Wave Propagation Along a Magnetically- 

 Focnsed Cylindrical Electron Beam 



By W. W. RIGROD and J. A. LEWIS 



(Manuscript received August 24, 1953) 



This paper analyses the nature of wave propagation along a cylindrical 

 electron beam, focused in Brillouin flow by means of a finite axial magnetic 

 field. Two different types of conducting boioidaries external to the beam are 

 treated: (/) the concentric cylindrical tube, forming a drift region; ami {2) 

 tlie sheath helix, forming a model of the helix traveling-wave tube. The field 

 solution of the helix problem is used to evaluate the normal-mode parameters 

 of an equivalent circuit seen by a thin beam, thereby permitting computation 

 of the gain constant of growing waves. The gain constant of the cylindrical 

 beam with Brillouin flow is fourui to exceed that of a similar beam with 

 rectilinear flow, presumably because of the transverse component of electron 

 motion in the former. 



IXTRODUCTIOX 



The theory of the heUx traveHng-wave has been treated in previous 

 l^apers/"^ for cases in which the electrons move along straight lines paral- 

 lel to the axis of the helix, as though immersed in an infinitely strong 

 magnetic field. In practice, however, the electron beam is focused by a 

 magnetic field of finite intensity,^" ^ such that the electrons follow spiral 

 paths alxjut the common axis. The purpose of this paper is to extend 

 traveling-wave tube theory to the case of such focused beams, and to 

 compare the gain constants for the two types of electron motion. The 

 motion of the beam in an infinite field is usually described as rectilinear 

 flow; that in a finite focusing fi(>ld, as Brillouin flow. 



The gain constant of the dominant mode in a traveling-wave tube 

 may be computed from the field solution for the electron beam in the 

 presence of its circuit structure. This procedure, however, recjuires the 

 solution of cumbersome transcendental equations for each particular set 

 of dimensions and operating conditions. A more flexible method of anal- 

 ysis has been provided by Pierce,^ based on an expansion in terms of 



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