Eigenmode analysis of optical guiding in free-electron lasers

The spatial properties of the optical field and hence the performance of a free-electron laser depend on the fact that the electron beam, which acts as both an amplifying and a refractive medium, is transversely nonuniform. Under certain circumstances, optical guiding may be realized, where the optical field is stably confined near the electron beam and amplified along the beam over many Rayleigh ranges. We show that the three-dimensional evolution of the optical field through the interaction region can be determined by a guided mode expansion before saturation. Optical guiding occurs when the fundamental growing mode becomes dominant. The guided mode expansion is made possible by implementing the biorthogonality of the eigenmodes of the coupled electron-beam optical-wave system. The eigenmodes are found to be of vectorial form with three components; one specifies the guided optical mode and the other two describe the density and the energy modulations of the electron beam.