Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder
The interaction between a tubular beam of charged particles and a nonmagnetic anisotropic dispersive medium of cylindrical configuration has been investigated. It has been found the absolute instability of bulk-surface waves that occurs because of peculiarities of the anisotropic cylinder properti...
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| Zitieren: | Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder / Yu.O. Averkov, Yu.V. Prokopenko, V.M. Yakovenko // Вопросы атомной науки и техники. — 2018. — № 4. — С. 3-12. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1473202019-02-15T01:24:16Z Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder Averkov, Yu.O. Prokopenko, Yu.V. Yakovenko, V.M. Нерелятивистская электроника The interaction between a tubular beam of charged particles and a nonmagnetic anisotropic dispersive medium of cylindrical configuration has been investigated. It has been found the absolute instability of bulk-surface waves that occurs because of peculiarities of the anisotropic cylinder properties. The resonance behavior of the permittivity frequency dependence causes the emergence of the sections of dispersion curves of the E-type bulk-surface eigenmodes with negative group velocity. It has been shown there are the E-type surface eigenmodes and pseudo surface eigenmodes of E- and H-types in the cylinder. Вивчено взаємодію нерелятивістського трубчастого потоку заряджених частинок з немагнітним анізотропним диспергуючим середовищем циліндричної конфігурації. Виявлена абсолютна нестійкість об'ємноповерхневих хвиль, що обумовлена особливостями властивостей анізотропного циліндра. Резонансний характер частотних залежностей діелектричної проникності циліндра призводить до появи ділянок дисперсійних кривих власних об'ємно-поверхневих хвиль E-типу з негативною груповою швидкістю. Показано існування в циліндрі власних поверхневих хвиль E-типу і псевдоповерхневих хвиль E- та H-типів. Изучено взаимодействие нерелятивистского трубчатого потока заряженных частиц с немагнитной анизотропной диспергирующей средой цилиндрической конфигурации. Обнаружена абсолютная неустойчивость объёмно-поверхностных волн, обусловленная особенностями свойств анизотропного цилиндра. Резонансный характер частотных зависимостей диэлектрической проницаемости цилиндра приводит к появлению участков дисперсионных кривых собственных объёмно-поверхностных волн E-типа с отрицательной групповой скоростью. Показано существование в цилиндре собственных поверхностных волн E-типа и псевдоповерхностных волн E- и H-типов. 2018 Article Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder / Yu.O. Averkov, Yu.V. Prokopenko, V.M. Yakovenko // Вопросы атомной науки и техники. — 2018. — № 4. — С. 3-12. — Бібліогр.: 25 назв. — англ. 1562-6016 PACS: 03.50.-z, 52.40.-w, 52.59.-f, 85.45.-w http://dspace.nbuv.gov.ua/handle/123456789/147320 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| language |
English |
| topic |
Нерелятивистская электроника Нерелятивистская электроника |
| spellingShingle |
Нерелятивистская электроника Нерелятивистская электроника Averkov, Yu.O. Prokopenko, Yu.V. Yakovenko, V.M. Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder Вопросы атомной науки и техники |
| description |
The interaction between a tubular beam of charged particles and a nonmagnetic anisotropic dispersive medium
of cylindrical configuration has been investigated. It has been found the absolute instability of bulk-surface waves
that occurs because of peculiarities of the anisotropic cylinder properties. The resonance behavior of the permittivity
frequency dependence causes the emergence of the sections of dispersion curves of the E-type bulk-surface
eigenmodes with negative group velocity. It has been shown there are the E-type surface eigenmodes and pseudo
surface eigenmodes of E- and H-types in the cylinder. |
| format |
Article |
| author |
Averkov, Yu.O. Prokopenko, Yu.V. Yakovenko, V.M. |
| author_facet |
Averkov, Yu.O. Prokopenko, Yu.V. Yakovenko, V.M. |
| author_sort |
Averkov, Yu.O. |
| title |
Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder |
| title_short |
Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder |
| title_full |
Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder |
| title_fullStr |
Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder |
| title_full_unstemmed |
Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder |
| title_sort |
іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2018 |
| topic_facet |
Нерелятивистская электроника |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147320 |
| citation_txt |
Іnteraction between a tubular beam of charged particles and an anisotropic dispersive solid-state cylinder / Yu.O. Averkov, Yu.V. Prokopenko, V.M. Yakovenko // Вопросы атомной науки и техники. — 2018. — № 4. — С. 3-12. — Бібліогр.: 25 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT averkovyuo ínteractionbetweenatubularbeamofchargedparticlesandananisotropicdispersivesolidstatecylinder AT prokopenkoyuv ínteractionbetweenatubularbeamofchargedparticlesandananisotropicdispersivesolidstatecylinder AT yakovenkovm ínteractionbetweenatubularbeamofchargedparticlesandananisotropicdispersivesolidstatecylinder |
| first_indexed |
2025-07-11T01:52:10Z |
| last_indexed |
2025-07-11T01:52:10Z |
| _version_ |
1837313545154330624 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2018. №4(116) 3
NONRELATIVISTIC ELECTRONICS
INTERACTION BETWEEN A TUBULAR BEAM
OF CHARGED PARTICLES AND AN ANISOTROPIC DISPERSIVE
SOLID-STATE CYLINDER
Yu.O. Averkov1,2, Yu.V. Prokopenko1,3, and V.M. Yakovenko1
1A.Ya. Usikov Institute for Radiophysics and Electronics
of National Academy of Sciences of Ukraine, Kharkov, Ukraine;
2V.N. Karazin Kharkiv National University, Kharkov, Ukraine;
3Kharkiv National University of Radioelectronics, Kharkov, Ukraine
E-mail: yuriyaverkov@gmail.com; prokopen@ire.kharkov.ua; yavm@ire.kharkov.ua
The interaction between a tubular beam of charged particles and a nonmagnetic anisotropic dispersive medium
of cylindrical configuration has been investigated. It has been found the absolute instability of bulk-surface waves
that occurs because of peculiarities of the anisotropic cylinder properties. The resonance behavior of the permittivity
frequency dependence causes the emergence of the sections of dispersion curves of the E-type bulk-surface
eigenmodes with negative group velocity. It has been shown there are the E-type surface eigenmodes and pseudo
surface eigenmodes of E- and H-types in the cylinder.
PACS: 03.50.-z, 52.40.-w, 52.59.-f, 85.45.-w
INTRODUCTION
Investigation of the generation mechanisms of elec-
tromagnetic waves that are excited when charged parti-
cles move in various electrodynamic systems is im-
portant in microwave electronics. To create sources of
electromagnetic radiation in the millimeter and submil-
limeter ranges, the beam instabilities occurring in elec-
trodynamic systems of various kinds are of great inter-
est. Currently, special attention is given to multiwave
Cherenkov sources of surface waves [1, 2] and auto-
oscillatory systems based on dielectric resonators [3 -
5], and dielectric Cherenkov masers [6]. Besides, the
beam instabilities that occur in electrodynamic systems
containing dispersive media are of special interest. In
particular, the instabilities of the tubular electron beam
that interacts with a plasmalike medium and a left-
handed dispersive medium of cylindrical configuration
were studied in [7] and [8], respectively.
