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In this paper, a nonlinear analytical solution for a cold plasma-bunched beam system based on the Hamiltonian formalism where a = ndno and fi+ = U& have been taken as parameters matching between zero and unity is given. The oscillation limiting energies, frequencies and transformer ratios have been carried out in general for both the one-dimensional and the case where a small transverse component of motion is included. The plasma wakefield accelerator has been treated as a special case.

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An analysis of the excitation of a space charge wave and a predominantly transverse magnetic mode in a cylindrical metallic waveguide filled with a relativistic electron beam is presented. The beam is guided by a uniform, axial magnetic field of arbitrary magnitude and passes through a spatially periodic, longitudinal electric (wiggler) field. A non-linear wave equation is derived in a form which includes the coupling of TM and TE modes due to the finite axial magnetic field. For the case of a weak and purely axial wiggler field, a solution of this wave equation is developed with each of the two excited waves expressed as a sum of three partial waves. A spatial growth rate formula is derived in a form appropriate for operation infinitesimally above cut-off of the TM mode. Numerical calculations indicate that the axial magnetic field magnitude should be somewhat larger than the value which maximizes the growth rate in order to achieve high efficiency.

Excitation of a slow space-charge wave and a TM waveguide mode by passage of a relativistic electron beam through a static, spatially periodic, longitudinal electric field is investigated. The non-linear wave equation, free-electron-laser dispersion relation and spatial growth rate formula are derived in the laboratory reference frame. A formula for the energy-conversion efficiency based on saturation by electron trapping in the space-charge wave is also derived. Equations for transformation to the beam reference frame are presented. The results of a numerical study indicate that efficiencies of 25 to 60% and radiation wavelengths in the millimetre range are possible by operation near cut-off of the TM mode. Radiation wavelengths of the order of 100 mu m or less with somewhat lower efficiencies and growth rates may be obtained by operation above cut-off using a 1 cm pump wavelength

The spatial variation of a non-uniform glow discharge due to the different diameters of the same discharge tube on the plasma parameters is attracting a considerable attention. For this reason the fundamental spatial plasma parameters of an argon glow discharge have been measured by using fast three couple of Langmuir double probe (TCDP) technique. The orbital motion limited (OML) theory has been used, since the probe radius is smaller than Debye length ξ < 1. The axial and radial variation of the electron temperature and density have been investigated. The results can be summarized as follow: due to the rise in the space-charge confinement with decreasing the tube radius and discharge current, the space-charge density increases, the electron temperature decreases.