It is important for amateurs to be able to test their own optics. The most common question when a new telescope arrives is, "How good are the optics?". When the images are bad, knowing how to distinguish between poor optics, bad atmosphere, bad eyepieces, bad collimation, and so forth, is important.
Unfortunately the vast majority of amateurs do not know how to test mirrors, or interpret the results.
In addition, most mirror makers do not realize that the graphical tornado method of plotting zonal measurements is invalid.
Valid ways to rate mirrors include RMS, Strehl, and the star test.
Danjon-Couder condition 2 or Raleigh Criteria: Maximum wavefront error must not exceed a quarter wave and, for the major part of the mirror surface, the defects should be appreciable less. Some experienced planetary observers feel that 1/10 wave gives a slightly perceptibly better image, but since most mirrors are over-rated it's not possible to make this statement unequivocally.
Danjon-Couder condition 1: The radius of the circle of least aberration should be comparable with that of the theoretical diffraction disk and, on the average, the transverse aberrations should not exceed the diffraction disk radius. This condition is misleading - see http://www.halcyon.com/burrjaw/atm/t_verse.lwp/t_verse.htm. The popular Milles-Lacroix's tolerance tornado is based on this condition, thus should be discarded.
Marechal Limit: 1/14 wavefront RMS.
Strehl ratio: 80%.
Star test: Slight differences between intra and extra focal images, no
turned edge.
surface P-V: Measured on the mirror's surface: the peak to valley (P-V) distance. An error here is doubled at the wavefront so a mirror may 'appear' twice as good as it really is. For instance, a surface P-V of 1/8 wave is actually only 1/4 wavefront. As in the wavefront P-V, this means of rating a mirror does not say near as much as we would like.
+- surface P-V: Above and below error on the mirror's surface from a median value. This number will be half the surface P-V and a quarter the wavefront P-V so a mirror may 'appear' to be four times better than it really is. For instance, our example mirror will be penciled in at += 1/16 wave!
RMS wavefront: Root mean square error, or a statistically averaged error: The RMS is computed by finding the wavefront error at a bunch of uniformly spaced points on the mirror, computing the average error, then difference each individual value from the average, square each difference, sum the squares and divide the sum by the number of data points, and then take the square root. (If you're familiar with statistics, it's the same equation as for a sample standard deviation.) Because RMS says something about many points on a mirror, it is a much more meaningful measurement than peak to valley.
Oftentimes, P-V and RMS are related by a 1 to 3.5 ratio. In our example mirror, if the mirror had but a single broad smooth defect, the 1/4 wavefront P-V could be converted by 1/(4*3.5) or 1/14 wavefront RMS. Unfortunately, this is only true if the rare condition that the mirror surface is a smooth and true conic. Therefore RMS cannot be derived simply by taking the peak-valley measurement and multiplying it by 3.5. A mirror with a RMS that just happens to be 3.5x to 4x worse than P-V should be taken skeptically. Chances are the optician measured peak to valley then made a best case guess of the RMS error. See: http://zebu.uoregon.edu/~mbartels/weighted-rms/paraboloid.html for much more on this and other related topics.
RMS surface: RMS at the surface. Hence, in our example mirror of 1/14 wavefront RMS, the mirror would be rated at 1/28 wave surface RMS. Wave inflation is the derisive term given to making a mirror seem better than it really is by emphasizing inflated numbers such as the RMS surface. For instance, in red light this 1/4 wavefront mirror becomes 1/40 wave surface RMS!
adjacent RMS: Larger mirrors may be spec'd where adjacent areas of the mirror must have better or lower RMS values than distant areas of the mirror. For instance, areas 4-6 inches apart may be required to be within 1/20 wavefront RMS whereas areas 8-12 inches apart only need to be within 1/14 RMS. Atmospheric seeing, thanks to the atmosphere's poor mixing qualities, mimics adjacent RMS. Rating a mirror by adjacent RMS is saying that the mirror has but broad smooth gentle errors, and admitting that it will be used at the bottom of the ocean of air.
