The electron trajectories in the external magnetostatic fields in free-electron lasers are fundamental to any understanding of the operational principles and have been the object of study for a considerable time. The concept relies upon a spatially periodic magnetic field, called either a wiggler or undulator, to induce an oscillatory motion in the electron beam, and the emission of radiation is derived from the corresponding acceleration. The specific character of the wiggler field can take on a variety of forms exhibiting both helical and planar symmetries. The most common wiggler configurations that have been employed to date include helically symmetric fields generated by bifilar current windings and linearly symmetric fields generated by alternating stacks of permanent magnets. However, wiggler fields generated by rotating quadrupole fields (helical symmetry) and pinched solenoidal fields (cylindrical symmetry) have also been considered. This chapter deals with the single-particle trajectories in both helical and planar wigglers in both one and three dimensions. Detailed expressions for the trajectories are given, and concepts such as betatron oscillations are discussed.
Wiggler Undulator Helical wiggler Planar wiggler Flat-pole-face wiggler Parabolic-pole-face wiggler Lorentz force equations Steady-state trajectories Stability Betatron period Betatron oscillation Group I orbits Group II orbits Negative-mass trajectories Tapered wiggler
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