The axial-velocity distribution function and the longitudinal susceptibility of an e-beam in a planar wiggler FEL

This paper introduces a development of a normalized axial velocity distribution function and a susceptibility term for an electron beam in a planar wiggler FEL. The model includes the independent contributions of the energy spread and the angular spread, the emittance and the betatron motion, to the axial velocity distribution function. In the case of an emittance dominated spread, the resulting distribution function is a skewed, asymmetrical function. The e-beam susceptibility is described in this case by the first derivative of the complex error function (the plasma dispersion function), convolved with a decaying exponent. The "exact" distribution function and the susceptibility integral that are represented in this paper may be used in any linear, kinetic FEL model to improve its accuracy in cases where the angular spread, the emittance or the betatron motion, are dominant spread sources.