[ATM] A two way null test for convex hyperboloids

[Mauritz Andersson]

Regarding the latest discussion on testing convex hyperboloids.
Maybe someone could comment on the following idea on a null test or
point to previous identical or similar schemes.

As an example for discussion, consider the secondary for a 400 mm
diam, f8.4 RC telescope. The secondary has a RoC of 1374.6 mm and a
conic constant of -5.907.

Assume that the back side of this glass is figured to a convex surface
of RoC 1277 mm and conic constant of -4.91.

So the prescription is
R1 1374.6 mm   CC -5.907   (Surface to be used in telescope)
R2 1277.0 mm   CC -4.910
Center thickness 20 mm, diameter 160 mm

When testing the surface 1 from the back through surface 2 using a
Foucault test you will have a very good null.
Now the interesting thing is that when you turn the lens around to
test surface 2 through the front surface 1 you _also_ have a null. So
you see a null from both sides of the glass.

See OSLO files here:
http://hem.bredband.net/mauritzandersson/convextesting_rc_null_asphback.len
http://hem.bredband.net/mauritzandersson/convextesting_rc_null_asphback_reverse.len
(I'm assuming testing in HeNe light and a quartz mirror subtrate here.)

OK, fine but how do one get the R2 hyperboloidal surface in the first place?
Turns out that if you start with both surfaces spherical with the
correct RoC and perform the following iterative procedure, you end up
with both surfaces having the correct hyperbolic constant.

Test the second surface through the first, you will see a deviation
from null (about 20 waves). Now polish the second surface until you
get closer to a null, about 10-5 waves asph remaining.

Now turn the lens around and test the first surface through the second
surface. Polish the first surface until you get close to a null. Go
back to testing and polishing the second surface and reiterate this
until you see a good null on both sides. This procedure converges to a
k=-5.907 for the 1374 mm surface and to a k= -4.91 for the 1277 mm
surface. The convergence is in theory exponential with the number of
iterations. After 4-5 iterations you should be below 0.1 waves PV
error.

Convergence occurs because one tests and polishes the _reflecting_
surface in each iteration, the reflecting surface contributes with
approx twice the wavefront error compared to the refracting surface.
(You can try the iterations by resetting the conic constants to 0 in
the OSLO files and do the stepvise "figuring" by introducing a cc on
the reflecting surface in each case.)

What do you think, is this method also feasible in _practice_?

Regards,
Mauritz Andersson