Re: ATM Polishing/Figuring Done!

[epabcc@epa.ericsson.se (Bratislav Curcic)]

Rob O'Toole wrote :

>   In addition to using TEX, I adapted Richard Berry's method (from his 
> book) to a spreadsheet and looked at the results this way as well. He is 
> essentially generating the Millies-Lacroix plot. What's different is 
> that Berry sets two tolerance envelopes, one he describes as roughly 1/4 
> wave (wavefront) and one roughly 1/8 wave. The 1/4 wave envelope seems 
> to be identical to the Milles-Lacroix envelope in TEX. This raises the 
> following question with me:  how does TEX come up with a max. wavefront 
> error of about 1/8 wave in my case, when the data just satisfies the 
> "1/4" wave tolerance envelope? Is TEX off by a factor of two or is 
> Berry?

Well, neither really ... It is difficult to explain everything about
wavefronts without a full fledged article (and by the way, that article 
has already been written in TM #32 by Suiter), but in short :
There is an infinite number of parabolas that will closely match the 
actual figure of your mirror. What differs them is a paraxial radius
("focal length"). Depending on the method that you adopted, you can
pick one of them on merit of 

a) minimal transverse aberration
b) minimal wavefront error
c) maximal overlap between a) and b)
d) any other criteria you can think of

(Funny thing is that all of us do this every time we use telecope - it
is called focusing !)

Now, in order to SIMPLIFY testing methods people have adopted various
approximations. One is for examle that surface of the finished mirror
shouldn't depart from an ideal (best matching) parabola by 1/8 wave.
This is in essence Milles-Lacroix method (but they derive surface
"bumps" and "valleys" into the "permissible" longitudinal aberration at
mirror's radius).  What is wrong with that ? Well, the wrong thing is
that this method doesn't take fully into account SLOPES of the error;
It is possible to have surface that is well within this 1/8 wave
envelope and STILL not be a good mirror. (I think that Suiter's
Aperture has one example with sawtooth surface that will show this
better than I can do with another few hundred paragraphs of hopeless
explanations !)

Texerau's method searches for minimal wavefront error, but
unfortunately the Tex program omits the very important graph that is
the essential part of this method (transverse aberration). It is in the
book of course, (+/- "rho" diagram, just above the wavefront plot). The
data needed for construction is of course produced by the program, so
you only need to check one row in the Tex output (can't remember the
row name/number) and conform that transverse aberrations are less than 1,
compared to "rho".
BOTH of these conditions must be fulfiled in order to have a good
mirror.  Contrary to popular belief, it is quite possible to find the
place where wavefront is well within the required, but have the
transverse aberrations larger than theoretical Airy disc.

So which of these methods is foolproof ? None, I'm afraid. So which one
to believe ? If you aren't quite sure what's hapenning, play it safe -
take the worst case. As Texerau says, it is more difficult to fulfill
the transverse aberration requirement for small mirrors. As result of
that, methods that depend on this qualification (Milles-Lacroix for
example) may look worse for a particular mirror, compared to methods 
that search for the minimal wavefront error.
But this isn't a hard rule in any case ! Test you mirror with more than
one method and keep in mind that all the numbers that they are coming
with are just APPROXIMATIONS.  The only way to really find out about
the real wavefront quality of your mirror is interferometry. And that
is outside the reach for most amateurs.  But star test doesn't cost
anything, and it is very sensitive (in fact more sensitive than most of
us will ever need). THAT one you can rely on.

Bratislav