In the present paper, the interaction between a tubular
beam of charged particles and eigenmodes of nonmagnet-
ic cylindrical solid-state waveguide, in which the compo-
nents of permittivity tensor have frequency dispersion, is
theoretically investigated. This crystal-like medium of the
waveguide may have the permittivities with the different
signs in perpendicular and parallel directions to the opti-
cal axis in a certain frequency range. Our goal is to de-
termine the conditions for the excitation of eigenmodes
with anomalous dispersion. It will be shown that the in-
teraction of an electron beam with the waveguide
eigenmodes gives rise to the absolute instability of the
bulk-surface electromagnetic waves, which are the prop-
agating waves in the waveguide and, at the same time, are
evanescently confined along the normal to the lateral
cylinder surface in vacuum. This means that the aniso-
tropic dispersive media can be used as the delaying struc-
tures with "natural feedback" for generation of electro-
magnetic waves in backward-wave tubes. Besides, the
possibility of excitation of weakly damped whispering
gallery waves in an anisotropic cylinder [9] will allow the
generation of electromagnetic waves in the submillime-
ter region of the spectrum.
1. STATEMENT OF THE PROBLEM
AND BASIC EQUATIONS
Consider a nonmagnetic cylindrical solid-state
waveguide with the radius 0ρ occupying the region
00 ρ ρ≤ ≤ , 0 2ϕ π≤ ≤ , and z−∞ ≤ ≤ +∞ (Fig. 1).
The cylinder is made of an anisotropic single crystal, the
optical axis of which orientates parallel to the symmetry
axis Z of cylinder. A tubular electron beam with the
radial thickness a and density 0 ( )N ρ moves in vacu-
um at a distance of bρ from the cylinder axis at a veloc-
ity 0v . We assume that the charges of electrons are
compensated by the background of positive charges and
the thickness of the beam a is much smaller than the
other spatial scales of the electrodynamic system under
consideration. Hence, the undisturbed beam density can
be represented as 0 0( ) ( )bN N aρ δ ρ ρ= − , where 0N is
the equilibrium beam density and ( )bδ ρ ρ− is the Di-
rac delta function.
Fig. 1. Geometry of electrodynamic system
In case of linear approximation, the disturbed beam
current density at a point with the radius-vector r at a
moment t has the form
0 0( , ) ( ) ( , ) ( , )t eN t e N tρ= +j r v r v r ,
where e is the electron charge, and ( , )N tr and ( , )tv r
are the variable components of the beam density and the
electron velocity, respectively. Hereafter, we will sup-
pose that the radial component of the beam current den-
sity is equal to zero because of the chosen model of the
electron beam.
mailto:yuriyaverkov@gmail.com
mailto:prokopen@ire.kharkov.ua
mailto:yavm@ire.kharkov.ua
ISSN 1562-6016. ВАНТ. 2018. №4(116) 4
System of equations, which describes the interaction
between the electron beam and the cylinder eigenmodes,
represents the Maxwell equations supplemented with
the linearized continuity and motion equations for the
beam electrons:
1 4rot ( , ) ( , ) ( , )t t t
c t c
π∂
= +
∂
H r D r j r , (1)
1rot ( , ) ( , )t t
c t
∂
= −
∂
E r H r , (2)
div ( , ) 4 ( , )t eN tπ=D r r , (3)
div ( , ) 0t =H r , (4)
( , ) div ( , ) 0N te t
t
∂
+ =
∂
r j r ,
0 0
( , ) ( , ) 1( , ) [ , ( , )]t t ev t t
t z m c
∂ ∂ + = + ∂ ∂
v r v r E r v H r ,
where m is the electron mass, c is the velocity of light
in vacuum, ( , )tE r and ( , )tH r are the electric and
magnetic field vectors, and ( , )tD r is the electric dis-
placement vector that is related with the ( , )tE r -vector
by the constitutive equations
ˆ( , ) ( ) ( , )
t
i ij jD t t t E t dtε
−∞
′ ′ ′= −∫r r ,
where ˆ ( )ij t tε ′− is the influence function that character-
izes the efficiency of the field action in time. Indices i
and j correspond to one of the directions along the
coordinate axes ρ , ϕ , and z . Here, the summation by
the index j is carried out with a search of all directions.
In vacuum we have ˆ ( ) ( )ij ijt t t tε δ δ′ ′− = − , where ijδ is
the Kronecker symbol. Note that the difference nature
of the kernels of the integrals is due to the homogeneity
of the waveguide properties in time.
In order to derive the dispersion equation for the
electromagnetic waves in the electrodynamic system
under consideration, it is necessary to satisfy certain
boundary conditions at 0ρ ρ= and bρ ρ= . These con-
ditions are as follows. First, the tangential components
of the electric and magnetic fields are continuous at
0ρ ρ= . Second, at bρ ρ= the tangential components
of the magnetic field have to be discontinuous because
of the beam current, whereas the tangential components
of the electric field are continuous. Note that the normal
component of the magnetic field vector remains contin-
uous, whereas the normal component of the electric
displacement vector suffers discontinuity because of the
disturbed beam charge.
We determine the discontinuities of the tangential
components of the magnetic field and the normal com-
ponent of the electric displacement [in vacuum
( , ) ( , )D t E tρ ρ≡r r ] by integrating (1) and (3) over the
infinitesimally small beam thickness. As a result, we
have
0 0
0
4( , ) ( , ) ( , )lim
b
b b
b
z
b
H t H t j t d
c
ρ ρ
ϕ ϕρ ρ ρ ρ
ρ ρ ρ
π ρ ρ
ρ
+∆
= + = −
∆ → −∆
− = ∫r r r ,
0 0
0
4( , ) ( , ) ( , )lim
b
b b
b
z zH t H t j t d
c
ρ ρ
φρ ρ ρ ρ
ρ ρ ρ
π ρ
+∆
= + = −
∆ → −∆
− = − ∫r r r ,
0 0
0
4( , ) ( , ) ( , )lim
b
b b
bb
eE t E t N t d
ρ ρ
ρ ρρ ρ ρ ρ
ρ ρ ρ
π ρ ρ
ρ
+∆
= + = −
∆ → −∆
− = ∫r r r .
