Theory of beam-plasma instability in a periodic plasma-filled waveguide

Abstract

The beam-plasma wave interaction in a periodic plasma-filled waveguide is treated in a mathematically correct manner on the basis of the integral equation (IE) method. It has been shown that the relevant boundary-value problem could be reduced to an IE with a singular kernel for the longitudinal component of the electric field on the waveguide axis. The regularization of the IE was performed by extracting the static part of the kernel. The resulting IE of the second kind with a regular kernel, being rather convenient for a numerical analysis, is treated in a quasistatic approximation as a spectral problem. First-order expressions for eigenfunctions, and an infinite set of dispersion relations linking a wave number and frequency of plasma oscillations which separate radial branches of plasma oscillations from axial ones, have been obtained in the closed analytical form, thus enabling us to avoid the problem with the so-called "dense" spectrum. The solutions of the relevant "cold" dispersion relations establish a periodical dependence of the frequency on the wave number over several periods within the accuracy of order of the neglected terms. In the presence of an electron beam they turn out to be unstable near frequencies providing the resonances of the beam with spatial plasma harmonics. Evaluations of the instability saturation level predict a more efficient beam-plasma wave energy transfer compared with those following from a conventional theoretical analysis based on the formulation of a dispersion relation in terms of an infinite determinant, with following truncation of the latter to the finite sized relation.

Complex eigenvalues of natural TM ‐oscillations in an open resonator formed by two sinusoidally corrugated metallic strips

“…In the limiting case, when L → ∞, one can transform (5) to the well-known formula for parallel-plate waveguide (γ n ¼ k n d ¼ nπ, n ¼ 1; 2; …). The expanded view of condðA N tr ÞðkdÞ, shown in Figure 4b, allows us to find the correspondence between the results of numerical computation and those obtained by the usage of the formula (5), when we set d=2L ¼ 0: Here, the first value is calculated by (5), and the second (in brackets) is extracted from the graph of the dependence condðA N tr ÞðkdÞ. This observation signals the emergence of standing waves, even when the rectangular resonator is open.…”

“…where indices n and m stand for the number of electromagnetic field ðEMFÞ variations along the length of the resonator and along the transverse direction, respectively. In the limiting case, when L → ∞, one can transform (5) to the well-known formula for parallel-plate waveguide (γ Here, the first value is calculated by (5), and the second (in brackets) is extracted from the graph of the dependence condðA N tr ÞðkdÞ. This observation signals the emergence of standing waves, even when the rectangular resonator is open.…”

“…Modelling the effect of plasma on the diffraction radiation of a relativistic beam moving over a grating of finite extent is highlighted in [6]. It should be noticed, in [5,6], that the authors treated the electromagnetic part of both problems in a rigorous fashion, applying the idea of analytical regularization to surface integral equations.…”

“…In general, applications where use of periodic structures is of paramount significance cover diverse aspects of theoretical and technical physics. One example is the theory of beam‐plasma instability in a periodic plasma‐filled waveguide [5], where a planar symmetrical metallic waveguide with arbitrary periodic walls is considered as a basic element. Modelling the effect of plasma on the diffraction radiation of a relativistic beam moving over a grating of finite extent is highlighted in [6].…”

Fractal Properties of Trivelpiece-Gould Waves in Periodic Plasma-Filled Waveguides

“…The generally accepted approach which is traditionally used for periodic media and associated with the field representation as a superposition of a finite number of spatial harmonics seems to be useless even in this simplest case. It yields the so-called dense spectrum [2,3] which contains spurious information [4][5][6][7] and leads to the divergence of numerical results. Recently, new approaches have been developed [4 -7] which allow us to get rid of spurious solutions and to reduce the problem to the functional equation describing adequately the real spectrum of TG modes in periodic plasma-filled waveguides.…”