Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field
The purpose of this paper is to study the electrodynamic properties of eigen dipolar electromagnetic waves of coaxial metal waveguide filled by slightly axially non-uniform and strongly radially non-uniform cold dissipative plasma immersed in non-uniform azimuth external magnetic field. The influenc...
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| Cite this: | Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field / N.A. Azarenkov, V.P. Olefir, A.E. Sporov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 77-80. — Бібліогр.: 8 назв. — англ. |
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Azarenkov, N.A. Olefir, V.P. Sporov, A.E. 2015-05-25T05:45:24Z 2015-05-25T05:45:24Z 2015 Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field / N.A. Azarenkov, V.P. Olefir, A.E. Sporov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 77-80. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 52.35g, 52.50.Dg https://nasplib.isofts.kiev.ua/handle/123456789/82093 The purpose of this paper is to study the electrodynamic properties of eigen dipolar electromagnetic waves of coaxial metal waveguide filled by slightly axially non-uniform and strongly radially non-uniform cold dissipative plasma immersed in non-uniform azimuth external magnetic field. The influence of external azimuth magnetic field, geometric parameters of the waveguide structure and the plasma electron collisions on the dispersion properties, spatial attenua-tion and radial wave field structure of these waves for different radial plasma density profiles is studied. Исследованы электродинамические свойства собственных дипольных электромагнитных волн, распространяющихся в коаксиальном металлическом волноводе, заполненном слабо неоднородной в аксиальном и сильно неоднородной в радиальном направлениях холодной диссипативной плазмой, находящейся во внешнем неоднородном азимутальном магнитном поле. Изучено влияние величины внешнего азимутального магнитного поля, геометрических параметров волноводной структуры, частоты столкновений электронов на дисперсионные свойства, пространственное затухание и радиальную структуру поля волны для различных радиальных профилей плотности плазмы. Досліджено електродинамічні властивості власних дипольних електромагнітних хвиль коаксіального металевого хвилеводу, заповненого слабко неоднорідною в аксіальному напрямку та сильно радіально неоднорідною холодною дисипативною плазмою, що знаходиться в зовнішньому неоднорідному азимутальному магнітному полі. Вивчено вплив величини зовнішнього азимутального магнітного поля, геометричних параметрів хвилеводної структури, частоти зіткнень електронів на дисперсійні властивості, просторове загасання і радіальну структуру поля хвилі для різних радіальних профілів густини плазми. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Фундаментальная физика плазмы Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field Собственные дипольные электромагнитные волны коаксиальных плазменно-металлических волноводов с неоднородной плазмой во внешнем азимутальном магнитном поле Власні дипольні електромагнітні хвилі коаксіальних плазмово-металевих хвилеводів з неоднорідною плазмою в зовнішньому азимутальному магнітному полі Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field |
| spellingShingle |
Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field Azarenkov, N.A. Olefir, V.P. Sporov, A.E. Фундаментальная физика плазмы |
| title_short |
Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field |
| title_full |
Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field |
| title_fullStr |
Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field |
| title_full_unstemmed |
Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field |
| title_sort |
eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field |
| author |
Azarenkov, N.A. Olefir, V.P. Sporov, A.E. |
| author_facet |
Azarenkov, N.A. Olefir, V.P. Sporov, A.E. |
| topic |
Фундаментальная физика плазмы |
| topic_facet |
Фундаментальная физика плазмы |
| publishDate |
2015 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Собственные дипольные электромагнитные волны коаксиальных плазменно-металлических волноводов с неоднородной плазмой во внешнем азимутальном магнитном поле Власні дипольні електромагнітні хвилі коаксіальних плазмово-металевих хвилеводів з неоднорідною плазмою в зовнішньому азимутальному магнітному полі |
| description |
The purpose of this paper is to study the electrodynamic properties of eigen dipolar electromagnetic waves of coaxial metal waveguide filled by slightly axially non-uniform and strongly radially non-uniform cold dissipative plasma immersed in non-uniform azimuth external magnetic field. The influence of external azimuth magnetic field, geometric parameters of the waveguide structure and the plasma electron collisions on the dispersion properties, spatial attenua-tion and radial wave field structure of these waves for different radial plasma density profiles is studied.