We represent all variables in the form of the set of
space-time harmonics, for instance:
( , ) ( , , ) exp[ ( )]n z z z
n
t q i q z n t dq dρ ω ϕ ω ω
∞ ∞∞
=−∞ −∞ −∞
= + −∑ ∫ ∫E r E , (5)
where ω , zq , and n are the frequency, longitudinal
wave number, and the number of the spatial harmonic
(coinciding with the azimuthal mode index), respective-
ly; 2 1i = − . Then we have
( , , ) ( ) ( , , )i z ij j zD q E qρ ω ε ω ρ ω= ,
where
0
( ) ( ) exp( )ij ij i dε ω ε τ ωτ τ
∞
= ∫ is the permittivity
tensor of medium.
Consider the medium inside the cylinder, which
consists of anisotropic oscillators characterized by a set
of eigenfrequencies Lω , rω , and sω . Such a medium
corresponds to a crystal, whose permittivity tensor has a
diagonal form with components ε⊥ and ||ε , where the
indices " ⊥ " and " || " indicate the material properties in
the perpendicular and parallel directions to the optical
axis of the crystal, respectively. We assume that the
frequency dependences ( )ε ω⊥ and || ( )ε ω have the
form [10 - 13]
2
0 2 2( ) L
r
ω
ε ω ε
ω ω⊥ = −
−
,
2
|| 0 2 2( ) L
s
ω
ε ω ε
ω ω
= −
−
,
where 0ε is the background value of the dielectric con-
stant of the crystal determined as the high-frequency
limit of ( )ε ω⊥ and || ( )ε ω . It is clear that there are such
frequency bands in which ( )ε ω⊥ and || ( )ε ω have nega-
tive values.
In particular, these dependences ( )ε ω⊥ and || ( )ε ω
characterize the magnetized collisionless semiconductor
medium, in which
22
0 2
1
( ) L
r
α
α α
ω
ε ω ε
ω⊥
=
= +∑ ;
22
|| 0 2
1
( ) Lα
α
ω
ε ω ε
ω=
= −∑ ,
where 2 24 /L e n mα α α αω π= and 0 /r e H m cα α αω = , eα ,
mα and nα are the charge, mass and the majority-
carrier concentration of the α -kind: electrons (α = 1)
and holes (α = 2), respectively, 0H is the induction of
an external magnetic field (whose vector orientates par-
allel to the symmetry axis Z of cylinder). At the same
time 0sω = and r rαω ω≡ , and rαω ω>> because of
0H →∞ .
If we take into account (5), we can rewrite the origi-
nal equations (1) - (4) for the axial spectral components
of the field in the region inside the cylindrical solid-
state waveguide ( 0ρ ρ≤ ) in the following form:
ISSN 1562-6016. ВАНТ. 2018. №4(116) 5
2
2
2
1 ( , , ) 0E zn z
nq E qρ ρ ω
ρ ρ ρ ρ
∂ ∂
+ − = ∂ ∂
, (6a)
2
2
2
1 ( , , ) 0H zn z
nq H qρ ρ ω
ρ ρ ρ ρ
∂ ∂
+ − = ∂ ∂
, (6b)
where 2 2 2 2/H zq c qε ω⊥= − and 2 2
|| /E Hq q ε ε⊥= are the
square of the transverse (radial) wave number of elec-
tromagnetic wave of H- and E-types, respectively. The
corresponding equations for the axial spectral compo-
nents of the field in vacuum ( 0ρ ρ> ) outside the elec-
tron beam ( bρ ρ≠ ) are
2
2
2
( , , )1 0
( , , )
zn z
zn z
E qnq
H q
ρ ω
ρ
ρ ωρ ρ ρ ρ
∂ ∂
+ − = ∂ ∂
, (6c)
where 2 2 2 2/ zq c qω= − . If 2 2 2, , 0H Eq q q > , the equations
(6) have the form of the Bessel equations, whereas if
2 2 2, , 0H Eq q q < they are the modified Bessel equations.
We are only interested in the waves, which have sur-
face behavior in vacuum. For these waves the condition
2 0q < is satisfied. Exactly, these waves are excited by
the beam of charged particles provided the Cherenkov
resonance 0zq vω = [14]. Indeed, for the nonrelativistic
electron velocities ( 1β << , where 0v cβ = is the di-
mensionless electron velocity) considered herein, we
have 2 2 2
zc qω << and 2 0q < . Taking into account the
aforesaid, we represent the expressions for the spectral
components of the electromagnetic field ( , , )zn zE qρ ω
and ( , , )zn zH qρ ω in the following form:
2
02
0
( ), 0
,
(| | ), 0
( , , )
(| | ) (| | ), ,
(| | ), ,
E
n n E E
E
n n E E
zn z E E
n n n n b
E
n n b
A J q q
A I q q
E q
B K q C I q
D K q
ρ
ρ ρ
ρ
ρ ω
ρ ρ ρ ρ ρ
ρ ρ ρ
>
≤ <= + < <
>
2
02
0
( ), 0
,
(| | ), 0
( , , )
(| | ) (| | ), ,
(| | ), ,
H
n n H H
H
n n H H
zn z H H
n n n n b
H
n n b
A J q q
A I q q
H q
B K q C I q
D K q
ρ
ρ ρ
ρ
ρ ω
ρ ρ ρ ρ ρ
ρ ρ ρ
>
≤ <= + < <
>
where ( )nJ u is the n th order Bessel function of the
first kind; ( )nI u and ( )nK u are the modified functions
of the first kind (Infeld function) and the second kind
(Macdonald function), respectively; ,E H
nA , ,E H
nB , ,E H
nC ,
and ,E H
nD are the arbitrary constants. The choice of the
solution is due to the fulfillment of finiteness conditions
for ( , , )zn zE qρ ω and ( , , )zn zH qρ ω at 0ρ → and
ρ →∞ . At 2 1ε β⊥ > the radial distribution of the field
component ( , , )zn zH qρ ω inside the cylinder is de-
scribed by Bessel functions ( )n HJ q ρ , and at 2 1ε β⊥ <
it is described by modified Bessel functions
(| | )n HI q ρ . Using the Maxwell equations, we express
transverse spectral components of the electromagnetic
fields in the cylinder region ( 0ρ ρ< ), as well as in the
annular gap ( 0 bρ ρ ρ< < ), and on the other side of the
beam ( 0ρ ρ> ) via the components ( , , )zn zE qρ ω and
( , , )zn zH qρ ω .
We note that in the nonrelativistic case, if 2 1β << ,
but 2 1ε β⊥ > , the discontinuities of the tangential mag-
netic field components ( , , )n zH qϕ ρ ω and ( , , )zn zH qρ ω
at the beam surface ( bρ ρ= ) are small values of the
order of ( )O β . Therefore, in what follows, in the
boundary conditions at the beam surface ( bρ ρ= ), we
suppose these components are continuous, and take into
account only the discontinuity of the normal (radial)
electric field component ( , , )n zE qρ ρ ω .
