ATM Immersion Null Test for Aspheres

[Anthony Stillman <atmer@flash.net>]

Recently while thumbing through old optics journals I came across a letter
by Robert T. Holleran originally printed in Applied Optics Vol. 2(12)
1336-1337 (Dec.  1963) entitled "Immersion Null Test for Aspherics".  The
author presented what he believed to be a novel null test for aspheric
surfaces.

Nominally the test is intended for use with a knife edge or grating
(foucault or ronchii).  The nulling apparatus consists of a pan and an
immersion fluid.  The pan must be deep enough to fully immerse the mirror
and provide for containment of the liquid to a specified depth above the
test surfaces vertex (center of the mirror).  Though not mentioned in the
letter, the pan should be sufficiently larger than the mirror that boundary
effects at the pan edge do not overly effect the test.

The letter presents the mathematical derivation for the relationship of the
thickness t (the distance from the surface of the liquid yo the tested
surfaces vertex) and the refractive index n of the immersing liquid.  When
this relationship is met then, to first order the tested asphere of
eccentricity e is nulled.  That is, it tests out like a sphere.
Specifically equation eight from the letter is:

n^2 = 1 + (e^2 / (1-c))

where

n    index of the immersion fluid
e    eccentricity of the asphere
c    = t / R
t    the thickness of the immersion fluid
R    the paraxial radius of curvature of the asphere  (the letter refers to
this as the vertex radius)

It is noted that in the limit when t goes to zero (c=0), a reasonable
approximation for large shallow mirrors that are just immersed, the
equation for n reduces to

n = ( 1+e^2 )^0.5

For a paraboloid where e = -1 this translates to a value of 1.414 for n.
Some oils have indices near this, however the letter notes that this
nulling method, when used on shallow curves, can tolerate wide parameter
errors.   Specifically stating that using water will result in a null when
a paraboloid is 77% corrected.  This corresponds to an eight wave of error
for a 6 inch f/8.  Please note that result is based on two assumptions.
First, that the immersion thickness is zero.  And second, that both the
light source and knife edge are at the radius of curvature.  Using water
with a correct thickness value will take this methodology to its limit.
Unbalancing the conjugate distance ratio can extend it.

The letter goes on to analyses errors,  noting that they increase for
faster surfaces.  Specifically, if a surface is fabricated to null using
this method, then for small values for c the sagittal departure delta s can
be approximated by

delta s = r^6 * e^2 / (16*R^5)

where

delta s      the departure of the surface from the desired surface
r            the radius of the zone (max. zone=mirror diameter / 2)
e            the eccentricity of the asphere
R            the paraxial radius of curvature of the asphere

The letter then, I believe incorrectly, presents a computed error value of
0.11 waves for an f/2.5 100cm ROC paraboloid figured in this manner.  The
value I calculate is approximately 1.4 waves.  I have yet to search for
errata in subsequent journals.

The paper also discusses testing convex surfaces.


A few observations:

An alternate form for equation 8 above is:

c = 1 - (n^2 - 1)/e^2

For a paraboloid in water c = 0.2311
Hence, t = cR = 0.2311 * R

For a 6 inch f/5 R= 60 inches.  Using water the thickness t would be nearly
14 inches.  The residual error would be no less than 0.003 waves.  Using an
aqueous solution of hydrochloric acid with an index of 1.411 would be
nearly perfect ;-)  Alternatively a solution of glycerol and water or sugar
and water might be easier.  With a well selected index an inch or less of
immersion thickness is all that is needed.  Of course water evaporates,
thus altering index, so it would be a good idea to keep the environment
cool and humid when using water based solutions.

Given the vertical nature of this nulling method it seems impractical for
large slow surfaces.  I do however plan to rig an apparatus and test a few
small mirrors.  I'll post again on the immersion fluids I tried afterwards.

Anthony


Some miscellaneous notes:
The copyright for this letter is claimed by Optical Society of America.
Robert T. Holleran was working for American Optical Company of Keene, New
Hampshire at the time of  its writing.  My presentation is in accordance
with the copyright act of 1976.

Equation 12 from the letter relating nulling error to other parameters is
given as:

delta s = R * k^6 * e^2 / 16  = (y^6 * e^2 / 16 * R^6 * lambda) wavelengths

where y = k*R

It didn't make sence to me either.