Исследованы электродинамические свойства собственных дипольных электромагнитных волн, распространяющихся в коаксиальном металлическом волноводе, заполненном слабо неоднородной в аксиальном и сильно неоднородной в радиальном направлениях холодной диссипативной плазмой, находящейся во внешнем неоднородном азимутальном магнитном поле. Изучено влияние величины внешнего азимутального магнитного поля, геометрических параметров волноводной структуры, частоты столкновений электронов на дисперсионные свойства, пространственное затухание и радиальную структуру поля волны для различных радиальных профилей плотности плазмы.
Досліджено електродинамічні властивості власних дипольних електромагнітних хвиль коаксіального металевого хвилеводу, заповненого слабко неоднорідною в аксіальному напрямку та сильно радіально неоднорідною холодною дисипативною плазмою, що знаходиться в зовнішньому неоднорідному азимутальному магнітному полі. Вивчено вплив величини зовнішнього азимутального магнітного поля, геометричних параметрів хвилеводної структури, частоти зіткнень електронів на дисперсійні властивості, просторове загасання і радіальну структуру поля хвилі для різних радіальних профілів густини плазми.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/82093 |
| citation_txt |
Eigen dipolar electromagnetic waves of coaxial non-uniform plasma-metall waveguide with external azimuth magnetic field / N.A. Azarenkov, V.P. Olefir, A.E. Sporov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 77-80. — Бібліогр.: 8 назв. — англ. |
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ISSN 1562-6016. ВАНТ. 2015. №1(95)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 77-80. 77
EIGEN DIPOLAR ELECTROMAGNETIC WAVES OF COAXIAL
NON-UNIFORM PLASMA-METALL WAVEGUIDE WITH EXTERNAL
AZIMUTH MAGNETIC FIELD
N.A. Azarenkov, V.P. Olefir, A.E. Sporov
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
E-mail: vpolefir@gmail.com
The purpose of this paper is to study the electrodynamic properties of eigen dipolar electromagnetic waves of coaxi-
al metal waveguide filled by slightly axially non-uniform and strongly radially non-uniform cold dissipative plasma
immersed in non-uniform azimuth external magnetic field. The influence of external azimuth magnetic field, geometric
parameters of the waveguide structure and the plasma electron collisions on the dispersion properties, spatial attenua-
tion and radial wave field structure of these waves for different radial plasma density profiles is studied.
PACS: 52.35g, 52.50.Dg
INTRODUCTION
At present time the intensive theoretical and experi-
mental studies of plasma, produced and sustained by the
eigen traveling along the waveguide structure electro-
magnetic waves and properties of these waves are car-
ried out in the leading scientific laboratories of the
world [1]. These studies are stipulated by the fact that
such waveguide systems of different radial structure are
widely used in the devices of plasma electronics [2] and
also as the discharge chambers in plasma-technological
processes [3]. The properties of eigen waves and plasma
are simultaneously determined by different factors but
the external magnetic field and the azimuth structure of
the electromagnetic wave considered exert substantial
influence on it [2, 3]. The eigen dipolar waves with
azimuth wave number m 1 are often used for vari-
ous technological applications in plasma electronics and
for sustaining discharges in cylindrical waveguides [4].
At the same time coaxial waveguide structures are wide-
ly used in different technological applications [5]. The
properties of eigen dipolar waves in cylindrical wave-
guide structure are studied well but propagation of such
waves in long strong radially non-uniform coaxial struc-
tures are studied insufficiently. These facts determine
the urgency of the presented study.
1. BASIC EQUATIONS
The considered waveguide structure is composed of
the central cylindrical metal conductor of radius 1R that
is immersed in the cylindrical plasma layer with outer
radius 2R . The vacuum gap 2 3R r R separates the
plasma layer from outer metal wall with radius 3R . The
direct current zJ flows along the central conductor and
produces the radially non-uniform azimuth magnetic
field )(0 rH . Plasma is considered in the hydrodynamic
approach as cold slightly dissipative medium with con-
stant effective collision frequency . It is supposed that
plasma density n varies slightly along the plasma col-
umn at the distances of wavelength order [1, 3, 4, 6]. It
is also considered that plasma density radial profile rn
along all plasma column has the form:
22
maxmax /exp)( rrrrnrn . The non-uniformity
parameter describes the plasma density shape and
varies from 0 (radially uniform plasma) to 1
(completely radially non-uniform plasma). The parame-
ter maxr is radial coordinate, where plasma density
reaches the maximum, and the parameter r characte-
rizes the width of profile [7]. In this research it is sup-
posed that maxr corresponds to the centre of plasma
layer and 2 10.1r R R . Under these assumptions
the permittivity tensor of collisional plasma in azimuth
magnetic field i , j was obtained in [8] with compo-
nents 3,2,1 which depend on radial position r [2] and
slightly depend on axial coordinate.
