Parametric interaction in electron beam waves—A system function characterization

In this paper we develop a technique for characterizing parametric interaction in electron beam waves in terms of a system transfer function. We consider systems in which the parametric interaction is described by a matrix equation of the form (∂/∂t + u
o
∂/∂
z
)a = -iH a where a is a state vector of the system and H is the matrix characterizing the interaction. Solutions in the time domain can be expressed in the form a(z, t) = M(z, t) a(O, t - z/u
o
) where M (a matrix) is called the system function in the time domain. Solutions in the frequency domain can be expressed in the form â(z, ω) = ∫ M(z, ω - ω')â(O, ω')dω' where â(z, ω) is the Fourier transform of a(z, t) and M(z, ω) is the transform of M(z, t). Solutions for parametric interaction in transverse beam waves are worked out for two cases: In Case I we derive the well-known result that describes the behavior of a single pump wave, electron beam parametric amplifier. In Case II we consider a parametric interaction involving two pump waves in which an infinite number of frequencies are parametrically coupled.