[ATM] Re:Satisfying "Millies-Lacroix" is neither necessary norsufficient for "diffraction-limited" performance

[Vladimir Sacek <vladis.2@juno.com>]

In general, it can be either unnecessary, or insufficient. Geometric blur
size has no common "diffraction-limited" value for major aberrations. In
fact, none of them becomes diffraction limited at the blur size nearly
equal to the Airy disc size. In terms of the RMS wavefront error "w", the
blur size in units of Airy disc diameter is given by 48(sq.rt.5)w/2.44
for spherical aberation at the best focus, 36(sq.rt2)w/2.44 for coma, and
with 8(sq.rt.6)w/2.44 for astigmatism (also best focus location). With
the conventional "diffraction-limited" wavefront RMS of 1/13.4 wave, this
gives "diffraction-limited" blur size as 3.3, 1.6 and 0.6 Airy disc
diameters, respectively.

The anatomy of s.a. defocus alone illustrates how meaningless is
geometric blur size by itself. Blur at the paraxial focus is 2.5 times
the blur at the marginal focus; yet, both locations have identical RMS
wavefront error. Best (or diffraction) focus, right at the midway between
those two, has the blur twice smaller than the paraxial, yet the
wavefront RMS error is four times smaller. And at the location of the
smallest blur (midway between marginal and diffraction foci), the RMS
wavefront error is double that at the diffraction focus, despite the blur
being twice smaller.

Of course, the best focus "secret" is that its blur has highest
concentration of rays within a relatively small central blur area. 

With most mirrors, the goal should be to obtain smooth conic as close to
parabola as possible. This implies a goal of having as pure form of
spherical aberration at the c.o.c. as possible. And that implies the
appropriate change of the TA standard with zonal hight. For instance, if
the residuals are reduced to the center zone, the RTA standard for the
edge zone for the "diffraction-limited" level should be ~6, exponentially
dropping to a fraction of 1 for the center zone. This is because here we
consider blur size at the paraxial focus, and for 1/4 wave s.a. level it
is at this location over 6 times larger than the Airy disc, with its
boundaries formed by the marginal rays. 

If the residuals are reduced to 71% zone, the corresponding focal
location is that of the diffraction focus, and the RTA standard for the
edge zone for 1/4 wave s.a. level should be ~3, dropping to zero towards
71% zone, then increasing to ~1 for 50% zone, and dropping somewhat again
for the center zone. The amount of allowable TA varies with the reduction
zone, because the corresponing location within defocus zone at the
infinity focus varies with it.

Texereau picked different route. He set the max. RTA  at about the Airy
disc size uniformly for all zones. This sacrifices the principle of
maximally smooth conic, but preserves more important principle which is
that the error over most of the wavefront is substantially less than 1/4
wave - at least according to the measurements. It is not unimportant that
it is more likley to result in a good mirror even with (expected)
moderate measurement errors, which wouldn't be the case if the TA
standard would be set at about 1/4 wave s.a. level. Since longitudinal
aberration at the focus is 1/4 of the "excess" longitudinal aberration
(positive or negative) at the c.o.c. vs. perfect parabola,
the max excess longitudinal aberration is given by DF^2/372h, "h" being
the zonal hight. 
I think it's good to have a higher standard, although  it is not to be
taken as strictly as it usually is.

An alternative route is to go directly with the conic. It may be even
simpler, and the advantage is that is more directly related to the actual
surface.

Vlad



_______________________________________________
ATM mailing list http://www.atmlist.net/