ATM A long post about diffraction limited optics

["Nils Olof Carlin" <nilsolof.carlin@swipnet.se>]

Recently there has been discussion on how best to fit a reference parabola
to the data obtained from a Foucault test (Scott Rychnovsky & others). Now
that I have got me a blank, it made me think again about what one really
wants to to achieve when testing (or for that matter, figuring) a mirror (I
stick to the common or garden paraboloid here). Here are some of my
thoughts, hoping for a long, hot thread.

I started by asking what kind of performance do we want from the mirror?
The buzzword is diffraction limited, of course, but what could it mean?
 
Experience as well as wave theory tells us that any reasonably good mirror
will, when focused and seen at high magnification (bad seeing is not
considered), give a star image in the form of a diffraction pattern of a
central spot, the "Airy disk" surrounded by a system of rings. A telescope
with a good mirror, well focused, gives a brighter spot and fainter rings
-a poorly focused and/or figured mirror (same size) gives a fainter spot
and brighter rings. Direct the mirrors to a planetary surface, and the
better one gives a "crisper" image of better detail contrast, even if the
minimum size visible of high contrast details (like moon crater shadows)
doesn't differ much (unless the poor mirror is terribly poor.) 

It appears that the disk is what contributes to the image we want - the
rings do not go very far out but they form "fuzz" around the disk. A real
image is built up of point images (disk and rings) from each point of the
object. I like to think of the real image as a superposition of a bright,
sharp image from the disks, and a fuzzy image that doesn't stray far but
tends to wash out the contrast of details, particularly on planet images.
The intensity that is not in the rings will be in the disk and vice versa
-the width of the disk is mainly decided by the aperture, so what we want
is as much "height" - brightness - as we can reasonably manage.

Theory tells us that at most some 84 percent goes into the disk and the
rest into the rings (no obstruction, no apodization). Just about any
reasonable optical defect of the mirror "shifts light from the disk to the
rings", and lowers the intensity. So if the disk is 80% of what it could
be, about 67% of all light is in the disk (Not exactly - the shape may not
be identical, but not very different either) and 33% in the rings , twice
the 16% of the perfect aperture. Could we call this diffraction limited?

Well, the ratio actual/ideal peak intensity in the disk is known as the
Strehl ratio - it is nothing new but I had not until now quite grasped the
significance. And a Strehl ratio of 0.8 is accepted as the lower limit for
"diffraction limited" performance. But is it useful, in the sense that you
could calculate it from the mirror profile, as found by Foucault testing?

Let me discuss a *very* simple model of how the image is built from light,
according to the wave optics (founded by Huygens in the 17th century and
refined by 19th century physicists): The light from a star will reach the
earth´s atmosphere as light waves of plane wavefronts. A perfect telescope
will then shape that wavefront into a spherical surface centered on the
focal point. Any optical defect will deform the sphere, and different parts
of it will be more or less out of phase when reaching the point of focus.
The intensity in each point will be the square of the vector sum of
contributions. With the Fraunhofer approximation, we can "flatten" the
sphere, and see how much light goes in different directions. Seen straight
on, all waves will add in phase, but at a little angle, the light from the
far and the near parts of the aperture are more or less out of phase and
will cancel. We have the first dark ring here. A little farther out, the
sum is again non-zero - we have a bright ring, etc. This way we can
reconstruct the image - this is more or less how Suiter's images in "Star
testing astronomical telescopes" were made. And if we know the abberration
of the wavefront from mirror surface defects, we can take them in account
when we sum up the vectors (wavefront abberrations are twice the surface
ones, 1/4 wave is equal to an angle of pi/2 or 90 degrees).

Nice, but a fat lot of computing, what for? Well, to get the Strehl ratio
we only need to calculate *one* point, the center, remember? And because of
symmetry (we assume it here), only cosines need be added. And if we have a
Foucault test measuring the slopes in a few zones, we need not consider
many points either. (Remember the Nyquist theorem? With 4 sample points or
so, only very broad abberrations can be modeled, not narrow zones or
turned-down edge, or ripple). 

What about transverse abberrations? The first Danjon-Couder condition
requires that "... the wave front has a mild slope and does not divert
light rays outside the diffraction disk". This is a terribly misleading mix
of metaphors - wave and ray optics. Of course, if we have no steep slopes
the hills and valleys can't be very pronounced, and since the diffraction
disk is smaller, there is a convenient scale factor built in. So by all
means keep slopes mild. But a slope *does not* "divert light rays outside
or inside the diffraction disk" - the concept of light rays is not valid
unless the diffraction disk can be regarded as negligibly small.

So what *does * divert light rays outside the Airy disk? Wavefront, that is
phase, errors do. The light from a small zone 1/4 wave out of phase with
the rest will contribute nothing to the center and Strehl ratio (actually,
it will fill the original dark rings with some light), and light 1/2 wave
out of phase will actually detract from the center!

The second D-C condition is a watered-down version of what lord Rayleigh
said: "Abberration begins to be decidedly prejudicial when the wave-surface
deviates from its proper place by about a quarter of a wave-length."
(quoted from Philosophical Magazine 1879-80 by RWS of S&T Oct 1983 p 356).
This is a far cry from what we usually think he said - that if all parts of
the wave-front are within 1/4 wavelength, the optics are
diffraction-limited, that is nearly perfect.

Another measure of optical quality is the (area-weighted) RMS error - the
square of the deviations are weighted according to their part of the area,
and averaged - and the square root of the average is the RMS. 

So what I thought was that you would want to maximize the cosine terms -
not minimize the RMS. As Jim B and Bratislav pointed out to me, since the
cosine for small angles is approx=1-x^2, the Strehl ratio is 
1-(2*pi*RMS)^2 ! (and for what small angles should mean, we can use the
Rayleigh criterion). So for a Strehl ratio of 0.8, the RMS=1/14 wavelength,
also close to what is considered the upper limit for "diffraction limited
performance". And, (as Mike Peck has shown elegantly, see his piece on
Mel's pages if you are not allergic to maths) if the error curve is simple
and "smooth", the 1/4 wave criterion actually is closely equivalent to both
the RMS and Strehl criteria.

Conclusion - the Strehl ratio is an intuitive and useful number as a simple
quality indicator - and equivalent to (or for large deviations somewhat
better than) the RMS, and with some restrictions, also the 1/4 wave
criterion. Moreover, it should be computable when the Foucault data are to
be reduced (Jim B's program computes the RMS already, but would Larry P
consider it for TEX?).

While my tile tool base is curing,

Nils Olof