The dipolar wave propagation is governed by the
system of Maxwell equations that in cylindrical coordi-
nates , ,r z possesses the solutions in the form:
3
0
( , )exp [ ' ' ]
z
z
A r A r z i k z dz m t
, (1)
where A denotes the electric and magnetic wave field
component; is given wave frequency; 3k is axial
wave number. Due to slight plasma density changing in
axial direction at the distances of wavelength order we
follow the authors of [6] and neglect all the terms propor-
tional to 1
z
and higher derivatives, where
denotes all the quantities which slightly depend on z .
For the plasma layer 1 2R r R one can obtain
the system of ordinary differential equations that de-
scribes the radial distribution of tangential wave field
components and two algebraic equations which describe
the radial wave field components [8]. For arbitrary pa-
rameters of plasma region and waveguide structure the
solution of this system can be found with the help of
special numerical methods.
In the vacuum region 32 RrR the correspond-
ing system of Maxwell equations can be solved analyti-
cally [8] and wave field components can be expressed in
terms of linear combination of modified Bessel func-
tions [2]. Values 1,2,3,4C which are present in the expres-
sions for wave field components can be obtained with
the help of boundary conditions consisting in the conti-
nuity of tangential wave field components at plasma-
80 ISSN 1562-6016. ВАНТ. 2015. №1(95)
vacuum interface. The values of plasma wave field
components at plasma-vacuum interface ( 2r R ), can
be obtained by the direct numerical solution of the sys-
tem of differential equations [8].
The analogue of the local dispersion equation that con-
nects and 3k can be obtained from the boundary condi-
tions at 3r R and can be written in the following form:
0)()(
0)()(
3
'
43
'
3
3231
RKCRIC
RKCRIC
vmvm
vmvm
, (2)
where 22
3
2 kkv , ck / and iC are the values
which are present in the expressions for vacuum wave
field components and are connected with wave field
components in plasma due to boundary conditions.
2. MAIN RESULTS
The dipolar wave 1m possesses all six compo-
nents of electromagnetic wave field, so the solution of
the problem is rather hard and bulky. First the influence
of direct current value and waveguide geometric para-
meters on the phase properties of the wave for the case
of collisionless plasma ( / 0 ) is studied. The
dependence of the normalized parameter
max/ p r ( maxp r is electron plasma fre-
quency) that depends on the plasma density that slightly
varies along the cylindrical plasma column on the nor-
malized axial wavenumber 3 2Rex k R for different
normalized direct current values )2/( 3mceJj z is
shown on the Fig. 1. In the considered case the disper-
sion equation (2) possesses the number of solutions.
Three solutions of the eq. (2) with the larger values
for different direct current values are shown on the
Fig. 1.
Fig. 1. The solutions of the equations (2) p / on
the dimensionless wave number 3 2Rex k R under
the parameters values: 1.0/ 21 RR ; 5.0/2 cR ;
3 2/ 2.0R R . The numbers at the upper right corner of
the graph correspond to such current values:
1 1.5j ; 2 2.0j ; 3 4.0j
The presented results correspond to the eigen waves with
p . It's necessary to mention that the presented solu-
tions don't intersect with each other. These solutions
come close to each other, but they are essentially different
modes with different radial wave field structure.
Solutions which are represented on Fig. 1 correspond
to the eigen modes which have different radial wave
field structure especially in plasma region (Figs. 1,a,b)
and different dependence of phase and group velocities
on the wavenumber. The mode with the lower dimen-
sionless frequency under the fixed axial wavenumb-
er value possesses smaller scale length of spatial wave
field oscillations in radial direction (see Figs. 1,a; 1,b).
Fig. 1,a. The electric wave field components (norma-
lized on the 1H R ) for the first tree solutions of the
equations (2). The score parameters and numbering of
the curves are the same as for Fig. 1,
2/r R
Fig. 1,b. The magnetic wave field components (norma-
lized on the 1H R ) for the first tree solutions of the
equations (2). The score parameters and numbering of
the curves are the same as for Fig. 1,
2/r R
The radial distribution of the normalized wave field
components for the first three solutions for 2.0x are
presented on the Fig. 1,a (electric field components) and
Fig. 1,b (magnetic field components). The presented
wave field components are normalized on the 1H R .