Strehl ratio: Thanks to the wave nature of light, even if all the light is brought to a single focus point, the light will actually form a small disk surrounded by ever fainter rings. This is the Airy disk and rings. A perfect mirror results in 84% of the light in the Airy disk and 16% of the light in the rings. A less than perfect mirror places less light in the Airy disk and more in the rings. (Incidentally, the Airy disk can shrink for a bad mirror, resulting in better resolution in certain cases.) The Strehl ratio is defined as the intensity of the image spot at its central brightest point divided by the same image intensity without aberration. A Strehl ratio of 100% means a perfect mirror - a mirror that is putting 84% of its light into the Airy disk.. Since all parts of the mirror should contribute to the light that gets into the Airy disk, the Strehl ratio is a measure of total surface quality. The Strehl ratio can be calculated from 1/exp((4*pi* (RMS in nanometers/500)^2). As with RMS, any rating that measures all of the surface quality is a good valid method.
For instance, Univ. of Arizona's Spinning Mirror Lab's Dean Ketelsen says that the 6 meter mirror was pushed out the door with 14 nm RMS. If the secondary is perfect, the 6 meter system will have a Strehl ratio of 1/exp((4*pi*14/500)^2) ~ 90%.
encircled energy: A measurement often used by professional opticians specifies the circle diameter that contains a certain percentage of the light rays. For instance, 90% of the reflected light rays shall pass through a 1 arcsecond circle.
star test: Finally we come to a method that is qualitative only. Popularized by John Dobson and his protégé Bob Kestner (head optician for the COBE Hubble correction optics at Tinsley Labs), this method relies on slightly defocusing a test star, first inside of focus, then outside of focus. By noting how the light from the entire mirror focuses, a rating in the form of: excellent, good, average, poor, or unacceptable, can be given to the mirror.
Here is Bratislav's scale:
1. Can't find anything wrong with it, absolutely perfect: '<expletive>'
Yet to see one after ~25 yrs
2. Defects visible only in extrafocal images, and only after extensive star
testing in best seeing conditions ( << 1/10 wf): 'You lucky b@$#@rd!' Can
count these on fingers of one hand
3. Extrafocal defects readily visible, but really minor ( < 1/10 wf):
'Excellent' Best examples of best commercial telescopes (Zeiss,AP,Tak etc)
Best examples of homemade optics
4. Extrafocal defects fairly obvious, but in focus image still essentially
perfect ( 1/10 - 1/6 wf): 'Very good' Majority of current 'best commercial
telescopes'; best examples of mass produced scopes
5. Large defects visible on extrafocal images, in focus image suffers only
slightly ( 1/6 - 1/4 wf): 'Good' selected examples of mass produced telescopes,
most well made amateur optics; some examples of 'best commercial scopes' can
still be found here
6. In focus image visibly suffers ( ~1/4 wf): 'Acceptable' good mass produced
scope, most good large/fast mirrors I've seen
7. Image deterioration serious, clearly beyond 1/4 wavefront: 'Light bucket'
majority of older generation mass produced scopes, special purpose telescopes
(astrographs)
8. It's difficult to determine when scope is in focus at all ( 1/2 - 1 wf): 'If
you're happy with it ...' unfortunately, not that difficult to find !
9. Usable only at very low magnification ( ~1 wf): 'I don't want to have
anything to do with this one'
10. Absolutely useless: '<expletive>' unlike 1, I've seen these :-)
99% of all scopes I've seen fall into '4-10' bracket.
magnification roll-off: Another qualitative method. A low contrast object such as Jupiter is selected, and a series of magnifications are run through. At some point, the image will roll-off and begin to lose it sharpness. Dividing this magnification by the aperture gives a rating such as, this mirror is good to 50x per inch of aperture.
Here is my scale:
Mirrors that can sustain 35x to 50x per inch of aperture I rate as excellent.
Mirrors that sustain 25x to 35x per inch of aperture I rate as good.
Mirrors that sustain 25x per inch of aperture are acceptable.
Mirrors that sustain 15x to 25x per inch of aperture are poor, usable only at
lower powers.
Mirrors that fail at 12x per inch of aperture are plain just not finished.
For smaller mirrors, push these numbers a little higher, for gargantuan mirrors,
push the numbers a little lower.
Optics limited by laws of diffraction. Good and bad optics are equally limited by the laws of diffraction, so this definition applies equally well to all optics.
The optic's resolution no worse than airy disk or put another way, the output image is limited in its quality only by the aperture of the instrument and the effects of the central obstruction (a 1/3 obstruction is equivalent to a 1/4 wavefront P-V degradation).