Assuming the beam is nonrelativistic, and satisfying
the above-mentioned boundary conditions at the cylin-
der and electron beam surfaces, we obtain the following
dispersion equation for the beam-cylinder coupled
waves:
2 2 2
0Δ [( ) Γ( , ) ]n z z b bq v q nω ω αω− − = , (7)
where 2
04 /b e N mω π= is the plasma frequency of
beam electrons, zΓ(q , )n is the depression factor of
space-charge forces [15], found to be
2 2 2
z
0
0
Γ(q , ) ( ) (| | ) (| | )
(| | ) (| | )
1 .
(| | ) (| | )
z b n z b n z b
n z n z b
b n z b n z
n n q I q K q
I q K qa
I q K q
ρ ρ ρ
ρ ρ
ρ ρ ρ
= + ×
× −
The value α is the coupling factor of the beam with
cylinder eigenmodes that has the form
2
2 2 2
2 2 2
0 0
(| | )
( )
(| | )
Hn z b
z b n
b z n z
K qa n q
q K q
ρ
α ρ
ρ ρ ρ
= + ∆ ,
2 H E
n n n na∆ = −∆ ∆ ,
2
2
2 2 2
0
( 1)z
n
H
nqa
q q c
ω ε
ρ
⊥ −
=
,
||0 0
0 0 0 0
(| | ) ( )1Δ
| | (| | ) ( )
E n n E
n
n E n E
K q J q
q K q q J q
ερ ρ
ρ ρ ρ ρ
′ ′
= + ,
0 0
0 0 0 0
(| | ) ( )1 1Δ
| | (| | ) ( )
H n n H
n
n H n H
K q J q
q K q q J q
ρ ρ
ρ ρ ρ ρ
′ ′
= + .
The primed cylindrical functions denote their deriva-
tives with respect to the argument. Note that equation
(7) has the form analogous to the characteristic equation
of a traveling-wave tube [15]. In our case, it describes
the interaction of the beam space-charge waves (SCWs)
with the cylinder eigenmodes. Dispersion equations for
the beam SCWs and the cylinder eigenmodes are de-
scribed by the following equations:
2 2
0( ) Γ( , ) 0z z bq v q nω ω− − = , and 0n∆ = .
The solutions of the equation 0n∆ = determine the
eigenfrequencies ns ns nsiω ω ω′ ′′= − , 0nsω′′ ≥ , of the cylin-
drical waveguide with the hybrid E- and H-type waves.
The azimuthal mode index n = 0, 1, 2, 3, … corre-
sponds to half the number of field variations in the angle
ϕ . The radial index s represents the number of field
variations along the radial coordinate ρ and corre-
sponds to the pair of roots order number of the equation
0n∆ = , whose solutions determine the frequencies nsω
of the cylinder eigenmodes with the longitudinal wave
ISSN 1562-6016. ВАНТ. 2018. №4(116) 6
number zq . In the case of azimuthally-homogeneous
symmetric ( n = 0) waves and axially-homogeneous
( zq = 0) oscillations, the indices s correspond to the
root order numbers of the homogeneous dispersion
equations 0H
n∆ = and 0E
n∆ = , on which the equation
0n∆ = splits. In the dispersion equation 0n∆ = the
value na plays the role of the coupling constant be-
tween the E- and H-waves.
The dispersion dependences 0 ( )s zqω of the symmet-
ric eigenmodes H0s and E0s of a solid-state cylinder are
determined by the solutions of the dispersion equations
0 0H∆ = and 0 0E∆ = , respectively. The solutions of the
equation 0n∆ = at n ≠ 0 determine the dispersion de-
pendences ( )ns zqω of the hybrid EHns (H-type) or HEns
(E-type) eigenmodes of the waveguide. A unique corre-
spondence of these equation solutions to a specified
type of wave (H- or E-type) can be identified only after
determining the dominant longitudinal field component,
in other words after comparing the maximum values of
the moduli ( , , )zn z nsH qρ ω and ( , , )zn z nsE qρ ω [9]. In
the case of HEns mode the constant H
nA is determined
through the constant E
nA , and vice versa, in the case of
EHns mode the constant E
nA is determined through the
constant H
nA .
The quantities contained in equation (7) correspond
to the cylinder eigenmodes with transverse wave num-
bers for which the conditions 2 0Hq > and 2 0Eq > are
satisfied. In the case of 2 0Hq < and 2 0Eq < , the terms
0 0 0( ) / ( )n H H n HJ q q J qρ ρ ρ′ and || 0 0 0( ) / ( )n E E n EJ q q J qε ρ ρ ρ′
in H
n∆ and E
n∆ in the equation (7) acquire the form
0 0 0(| | )/ | | (| | )n H H n HI q q I qρ ρ ρ′− and
|| 0 0 0(| | )/ | | (| | )n E E n EI q q I qε ρ ρ ρ′− , respectively.
Depending on the signs of 2
Hq and 2
Eq , the
eigenmodes of the waveguide have different types (Ta-
ble). In Table, the type classification of eigenmodes is
given in accordance with the terminology in [16 - 18].
Types of eigenmodes of a solid-state cylinder located in vacuum
The sign of the square of the
transverse wave number
The sign of the
permittivity Type of eigenmodes Reference
2
Hq 2
Eq ( )ε ω⊥ || ( )ε ω
+ + + + bulk-surface [16]
+ – + – surface and/or bulk-surface symmetric,
pseudo-surface hybrid
[16 - 18]
– + + – bulk-surface [16]
– + – + bulk-surface [16]
– – – – surface [17, 18]
– – + + do not exist (forbidden zone)
We note that the pseudo-surface axial-homogeneous
( zq = 0) eigenmodes and pseudo-surface azimuthally
homogeneous symmetric ( n = 0) eigenmodes do not
exist because they are hybrid. The absence of cylinder
eigenmodes is determined by the absence of solutions of
the dispersion equation 0n∆ = . In this case, the corre-
sponding frequency and wavenumber regions form for-
bidden zones in the spectra of the waveguide waves.
When the cylinder is absent in the electrodynamic
system, i.e. in the case of 0 0ρ → , we have Δ 0nα → ,
and the solutions of the dispersion equation (7) deter-
mine the frequencies of the slow (ω− ) and fast (ω+ )
beam SCWs: 0 0 ( )z z bq v R q ,nω ω− = − and
0 0 ( )z z bq v R q ,nω ω+ = + , where 0 0( ) Γ ( )z zR q ,n q ,n=
is the reduction factor [15], and
0
0 0
2 2 2
Γ ( ) Lim Γ( )
( ) (| | ) (| | ).
z z
z b n z b n z b
b
q ,n q ,n
a n q I q K q
ρ
ρ ρ ρ
ρ
→
= =
= +
Consequently, the phase velocities of the slow
( / zqω− ) and fast ( / zqω+ ) SCWs are less and greater
than the beam velocity 0v , respectively.