The numerical study shows that the considered eigen
dipolar wave 1m is neither pure surface one nor pure
volume wave. This wave demonstrates complex radial
structure that corresponds to the pseudo-surface wave
according to the classification presented in [3]. This
means that wave field is composed from the surface and
78
ISSN 1562-6016. ВАНТ. 2015. №1(95) 79
volume radial mode. In our study we cannot separate this
radial modes but the effect is analogous to those in [3].
It is obtained that each of three displayed solutions
has different type of dependence of the frequency on the
normalized direct current value j (see subplots 1-3 on
Fig. 1). While the direct current j increases its value
from 1.5j up to 4.0j the first solution (solid curve
on Fig. 1) and the third solution (dashed curve) increases
its phase velocity on the whole range of wavenumber x
(this can be obtained from subplots 1-3). The second so-
lution (dotted curve) has different type of dependence. In
the range of small ( 1.2x ) and large ( 7.4x ) wave-
number values the increase of j value from 1.5 up to
4.0 leads to the increase of for the fixed x value. In
the range 1.2 7.4x the increase of the current j
leads to the decrease of wave phase velocities.
It is necessary to mention that the increase of direct
current j leads to the essential changing of the solu-
tions behavior. So, when j raises up to 4.0 the first
solution (solid curve on Figs. 1; 1,a; 1,b) becomes well
separated from the others.
Fig. 2. The dependence of dimensionless attenuation
coefficient on the dimensionless wave number x for
the first root (solid curve on Fig. 1). Numbers just near
the curves correspond to the different values:
1 0 ; 2 - 0.005 ; 3 0.01 ; 4 0.05 ;
5 0.1 . Other parameters are the same
as for Fig. 1, except 4.0j
The influence of geometric parameters of the wave-
guide structure on the dipolar wave dispersion and at-
tenuation is studied as well. The width of vacuum gap
that is characterized by the parameter 3 2/R R
strongly influences mainly the wave dispersion in the
range of small and moderate values ( 2.0 ). When
parameter grows up to rather large values ( 2 ) it
has negligible influence on the dispersion. The numeri-
cal calculations show that the variation of parameter
slightly influences on the attenuation coefficient .
The influence of effective electron collision fre-
quency on the spatial attenuation coefficient
13)Im( Rk is studied for the first solution (solid
curve on the Fig. 1) and is presented at the Fig. 2. It is
obtained that the increase of the value leads to the
increase of the absolute value of wave attenuation coef-
ficient. It is necessary to mention that characteristic fea-
ture of this solution is negative value of spatial attenua-
tion coefficient . This fact means that this wave can-
not be effectively used for gas discharge sustaining, but
the waves of such type are widely used in plasma elec-
tronics [2].
Fig. 3. The dependence of frequency on the wave
number x for the first root (solid curve on fig. 1). Num-
bers of the curves correspond to the non-uniformity pa-
rameter value: 1 0 ; 2 0.2 ; 3 0.4 ;
4 0.6 ; 5 0.8 ; 6 1 . Other parameters
are the same as for Fig. 1,
except 4.0j and 0.001
The influence of plasma density non-uniformity on
the dispersion and attenuation properties at gradually
increase of the non-uniform parameter from 0 up to
1 is studied. The results of numerical study of plasma
density non-uniformity on the wave dispersion for some
parameter values are presented on the Fig. 3. It is
shown that the dispersion has different behavior in the
case of uniform (curve 1) and non-uniform plasma
(curves 2-6). For the non-uniform plasma the increase
of the non-uniformity parameter leads to the decrease
of the dipolar wave phase velocity in the region of short
wave lengths (curves 2-6 in the range 6x ). In the
region of long wave lengths (curves 2-6 in the range
5x ) the increase of parameter leads to the increase
of the phase velocity.
Fig. 4. The dependence of spatial attenuation coefficient
on the wave number x for the first root (solid curve
on Fig. 1). Numbers near the curves and parameters
values are the same as for Fig. 3
The results of studying the influence of radial plasma
density non-uniformity on the wave attenuation are pre-
sented on the Fig. 4. Similarly to the dispersion of dipo-
lar wave the spatial attenuation coefficient has dif-
ferent behavior in the case of uniform (curve 1) and
non-uniform plasma (curves 2-6). The absolute value of
80 ISSN 1562-6016. ВАНТ. 2015. №1(95)
the coefficient reaches the minimum at axial wave
number 2x .
