The Polarization Response Function and Electrostatic Modes of Focused Beams

It is shown that the method of percussion for obtaining the spatial-temporal representation of the polarization response function directly [1] can also be used to study stability problems in nonuniform plasmas, thus providing a physical and mathematical alternative to the traditional approach. As an example, the stability analysis of particle beams is given, focused with cylindrical symmetry through an axis. The integral equation for electrostatic modes is set up with the polarization response, and solved with a technique that provides both the characteristic values and an integral representation for the field. The instability of this model, which is the cylindrical counterpart of the plane symmetric counterstreaming configuration, is then discussed.