When the beam electrons move along the lateral sur-
face of a cylindrical solid-state waveguide ( 0bρ ρ= ) or
close to it ( 0bρ ρ≠ ) under the condition that the re-
duced plasma frequency of the beam
( , ) | |z bR q n ω δω<< , where ( , ) ( , )z zR q n q n= Γ and
δω are small additions to the frequencies nsω that arise
due to interaction between the beam and cylinder
eigenmodes, the Cherenkov effect, under which
0ns zq vω = , is realized in an electrodynamic system [8].
The instability increments of the beam-cylinder coupled
waves are expressed as follows [8]:
1 3
2 3( )3Im
2 Δ ( )
ns
b
n nsω
α ω
δω ω
ω
=
′
, (8)
where ( )nsα ω is the coupling factor α at the resonance
frequency nsω . We note that 1/3
0Im Nδω ∝ . Conse-
quently, the excitation of the cylinder eigenmodes by
resonance beam particles (whose velocity satisfies the
condition 0ns zq vω = ) is coherent [19].
If the electron beam is transported at a considerable
distance from the cylindrical surface of the waveguide
( 0bρ ρ> ), the anomalous Doppler effect is realized in
the system. In this case, the resonance interaction of the
beam with the cylinder eigenmodes is realized at fre-
quencies 0 ( , )ns z z bq v R q nω ω ω±
±= = ± [8]. The instabil-
ity arises only in the interaction of slow space-charge
waves with the cylinder eigenmodes. The instability
increments are determined as follows [8]
ISSN 1562-6016. ВАНТ. 2018. №4(116) 7
1 2
( )
Im
2 ( , )Δ ( )
ns b
z n nsR q n ω
α ω ω
δω
ω
−
−
= ′
. (9)
It follows that 1 4
0Im Nδω ∝ .
For a fundamental understanding of the interaction
mechanism between the charged particles of a tubular
beam and cylinder eigenmodes, below we present the
results of numerical analysis of the dispersion equation
(7), and the expressions for the instability increments
(8) and (9).
2. NUMERICAL ANALYSIS
OF THE DISPERSION EQUATION
The dispersion equation 0n∆ = has dimensionless
form, which emphasizes its universality. The dimen-
sionless form of the waveguide eigenfrequencies is pro-
vided by their normalization to the characteristic fre-
quency 0 0/cω ρ= , taking into account the identity of
the cylindrical waveguide configuration.
We suppose that the cylindrical solid-state wave-
guide under study has the characteristic frequency
0ω = 6⋅1010 s-1, which corresponds to the radius
0ρ = 0.5 cm, and is made of an artificial material with
following parameters: 0ε = 2, 0/Lω ω = 3.5,
0/rω ω = 4, 0/sω ω = 6. The values of the equilibrium
beam electron density 0N , the radial thickness of the
beam a , and the directed motion velocity of the beam
electrons are chosen as follows: 0N = 7.6⋅1010 cm-3, and
a =0.05 cm, and 0v = 0.3 c , respectively. For the se-
lected system parameters, we have 2 2
0bω ω ≈ 0.07. The
normalized frequencies || 0/ω ω and 0/ω ω⊥ , at which
|| ( )ε ω = 0 and ( )ε ω⊥ = 0, have values 6.49 and 4.7,
respectively.
2.1. SPECTRA OF THE CYLINDER
EIGENMODES
Fig. 2 shows the spectra of the cylinder symmetric
( 0n = ) and the unsymmetrical ( 0n ≠ ) eigenmodes.
a b
Fig. 2. Dispersion dependences of the symmetric (a)
and the hybrid (b) eigenmodes of the cylinder
Lines 1 correspond to the frequencies and the longi-
tudinal wave numbers at which the transverse wave
numbers Hq vanish. Line 2 refers to the light line in
vacuum 0 0/ zqω ω ρ= when q = 0. We are only inter-
ested in the ranges of frequencies and longitudinal
wavenumbers of the waveguide eigenmodes where the
condition 2q < 0 is satisfied. Lines 3 and 4 correspond
to the frequencies || 0/ω ω and 0/ω ω⊥ . Lines 5 and 6
are for the normalized eigenfrequencies of the oscilla-
tors of artificial material in perpendicular ( 0/rω ω ) and
parallel ( 0/sω ω ) directions to the symmetry axis of the
cylinder, respectively. Lines 7 and 8 represent the spec-
tra of the H- and E-type bulk-surface waves, for which
2
Eq > 0 and 2
Hq > 0, namely, the symmetric modes H0 1,
E0 1, H0 2, H0 3 and E0 2 (Fig. 2,a), and the hybrid modes
EH1 1, HE1 1, EH1 2, EH1 3 and HE1 2 (Fig. 2,b) that are
arranged in ascending order of frequencies nsω . Note
that the density of the dispersion curves of the H-type
eigenmodes increases with radial index s in the fre-
quency range 0 rω ω< < at 0ns rω ω→ − (from be-
low). In doing so, the wave number Hq , which enters
into the argument of the Bessel function 0 0( )HJ q ρ in
the dispersion equation 0 0H∆ = , changes from 0Hq =
at 0ω = and 0zq = to Hq →∞ at rω ω= . The num-
ber of the E-type eigenmodes remains finite in the same
frequency range. This is because the transverse wave
number Eq , which enters into the argument of the Bes-
sel function 0 0( )EJ q ρ in the dispersion equation
0 0E∆ = , changes from 0Eq = at 0ω = and 0zq = to
|| ( ) /E r rq cω ε ω= at rω ω= . Curves 9 correspond to
the dispersion dependences of the bulk-surface symmet-
ric E0s (see Fig. 2,a) and the hybrid HE1s (see Fig. 2,b)
modes with s = 3, 4, 5, 6 in the frequency range
rω ω ω⊥< < . The density of the dispersion curves of
the waveguide eigenmodes increases with radial index
s when their frequencies nsω tend to the frequency ω⊥
from below ( 0nsω ω⊥→ − ). In the frequency range
||sω ω ω< < the dashed parts of the dispersion branches
10 correspond to the surface symmetric E-type waves in
Fig. 2,a and the pseudo-surface hybrid HE1 1 waves in
Fig. 2,b for which 2
Eq < 0, 2
Hq > 0 and || ( )ε ω < 0,
( )ε ω⊥ > 0. In Fig. 2,a, the branch of the surface waves
(curve 10) intersects the curve Hq = 0 (curve 1) and
converts to the branch of the bulk-surface E0 1 waves for
which 2
Eq > 0. Note that the dispersion equation 0E
n∆ =
has no solutions at the very point of intersection. In
Fig. 2,b the branch of the pseudo-surface hybrid HE1 1
wave converts continuously to the branch of the bulk-
surface HE1 1 wave at ||ω ω> . Lines 11 represent the
spectra of the bulk-surface symmetric E0s (see Fig. 2,a)
and the hybrid HE1s (see Fig. 2,b) modes. The E0 1 and
the HE1 1 modes have the lowest frequencies in the fre-
quency range sω ω ω⊥ < < , whereas the E0 2 and the
HE1 1 modes have the highest frequencies in the fre-
quency range ||sω ω ω< < . In the frequency range
sω ω ω⊥ < < the density of the dispersion curves of the
waveguide eigenmodes increases with radial index s at
0ns sω ω→ − when || ( )ε ω → +∞ , whereas in the fre-
quency range ||sω ω ω< < the density of the corre-
ISSN 1562-6016. ВАНТ. 2018. №4(116) 8
sponding curves increases with radial index s at
0ns sω ω→ + when || ( )ε ω → −∞ . The dispersion
curves go from one frequency band to another through
the zero forbidden zone (the point of intersection of the
branches and the curve Hq = 0 in Fig. 2) [20]. Under
the transition of the branches of the symmetric modes
between frequency bands, the radial indices are in-
creased by one in the direction of increasing frequencies
0sω . The series of curves 12 in Fig. 2,a show the spectra
of the bulk-surface symmetric eigenmodes H0s with
s = 1, 2, and they are arranged in order of increasing
frequencies 0sω . In Fig. 2,b the dispersion branch 12
refers to the pseudo-surface waves (dashed line) in the
frequency range ||sω ω ω< < , whereas at frequencies
||ω ω> it is for the bulk-surface hybrid EH1 1 waves
(dash-and-dot line). Note that the conversion of the
pseudosurface waves into the bulk-surface ones at the
frequency ||ω , when ||ε = 0, accompanies the above
mentioned continuous transition of the dispersion
branch from one frequency range to another. Lines 13
are the dispersion curves of the bulk-surface symmetric
E0s waves ( s = 1, 2 in order of increasing frequencies
0sω ) in Fig. 2,a and the hybrid EH1 2 waves in Fig. 2,b.
