Theory of Small Signal Bunching in a Parallel Electron Beam of Rectangular Cross Section

Abstract

The phenomenon of bunching which takes place in a velocity modulated electron beam can be described in purely kinematical terms if space charge effects are negligible. However, the non‐uniform distribution of charge in the bunched beam gives rise to a field which opposes the bunching process so that the kinematic solution may be validly applied only for a limited length of drift space. An accurate solution of the bunching process requires the integration of the dynamical and field equations. These reduce to a linear homogeneous system under the assumption of ``small signal'' conditions. The device of a high frequency ``surface charge'' is employed in formulating the boundary conditions at the surface of the beam. In the non‐relativistic approximation the dynamical and field equations yield solutions which are classified as non‐solenoidal or solenoidal accordingly as the motion produces or does not produce a high frequency charge density within the beam. There exists a non‐solenoidal type of motion which generates no field outside the beam. The actual physical problem is solved by taking a suitable linear combination of both solutions. Under conditions met with in practice, a large part of the actual solution may be of the solenoidal type. The high frequency component of beam current appears as a sum of a volume current and a ``surface current,'' the latter term arising from the longitudinal motion of the high frequency ``surface charge.'' Again, under conditions met with in practice, a large part of the high frequency component of beam current may occur in the form of ``surface current.'' The theory contains three debunching wave numbers: one identical with that which occurs in Webster's debunching theory for a beam of unlimited cross section, and two which depend upon the transverse dimensions of the beam and drift tube, the current density, frequency, and beam velocity.