Proceedings of ITC/ISHW2007

Instability of Thin-Walled Annular Beam in Dielectric-Loaded
Cylindrical Waveguide
S. Tamura, M. Yamakawa, Y. Takashima and K. Ogura
Graduate School of Science and Technology, Niigata University, 950-2181, Japan(Received 15 October 2007 / Accepted 3 August 2007)
Cherenkov and slow cyclotron instabilities driven by an axially injected electron beam in cylindrical waveguideare studied by using a new version of self-consistent liner theory considering three dimensional beam perturbations.
There are three models of beam instability analysis, which are based on solid beam, infinitesimally thin annularbeam, and thin-walled annular beam. Among these models, a model properly representing the often used actual
annular electron beams is thin-walled annular beam. We develop a numerical code for a cylindrical waveguide with athin-walled annular beam having a finite thickness. Our theory is valid for any beam velocity. We present eigen
modes with plasma and beam. And instabilities driven by thin-walled annular beam in dielectric-loaded waveguideare examined.
Keywords: Thin-walled annular beam, low-frequency surface wave mode, high-frequency surface wave
mode, Cherenkov instability, slow cyclotron instability

1. Introduction
The backward wave oscillator (BWO) or travelingwave tube (TWT) is one of the high power microwave
sources and can be driven by an axially injected electron
beam without initial perpendicular velocity. For relatively
low power in some 10 kW level or less, Pierce type
cathodes are commonly used and beam shape isapproximated by a solid beam. In many high power
experiments, cold cathodes are used and the shape ofelectron beam is a thin-walled annulus. A new version ofself-consistent field theory considering three-dimensional
beam perturbations are developed based on a solid beam
[1-2]. For infinitesimally thin annular beam, the boundary
is modulated due to the transverse modulation of annularsurface. Analyses of infinitesimally thin annular beam need
to be based on a different theory from thin-walled annularand solid beam. A pioneering work can be seen in Ref. [3].
Recently, instabilities of eigen mode due to the surface
modulation are analyzed in Ref. [4], presenting aself-consistent field theory considering the moving surface
modulation.
In this work, we develop a numerical code for acylindrical waveguide with a thin-walled annular beam.
Note that the annular beam thickness is finite. Theboundary condition at the beam surface is different fromthe infinitesimally thin annular beam. Solid beam andthin-walled annular beam are based on the same boundary
condition, but the number of boundary is different.
author’s e-mail: f06e072j@mail.cc.niigata-u.ac.jp

Thin-walled annular beam has outside and inside surface,and solid beam has only outside surface. Our numericalcodes are valid for any beam velocity v. We present theeigen modes for waveguide with plasma (v=0) and beam(v>0) modes.
The organization of this proceeding is as follows. InSec. 2, we describe numerical method of thin-walled
annular beam and solid beam. In Sec. 3, dispersion carvesof thin-walled annular plasma and solid plasma areexamined. In Sec. 4, we examine Cherenkov and slow
cyclotron instabilities driven by thin-walled annular beamin dielectric-loaded waveguide. Discussion and conclusionof this paper are given in Sec. 5.

2. Numerical method
For slow-wave devices, the periodically corrugatedslow-wave structure (SWS) is used [5, 6]. However,analyses of these devices are very complex. Therefore, weanalyzed basic electromagnetic characteristics of
cylindrical waveguide with straight wall. We consider adielectric-loaded waveguide (Fig.1). The wall radius
Rw=1.445 cm, dielectric radius Rd=0.85 cm. In the case ofplasma (v=0), relative permittivity of dielectric εr=1.0. For
analyzing beam interactions, εr=4.0. We used this model toexamine basic characteristics of the electromagnetic wave
and the beam interaction. For thin-walled annular beam,
average beam radius Ra=0.75 cm, and beam thickness
Δp=0.1 cm. For solid beam, beam outside radius Rb=0.8 cm.

Proceedings of ITC/ISHW2007
A guiding magnetic field B0 is applied uniformly in axial
direction. An electron beam is propagating along theguiding magnetic field. The temporal and spatial phasefactor of all perturbed quantities is assumed to be
exp[i(kzz+mθ-ωt)]. Here, m is the azimuthal mode number
and kz is the axial wave number.
In the system with magnetized plasma or beam suchas Fig.2, electromagnetic modes are the hybrid mode oftransverse magnetic (TM) and transverse electric (TE)mode. Two letters of EH and HE is used, to designate thehybrid mode. In this paper, TM is dominant in EH modeand TE is dominant in HE mode.

An electron beam surface is modulated as beam ispropagating. For thin-walled annular beam and solid beam,
the transverse moderation appears as the surface electriccharge at the fixed boundary as shown in Figs.2 and 3.
Solid beam has one surface. Thin-walled annular beam has
another surface inside beam because there is a vacuum
region inside the beam.

3. Dispersion curves of plasma
We present results of thin-walled annular plasma andsolid plasma analyses (v=0). Here ωp and Ω are plasma
frequency and cyclotron frequency. In this section,ωp=3×1010 rad/s. Figure 4 shows the dispersion curves ofthin-walled annular plasma and Fig. 5 shows the dispersion
curves of solid plasma in the absence of magnetic field.
Axisymmetric electromagnetic modes are the TM0n andTE0n mode. Here, n is any positive integer. Dispersion
curves of thin-walled annular plasma shows two surface
waves due to the inner and outer surface space charges:
high- and low-frequency surface wave modes. Solidplasma has only one surface wave mode due to the outersurface space charge.

Fig.1 Model of thin-walled
annular beam analysis.

Fig. 4 Dispersion curves of thin-walled annular
plasma in absence of magnetic field.

Fig. 2 Beam surface of thin-walled
annular beam.