As seen from Fig. 2, the E-type bulk-surface waves
(curves 9) in the frequency range rω ω ω⊥< < have
negative group velocities and their dispersion depend-
ences approach asymptotically the straight line
0 0/ / 0rω ω ω ω= + at 0zq ρ →∞ . It is worthwhile to
emphasize that the H0s eigenmodes do not exist in this
frequency range because the dispersion equation
0 0H∆ = has no solutions there. The frequencies and the
longitudinal wave numbers of the symmetric E0s modes
are determined by the solutions of the dispersion equa-
tion 0 0E∆ = . Beyond the range rω ω ω⊥< < , the wave-
guide eigenmodes possess the positive group velocities.
In the frequency range rω ω< at 0zq ρ →∞ the disper-
sion curves of the bulk-surface eigenmodes approach
asymptotically the straight line 0 0/ / 0rω ω ω ω= − ,
whereas in the frequency range ||sω ω ω< < they ap-
proach the straight line 0 || 0/ / 0ω ω ω ω= − . It is inter-
esting to note that in the frequency range ||sω ω ω< <
the surface, the pseudosurface, and the bulk-surface
waves exist simultaneously at one and the same fre-
quency, but have different wavenumbers.
The shaded areas in Fig. 2 show the regions of fre-
quencies and wave numbers where the eigenmodes do
not exist in the waveguides under consideration (so-
called, forbidden zones). In these regions the corre-
sponding dispersion equations 0n∆ = have no solu-
tions. In addition, the frequency band rω ω ω⊥< < is
forbidden for the H-type waves.
Note that the qualitative behavior of the dispersion
dependences of cylinder eigenmodes with 1n > is simi-
lar to the dependences for the modes with 1n = .
In Fig. 3, the radial distributions of the field compo-
nents | ( , , ) |zn zE qρ ω of the bulk-surface symmetric E0s
(see Fig. 3,a) and hybrid HE1s (see Fig. 3,b) eigenmodes
with the indices s = 1, 3 are shown as an example. The
distributions of the field axial components are normal-
ized to their maximum values. Dependences 1 corre-
spond to the E0 1 and HE1 1 waves with frequencies
rω ω< and positive group velocities. Dependences 2
correspond to the E0 3 and HE1 3 waves with frequencies
in the range rω ω ω⊥< < . Note that the radial indices
represent the number of the total field variations along
the radial coordinate, reading from the symmetry axis of
the waveguide.
a b
Fig. 3. Normalized field distributions of the spectral
axial components of the symmetric E0s (a) and hybrid
HE1s (b) eigenmodes ( s = 1, 3) of the cylindrical
waveguide along the radial coordinate
It should be noted that in practice the finite energy
losses in the cylinder material cause the predominant
existence of the eigenmodes with the radial indices
s = 1, 2, 3, whereas the eigenmodes with s > 3 are
decayed [9, 21]. In fact, a weak decaying of the wave-
guide eigenmodes is provided by the concentration of
their fields near the waveguide cylindrical surface. Such
properties are inherent in the modes with azimuthal in-
dices 1n >> , for example, in the whispering gallery
modes in quasi-optical structures [9].
The fact that the E-type bulk-surface eigenmodes of
the cylindrical waveguide under consideration possess
the negative group velocities in the frequency range
rω ω ω⊥< < is very important to practical applications
because the interaction of these waves with a tubular
beam of charged particles results in the absolute insta-
bility [7]. It is important to stress that these waves exist
in an anisotropic waveguide with permittivities || 0ε >
and 0ε⊥ < that provides 2 0Eq > and 2 0Hq < .
2.2. SPECTRA OF COUPLED WAVES:
ABSOLUTE AND CONVECTIVE INSTABILITIES
Let us ascertain the nature of the instability that oc-
curs in the Cherenkov resonant interaction between the
electron beam and the bulk-surface symmetric
eigenmodes of the cylindrical waveguide ( 0 0z sq v ω= )
under the conditions ( , ) | |z bR q n ω δω<< and an ex-
tremely small distance of the beam from the cylinder.
Henceforward, we suppose that 0bρ ρ= . To this end,
we will use the well-known Sturrock method [8, 19, 22]
in the small areas in the vicinities of intersection points
of the dispersion dependences of the cylinder
eigenmodes with the beam waves 0 0/ zqω ω β ρ= (of
the so-called resonance points).
ISSN 1562-6016. ВАНТ. 2018. №4(116) 9
We note that only the E-type eigenmodes are unsta-
ble because only their fields have a nonzero component
of the electric field 0 ( , , )z zE qρ ω with which the non-
relativistic beam electrons interact. All conclusions
about the nature of the instabilities remain valid also for
the excitation of bulk-surface unsymmetrical modes of
the cylindrical waveguide in the small areas in the vicin-
ities of the corresponding resonance points.
