And since Superman's eyes aren't on giant stalks projecting out of his face, I think we can rule that out as a possible mechanism, too. The Chandra satellite is about 30 feet long, because that's how much space it needs to focus the x-rays with grazing-incidence reflections. It does have one major drawback, though, namely that it takes a lot of space.
This grazing-incidence technique works well- the images produced by Chandra are spectacular. The reflection of light at any wavelength is more efficient at small angles ("grazing incidence")- it's part of why if you're standing in a clear lake you can look down and see your feet, but the surface some distance out just reflects what's above it.
These are designed to accept x-rays coming along the line of sight of the telescope and bend them to a focus through two reflections at a very slight angle to the mirror surface. X-rays don't bounce nicely off a parabolic mirror the way visible light does in the Hubble telescope, but instead through a set of concentric metal shells coated with iridium. This does, in fact, focus incoming light to make an image, but not in the same way as an ordinary visible-light telescope. This, is, for example, how the double-helix structure of DNA was discovered: from a famous x-ray diffraction image made by Rosalind Franklin and her student Raymond Gosling.) The pattern of diffracted beams lets you deduce the crystal structure and spacing for whatever target you're looking at. (This x-ray diffraction is one of the main sources of information we have about how different structures are put together. You don't get a single reflected beam, but a collection of beams at different angles. The most perfectly smooth surface you can create will look to an x-ray coming in at normal incidence like a big lattice of atoms, and thus you need to worry not just about ordinary reflection, but diffraction off different planes of atoms. And even reflection is complicated by the fact that typical x-ray wavelengths are comparable to the spacing between atoms. There aren't any materials that nicely refract x-rays in the way that glass refracts light, so ordinary lenses are right out. With x-rays, however, neither of those things works very well. Cameras and human eyes use transparent lenses to focus light into a clear image, while the telescopes used by astronomers use large curved mirrors. Visible-light optics is mostly based around the refraction of light by transparent materials and reflection at angles not too far from head-on ("normal incidence" in physics terms).
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