# A telescope with a bunch of small mirrors

Together with colleagues we got this question. Imagine to take small mirrors, the size for example of a dentist mirror, and stick them to a wooden frame with a parabolic shape. Each mirror is flat, and once set of the frame it can be aligned properly with screws so that it redirects its reflection towards the focus of the parabola. All the parabolic surface is covered with these small mirrors.

Now, if you build a Newtonian telescope out of this system, would you get a decent telescope? I expect it to be affected by various aberrations, but would it work ?

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Though I don't have time to research a proper answer right now, I am confident that the arrangement could work. There are a number of large reflecting telescopes that use similar arrangements though in my recollection they have computer controlled servomotor adjusters. It seems to me that the difficulty of correcting the aberrations would outweigh the benefit of the increased collecting surface. –  AdamRedwine Oct 5 '11 at 11:53
In those segmented mirror telescopes, the individual mirror segments are still curved in the same way as the corresponding portion of a single large mirror would be curved. They are most definitely not just flat mirrors! –  Colin K Oct 5 '11 at 12:57
Doesn't a point source of light (like a star) need to be focussed to a point on the image plane? A dentist mirror will not do that, no matter how many of them you have. –  Mike Dunlavey Oct 5 '11 at 16:30
Many, small flat mirrors arranged tangentially on a parabaloid would generate a poorly focused image (that's one of Stefano's "various aberrations"), but they would generate a image. –  dmckee May 15 '13 at 17:16

The telescope you describe would work in some sense, but it depends strongly on how you define what it means for a telescope to work.

Combining many mirrors to behave as a single large mirror is possible, but those individual mirror segments must still be appropriately curved to achieve good optical performance. When building telescope mirrors for modern astronomy, we are concerned with deviations of the mirror shape from the ideal curve by distances on the order of nanometers. If the whole mirror surface were composed of small flat disks, that would be like using a mirror that deviates from the ideal shape by many millimeters, over it's whole surface!

This sort of thing would be fine for collecting light, and in fact similar systems are used in solar power generation, where one only needs to collect light at some central location, rather than produce an image. In a telescope however, you need very high optical quality to do much of anything.

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ok, but if the mirrors are flat, what effect would this deviation have ? less resolution, coma effects, chromatism, or would the telescope would just plain not work (optically, it would not make a focused, observable image) –  Stefano Borini Oct 6 '11 at 12:10
Also, the larger is the parabolic structure (so the larger is the telescope), the less the deviation would be important, so for the limit of a very, very large telescope, a flat one would be a reasonable approximation. Right ? –  Stefano Borini Oct 6 '11 at 12:11
A distorted mirror would produce an image with a poor point spread function. The exact form of the deteriorated PSF depends on the exact shape of the mirror. Coma is just the name for a particular type of mirror shave (relative to the ideal). Any deviation from perfection reduces resolution to some degree, and the deviation you propose is a huge error in mirror shape by the standards of astronomy. –  Colin K Oct 6 '11 at 15:19
@StefanoBorini: Also, you are wrong the deviation becomes less important as the mirror becomes larger. In fact the deviation becomes much more important as the mirror becomes larger (sort of... The critical parameter is actually the focal ratio). –  Colin K Oct 6 '11 at 15:21

Visit http://www.tmt.org/

You'll find an amazing telescope being build ....

Segmented mirrors are the only feasible way of constructing telescopes with apertures significantly in excess of 8 m, as monolithic mirrors would become extremely expensive and ultimately impossible to manufacture, transport, install and maintain. The disadvantage of segmented mirrors is that they often require asymmetric profiles, making them difficult to manufacture. The active optics systems required to support them is also complex, and the gaps between the segments (typically a few mm) can cause diffraction effects and increased infrared background in the final image.

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This is good information, but it must be made clear that each segment of the mirror in these telescopes is curved, not flat. –  Colin K Oct 5 '11 at 16:50
@Colin That was the reason for including the last paragraph. –  belisarius Oct 5 '11 at 16:52
Yeah that does mention it, but the OP seemed confused about the issue so I just wanted to make it clear. The downvote wasn't from me; there is nothing wrong or inappropriate about this answer. –  Colin K Oct 5 '11 at 17:41

(I have some experience with telescope mirror making.)

It would not work as described. It's not even close.

In a reflector telescope, the primary mirror needs to satisfy the Rayleigh criterion, which states that the surface error must not exceed λ/4. For visible light, that means 100 nanometers.

The ideal shape of the primary mirror in regular newtonian scopes is a paraboloid, so no matter what technique you use to make the mirror, the resulting surface needs to follow the ideal parabola with an error no greater than 100nm (due to Rayleigh).

