't read this part of the book recently myself, so I
can't say if the quotation is accurate, but I can say he should have know
better.)

2. Errors of the primary and diagonal in a Newtonian cause a less than
optimum image.  That less than optimum image gets magnified by the eyepiece
and your eye
lens before projection onto your retina.  Your retina is not finely enough
populated with receptors to tell much difference between a nearly perfect
image and say a 1/4 wave error image by itself.  The eyepiece and your eye
lens have to magnify the primary image for you to see the worst effects of
error. (Unless the error is really bad.)  A corollary is that your eye
lens, and at least some of the eyepiece lenses don't have to be quite as
good as the primary and diagonal, because the errors they introduce to the
image will not be magnified so strongly.  This is lucky for us, because
none of us has a diffraction limited
eye lens, not at least when the pupil is dilated.  Of course there are
limits. A really bad eye lens, for example a strongly astigmatic one, will
mess up the final image enough that, without correction, a good image is
impossible.

3. Analyzing the error contribution of a diagonal lens compared to the
error contribution of the primary is somewhat complicated.  Part of the
complication comes from the 45 degree tilt.  A more serious complication
arises because each point on the diagonal is receiving light from most or
all of the primary.  It is
not easily possible to say, for instance that a low edge on the primary
will be compensated by a high edge on the secondary. (Not counting the
difficulties of accounting for the 45 degree tilt here.)  This is a case
where distance does matter some.  For example, if you have a not too strong
lens, such as is typically found in a telescope objective lens, you could
effectively figure the second surface to compensate for errors in the
first.  In an achromat, you could
even figure the third or fourth surface.  This assumes the errors are not too great.

Because of 3, it is best to consider the error contributions of the primary
and diagonal as separate, and only additive.  That is, do not think that
errors in one will cancel errors in the other.  There is a statistical
sense in which errors in one may partly cancel errors in the other.  This
statistical interpretation makes it most likely that adding the square root
of the sum of squares of the RMS errors of each surface will give the most
likely final rms error.  Remember that this is a statistical sense.  For
example, a small spherical error of the flat will cause mostly astigmatism
at the image.  Simple (toroidal) astigmatism of the primary could,
depending on orientation, either add to or partly compensate for, the
astigmatism introduced by the spherical diagonal.  Most likely they will
not be lined up appropriately to subtract maximally.  The astigmatisms will
most likely partly cancel.  That is the statistical result predicted by the
root sum of squares calculation.  Remember, that this sort of rms
calculation, as opposed to the rms error of a single surface, is a
statistical calculation.  There is a finite, and not negligible,
probability that the errors of two surfaces will add in a detrimental way.
Therefore, wisdom suggests that both surfaces should have errors held to
somewhat less than the error one would tolerate in the final image.  The
very worst case is that the errors in surface 1 will directly add to the
errors in surface 2.  This suggests that the maximum error one should allow
in each of two
surfaces is 1/2 the error one would tolerate at the image.  The very worst
case is not too likely, so a betting man might choose to go the rms route
which puts the limit for each surface at 0.7071 times the error one would
accept at the image.  A real optimist might hope that errors in one would
exactly cancel the errors in the other, so that one could allow the same
amount of error in each surface as the maximum to be tolerated at the
image.

4. If you are considering only a single "point" in the final
image, say the image of a single star, it is likely that your diagonal is
somewhat larger than the minimum needed to catch all the light from the
primary headed for that point.  In this case, the portions of the diagonal
not illuminated by light from
that star do not enter into the image formation. (Not unless you are trying
to figure this out by quantum mechanical methods!)  Therefore, the
effective clear aperture of a diagonal, for the purpose of calculating its
wavefront error is somewhat smaller than its full size.  This is somewhat
of an oversimplification,
but I think, approximately true.  A smoothly curved diagonal might have 1/8 wave
P-V error over its whole surface, and only 1/10 over the part illuminated by any
one star.  To get much mileage out of this observation, one has to know
something about the shape of the error on your diagonal.  If your diagonal
is somewhat oversized as most are, and if it is smoothly curving, the
effective error can be reduced from the total by approximately the
proportion of the diagonal that is illuminated for a small image element. 
If the diagonal error is non-uniform, for example if the edge is turned,
but the rest of the diagonal is flat, one could say that the inner portions
of the image would have little aberration, but the outer portions would be
strongly affected.

Mark Holm
mdholm{at}telerama.com

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