ATM An $18.00 Null Test

[Aplanatic@aol.com]

I have discovered that one can null test almost any paraboloidal (or other 
conic section) mirror using a single lens in front of the knife edge and a 
good quality diagonal mirror.  The idea is the following, best told by an 
example:

We wish to null test a 10" f/6 paraboloidal mirror.  In our garage or 
basement we can move the point source 20 feet from the mirror, so we do that. 
 Next, we select a reasonable lens [1], an Edmund plano-convex lens, number 
P32,475.  It is a 20 mm diameter BK7 lens with one side flat (plano), the 
other side convex with a radius of curvature of 51.68 mm, and a center 
thickness of 4.3 mm.  Placing this lens 1.80" in front of the knife edge 
during Foucault testing completely cancels the spherical aberration induced 
by not having the point source at infinity. In fact, this same lens will 
cancel spherical aberration induced by any *reasonable* point-source distance 
for a large variety of mirror focal lengths by simply moving the lens with 
respect to the knife edge.  Amazing!

Because one's head would block the light from the point source in the 
traditional Foucault geometry, the image must be inspected by the use of a 
diagonal, which will block the central part of the mirror.  This diagonal can 
be the one that will be used in the final telescope, or it can be purchased 
specifically for this test.  Returning to our example, the 10" f/6 mirror 
will require a 1.8" diagonal in the actual telescope, which will block about 
2.7" of the mirror during the Foucault test if the source is 20 feet from the 
mirror.  This is very acceptable since the tolerance envelop near the center 
of the mirror is huge, and since the central 2.7" of the mirror accounts for 
only 7.3 percent of the light, some of which is blocked by the diagonal 
anyway in the actual telescope.

An added benefit of this test is that the diagonal itself can be tested for 
astigmatism.  If the diagonal produces astigmatism the Foucault shadows will 
show it rather clearly.

Interestingly, for testing a paraboloid, almost any converging lens will 
work, the only issue being the distance from the knife edge (KE) to the lens. 
 To minimize the diagonal size, it is desirable to have the lens no more than 
two or three inches from the KE.  Depending on the lens configuration 
(plano-convex or double-convex or meniscus) the distance to the KE will 
change, since the SA induced by a single lens will vary according to both its 
power and how much it is "bent".  See R and vV for a discussion of this.

Since we are using a single, uncorrected lens, the Foucault test should be 
performed in approximately monochromatic light, and the system should be ray 
traced at this wavelength.

So, given that you know the RoC of your mirror and the distance to the point 
source, how do you determine what lens to use and how far to place it from 
the KE?  Well, I'm sure that one could work out the Seidel coefficients for 
this system and find an analytic expression for the distance based upon the 
mirror and lens parameters, etc.  But with ray tracing programs, simply 
select a *reasonable* lens and move it with respect to the focal plane until 
the rays form a point.  The focal plane is then the null position for the KE.

Regarding accuracy of placement, etc, it turns out that this test is very 
forgiving, at least for the cases that I have analyzed.  One must know the 
distance to the point source to perhaps 0.5%, which is easily done with a 
tape measure.  The paraxial radius of curvature of the mirror should be 
measured to about 0.5%, once again easily done with a tape measure and the 
standard Foucault test, and the distance from the lens to the KE should be 
set to about 10 mils, easily done with a caliper.  If the lens is perfectly 
characterized then the accuracy of the null test is better than 20th wave 
under the above tolerances in the aforementioned example.  The lens parameter 
tolerance analysis is best left to the ray tracer, but it does not seem 
severe to me for the cases that I have looked at.

I have not yet tried this method on the bench, but I will soon.  This may not 
be a "new" null test, but if it exists in the literature I can't find it.

Comments are most welcome, as always.

Dave Rowe
Torrance, CA
Medium Format Astrophotography:
http://members.aol.com/aplanatic/photos/astro.html

[1] A reasonable lens would be large enough in diameter to pass all the rays 
from the mirror when placed at its correct location, and would have a focal 
length somewhere between 100 and 200 mm.  A plano-convex or double-convex 
lens is probably better than a meniscus.  The index of refraction, center 
thickness and radii of curvature must be known to good precision.  For 
testing a paraboloid, a converging lens is necessary.