ATM "Flats" for Autocollimation Null Testing

[DROWESMI@aol.com]

I have been wondering for a long time about "flats."  By direct calculation (a
mess) and by ray tracing, I find that for reasonable focal lengths and mirror
diameters, a slightly spherical "flat" when used for autocollimation null
testing, will work quite well to the levels of accuracy even the most
demanding mirror maker would require.  Hereafter, I call this spherical "flat"
the autocollimation mirror (AM).

For instance, I ray traced a 10" f/6 paraboloidal mirror with an AM placed 100
inches from the primary (the location of the AM does not make any significant
difference).  Rays were stared from a point source at the focal plane of the
paraboloid, allowed to reflect from this mirror, propagated to the AM,
reflected back to the paraboloid and finally brought to a focus at (or close
to) the point of origin.  You will recognize this as the standard
autocollimation null test.  My tracing software calculates the P-V deviation
of the path length for all rays started.  In this way, I can get the maximum
wavefront error induced by the non-flat AM.  I varied the radius of curvature
of the AM and noted the point where the system's P-V wavefront error was 1/8
wave. Surprisingly, the flat can have a radius of curvature of about 200
meters before the rays at the edge of the mirror differ in path length by more
than an 1/8 wave from those at the center.

Now, it turns out that a 10" diameter AM with a radius of curvature of 200
meters would have a sagitta over 0.001" (over 50 waves).  Using a decent
spherometer, I can grind two pieces of glass together so that each one has a
sagitta of less than 1/10th this much, i.e., less than 100 microinches.  After
fine grinding, the two pieces of glass should be quite spherical if good
contact has been maintained throughout.

Now, the real questions.  A)  Is it possible to maintain the spherical surface
during polishing?  B) How can the surface accuracy be tested?  If the surface
of the AM is truly spherical and has a very long radius of curvature, then it
works fine in an autocollimation null test.  On the other hand, if it picks
up, during polishing, a turned edge, a hole in the center, or any one of the
numerous other possible defects, then it will not work as an AM.  How can one
tell?  Is this the real reason that people make flats for autocollimation null
testing?  Because a flat can be tested against two other flats, cyclically, to
quantitatively determine flatness?

Dave Rowe
Torrance, CA