Coherent radiation in an undulator

This paper is concerned with the synchrotron radiation from an undulating electron beam in a rectangular waveguide. The analysis is based on the dyadic Green's function approach to solve Maxwell's equations in terms of the vector potential. It is shown analytically and numerically that the radiated energy spectrum may differ significantly from the free space results when the undulator length divided by the Lorentz factor of the electron beam is larger than the transverse size of the waveguide. Then, the appearance of the spectrum is changed into a small number of sharp peaks, each corresponding to an excited waveguide mode. The undulator radiation is identified with the wake field in beam instabilities. The concepts of wake function and impedance are introduced to formulate the present problem in the same manner as the beam instability problem, so that the accumulated techniques of the latter can be applied. It is shown that the obtained impedances satisfy the Panofsky-Wenzel theorem and other properties inevitable for wake fields. 4 refs., 2 figs