This is beyond unobtainium. A one-way mirror violates the Second Law of Thermodynamics. The mirrors that they use in interrogation rooms and the like actually transmit light equally well in either direction; the only reason you can’t easily see into the observation room is that the observation room is darker.
If we’re treating this as a problem in idealized ray optics, so we can ignore all of the real-world problems inherent with light that CalMeacham mentions, then there are a few ways to do it. CJJ*'s irrational method is probably the simplest, but Robot Arm’s light funnel can also be made to work (and it doesn’t even need to be a spiral, so the mirrors don’t have to be infinitely thin).
I suspect that there is an answer if you add the element of gravity (and the previous posts which allow for impressively long paths) then you could solve it. Seeing as light is, to some extent, affected by gravity you would actually need to figure for this if you were able to achieve a loop or else the beam would slowly move towards the nearest large body of matter. (And of course you would need to stop all motion in the universe so that they didn’t, similarly, throw it off eventually and so that it could be calculated for…)
Obviously, this would be a real world solution rather than a mathematical one.
This is the way I’ve thought about this, and it seems theoretically possible. You have a giant room (a box) shape liked a googlehedron (a shit load of sides). A pulse of laser light is sent into a tiny opening in the this room. It takes a month for the light to reach the hole from whence it came. Within that time the hole is sealed with another mirror, so when the light pulse does come to exit, it just bounces back into another mirror. Seems logical, no?
It had better. I’m building one on my way to world domination.
Don’t light rays right at the event horizon of a black hole move in an infinite “orbit” of sorts? That is, rays that didn’t start with velocity component radial to the singularity end up being in equilibrium right there. The event horizon is the boundary where rays just beyond it have a chance to escape, and those inside it are doomed, but right at the boundary there is an infinitesimal region of balance. Obviously, we’re dealing with exactitudes, and only theoretically could you expect a light ray to stay in such a path for a long time.
So I think if you had a black hole in the middle of the box, and your laser beam was aimed to hit perfectly tangential to the event horizon, you could trap the beam in a loop along the “surface” of a black hole.
This, of course, doesn’t really fit in the spirit of the OP. But it is one way to “trap” a laser beam coming in from a point outside the loop.
Thank you, ultrafilter and Amazon “search inside”! Search for “idealized pulse laser” and you’ll find he says that in the idealized situation (perfectly square box, perfectly reflective mirror, etc.), the beam returns to the hole only if the angle of the pulse is rational. The proof, however, is left as an exercise for the reader.
Aw, man, I thought that guy was awesome – he wrote The Planiverse, a more “realistic” (so to speak) version of Flatland. My friends and I thought that book was really cool back in junior high and high school.
The key, of course, is to tell your investors you’ve trapped the beam, and if they try to check by opening the box and looking, they’ll just let the beam escape.
Does the rotation of the Earth or the orbit of the Earth around the sun affect the path of the light beam? Or do those pale in comparison to gravitational effects?
Probably the OP meant “infinite path” and not infinite LOOP. Clearly any loop would by definition come back out the entrance hole. But depending on the shape, could there be an infinite path that never retouches it’s origin point? I wonder if the answer is the same in both 2D and 3D? I wonder also if as has been hinted here, that some “sub-shape” might exist with a small hole such that entrance is possible but never exit given a certain range of initial angles…
If we truly want to go with a mathematical solution where the “laser beam” is just a line, then we can use (perfectly) curved mirrors. And further, using infinitessimals, we can create a situation where the laser would bounce between two mirrors which decrease the angle of reflection (i.e. move it towards 180 degrees) infinitely.