The most common definition is 1/4 wavefront P-V. Light rays of optics are ray traced with the goal that the optics bring all the light rays to a point, or more precisely, a circular area called the circle of energy or focus spot. As the light rays are brought ever closer together, improvement is seen, but only up to a point. No matter how tightly the light rays are traced, a dot and not a point remains, thanks to the diffractive nature of light. When the improvement ceases to be significant, the optic is called 'diffraction limited'. This is a somewhat arbitrary judgment. Professional opticians most relate this to 1/4 wavefront P-V.
Schroeder's ASTRONOMICAL OPTICS says that the optics are diffraction limited if the Strehl ratio is greater than 0.80, which matches the 1/4 wavefront Rayleigh tolerance for spherical aberration.
Assuming the somewhat rare case of a smooth conic mirror surface, this equates to the Marechal Limit of 1/14 wavefront RMS.
Yet another definition is considering that the wave nature of light fundamentally limits the resolving power of the optics, then diffraction limited means that the errors in the optics don't materially degrade the resolving power.
Danjon-Couder condition #1 is met. As discussed earlier, this is an invalid test. See http://www.system.missouri.edu/ics/staff/andy/ATM/ARCHIVES/AUG97/msg00241.html.
McGraw-Hill Dictionary Of Scientific and Technical Terms, 5th ed. gives the following definition: "Capable of producing images whose separations are as small as the theoretical limit imposed by diffraction effects."
If a mirror gets 80% of the theoretical amount into the Airy disk,
it's considered diffraction limited
For more see:
http://www.jlc.net/~force5/Astro/ATM/atm.html
http://www.iap.fr/saf/
http://www.halcyon.com/burrjaw/
http://users.uniserve.com/~victorp/knife.htm
http://users.uniserve.ca/~victorp/testing.htm
http://www.jlc.net/~force5/Astro/ATM/Foucault/FoucaultTester.html
http://www.fred.net/bdavy/FoucaultTester.htm
http://victoria.tc.ca/~rasc/foucault.html
http://home.earthlink.net/~ckovacs/atm/foucault.html
http://www.geocities.com/CapeCanaveral/9601/ftester.htm
http://www.atmpage.com/foucault.html
http://web.hal.com/users/elvey/foucault.html
http://www.alltel.net/~microsys/astro.htm
http://lamar.colostate.edu/~field/foucault/
http://home.att.net/~mikel/
http://www.seds.org/billa/tester/images.html
Caustic: The returned light rays from the mirror actually do not come to a focus on axis, instead they form caustic horns to each side. The Caustic test measures not only in the 'Y' axis as the Foucault test does, but also measures in the 'X' axis. This can result in extremely precise results. Unfortunately, the X-Y testing stage is intimidating to build.
poor man Caustic: Here the returning light rays are intercepted to either side of axis, but only the 'Y' axis is measured. I feel this test is more accurate and consistent than the Foucault, plus since no Couder mask is used, the entire surface can be seen at once. For more see: http://zebu.uoregon.edu/~mbartels/caustic/caustic.html
Ronchi: A grating of about 100 lines per inch is placed in front of
the light source and the eye. The mirror will distort the returning lines
into wavy bands. By carefully comparing to computer generated ideal bands,
the state of the mirror may be deduced. I use and recommend this test -
it's incredibly easy to use. For more see: http://zebu.uoregon.edu/~mbartels/ronchi/ronchi.html
http://www.geocities.com/CapeCanaveral/9601/ronchi.htm
star: Every amateur should learn this test. Since it is performed with the diagonal, mirror mount, and the rest of the scope in place, it is the final test. But it can be difficult to discern multiple errors on the mirror.
the past and the future: (CCD) Hartmann: See http://members.aol.com/VirgilAJ/Hartmann.html
The best is a combination of tests: see Jim Burrow's mirror math http://www.halcyon.com/burrjaw/atm/atm_math.lwp/odyframe.htm and his sixtests
The thermal properties of pyrex as it constantly tries to equilibrates to
slight changes in the air temperature probably limits quality to 1/10 wavefront
anyhow.
The ATM list archives are at: http://www.system.missouri.edu/atm/
The ATM list FAQ is at: http://www.netacc.net/~poulsen/atm-faq.htm
book references at end of: http://www.halcyon.com/burrjaw/atm/atm_math.lwp/odyframe.htm