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

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

Nils,

>One thing to keep in mind is that what Rayleigh said (about low-order
sph.
aberration) of the 1/4 wave P-V is that this was when aberrations begin
to
be "decidedly prejudicial" - as I interpret it, a criterion to tell lousy
from mediocre, not to tell excellent from good. So, a wavefront error of
say
40 nm RMS would satisfy the Maréchal criterion but not the amateur that
will
work, or pay, for excellence.<

This sounds more like Conrady than lord Rayleigh. As I gather, the
original Rayleigh's criterion was derived from  a number of empirical
test for spherical aberration *at the paraxial focus", coma and
astigmatism (Applied Optics II, p626). The 1/4 wave wavefront error
criterion was generalized, and it was assumed that it causes
approximately 20% fainter central disc. After Airy made it possible to
calculate the actual values, it turned out that actual energy losses
under the original Rayleigh formulation are widely different for 1/4 wave
of various aberrations, as well as for different locations within the
s.a. defocus.  For the coma, the p-v error "p" translates into RMS
through RMS=p/4(sq.rt.2), which for 1/4 wave p-v error gives 1/22.6 wave
RMS and corresponding 0.926 Strehl: less than 8% loss in central
intensity. For the astigmatism, 
RMS=p/2(sq.rt.6), so that 1/4 wave p-v error translates into 1/19.6 wave
RMS, and 0.90 Strehl, with 10%  central intensity loss.

For spherical aberration (also primary), RMS=2p/3(sq.rt.5) for the p-v
wavefront error at  best focus, which is four times smaller than the
error at the paraxial focus. If lord Rayleigh was counting with the p-v
error at the paraxial focus (which is double the surface error for near
perfect conic), while testing - inevitably - at the best focus - his
actual toerance for spherical aberration (assuming near-perfect conic)
would be extraordinary low: RMS~p/6(sq.rt.5), which for p=1/4 wave comes
to ~1/50 wave RMS,  0.984 Strehl and less than 2% central intensity loss!

Of course, this part is little hard to swallow, besides it all being
rather speculative not knowing the exact details. But indications are
there that the original Rayleigh criterion was more demanding than
nowdays accepted conventional "diffraction limited" standard of 0.80
Strehl. Most likely, the approximate level it was originally set at was
appropriate to 0.90 Strehl, possibly better. 

>My simplified view is that the RMS says how much of the light is within
the
Airy disk and how much in the rings, but the RTA says something about how
wide the light is scattered in the rings. A small amount that is widely
scattered may be objectionable in much the same sense that the spikes
from
the spider vanes are.<

Yes, it is a piece of information that gives some general implications.
Hope my categorization of it as "meaningless" was clearly enough in the
context of  the size of aberration, not its particular effect on image
contrast. However, assuming nearly identical amount of energy out of the
Airy disc, there won't be significant difference in performance - in
general terms - between good mirror producing larger, fainter blur, and
one producing smaller brighter blur. The reason is that they both have
nearly identical RMS and Strehl, which means that their average contrast
drop over all frequencies will be also neraly identical. In other words,
if they both are 1/20 wave RMS and 0.9 Strehl, they will both have 10%
average contrast drop over the frequency range. The one with larger,
fainter blur, will have slightly lower contrast for larger details; the
other will have it slightly lower for smaller details. Quid pro quo.
Actually, larger and fainter blur is more desirable for general
observing, since it likely less affects contrast of smaller details near
the limit of low-contrast resolution. But, again, the difference is
rather slight for good (or better) mirrors.

In fact, real effect of an aberrated blur can't be reliably predicted -
even in general terms - based on its size alone. What matters is also
intensity distribution within the blur, as well as type of defocus. For
instance, when smallest astigmatic blur is about equal to the Airy disc,
astigmatic p-v error is 1/1.64 wave (comparable to 1/2.4 wave s.a.
contrast-wise), and cross-like intensity redistribution is readily
visible in the pattern seemingly out of the reach of the blur itself.
Mother Nature keeps on being tough on us: it would be too easy to know it
just by the geometric blur size; instead, we really have to do
diffraction calculation and MTF to find out.

Vlad
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