Formulation of normalized nonlinear free electron laser equations

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Abstract
The general nonlinear self-consistent equations describing the FEL in the steady-state amplifying configuration is derived with the space-charge effects and all the efficiency enhancement schemes: (i) a spatially varying wiggler amplitude and/or period and (ii) a dc electric field Edc (z) = [−∂φdc /∂z]êz. All the efficiency enhancement schemes are shown to be somewhat equivalent. For a variable period magnetic wiggler, the FEL equations can be appropriately normalized to a three-parameter set of coupled equations. All operating regimes are covered by these normalized equations. The condition for space charge to be unimportant is given. In the absence of the space-charge effects, the three-parameter set of equations reduces to a two-parameter set. The inclusion of additional efficiency enhancement schemes increases the number of parameters to describe the FEL. Curves showing the efficiency, radiation amplitude, and growth rate will be presented.