Wave equation for the optical guiding in a free-electron laser with full account of electron wiggling

The driving current of a free-electron laser is calculated by solving a linearized Boltzmann equation for a relativistic electron beam. Particular attention is paid to terms, generally neglected, which represent the effect of the electron-beam bending due to the finite radius of the electron wiggling. In solving the three-dimensional Helmholtz equation for a laser driven by a source varying periodically with the period of the wiggler, the laser field is represented as a superposition of space harmonics whose longitudinal propagation constants differ by multiples of the wiggler period (Bloch waves). In addition, the transverse distribution of the field is analyzed by expanding it as a series of Gauss–Laguerre modes.