For a mirror of reasonable aperture and focal length, your flat segments would diverge from the parabolic surface by an error far greater than 100nm, more like fractions of millimeter or so, which is about 1000x bigger than the acceptable value.

There's just no way around it. Each part of the mirror needs to track the ideal parabola with no more than 100nm error. Otherwise the mirror is not usable for visible light.

The technique you describe would be usable for microwaves. Since the wavelength in that case is on the order of 1cm or more, λ/4 would be a few millimeters. In that case, a wide shallow parabolic bowl, having multiple flat metal disks glued on the inside might work well enough. If the size of each metal disk is comparable to the wavelength, then it would definitely work well. Just make sure the error in each point is no greater than λ/4.

(Okay, you would then get some resonance effects in each disk itself, which would change the overall result, but let's not go there. Bottom line: it's feasible with microwaves, not feasible with visible light.)

If you don't need to create an image at focus, but just collect the energy of a microwave beam, then the error could be bigger than λ/4; the only ill effect would be that your reflector would become less and less efficient as the error increases.

One more thing: In theory, a mirror with an overall surface error of exactly λ/4 could have a total efficiency of exactly zero, but that's a pretty pathologic (unrealistic) case. In most real life situations, a few λ/4 zones here and there simply reduce the overall energy-gathering power. So keep the error down and it will work well.

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There is a lot wrong here. The $\lambda \over 4$ criterion is a rule of thumb for an optical system to be diffraction limited. Beyond that point it would still "work" but the image quality would be reduced; but reduced doesn't necessarily mean it would be useless. At 1/4-wave, plus just a little, the telescope would be almost as good as diffraction limited! –  Colin K Dec 17 '11 at 3:02
Also, resonance effects in a microwave reflector may affect the spectral reflectivity, but not the image forming abilities. And, what you claim about the "efficiency" of a mirror versus surface quality is simply false. For the same reflecting surface, the reflection efficiency is totally unrelated to the surface shape. I'm not sure what you even mean by " a mirror with an overall surface error of exactly λ/4 could have a total efficiency of exactly zero". This is absolutely false. –  Colin K Dec 17 '11 at 3:07
I don't doubt your theoretical knowledge, but you may want to spend some time with the practical craft of telescope making before quickly applying the "wrong" stamp. At λ/4 a mirror is already pretty sketchy; you may sell it in a cheap mass-produced telescope, but it's less than adequate for any serious work. A small-ish aperture scope which shows the Cassini division in Saturn's rings at λ/12, may fail to do so at λ/4. I would not use the word "good" for any optical surface near λ/4 if it's for astronomy; "barely acceptable" is more appropriate. –  Florin Andrei Dec 18 '11 at 5:41
Also, there are all sorts of different surface errors. A mirror with a single wave of λ/4 error across its entire surface (let's say the conic curve is a bit deeper than a parabola, let's say it's a very slight hyperbola, tangent to the parabola at the edge, but different from it at the apex by λ/4) would be okay-ish for casual work. A mirror full of many tiny streaks and bumps λ/4 deep would appear terrible even to the casual observer - the stars would be soft, planetary images would be very low contrast, etc. –  Florin Andrei Dec 18 '11 at 5:45
"what you claim about the "efficiency" of a mirror versus surface quality is simply false. For the same reflecting surface, the reflection efficiency is totally unrelated to the surface shape" - I was thinking of a microwave collector. Have you ever built such a thing? The field the LNB registers at the focus is dependent of the quality of the dish. Use a poorly formed dish and you get a very weak field; use a high quality dish, that tracks the ideal surface very precisely, and you get a high field amplitude at the focus. The shape does determine the quality of the reflector, as seen at focus. –  Florin Andrei Dec 18 '11 at 5:49

Somehow all answers so far failed to mention that a segmented mirror consisting of flat segments could in principle yield a high optical quality (diffraction-limited) telescope provided the mirror segments are small enough.

Spherical (or hexagonal or any other compact shape) flat mirrors would work perfectly provided:

1) the flat segments are positioned so as to deviate no more than a fraction of the wavelength from a perfect parabolic shape, and

2) the surface area of the segments doesn't exceed (by more than a few times) the product of the focal length of the mirror times the wavelength of the light.

Practically, you will need many flat segments (a large telescope with 30 m focal length would need centimeter-size flat segments) and it will be a nightmare to position all these flat segments more accurately than a wavelength (0.5 micrometer), and to keep them in these positions and orientations despite mirror orientation and temperature changes.

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## protected by Qmechanic♦May 15 '13 at 6:55

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