Synchrotron radiation in the presence of a cylindrical mirror. Note on   the interference of the direct radiation and the high-order   whispering-gallery modes
  Exact solutions are given for the Maxwell equations driven by a gyrating ultrarelativistic electron inside a reflecting cylindrical boundary. The axis of the electrons's trajectory and the axis of the surrounding cylinder are supposed to coincide. It is found that, at certain values of the ratio of the cylinder's and of the trajectory's radii, the field amplitudes diverge as functions of time. The physical reason for this resonance is the constructive interference between the radiation emitted earlier and fed back by reflection to the actual position of the radiating electron. 
Sandor Varro
physics.acc-ph math-ph math.MP