Abstract
It is shown how the usual first-order equations describing the one-dimensional free-electron laser dynamics can be obtained without relying on any a priori separation between fast and slow time and space variations in the radiation field (the well-known slowly varying envelope approximation). The exact first-order equations can be reduced to the approximate form and sufficient conditions are given for making this reduction. It is also shown that the start-up of the high-frequency signal is correctly described by the slowly varying approximation only when the electron bunch at t=0 has longitudinal density and/or parallel velocity fluctuations whose spectrum has significant values in the region comprising the wave number of the high-frequency wave.
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