Re: ATM stopped down apat. to 18 inches?

[Anthony Stillman <atmer@flash.net>]

In February 1988 a paper entitled "Aspherical sectional mirror of spherical
elements" by F.S. Dyadyukov was printed in the Soviet Journal of Optical
Technology 55 (2).

Skipping over the rational for the adopted criteria, the mathematical
derivation, and the exact solution, the presented empirical formula was:

  delta z bar = [0.025 * e^2 * h^2 * d^2 * (1 + 0.16*d/h)] / R^3

with the adopted criteria being:

  delta z bar <= lambda / 28  (the Marechal condition)

where:
  lambda  wavelength of operation
  e       eccentricity of asphere
  h       optical axis to center of spherical element
  d       radial extent of spherical element
  R       osculating ROC of asphere

For the given configuration and dimensions, and assuming the desired
asphere is a paraboloid.  Then using the archaic units of inches

  e = -1
  h = (48/2) - 18 + (18/2) = 15
  d = 18
  R = 18 * 10 * 2 = 360

  delta z bar = 2172.42 / 46656000 = 0.0000465625 inches

>From this we can see that excellent performance may be had with this
arrangement at a wavelength of 0.0013 inches.  Or in more conventional
units, 33 um.  The long end of the mid IR.   The atmosphere is basically
opaque at this wavelength.

For the same size element and distance off-axis, but with the intent to
operate in the visible.

  lambda = 0.000022 inches

  delta z bar <= 0.0000007857

and hence

  R >= 1403.5 inches

An element focal ratio of f/39.  This is slower than James Lerch's design
(f/23.6) because the Marechal condition is more constraining than the
Rayleigh criterion.


Anthony