Electron orbits in a free electron laser with a longitudinal magnetic wiggler

In studies of free electron lasers (FEL) systems, in which emission gain is induced through wiggler magnetic field modulation of the electron motion, detailed knowledge of the electron orbits is necessary in order to evaluate the critical features of the amplification scheme. These features include evaluating the frequency range for amplification, gain, rates, saturation mechanisms, and amplifying efficiencies. However, there exist very few magnetic geometries where the electron motion, even in the vacuum fields, can be obtained analytically with sufficient information to assess these critical features. Only recently has it been realized that in the on axis approximation the helical wiggler orbit problem can be reduced to quadrature, and that particular discovery has already led to significant progress in FEL amplification schemes. For FEL systems with strong magnetic guide fields, little or no insightful progress has been made except for this helical case. However, for just this class of magnetic systems, i.e. strong guide fields with a relatively low amplitude magnetic wiggler superimposed, an asymptotic mathematical formalism does exist where the electron equations of motion in the vacuum fields can be greatly simplified, and typically reduced to quadrature in parameter regimes of relevance to applications.