Coherent-instability-induced radiation

This paper is organized as follows: In Section 2, we start from the appropriate Vlasov equation in the microwave region. We show that the Vlasov equation together with an initial condition is equivalent to a non-linear integral equation. We then linearize the integral equation to the first order in the ''centered'' shot noise distribution function. In Section 3, we show that the linear integral equation derived in Section 2 is equivalent to a homogeneous linear integro-differential equation for the current density. We refer to this equation as the basic equation. The differential operator of the basic equation is second order in time. In Section 4, we introduce an orthonormal set of functions to a finite dimensional linear space in preparation for the discussion of a solvable model to be introduced in the next section. In Section 5, we introduce a model impedance and the corresponding basic equation for the current. We show that the basic equation can be diagonalized using the orthonormal set of functions discussed in the previous section. We also solve the initial value problem of the equation with the current and the induced radiation field expressed in terms of the initial perturbation current. In Section 6, we investigate how the radiation power grows in time because of the coherent instability. We assume that the initial perturbation of the instability is due to the grainy characteristic of the shot noise in the beam. The graininess of the shot noise is averaged over in the expression for the radiation power. (orig.)