Re: ATM Digital spherometer; how to

[Anthony Stillman <atmer@flash.net>]

All,

I have to wonder about the effacacy of a spherometer that can measure sag
to 0.5um.

For the sake of simplicity lets assume we are measuring a paraboloid.  Hence,

R = r^2/2s

Where R is the radius of curvature, r is the radius of the spherometer and
s is the saggita.

The derivative or R with respect to s is then

dR/ds = -r^2/2s^2
      = R/s
or

dR = (R/s) ds  (to all the math majors, its not Koesher, but it does work)

If you need to know R to 0.1% then

R +- dR  = R +- 0.001 R

dR = 0.001 R

hence

0.001 R = (R/s) ds

0.001 = (1/s) ds

Since s is typically between 0.1 and 0.5 inches.  ds needs to be no greater
than

0.0001 inches

or about 2.5um.  Five times the resolution referenced above.

Its true that for very small values of r, and/or large values of R, the
sagitta would be small enough to warrent greater accuracy.

Specifically, a value of ds of 0.5um correspondes to a 0.1% uncertainty in
R when the value fo s is

(1/s) ds = 0.001

s =  1000 ds

s = 1000 * 0.5um

s = 0.5mm

For a value of r of 50mm (a common enough spherometer radius) the
corresponding limiting value of R is,

R = r^2/2s

R = 50^2/(2*0.5)

R = 2500mm

Values of R greater than this, of course, suffer from greater uncertainty.

For a 25mm radius spherometer a 0.1% accuracy limits the value of R to 625mm.

I mention a 25mm radius spherometer, as I recently made such an instrument
to help in the manufacture of the Schiefspiegler secondaries on which I am
currently working.  However, not having $400 dollars to toss at an
electronic etched glass plunging dial indicator, I used my surplus Federal
mechanical dial indicator, manufactured in 1956.  Its repeatable to a
hundardeth and I believe it to two.  That is 0.00002" or 0.5um.  For the
radius in question, 127.5", I can measure, using this  spherometer, to an
uncertainty of about 0.5%.  This isn't suppose to be good enough for a
Schief, though actually it is, but it doesn't matter as I will fine tune
(its a matter or pride) the secondaries to match the primaries by contact
testing them under a Newton's ring interferometer.

I'm not saying that high accuracy spherometers don't have there place, only
that you should examine what is truely needed to do the job before
expending unneeded effort.  Also, remember that although errors in the
value of r and just how perpendicular the indicator motion is to the plane
of reference are less relavant, they do contribute to the overall
uncertainty.


Anthony