Nonlinear theory of short-wavelength free-electron lasers

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Abstract
The nonlinear evolution of free-electron laser (FEL) amplifiers is studied for infrared and shorter wavelengths. The configuration of interest consists in the propagation of an energetic electron beam through a drift tube in the presence of a periodic wiggler magnetic field with planar symmetry. A three-dimensional formulation is derived in which the electromagnetic field is represented as an expansion of Gaussian optical modes. Since the wiggler model is characterized by planar symmetry, the Gauss-Hermite modes are used for this purpose. A set of nonlinear differential equations is derived for the evolution of the amplitude and phase of each mode, and they are solved simultaneously in conjunction with the three-dimensional Lorentz force equations for an ensemble of electrons in the presence of the magneto-static wiggler, self-electric and self-magnetic fields due to the charge and current distributions of the beam, and the electromagnetic fields. It is important to note that no wiggler average is used in the integration of the electron trajectories. This permits the self-consistent modeling of effects associated with (1) the injection of the beam into the wiggler, (2) emittance growth due to inhomogeneities in the wiggler and radiation fields as well as due to the self-fields, (3) the effect of wiggler imperfections, and (4) betatron oscillations. The optical guiding of the radiation field is implicitly included in the formulation. This approach has important practical advantages in analyzing FELs, since it is necessary only to characterize the beam upon injection into the wiggler, and the subsequent evolution is treated self-consistently.