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The transition between metal dish and mirror happens around hundreds of microns. For example, the shortest wavelength the ALMA(metal dish) can detect is about 0.3mm, while the longest wavelength the Spitzer(optical mirror) can detect is about 160 micrometer.
My questions is what determines the surface material of the ALMA and Spitzer telescope? Could a metal dish observe shorter wavelength than a mirror telescope?
They're all mirrors, they just operate in different wavelength ranges. A wire fence could be a mirror if the wavelength is big enough.
The surface is an easy problem to solve. Pick any material and do vacuum deposition with it on the substrate. It's the substrate, the base material that matters a lot - the thick part that keeps the shape.
A metal mirror is easier to make. Standard engineering principles apply. That's why this technology is used for ALMA.
But there's a catch - precision. This is dictated by wavelength. Imagine all the radiation captured by the mirror, focused in one point. If the mirror is perfect, it's all good. If the mirror has deviations from perfection, parts of the wavefront arrive too soon or too late. The wavefront begins to interfere with itself, reducing the amplitude.
Interference is completely destructive (the signal is completely lost) when the difference between the paths is exactly half the wavelength: λ/2. Then the peak wave (maximum) from one side falls into the trough wave (minimum) from the other side, and they cancel each other.
How do you get a λ/2 difference between two parts of the wavefront? If the surface error on the mirror is λ/4. One part of the wavefront begins to be reflected immediately, while the other keeps traveling λ/4 to get to the mirror, and then wastes λ/4 again on its way back up.
Mirrors with λ/4 errors are bad - since the error is never exactly half-and-half, parts of the wavefront are not cancelled so the mirror keeps working somewhat, but poorly. Around λ/8 the mirrors become decent. λ/16 are very good, and beyond λ/20 are essentially indistinguishable from perfection. This is for optical mirrors (visible spectrum).
This is why optical telescope mirrors are made of glass. It takes such a smooth, rigid material to reach the precision necessary for optical observations.
The Spitzer mirror is actually made of beryllium, cooled to 5.5 K. This is fine. Precision requirements are a bit more relaxed, compared to visible light. In fact, solid metal mirrors have been made for visible light telescopes even by amateurs - but in these conditions glass is preferred.
The ALMA would have to be manufactured to a precision better than 75 micron, in reality much better. That's doable with regular metal parts.
Other materials used for visible light telescopes:
Granite. It takes an excellent polish, just like glass, it's very rigid, and the coefficient of expansion is similar to glass. As long as it doesn't have voids, it's a decent material.
Marble - same as above.
Speculum metal - back in the days of Herschel they were casting mirrors out of this bronze-like alloy. Easy to work with but it tarnishes very quickly.
Special ceramics such as Zerodur.
Quartz - like glass, but better (low thermal expansion). Takes more work to grind and polish because it's harder than glass.
TLDR: It's mostly about precision and ease of manufacturing.
Once the mirror is made, it could be coated with anything you like, so it could reflect any wavelength up to hard UV. Like I said, the surface, the reflective layer, is the easy part.
Simply make a list of all materials that provide good reflectivity in the intended range of wavelengths. Most of these materials turn out to be metals, unsurprisingly. Now compare them based on such criteria as: durability, price, ease of coating. Pick one and drop it in the tungsten "boat" in the vacuum chamber. Put the mirror in the vacuum chamber and run some amperage through the tungsten boat until the metal layer on the mirror is thick enough so it's not transparent anymore - but not thicker than that (for precision). Done.
An alternative that is sometimes used in infrared and visible light is dielectric coating. It's a stack of fairly transparent layers, and there are multiple reflections from all the surfaces between layers. The thickness is such that the path difference between reflections is exactly one wavelength, 1 λ. That strengthens all reflections (constructive interference).
Dielectric mirrors have extremely high reflection coefficients, and are used for lasers, etc. Also tend to have preferred wavelengths where they work well (though broadband dielectric mirrors also exist). They're quite expensive and are not commonly used for telescopes.
Finally, after all that, telescope mirrors sometimes get a protective coating to shield the reflective metal layer from corrosion. A commonly used material is silicon oxide (SiOx, not SiO2 which is quartz and it's different). If the mirror operates in a vacuum or it can be recoated often then this is less important.
The vast majority of amateur telescopes nowadays use:
Some mirrors have "enhanced coating" which is conceptually similar to dielectric mirrors, in addition to the metal layer.
+1 for a thorough answer!
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