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Exact solutions are given for the Maxwell equations driven by a gyratingultrarelativistic electron inside a reflecting cylindrical boundary. The axisof the electrons's trajectory and the axis of the surrounding cylinder aresupposed to coincide. It is found that, at certain values of the ratio of thecylinder's and of the trajectory's radii, the field amplitudes diverge asfunctions of time. The physical reason for this resonance is the constructiveinterference between the radiation emitted earlier and fed back by reflectionto the actual position of the radiating electron.

Synchrotron radiation in the presence of a cylindrical mirror. Note on  the interference of the direct radiation and the high-order  whispering-gallery modes