Three-dimensional numerical simulations of FEL's by the transverse mode spectral method

We present a numerical method for solving a three-dimensional radiation field by decomposing the radiation field into transverse modes that satisfy the free space wave equation and the boundary conditions. The formalism includes betatron oscillations in a realizable wiggler, finite emittance, and energy spread in an amplifier or oscillator configuration. This formalism can be generalized to calculate sidebands resulting from the synchrotron oscillations. The transverse mode spectral method is compared to the finite difference method, spectral method, and transform spectral method. Examples from a computer code using Gaussian-Hermite expansion are given. We show self-focusing of the radiation beam due to gain and refraction of the FEL.