Re: ATM Ellipsoidal mirror test

[Anthony Stillman <atmer@flash.net>]

Giuseppe,

After reviewing the literature I have come to the conclusion that the most
viable approach for testing during the fabrication of a small convex
prolate ellipsoid is a proof plate.  The relevant information you will need
is detailed below.

Another slightly more complicated approach which also utilizes a concave
prolate ellipsoid, but as a nulling element was presented by Abraham Szulc
in Applied Optics Vol. 36 No. 25 6274-6277.  I will detail this testing
arrangement in a forth coming post.  In short this configuration calls for
a concave ellipsoid approximately two to three times as large as the mirror
to be tested.  The light source and test stand (knife edge, ronchi, ...)
look through a hole in the mirror under test at the nulling element.
There, the light is reflected and returned to the mirror under test, nearly
normal to its surface, where it again reflects and returns to the nulling
element to be refocused at the knife edge, ronchi,...

Szulc's methodology is interesting and certainly appropriate to the highly
aspheric convex hyperbolic surfaces for which it was intended.  However, in
my opinion, a concave proof plate would be easier.

Proof plate approach

Zonal radius of a prolate ellipsoid

rh = r0 - ((epsilon * h^2) / (2 * r0))

r0 - osculating radius of curvature
rh - radius of curvature at a zonal height of h
h  - zonal height
epsilon - conic constant for the prolate ellipsoid  = -eccentricity^2

As epsilon is by definition negative, rh grows with zonal height.  That is,
the curve gets flatter.

If the initial sphere is fabricated to a radius equal to the desired
osculating radius, it will be necessary to remove material from the edge to
bring it to figure.  I suggest instead fabricating the initial sphere long
in radius and then driving down the center as is typically done to figure a
paraboloid.  An initial radius around or slightly longer than the
appropriate edge radius of the final ellipsoid should be a good starting
point.

This concave ellipsoidal proof plate can be tested in a nulling
configuration using separate conjugate distances for the light source and
knife edge or grating.  I suggest that you place the light source at the
medial conjugate and the tester distally.  Further I suggest using a fiber
optic as a pin hole source as this will minimize the obstruction.  The
conjugates for a prolate ellipsoid are located at:

s  = r0 / ( 1 + (-epsilon)^0.5)

s' = r0 / ( 1 - (-epsilon)^0.5)


The Szulc's paper

At this point in time I have extensively studied the aforementioned paper.
It is my intent to write up my findings and post them to the list,
hopefully soon.  Should others wish to review the paper before this, please
be aware that there are a few editorial oversights.  In section 5, the
subscripts associated with the conic constants used in the text are
incorrect.  The parenthesized statement is inappropriate and confusing .
The nomenclature used to describe the conjugate distances should have
differed from what was used earlier as they are not the same variables.
Lastly, reference 8 should have a date of 1957 not 1967.

Anthony