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I'm having some trouble figuring out how to get started on this question:

If I have a symmetric resonator with two concave mirrors of radii $R$ separated by a certain distance, after how many round trips through the resonator could I expect the ray to retrace its path? It seems to me that this implies some sort of periodic function should exist that models the behavior of a ray in such circumstances.

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You should consider what constraints you can place on the ray simply based on the information you have about the resonator. You should also remember what you already know about rays in a stable resonator.

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