Re: ATM atm ROC of lens

[Anthony Stillman <atmer@flash.net>]

John, others,

In a paper entitled "Determination of long radii of Curvature of Positive
Lenses"  written by T.S. Hugo and N. v.d. W. Lessing, which appeared in
"Applied Optics" Vol. 3. No 4. April 1964, the abstract reads

"A method for the determination of the radii of curvature of any positive
single lens of a weak curvature is described.  It consists of applying the
Foucault knife-edge method to the lens considered as a lens-mirror,
measuring the distance between the lens and the knife edge for the each
surface facing the knife edge together with thickness of the lens, and
calculating the radii of curvature be formulas derived paraxial equations.
The refractive index of the glass is required.  In an example the maximum
error on radius of curvature of approximately 4.5 meters is shown to be
1mm, if the accuracies in the measurement of the distances, the thickness
of the lens, and the refractive index are +-0.1mm, +-0.01mm, and
+-0.00001mm, respectively."

Why an index of refraction has a measurement accuracy in millimeters, I
don't know.  Must be a typo.

The paraxial approximation requires that a mask be used.   Think, f/200
light cones.  The known quantities needed to calculate the ROCs of the two
surfaces are, the two catadiopticly measured radii, L1 and L2, the index of
refraction of the glass N, and the lens thickness d.  L1 is associated with
the reflection off of r2.

The radius of curvature of one of the two surfaces "r2" is then found by
solving the quadratic equation.

    a * r2^2 + b * r2 + c = 0

where:

   a = N * (L1 - L2) - (L1 - d)
   b = ((2 * N - 1) * L1 - d) * (L1 - d)
   c = L2 * D * (N - 1) * (L1 - d)

finally "r1", the other radius of curvature, is derived from the equation

   r1 = (N - 1) / ( N/(r2 + d) - 1/L1)

The paper goes on to give an analytical error analysis and provides an
example.  The paper finishes up with some considerations and a rational for
using this means of testing.  Clearly there are considerations,
monochromatic light, the catadioptic images must be real, the paraxial
condition must be satisfied.  And what I find interesting is, before the
mid-70s, many papers in the optical literature included a paragraph or two
on why one should use the method presented.  But now, few do.  Is this
because we no longer need to be persuaded that the latest is the greatest
or, is it because there is no point in trying.  Anyone care to offer an
opinion?  Richard?

Apparently at some point in the distant past I solved for "r2."  I also
reduced the model by setting d, the thickness of the lens, to zero.  If
anyone wants these tangles of symbols I will endeavor to type them.
Hopefully, correctly.

The analytical solutions to the above quadratic equation are:

r2 = ( -b +- (b^2 - 4*a*c)^0.5 ) / 2*a


Anthony

cives censent, servi non facent