Fig. 3 Beam surface of solid beam.

Fig. 5 Dispersion curves of solid plasma in absenceof magnetic field.

Proceedings of ITC/ISHW2007
The high-frequency surface wave mode is attributed
to the inner boundary of thin-walled annular plasma (Fig.2).
The low-frequency surface wave mode is attributed to theouter surface of annulus and corresponds to the surface
wave mode of solid plasma. Their frequency is zero at kz=0,increases with increasing kz, and approaches an asymptotic
limit as kz→∞. This limit is ωp/√2. The frequency ofhigh-frequency surface wave mode for the annulus is ωp atkz=0, decrease with increasing kz, and approach an
asymptotic limit as kz→∞. This limit is also ωp/√2.
Te dispersion curves for the thin-walled annular andsolid plasma are respectively shown in Figs. 6 and 7 at
B0=0.1 T (ωp>Ω) and in Figs. 8 and 9 at B0=0.2 T (Ω>ωp).
With the finite strength magnetic fields, electromagnetic
modes are the hybrid mode of TM and TE modes:
axisymmetric EH0n and HE0n modes in the figures. And
cyclotron modes are appeared in addition to the spacecharge modes. For both of thin-walled annular and solid
plasma, the frequencies of cyclotron mode are between theupper hybrid frequency ωh=(Ω2+ωp2)1/2 and Ω or ωp,whichever is larger.

Fig. 8 Dispersion curves of thin-walled
annular plasma with B0=0.2 T.

Fig. 9 Dispersion curves of solid plasma with
B0=0.2 T.

Fig. 6 Dispersion curves of thin-walled
annular plasma with B0=0.1T.

Fig. 7 Dispersion curves of solid plasma
with B0=0.1T.

As for the space charge modes, surface wave modes
exist as in the case of B0=0 if Ω < ωp. With increasing kz,
the frequencies of low-frequency surface wave mode andsurface wave mode of solid plasma are increased andapproach an asymptotic limit as kz→∞. The limit isbetween ωp and ωp/√2. The frequency of high-frequency
surface wave mode is decreased with increasing kz, and has
the same asymptotic value as the low-frequency mode in
the limit of kz→∞
The presence of magnetic field yield plasma modes
due to the volume space charge perturbation, those aredenoted as plasma modes in Figs.6-9 [2]. The
low-frequency plasma mode of thin-walled annular plasmaand the solid plasma mode have frequencies that are zero atkz=0, increase with increasing kz, and approach an
asymptotic limit as kz→∞. This limit is ωp or Ω whicheveris smaller. The annular plasma has high-frequency plasmamode. Its frequency approaches an asymptotic limit
between ωh and Ω, as kz→∞ with Ω>ωp.

Proceedings of ITC/ISHW2007

5. Instabilities driven by thin-walled annularbeam
In this section, we examine instabilities driven bybeam (v>0). Figure 10 shows the dispersion curves for athin-walled annular beam with the energy 660 keV, beam
current 2.3 kA and B0=0.8 T. For an electron beam
propagating along the direction of an axial magnetic field,
there exist four beam modes. They are the fast and slow
space charge modes and the fast and slow cyclotron modes.
Slow space charge mode and slow cyclotron mode couple
with both of EH01 and HE01 mode, resulting in theCherenkov and slow cyclotron instabilities. Instabilities for
EH01 mode are predominating, compared with those for
HE01mode.
Figure 11 shows the dependence of the temporal
growth rate on beam thickness Δp. The beam outer radius isfixed to 0.8 cm, and beam inner radius has been changed
with a fixed line charge density. The growth rate ofCherenkov instability increases by decreasing Δp.

The growth rate of slow cyclotron instability also
increases by decreasing the beam thickness. But, in theregion of Δp < 0.035 cm, the growth rate decreases.
In the limit that the beam inner radius is zero, thegrowth rate of Cherenkov and slow cyclotron instabilitiesof thin-walled annular beam approaches the growth rate ofsolid beam, ▲ in Fig.11. In the other limit that Δp→0, thecorresponding growth rates are those of an infinitesimallythin annular beam model with Δp=0 [2] and are depicted by● in Fig.11. Two models based on finite and zero Δp givealmost the same results for the Cherenkov instability. For
slow cyclotron instability, the growth rates are different
between two models. This might be caused by thedifference of annulus structure: one has an internal
structure between the inner and outer surface and the otheris just a sheet without any internal structure.

5. Conclusion
We develop a numerical code for a cylindricalwaveguide with a thin-walled annular beam considering
three-dimensional beam perturbations, which is valid forany beam velocity v. Electromagnetic modes are the hybrid
mode of TM and TE mode. Cyclotron and space charge
modes corresponding to the solid case exist. Note that, forspace charge modes, surface wave modes exist in addition
to the volume wave mode if Ω < ωp. As characteristic
modes of annular case, the high-frequency space charge
modes appear due to the inner surface. A thin-walled
annular beam drives slow cyclotron as well as Cherenkov
instability. The Cherenkov instability increases bydecreasing Δp to zero. The slow cyclotron instability
increases by decreasing Δp, reaches the maximum at acertain value of Δp and decreases in the limit of Δp→0.

Fig. 10 Dispersion curves of thin-walled annularbeam in dielectric-loaded waveguide withbeam energy 660 keV, beam current 2.3
kA and B0=0.8 T.

Acknowledgments
This work was partially supported by a Grant-in-Aidfor Scientific Research from the Ministry of Education,
Science, Sports and Culture of Japan, and partiallyperformed under the Collaborative Research Program
between NIFS and University of Tsukuba in cooperation
with Japanese universities.

References
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Fig. 11 The dependence of the temporal
growth rate on beam thickness.

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