Abstract
The free-electron laser can be described by solving the Lorentz-Maxwell equations self-consistently in weak optical fields. The field evolution is determined by an integral equation that allows the inclusion of an arbitrary electron distribution function in a simple way. Contour maps are used to show the gain degradation due to an electron-beam energy spread and an electron-beam angular spread. In the limit of low gain, the gain spectrum is related to the spontaneous emission line shape through successively higher derivatives. In the limit of high gain, it is shown that the growth rate becomes less susceptible to degradation from the electron-beam quality.
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