ATM Measuring Long Radii

[Anthony Stillman <atmer@flash.net>]

As I wrote in my RTMC report, in a discussion with Gerry Logan, the topic
of how to measure long radii came up.   As there has been some interest I
have tried to chase down the reference.   This has been to no avail.
Likely I have remember the reference incorrectly.  In any case, the method
is simple enough, so I've done the analysis and checked it.

The method  uses a telescope, a flat, a foucault, a micrometer, and the
surface to be tested.  Ideally the foucault/micrometer should be
integrate-able with the focuser of the telescope.  It works like this.  Put
a knife edge/light source  (foucault) at the focus of a bench mounted
telescope.  Position a stand to hold the flat/test surface in front of the
telescope.  Place the flat on the stand in front of the telescope.  Move
the knife edge/light source until it is at the focal plane.  Note the
position of the knife edge.  Now remove the flat and replace it with the
surface to be tested.  Again move the knife edge/light source until it is
at the focal plane.  Again note the position of the knife edge.  The
difference in the noted knife edge positions may then be used to determine
the radius of curvature of the surface being tested.

The formula to convert knife edge translation to radii is simplest if the
flat/test surface is placed one telescope focal length from the telescope's
objective.  In this case the formula is:

      R = f^2/m

where
   R is the ROC of the test surface
   f is the focal length of the telescope
   m is the difference in knife edge positions

If m is negative, that is you had to rack the knife edge in (toward the
objective) when going from the flat to the test surface, the test surface
is concave.  If m is positive, then the test surface is convex.

If you hate math, stop here.  If placing the flat/test surface one focal
length from the telescope's objective isn't possible, then I'm afraid you
must read on.

The derivation of the formula is straight forward, using only first order
optics.  Namely:

  1/object distance   +   1/image distance    =   focal length

The arrangement of optical components is such that the image of the light
source/knife edge is coincident with the point at which the radius of
curvature of the test surface is located.  When this is the case, light
from the telescopes objective strikes the test surface perpendicularly, and
hence is directed back on itself.  For this to be true the image distance
must equal the separation between the telescope objective plus the radius
of curvature of the test surface.  That is:

    image distance = d + R

where
   d  is the distance between the telescope objective and test surface
   R is the radius of curvature of the test surface

With the object distance set equal to the telescope's focal length plus an
offset we have (using the above variables)

     1/(f + m)     +    1/(d + R)   =  f

A little bit of algebra yields

    R = -(m*d - m*f  - f^2) / m

This simplifies to the previous equation for R when d is set equal to f.

I've tested this equation in OSLO LT.  It works.

I was interested in the sensitivity of this test method.  Assuming d equal
to f, a small change in m is related to a change in R by:

     delta R = (-f^2 / m^2) * delta m
     delta R = R * delta m / m

So if :

  f = 1000mm
  R = 10000mm

then :

  m = 100 mm

if the uncertainty in m is +- 0.01mm, then the uncertainty in R is +-1mm.
That's pretty good for a 10 meter radius of curvature.

Next I was interested in what would happen if d wasn't exactly one focal
length from the objective, but I believed it to be.  As it turns out there
is a linear relationship between d and R.  If d is 1 mm too long, R is 1 mm
too short and visa versa.  For large values of R, small errors in d are
ignorable.

Finally, I was interested in the same situation with f.  Here there is some
concern.  For the above example, a 1 mm error in the believed focal length
will result in a 21 mm error in the calculated value of R.  Of course for a
10 meter ROC, this amounts to an error of 0.2 percent.  This can be reduced
by choosing a longer focal length telescope.  For instance a two meter
focal length, albeit erroneous by 1 mm would yield an error in the value of
R of 0.1 percent.

There are several other methods for measuring long radii.  The direct
method is the most obvious, but this solution is problematic for really
long radii.  Murty and Shukla describe this and several other means in a
paper entitled "Measurement of long radius of curvature" which appeared in
"Optical Engineering"  Vol. 22 No. 2   pp 231- 235  (March/April 1983).
Curiously this paper ignores the above described method.  This is
particularly interesting as it's convenience and achievable accuracy is as
good or better than that obtained with a Murty interferometer.  Other
methods employing moire fringes and the Talbot effect are also notable.  A
paper by Chang and Su entitled "An Improved Technique of Measuring the
Focal Length of a Lens" which appeared in "Optics Communications"  Vol. 73
no. 4  pp 257-262  (15 October 1989) discuss one such method.  If you don't
like math, don't bother with this paper.  An additional reference worth
noting is a paper by Hugo and Lessing entitled "Determination of Long Radii
of Curvature of Positive Lenses" which appeared in "Applied Optics"  Vol.3,
No 4 pp 483-485  (April 1964.)  I find this paper most interesting as it
describes a method previously presented by Anton Kutter in Gleanings in
"Sky & Telescope" p348 in April 1959.  No reference to Kutter's note is
given by Hugo and Lessing, yet the methods are identical.

Hope this helps someone.


Anthony