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?
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.
A hollow sphere with perfectly reflecting mirrors is a particular case of a cavity resonator
The others answers presume that the light ray (or a single photon if you prefer) travel along a plane that contains the centre of the sphere and the question set no constraint on this. Also the dimension of the sphere (radius) must be adequate to the wavelenght of the photon to trap it (total reflection).
There are certain configurations of angles where the ray will return to the same initial point, and I suspect that in any situation it will always return to the origin. Probably it is already proved.
Is there any arrangement of mirrors one can place inside the sphere such that the photon will never escape?
Yes - The number of mirrors, size, the shape and orientation are important and some constraints on these must be previously known. The solution is trivial if we can change the mirrors configuration after the ray is inside or with a proper rotation of the sphere. I am not visualizing a solution if we can not change the configuration.
A cavity resonator is a hollow conductor blocked at both ends and along which an electromagnetic wave can be supported. It can be viewed as a waveguide short-circuited at both ends (see Microwave cavity).
physicists-trap-light-in-a-bottle and bottle microresonators
Even a perfect reflection can not avoid the Evanescent_wave
so, there are no perfect mirrors. Some of the energy will evade.
In optics and acoustics, evanescent waves are formed when waves traveling in a medium undergo total internal reflection at its boundary because they strike it at an angle greater than the so-called critical angle. The physical explanation for the existence of the evanescent wave is that the electric and magnetic fields (or pressure gradients, in the case of acoustical waves) cannot be discontinuous at a boundary, as would be the case if there were no evanescent wave field. In quantum mechanics, the physical explanation is exactly analogous—the Schrödinger wave-function representing particle motion normal to the boundary cannot be discontinuous at the boundary.
In optics, evanescent-wave coupling is a process by which electromagnetic waves are transmitted from one medium to another by means of the evanescent, exponentially decaying electromagnetic field.
dielectric microsphere resonators
The whispering gallery modes (WGMs) of quartz microspheres are investigated for the purpose of strong coupling between single photons and atoms in cavity quantum electrodynamics (cavity QED). ...
In optics the treatment is done using only the wave properties of the light, and as a particle when it is absorbed by some atom/electron/...
EDIT add: The first sentence of this answer is not very interesting. In one situation the study uses 'geometric optics' because the wavelength is << sphere Radius and we must use EM pure treatment in a waveguide (the wavelength is in the order of transversal dimension of the waveguide).
If you're asking a mathematical question: does a line starting at a point on the inside of a sphere, and reflecting off the sides of a sphere, ever come back to its starting point, then the answer is yes if and only if the angle the line makes with the tangent to the sphere at the starting point is 90/n degrees, for integral n > 0.
If you're asking a physical question: does a photon going through a finite-size hole and bouncing around inside ever come out, the answer is yes -- even if you close the hole up after you shoot the photon inside.
A line entering a sphere will bounce around in the plane containing the original ray and a ray from the entering point through the center of the sphere. It will bounce around until it hits the entrance, then exit.
Your first example with theoretically perfect surfaces would work of course if you could perfectly close the entrance hole after you inject the photon, but I suspect this is not what you were looking for. Edit: noticed the other answers with the trapping arrangements so I won't try to answer that part here :)
Yes, it will always exit.
If you want to trap it - you will need to to be able to move mirror a little while photon is somewhere in the middle of your trap.