Fig. 4 shows the dispersion curves corresponding to
the symmetric eigenmodes of the cylindrical waveguide,
and to the waves being radiated by the beam electrons,
and to the space-charge waves of the beam. Lines 1, 2,
4, 5 and curves 7 - 9 have the same physical meaning as
those in Fig. 2. Line 3 is for the beam waves with fre-
quencies 0zq vω = . Lines 6 and 10 show the spectra of
the slow and the fast space-charge waves of the beam,
respectively. Points A and B correspond to the intersec-
tions of the dispersion dependence of the beam waves
with the dispersion curves of the bulk-surface waves
E0 3 and E0 1 in the frequency ranges rω ω ω⊥< < and
0 rω ω< < , respectively. The coordinates of these
points ( 0 ,z resqρ , 0/resω ω ) refer to the Cherenkov reso-
nances of the particle-wave type ( 0 0z sq v ω= ) [23]. The
group velocities of the symmetric electromagnetic
waves E0s are determined as follows [8]:
,
1
z z res
res
E E
n n
gr
z q q
v
q
ω ω
ω
−
=
=
∂∆ ∂∆
= − ∂ ∂
,
where the partial derivatives /E
n zq∂∆ ∂ and /E
n ω∂∆ ∂
are calculated at the resonance points ( ,z resq , resω ).
Fig. 4. Dispersion dependences of the cylinder symmet-
ric modes [curves (7)-(9)], and the beam waves (3),
and the slow (6) and fast (10) space-charge waves
of the beam
In Fig. 4, the intersection points of the dispersion
dependence of the slow SCWs (straight line 6) with the
dispersion curves of the bulk-surface waves E0s in the
frequency range rω ω ω⊥< < (curves 9) are of special
interest. These points with coordinates ( 0 ,z resqρ ,
0/resω ω ) refer to the resonances with anomalous Dop-
pler effect of the slow bulk-surface waves with the cyl-
inder symmetric modes ( 0 0( , )z z b sq v R q n ω ω−− = ).
Fig. 5 presents the dispersion dependences of the
waves excited by the beam in the small areas in the vicini-
ties of point A with coordinates 0 ,z resqρ ≈ 13.79,
0/resω ω ≈ 4.13 (see Fig. 5,a), and point B with coordinates
0 ,z resqρ ≈ 13.016, 0/resω ω ≈ 3.896 (see Fig. 5,b). These de-
pendences are the solutions of the following equation [8]:
,
1
2 2
0
0
( ) ( )
z z res
res
E
n
z gr z b
q q
av q v q
ω ω
δω δ δω δ ω
ρ ω
−
=
=
∂∆
− − = ∂
,(10)
where zqδ is the small variation of the corresponding lon-
gitudinal wave number ,z resq . Note that equation (10) is
the result of the transformation of equation (7) in the small
areas in vicinities of resonance points. Lines 1 and 2 refer
to the values 0zqδ = and 0δω = , respectively. Line 3 is
for the asymptote 0 0/ ( / )gr zv c qδω ω ρ δ= , and line 4 is
for the beam wave 0 0/ zqδω ω βρ δ= . Curves 5 and 6 are
for the bulk-surface modes E0 3 (see Fig. 5,a) and E0 1 (see
Fig. 5,b) excited by the beam in the frequency ranges
rω ω ω⊥< < and 0 rω ω< < , respectively.
a b
Fig. 5. Dispersion curves of the coupled bulk-surface
symmetric waves E0 3 and E0 1 excited by the beam
in the small areas in the vicinities of points A (a)
and B (b) in Fig. 4, respectively
Since the equation (10) is a cubic one, then, as
known, it has three different real roots or one real root
and two conjugate complex roots. As one of these com-
plex roots has positive imaginary part, the instability
develops. From Fig. 5, it follows that the instabilities
occur at values 0 0 ,0z zq qρ δ ρ δ< and hold up to values
0 zqρ δ → −∞ . It is also clearly seen that that asymp-
totes 3 and 4 are inclined in different directions in
Fig. 5,a and in the same direction in Fig. 5,b with re-
spect to line 2. The negative slope of asymptote 3 in
Fig. 5,a and the positive slope of analogous asymptote
in Fig. 5,b are caused by the negative and positive val-
ues of the group velocities of the E0 3 and E0 1 modes in
the frequency ranges rω ω ω⊥< < ( /grv c ≈ –1.35⋅10-2)
and 0 rω ω< < ( /grv c ≈ 1.7⋅10-2), respectively. In ac-
cordance with the Sturrock rule [19, 22], this signifies
the occurrence of the absolute and convective instabili-
ties in corresponding frequency ranges.
Note that the absolute and convective instabilities
are used for the generation and amplification of elec-
tromagnetic oscillations, respectively [15, 19, 24, 25].
2.3. ANALYSIS OF INSTABILITY INCREMENTS
Let us dwell on the dependences of instability in-
crements Imδω for the E-type coupled bulk-surface
waves on the values of azimuthal n and radial s mode
indices in the frequency range rω ω ω⊥< < . These in-
crement values are calculated using formula (8) under
the Cherenkov resonance conditions (when 0bρ ρ= )
and the formula (9) under the conditions of the reso-
nance with anomalous Doppler effect (when 0bρ ρ> ).
ISSN 1562-6016. ВАНТ. 2018. №4(116) 10
The values of the absolute instability increments
0Im /ω ω of the excited bulk-surface modes E0s and
HEns with azimuthal indices in the range n = 1…20 for
the radial indices s = 3, 4, 5 are shown in Fig. 6. Note
that the dispersion dependences for the modes with
n = 0, 1 (curves 9) are only shown in Figs. 2 and 4.
Fig. 6,a shows the increments of the waves excited by
the beam under the Cherenkov resonance conditions,
when the resonant interaction between the beam elec-
trons and the eigenmodes of the solid-state cylindrical
waveguide takes place. Fig. 6,b presents the increments
of excited waves under the conditions of the resonance
with anomalous Doppler effect, when the interaction
between the beam SCWs and the waveguide
eigenmodes holds. The increment values are grouped in
accordance with the radial index s of cylinder
eigenmodes. The dependences of the increment values
of the E0s and HEns modes with the radial indices s = 3,
4, 5 on the azimuthal index n are labeled by the num-
bers 1, 2, and 3, respectively. As evident from Fig. 6, in
the frequency range rω ω ω⊥< < the instability incre-
ment values of the E-type under the Cherenkov reso-
nance conditions are two orders of magnitude higher
than for the Doppler effect. The hybrid modes with
three field variations along the radial coordinate (HEn3
modes) have the maximum increments. In Fig. 6, the
presented dependences have extreme maxima. It is no-
table that these maxima belong to the coupled hybrid
HEns modes of the whispering gallery [9]. As seen from
Fig. 6, as the radial index s increases, the azimuthal
index n of the mode with the maximum increment de-
creases.
a b
Fig. 6. Increment values of the absolute instability
of the coupled bulk-surface waves E0s and HEns under
the Cherenkov resonance conditions (a)
and under the conditions of the resonance
with anomalous Doppler effect (b)
In Fig. 7, the dependences of the instability incre-
ment values of excited E03 and HEn3 modes on the azi-
muthal index n are shown for different radial distances
between the cylinder and the electron beam 0ρρ −b ,
when the anomalous Doppler effect is realized. The
dependences corresponding to the values 0ρρ −b = 0.1,
0.11 and 0.14 cm are labeled by the numbers 1, 2 and 3,
respectively. It is seen that the coupled whispering gal-
lery modes, which are excited by a tubular electron
beam moving at a minimum distance above the cylin-
der, possess the greatest values of the increments. The
increment values of excited waves decrease with the
increase of the distance between the electron beam and
the cylinder 0ρρ −b . In represent dependences, the
HE9 3 mode has the greatest increment. From Fig. 7 it
follows that the azimuthal index n of the mode pos-
sessed the maximum increment decreases with the in-
crease of 0ρρ −b . It is equivalently that the frequency
of the most nonstable wave decreases.
