How can we mirror (put shiny reflecting coating on) the surface of a very large parabolic antenna such as that of the world's largest telescope (305 m diameter) in Puerto Rico, USA.
Metals can be deposited on the surface of the mirror after making it very smooth by polishing.Silver or platinum may be used . The deposition can be undertaken in very large evaporators. May be a more easier method is to use electroless and electrolytic deposition in large electrochemical bath.
Evaporative deposition is a fairly standard process for this, as I understand it. About the only way you're going to get a mirror that size, however, is to do it in chunks. This is not just because of the nature of the process, which requires a vacuum chamber and a comparable electrical system to the size of the desired mirror, but because most any material one would go about making a mirror that large from would have considerable stresses involved. (And that's not to mention difficulty in placing, aiming, etc.) For some reference as to what I mean by "chunks", this is the direction the James Webb Space Telescope took in its construction.
The more fundamental question is, "Why would you want to?" Why would you need a mirror (optical) finish on a parabolic reflector for an antenna? What frequency are you operating at? The performance of the reflector is a function of its parabolic accuracy relative to the wavelength. Telescope reflectors need to be accurate to angstroms since they're used for visible light, but a large microwave dish reflector only needs accuracy of millimeters.
Note that this is different than talking about the reflectivity of the surface, which is generally a function of the conductivity and dielectric properties, again as a function of wavelength. Thin film metalization of a glass substrate is just fine for optical mirrors but isn't very good for a low frequency radio reflector since the metal has virtually no skin depth. At least a portion of the energy is likely to penetrate the reflector if the metal layer is thin relative to the wavelength. Thus thin film metal deposition on a dielectric substrate may only marginally improve the reflectivity of the surface vs. using a solid metal reflector.
About the the only value I can see in an optically polished reflector for a non-optical region (e.g. below the TerraHertz region) would be that it would give a visual indication of defects in the parabolic surface where any warping would be optically visible. However that's an excessive amount of added cost for something that can be evaluated electronically much more easily.