The interaction of a modulated electron beam and a narrow tape helix is considered using a confined flow model for the beam. With this model, the analysis is self-consistent in that it takes into account the first-order effects of the interaction on the beam. The interaction is studied by means of complex waves on the composite system of the beam and the helix. The role of these waves is established in part on the basis of their complex wave numbers which are solutions of the appropriate determinantal equation. Although coupling similar to that used in traveling wave tubes is considered initially, the investigation concentrates on coupling which can be used to produce Smith–Purcell radiation. The results of the numerical evaluation of the wave numbers demonstrate that for the latter type of coupling, the modulation changes very slowly along the length of the interaction. This establishes the heretofore questionable validity of the prescribed source model for evaluating the interaction of a modulated electron beam and a narrow tape helix. Similar results are expected to hold for other interaction geometries.
Derivation of the spatial-domain, closed-form Green's functions of the vector and scalar potentials are demonstrated for planar media, and their use in conjunction with the method of moments (MoM) is presented. As the first step of the derivation, the Green's functions are obtained analytically in the spectral domain for various sources viz., horizontal and vertical electric and magnetic dipoles embedded in a planar stratified media. The spatial-domain Green's function can be obtained from the Sommerfeld integral which is the Hankel transform of the corresponding Green's function in the spectral domain. The analytical evaluation of this transformation yields the closed-form, spatial-domain Green's functions which can be used in the solution of a mixed-potential integral equation (MPIE) via the MoM. This combi nation, i.e., the use of the closed-form Green's functions in conjunction with the MoM, results in a significant improvement in the fill-time of MoM matrices. In the conventional application of the spatial-domain MoM, the matrix elements are double integrals and they require the evaluation of the time-consuming Sommer feld integral for the spatial-domain Green's function. In the approach presented herein, the spatial-domain Green's functions are in closed forms, and the remaining double-integrals in the matrix elements are evaluated analytically. Thus, there are two factors in this approach that contribute to the improvement in the computa tion time: (i) elimination of the numerical integration to obtain the spatial-domain Green's functions; (ii) circumventing the need to carry out the numerical integration in the calculation of the MoM matrix elements.
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