Self-amplified spontaneous emission in Smith–Purcell free-electron lasers

We present an analysis of a Smith–Purcell system in which a thin current sheet of electrons moves above a grating surface in the direction perpendicular to the grating grooves. We develop a consistent theory for evolution of the electromagnetic field and electron distribution in the exponential growth regime starting from the initial electron noise and the incoming amplitude. The dispersion relation for the complex growth rate for this system is a quadratic equation.