Quasi-Linear Analysis of Smith-Purcell Free-Electron Amplifier

Calculations are presented of the scattering of an electro- magnetic wave from a periodic structure above which flows an electron beam. The reflected fields are computed and found to comprise two separate contributions: 1) the reflection in the homogeneous case (with-out the beam) and 2) the contribution of the beam. Both are shown to depend upon the structure by means of its reflection properties, as expressed by a reflection matrix. Moreover, the beam contribution is shown to be exponentially dependent on the longitudinal position. It also includes the exponential decay, which depends upon the distance between the beam and the structure and is a characteristic of Smith-Purcell devices. Expressions for the local and global gains are obtained. The local gain is found to he proportional to the first velocity derivative of the electron distribution function. Considerations of nonlinear effects introduce spatial dependence to the expression of the local gain. This dependence is determined by a nonlinear diffusion equation. Moreover, the gain in the nonlinear regime is found to depend not only on the first velocity derivative of the distribution function but also on the second velocity derivative.