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In optics one of "diffraction-limited" criteria is wavefront tolerance: a textbook example is is optical system with 1/4 wavelength limits producing image of a point source with 68% of the energy contained in the airy disk. Some optics books, however introduce a "roughness" idea, implying that certain mirror surface pattern would imply light scattering, and supporting this idea with naive ray-trace diagram. Does it make any sense, I would think one can made quite strong assertion, namely that any mirror surface shape within 1/8 wavelength tolerance of paraboloid would produce an of a point source with 68% of the energy contained in the airy disk area. Any proofs/refutations?

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No rigorous proof, but this is the experience of people making telescope optics.

Low-amplitude roughness does matter. This is called "dog biscuit" or "orange peel" or "ripple", depending on the particular variety. You can see it in the Foucault tester when checking the mirror during polishing, the shadows look rough, instead of smooth. In theory, a rippled mirror may still be a λ/4-compliant device. In practice, when you turn it towards the sky, the "dog biscuit" produces less sharp images, you never get the crisp tiny Airy disk at the eyepiece.

I am not sure why that happens, but I've a few ideas. Imagine the surface of the mirror, instead of flat or paraboloidal, it's a sawtooth shape. Each "tooth" is less than or equal to λ/4, but each one of them, within that limit, is very sharp, maybe a 45 deg at the top. What happens to the wavefront hitting that surface?

The naive answer is that it takes the same sawtooth shape. However, the size of the sawtooth cell pattern is less than λ/4, so it should not affect the wavefront too much.

Maybe some low-grade dispersion happens, maybe the wavefront gets muddled a little bit. Honestly, I don't know what's the causal explanation, only the practical reality at the eyepiece; an "orange peel" paraboloid never performs as well as a silky smooth one.

EDIT: Opticians are not relying so much on the Foucault test anymore. You could have a mirror that tests down to λ/12 on Foucault, yet it's ripply and rough. A different test has been created, called the Strehl test which is simply a measure of how much the diffraction figure deviates from ideal. The reason why some light misses the Airy is irrelevant to this test, all that matters is how big the difference is.

An orange peel mirror that tests at λ/12 on Foucault may indeed produce a weaker number on the Strehl tester than a λ/8 mirror that is free of microripple. A weaker Strehl number tends to correlate better with poorer image at the eyepiece than a weaker Foucault number. Yet even Strehl is not perfect. The ultimate tester is the human eye in real observing conditions.

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For example the HST mirror was wonderfully smooth - it was just perfectly smooth to the wrong shape –  Martin Beckett Jul 16 '11 at 2:15

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