  We study the multi-periodic oscillations in the spontaneous emission rate of
an atom in a medium with refractive index n sandwiched between two parallel
mirrors. The oscillations are not obvious in the analytical formula for the
rate derived based on Fermi's golden rule but can be extracted using Fourier
transforms by varying the system scale while holding the configuration. The
oscillations are interpreted as interferences and correspond to various
closed-orbits of the emitted photon going away from and returning to the atom.
This system provides a rare example that the oscillations can be explicitly
derived by following the emitted wave until it returns to the emitting atom. We
demonstrate the summation over a large number of closed-orbits converges to the
rate formula of golden rule.
