optics - Trapping a lightray

Given a solid whose interior is a hollow sphere with perfectly reflecting mirrors. A small hole is drilled in the sphere and a photon is sent in at some angle. Will it always eventually exit through the hole it entered?

Is there any arrangement of mirrors one can place inside the sphere such that the photon will never escape?

If not, is there a principle that explains it, or some way to prove it?

Answer

For the first problem, yes. Because you shoot in the photon from the boundary of the sphere, the trajectory of the photon, using elementary geometry, will stay within a fixed plane (the plane is defined by the center of the sphere, the point of entry, and the initial direction of the photon). So you reduce the problem to an essentially two-dimensional problem. It is well known that the billiards trajectory in a circle is integrable, and the trajectory is either periodic or hits a dense set of points on the boundary. In the former case the photon will quickly exit the sphere, even in the limit as the size of the initial hole tends to zero. In the latter case for every non-zero sized hole, you can find some finite time $T$ such that the photon will be guaranteed to exit the hole after $T$. But as you shrink the size of the hole, $T$ increases unboundedly.

For the second problem, it is possible to set up trapping obstacles inside your sphere. For an example, see this question/answer on MathOverflow. However, in general you can only trap trajectories for a small set of initial angles; I am inclined to say that you cannot do so for all angles simultaneously, though I don't have a proof for that.