Abstract
The paper touches upon the questions of the linear theory of an FEL amplifier with an axisymmetric electron beam. An FEL model is discussed wherein diffraction effects, space charge fields and energy spread of electrons in the beam are taken into account. The rigorous solutions of the eigenvalue problem have been found for the stepped and bounded parabolic electron beam profiles. The multilayer approximation method has been used to solve the eigenvalue problem for the beams with an arbitrary gradient profile of current density. For the stepped electron beam profile the initial problem is solved analytically with the Laplace transform technique and in the case of the arbitrary gradient profile with the numerical solution of the self-consistent field equations. The optimum conditions of the external radiation field focusing on the electron beam have been obtained.
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