Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets

Wave optics can limit the ways in which optical components can change light-ray fields. Optical components called METATOYs trade in the continuity of the phase fronts and the precision to which they change light-ray fields in return for additional possibilities when changing light-ray fields. Now only geometry limits the possible mappings between the positions of an object and its geometric image. Here, I study such limitations for the case of an infinite, planar, non-absorbing sheet that images the entire three-dimensional space. The most general case of such a sheet is equivalent to a thin lens with different object- and image-sided focal lengths. Special cases include ordinary thin lenses, confocal lenslet arrays, and negative refration with n2=-n1.