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Consider two spherical mirrors which reflect light from a source which is placed perpendicular to the line connecting their centres. Can there be infinite reflections between the mirrors, i.e., a ray bouncing infinitely many times between the spheres? How do we find such a ray?

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Not in ray theory: the source would have to be on the line connecting the two spheres. However, for a finite beam there will be some light diffracted normal to the sphere along this line. But since the reflecting surfaces are convex the energy propagating along this line will be spread exponentially with the number of bounces.

Tangentially, in order to get anything really interesting with rays you need a minimum of 3 scatterers (see Pierre Gaspard's book on chaotic scattering). Then you can get quasi-bound states where the dwell time in the scatterers can be chaotic. In fact if you plot the incident angle versus the exit angle you can see fractal behavior. Not what you asked, but pretty cool.

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