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The nonlinear evolution of the Cerenkov maser amplifier is investigated numerically for a configuration that consists of an energetic electron beam propagating through a dielectric-lined cylindrical waveguide. An axial guide magnetic field is included in the formulation in order to improve beam confinement. A set of coupled nonlinear differential equations is derived in three dimensions that governs the evolution of both the electromagnetic field and the trajectories of an ensemble of electrons. The system is assumed to be azimuthally symmetric, and the electromagnetic field is represented as a superposition of the TM0n modes of the vacuum waveguide. The initial conditions are chosen to model the simultaneous injection of either a solid or annular electron beam, and an electromagnetic wave of arbitrary input power. Thermal effects are treated under the assumption that the beam is initially monoenergetic but exhibits a pitch angle spread; however, the subsequent evolution of the beam is treated in a self-consistent manner. This class of distribution is appropriate to the treatment of diode-produced beams and describes a beam with an initial axial energy spread. This is the crucial determinant in the efficiency, since saturation occurs by means of an axial bunching mechanism that results in the phase trapping of the beam. The specific parameters used in the numerical analysis correspond to experiments conducted at Dartmouth College [J. Appl. Phys. 58, 627 (1985)], and good agreement is found between theory and experiment.

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Quantum transport through one-dimensional potential barriers is usually analyzed using either the transmission coefficient (TC) or the Wigner distribution function (WDF) approach. Fast, accurate, and efficient numerical algorithms are developed for each and are compared for (a) calculating current-field relationships for field-emission potentials with silicon parameters (and current-voltage relationships for resonant tunneling diodes), (b) their ability to accommodate scattering, self-consistency, and time dependence, and for (c) the behavior of their "particle trajectory" interpretations. In making the comparisons, the concern will be on the ability of each method to be incorporated into a larger ensemble-particle Monte Carlo simulation; it is argued that, in this regard, the WDF approach has significant advantages. Since the TC calculations rely on the Airy function approach, a detailed comparison of this method is made with the widely used Wentzel-Kramers-Brillouin and Fowler-Nordheim approaches for the general problem of field emission from a material into the vacuum.