In the region of long wave lengths ( 2x ) it is ob-
served the strong dependence of the increase of absolute
value coefficient on the wave length. In the region of
short wave lengths ( 2x ) has smooth dependence
on the wave length. It is obtained that the takes on
some minimum value at some 0 value. This can be
explained by the fact that under given parameters the
electric and magnetic field strength raises its maximum
in the region where plasma density tends to zero. Fur-
ther growth of the parameter value leads to the gra-
dual concurrence of the radial positions of the maximum
values of plasma density and wave field amplitude. This
process causes the increase of spatial wave attenuation
due to the Joule wave energy losses under plasma densi-
ty non-uniformity parameter growth.
CONCLUSIONS
This paper presents the developed new method of
calculation of eigen waves of coaxial waveguides par-
tially filled by the dissipative non-uniform plasma im-
mersed in azimuth external magnetic field. It is shown
that the number of eigen dipolar modes can propagate in
the considered coaxial waveguide structure. These mod-
es differ by their phase and spatial attenuation properties
and by their radial wave field structure. It is shown that
the solution with the greatest value is backward
wave with poor trend for gas discharge sustaining in
long coaxial waveguide structures, but it has good pros-
pect of usage in wave amplifiers. Other solutions pos-
sess the ranges of forward propagation that depends
mainly on the external azimuth magnetic field value and
demands on further study.
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2. A.N. Kondratenko, V.M. Kuklin. Osnovy Plasmen-
noy Electriniky. M: "Energoatomizdat", 1988 (in Rus-
sian).
3. I. Zhelyazkov, V. Atanasov. Axial structure of low-
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4. M. Djourelova, T. Petrova, I. Ghanashev,
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Article received 05.01.2015
СОБСТВЕННЫЕ ДИПОЛЬНЫЕ ЭЛЕКТРОМАГНИТНЫЕ ВОЛНЫ КОАКСИАЛЬНЫХ
ПЛАЗМЕННО-МЕТАЛЛИЧЕСКИХ ВОЛНОВОДОВ С НЕОДНОРОДНОЙ ПЛАЗМОЙ
ВО ВНЕШНЕМ АЗИМУТАЛЬНОМ МАГНИТНОМ ПОЛЕ
Н.А. Азаренков, В.П. Олефир, А.Е. Споров
Исследованы электродинамические свойства собственных дипольных электромагнитных волн, распро-
страняющихся в коаксиальном металлическом волноводе, заполненном слабо неоднородной в аксиальном и
сильно неоднородной в радиальном направлениях холодной диссипативной плазмой, находящейся во внеш-
нем неоднородном азимутальном магнитном поле. Изучено влияние величины внешнего азимутального
магнитного поля, геометрических параметров волноводной структуры, частоты столкновений электронов на
дисперсионные свойства, пространственное затухание и радиальную структуру поля волны для различных
радиальных профилей плотности плазмы.
ВЛАСНІ ДИПОЛЬНІ ЕЛЕКТРОМАГНІТНІ ХВИЛІ КОАКСІАЛЬНИХ ПЛАЗМОВО-МЕТАЛЕВИХ
ХВИЛЕВОДІВ З НЕОДНОРІДНОЮ ПЛАЗМОЮ В ЗОВНІШНЬОМУ АЗИМУТАЛЬНОМУ
МАГНІТНОМУ ПОЛІ
М.О. Азарєнков, В.П. Олефір, О.Є. Споров
Досліджено електродинамічні властивості власних дипольних електромагнітних хвиль коаксіального ме-
талевого хвилеводу, заповненого слабко неоднорідною в аксіальному напрямку та сильно радіально неодно-
рідною холодною дисипативною плазмою, що знаходиться в зовнішньому неоднорідному азимутальному
магнітному полі. Вивчено вплив величини зовнішнього азимутального магнітного поля, геометричних па-
раметрів хвилеводної структури, частоти зіткнень електронів на дисперсійні властивості, просторове зага-
сання і радіальну структуру поля хвилі для різних радіальних профілів густини плазми.
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