Analysis of the eigenvalue equation of the FEL amplifier with axisymmetric electron beam and diaphragm focusing line

The paper presents analysis of the eigenvalue problem of the FEL amplifier with axisymmetric electron beam and diaphragm focusing line. This novel type of an FEL amplifier has perspective to be used for applications where high average and peak radiation power is required [E.L. Saldin et al., Nucl. Instr. and meth. A 361 (1995) 317; these Proceedings, Nucl. Instr. and Meth. 375 (1996) 389; Nucl. Instr. and Meth. A 361 (1995) 101]. An FEL model is discussed wherein diffraction effects, space charge fields and energy spread of electrons in the beam are taken into account. To take into account diffraction effects at the diaphragms we apply the rigorous impedance boundary conditions proposed by Veinstein. The rigorous solutions of the eigenvalue problem have been found for the stepped electron beam profile. Analytical expressions for eigenfunctions of active open waveguide and formulae of their expansion in the eigenfunctions of passive open waveguide, are derived, too. Asymptotic behaviour of the obtained solutions is studied in details.