Fig. 7. Instability increment values of the system
with the coupled bulk-surface waves E03 and HEn3
at different distances 0bρ ρ− between the cylinder
and the electron beam
Thus, the analysis of the absolute instability of the
system under consideration suggests that the instability
occurs in the frequency range rω ω ω⊥< < where
( ) 0ε ω⊥ < and || ( ) 0ε ω > , and the largest values of the
increments correspond to the bulk-surface hybrid whis-
pering gallery modes HEn3.
CONCLUSIONS
The instability of a nonrelativistic tubular electron
beam that moves in vacuum above an anisotropic solid-
state cylinder has been theoretically examined. It has
been assumed that an electron beam is infinitely thin in
the radial direction and the components of the cylinder
permittivity tensor possess the frequency dispersion.
The dispersion equations for eigenmodes of the cylinder
and for the coupled modes of the system have been de-
rived. The analysis of the eigenmode properties and the
classification of the eigenmodes have been performed.
The spectra of the cylinder symmetric and unsymmet-
rical eigenmodes have been determined. It has been
revealed that the bulk-surface waves of the E-type have
negative group velocities over a certain frequency
range. It has been shown that the existence of such
waves is caused by the anisotropic properties of the
waveguide with the components of the permittivity
0ε⊥ < and || 0ε > , which provide fulfillment of the
conditions 2 0Hq < and 2 0Eq > in a unique fashion. The
existence of the E-type surface symmetric waves and
the pseudo-surface hybrid waves of the E- and the H-
types in the cylindrical solid-state waveguide has been
shown. It has been found that there are frequency ranges
where the surface waves, the bulk-surface waves and
the pseudo-surface hybrid waves can exist at one and
the same frequency, but with different wave numbers.
The ranges of both the frequencies and the wave num-
bers where the eigenmodes in the waveguide under
study cannot exist have been specified (so-called for-
bidden zones). The increments of the instabilities caused
by both the Cherenkov and the Doppler effects have
been analyzed. It has been demonstrated by the Sturrock
rules that the absolute and convective instabilities of the
E-type bulk-surface waves occur in different frequency
ranges. The numerical analysis of the dependences of
ISSN 1562-6016. ВАНТ. 2018. №4(116) 11
the instability increments of the system under considera-
tion with the symmetric and the bulk-surface hybrid
waves of the E-type on the values of azimuthal and ra-
dial mode indices for different distances between the
electron beam and the cylinder has been performed. It
has been established that the coupled bulk-surface
whispering gallery HEn3 modes excited by the beam
have the largest values of the increments.
Thus, the use of the anisotropic dispersive material
with the permittivity components ( ) 0ε ω⊥ < and || ( ) 0ε ω >
as the delaying medium makes possible the generation
of the bulk-surface waves over a certain frequency
range and eliminates the need for creating artificial
feedbacks in slow-wave structures.
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ISSN 1562-6016. ВАНТ. 2018. №4(116) 12
ВЗАИМОДЕЙСТВИЕ ТРУБЧАТОГО ПУЧКА ЗАРЯЖЕННЫХ ЧАСТИЦ
С АНИЗОТРОПНЫМ ДИСПЕРГИРУЮЩИМ ТВЕРДОТЕЛЬНЫМ ЦИЛИНДРОМ
Ю.О. Аверков, Ю.В. Прокопенко, В.М. Яковенко
Изучено взаимодействие нерелятивистского трубчатого потока заряженных частиц с немагнитной анизо-
тропной диспергирующей средой цилиндрической конфигурации. Обнаружена абсолютная неустойчивость
объёмно-поверхностных волн, обусловленная особенностями свойств анизотропного цилиндра. Резонанс-
ный характер частотных зависимостей диэлектрической проницаемости цилиндра приводит к появлению
участков дисперсионных кривых собственных объёмно-поверхностных волн E-типа с отрицательной груп-
повой скоростью. Показано существование в цилиндре собственных поверхностных волн E-типа и псевдо-
поверхностных волн E- и H-типов.
ВЗАЄМОДІЯ ТРУБЧАСТОГО ПУЧКА ЗАРЯДЖЕНИХ ЧАСТИНОК
З АНІЗОТРОПНИМ ДИСПЕРГУЮЧИМ ТВЕРДОТІЛЬНИМ ЦИЛІНДРОМ
Ю.О. Аверков, Ю.В. Прокопенко, В.М. Яковенко
Вивчено взаємодію нерелятивістського трубчастого потоку заряджених частинок з немагнітним анізот-
ропним диспергуючим середовищем циліндричної конфігурації. Виявлена абсолютна нестійкість об'ємно-
поверхневих хвиль, що обумовлена особливостями властивостей анізотропного циліндра. Резонансний ха-
рактер частотних залежностей діелектричної проникності циліндра призводить до появи ділянок дисперсій-
них кривих власних об'ємно-поверхневих хвиль E-типу з негативною груповою швидкістю. Показано існу-
вання в циліндрі власних поверхневих хвиль E-типу і псевдоповерхневих хвиль E- та H-типів.
INTRODUCTION
1. STATEMENT OF THE PROBLEM AND BASIC EQUATIONS
2. NUMERICAL ANALYSIS OF THE DISPERSION EQUATION
2.1. Spectra of the cylinder eigenmodes
2.2. Spectra of coupled waves: Absolute and convective instabilities
2.3. Analysis of instability increments
CONCLUSIONS
references
ВЗАИМОДЕЙСТВИЕ ТРУБЧАТОГО ПУЧКА ЗАРЯЖЕННЫХ ЧАСТИЦ С АНИЗОТРОПНЫМ ДИСПЕРГИРУЮЩИМ ТВЕРДОТЕЛЬНЫМ ЦИЛИНДРОМ
Взаємодія трубчастого пучка заряджених частинок з анізотропним диспергуючим твердотільним циліндром
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