Abstract
In this paper the author derives a Green's function for an FEL in a waveguide. The radiation field is decomposed into transverse waveguide modes and an integral is derived for the evolution of each mode. The treatment is exact; there is no slowly varying envelope approximation and no averaging over an undulator period. Each transverse mode consists of a spectrum of longitudinal frequencies that obey the waveguide dispersion relation ω2c2 = γ12 + k2, where γ1 depends on the transverse mode. The waveguide confines the radiation to propagation in a single dimension. This reduces a 3-D electromagnetic problem to a 1-D problem. The integral formulation of a waveguide FEL will be applied to studying frequency sidebands in FELs. It should also be particularly useful in studying slippage effects in FELs with short electron bunches.
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