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[ATM] Satisfying "Millies-Lacroix" is neither necessary norsufficient for "diffraction limited" performance...
However, I conjecture that any optical system that satisfies RTA <= 1 has
an RMS wavefront error *after* refocusing to the correct focus no worse
than about 0.046 waves (or a Strehl ratio no worse than about 0.92).
This is apropos of nothing in particular. I've been saving up some math to
frighten off newbies with, but I don't want to type in ascii math any more
than you want to try deciphering it so I'll spare you. If you want to
follow along there's an OSLO len file at the end of this post. One of these
days I'll write up a properly formatted document with the math and post it
on my web site, maybe.
Here's the argument. We all know that a 6" (150 mm) telescope at about
f/8.2 just satisfies the Rayleigh 1/4 wave (or the more modern Marechal
1/14 wave RMS) criterion. However at that focal ratio it fails the M-L
criterion by a fair margin. Let's not argue about whether that should
properly be considered "diffraction limited". I'll say other things we can
argue about if you wish.
So, at what focal ratio does a 150mm spherical telescope just satisfy M-L?
With a bit of the math that I promised to spare you it turns out to be
f/9.56. I've created an OSLO file for you, and if you want to follow the
argument you should load it now.
If you click on the "Spot diagram analysis" graphic you will see that the
traced rays exactly fill the Airy disk (you may have to set an option to
have the Airy disk diameter displayed). In ray trace terms having the spot
diameter equal to an Airy disk diameter is equivalent to satisfying RTA<=
1, i.e. the "Millies-Lacroix" criterion.
Now click on the "Wavefront analysis" graphic. The RMS wavefront error is
about 0.098 waves (P-V = 0.34), well short of "diffraction limited".
Now, what we have just done is evaluate the wavefront at the "circle of
least confusion." That is what Texereau's procedure for finding the maximum
transverse aberration is meant to accomplish. But, that is a physically
meaningless focus position. If you were sitting at the eyepiece of a 150mm
f/9.56 spherical mirror telescope you would never choose that as the best
focus (assuming of course that your eyes are fully corrected).
So, you should refocus the system to determine it's true performance. In
OSLO 6.1 at least you can do that by clicking on the box next to the
"Thickness" column in the image surface row, and selecting "Autofocus -
minimum RMS OPD > On-axis (monochromatic)". Or just enter 0.244.
Now if you click on the "Wavefront analysis" graphic you'll find the RMS
wavefront error is about 0.046 waves, which by any standard that I'm aware
of is truly diffraction limited. On the other hand the geometric spot
diagram spills well outside of the Airy disk, so the "M-L" criterion is no
longer met when evaluated at the best focus.
Notice the following actions are equivalent:
1. Choosing a focus position.
2. Choosing a reference wavefront.
3. Choosing a reference surface.
Texereau gives a procedure for evaluating the maximum transverse aberration
in a mirror under test. That procedure is equivalent to finding the "circle
of least confusion", which is the smallest geometrical spot size. I don't
know, but I guess most software and spreadsheets intended to evaluate
optics according to the "M-L" criterion do the same thing.
The problem is, that implies the choice of a reference wavefront that is
physically meaningless in a system that's close to diffraction limited.
Now here's the conjecture part. I think that 0.046 waves is a hard upper
limit on the RMS wavefront error after refocusing to the minimum RMS
wavefront error for any optical system that satisfies RTA<=1. That is
semi-trivial to demonstrate for a system with only 4th order wavefront
errors + defocus (i.e. 3rd order spherical), which is basically what our
long focus 6" sphere has. It seems intuitively obvious to me that mixing in
higher order aberrations can only make the upper limit of RMS decrease,
because at a given RMS there should always be local slope defects that are
larger than are present with 3rd order spherical. It may be possible to
prove that, probably inductively. If not, there's always the option of
throwing CPU cycles at the problem.
I'm trying to stir up some controversy here. Anyone care to join me?
By the way I have no particular opinion about whether one should use
transverse aberration as a stopping criterion during mirror figuring. On
the one hand it is disconnected from the real physical behavior of light.
On the other hand, it's main practical effect appears to be that it forces
you to be relatively conservative. That's probably a good thing.
Mike Peck
******* OSLO file ********
// OSLO 6.1 46998 0 64702
LEN NEW "Long focus sphere" -1434 3
EBR 75.0
ANG 0.0000572957795
DES "OSLO"
UNI 1.0
// SRF 0
AIR
TH 1.0e+20
AP 9.9999999977e+13
NXT // SRF 1
RFL
RD -2868.0
PY 0.0
NXT // SRF 2
AIR
NXT // SRF 3
AIR
TH 0.3666715653111
WV 0.55 0.48613 0.65627
WW 1.0 1.0 1.0
END 3
SDAD 41.05
SDSA On
*************************************
_________________
Michael Peck
email mpeck1@ix.netcom.com
Wildlife photography page http://home.netcom.com/~mpeck1/index.html
Amateur telescope making http://home.netcom.com/~mpeck1/astro/astro.html
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