Mode theory of FEL amplifier with gradient profile of electron beam

Abstract

The paper touches upon the analytical treatment of a FEL amplifier with the gradient profile of an electron beam. An exact solution of the eigenvalue problem for an electron beam with Epstein profile is found. The dispersion relation is derived and the dependence of its roots on the problem parameters is analyzed. The distributions of the mode fields are found, the asymptotic behavior of the exact solutions is investigated. A multilayer approximation method of the eigenvalue problem solution is proposed for the electron beams with an arbitrary gradient profile when one fails to find the exact solutions in terms of the known functions. Testing of this method was carried out with exact solutions for a beam with Epstein profile. As an example of an application, the multilayer approximation method is used for the eigenvalue problem investigation in the case of an electron beam with gaussian profile.