Is there a way to direct light toward the same location regardless of the angle of incidence? E.g., picture a circular hole in the center of a roof of an otherwise opaque cubical room on a sunny day. As the sun moves across the sky the spot will move across the floor and walls of the room, appearing in the center of the floor only when the sun is directly overhead. With a mirror, one might direct the light to appear in the center of the floor from any single incident angle but not from every incident angle. Is there a way, perhaps with a series of prisms or maybe some kind of waveguide or something, to direct the light to appear in the center of the floor regardless of where the sun is in the sky? I.e., for light to exit the apparatus always in the same direction regardless of the angle from which it entered?
 A: It is impossible to do this perfectly with a static optical system. Think about the backwards-going rays: If they all originate from the same spot and are generally indistinguishable, they will always emit in the same distribution of directions. There is nothing time-dependent about the optical system, so a static input will result in a static output. The ideal solution for your general problem would use mirrors which track the location of the sun to focus the maximum power on the target at all times.
That said, if you constrain the possible location of the sun (which we can do from prior knowledge), and you consider it a success to hit a large target, then using a suitable parabolic mirror or some such would work much of the time. Alternatively, if you consider it a success that some light always hits the target, then you can make the room a big integrating sphere, resulting in a uniform illumination of the interior.
A: No, because this is forbidden by time reversal symmetry.
In other words, imagine the light travelling in the opposite direction - you cannot know where it is supposed to go without building in some sort of mechanical "time of day" mechanism.
