Axial symmetric surface waves in tubular magneto-active plasma column
This paper is devoted to the dispersion properties of high-frequency axial symmetric potential surface waves propagating in a cylindrical waveguide structure. The structure is supposed to consist of a radially non-uniform plasma layer, partially filling a metal waveguide and immersed in an externa...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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nasplib_isofts_kiev_ua-123456789-817982025-02-09T14:39:12Z Axial symmetric surface waves in tubular magneto-active plasma column Аксиально-симетричные поверхностные волны в трубчатом столбе магнитоактивной плазмы Аксиально-симетричні поверхневі хвилі у трубчатому стовпі магнітоактивної плазми Akimov, Yu.A. Olefir, V.P. Azarenkov, N.A. Basic plasma physics This paper is devoted to the dispersion properties of high-frequency axial symmetric potential surface waves propagating in a cylindrical waveguide structure. The structure is supposed to consist of a radially non-uniform plasma layer, partially filling a metal waveguide and immersed in an external axial magnetic field. The influence of the waveguide structure parameters, as well as the magnetic field value, on frequency, phase and group velocities, resonance damping of the surface waves is investigated both numerically and analytically. В данной статье изучены дисперсионные свойства высокочастотных аксиально-симметричных потенциальных поверхностных волн, распространяющихся в цилиндрической волноводной структуре, состоящей из радиально неоднородного пламенного слоя, частично заполняющего металлический волновод и помещенного во внешнее аксиальное магнитное поле. Численно и аналитически исследуется влияние параметров волноводной структуры и внешнего магнитного поля на частоты, фазовые и групповые скорости поверхностных волн, а также на декременты их затухания. В роботі вивчено дисперсійні властивості високочастотних аксиально-симетричних потенціальних поверхневих хвиль, що розповсюджуються в циліндричній хвилеводній структурі, яка містить радіально неоднорідний плазмовий шар, що частково заповнює металевий хвилевод і знаходиться у зовнішньому аксіальному магнітно- му полі. Досліджено вплив параметрів хвилеводної структури та зовнішнього магнітного поля на частоти, фазові та групові швидкості поверхневих хвиль, а також на декременти їхнього загасання. 2006 Article Axial symmetric surface waves in tubular magneto-active plasma column / Yu.A. Akimov, V.P. Olefir, N.A. Azarenkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 121-123. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.35.Fp https://nasplib.isofts.kiev.ua/handle/123456789/81798 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Basic plasma physics Basic plasma physics |
| spellingShingle |
Basic plasma physics Basic plasma physics Akimov, Yu.A. Olefir, V.P. Azarenkov, N.A. Axial symmetric surface waves in tubular magneto-active plasma column Вопросы атомной науки и техники |
| description |
This paper is devoted to the dispersion properties of high-frequency axial symmetric potential surface waves propagating
in a cylindrical waveguide structure. The structure is supposed to consist of a radially non-uniform plasma layer,
partially filling a metal waveguide and immersed in an external axial magnetic field. The influence of the waveguide
structure parameters, as well as the magnetic field value, on frequency, phase and group velocities, resonance damping
of the surface waves is investigated both numerically and analytically. |
| format |
Article |
| author |
Akimov, Yu.A. Olefir, V.P. Azarenkov, N.A. |
| author_facet |
Akimov, Yu.A. Olefir, V.P. Azarenkov, N.A. |
| author_sort |
Akimov, Yu.A. |
| title |
Axial symmetric surface waves in tubular magneto-active plasma column |
| title_short |
Axial symmetric surface waves in tubular magneto-active plasma column |
| title_full |
Axial symmetric surface waves in tubular magneto-active plasma column |
| title_fullStr |
Axial symmetric surface waves in tubular magneto-active plasma column |
| title_full_unstemmed |
Axial symmetric surface waves in tubular magneto-active plasma column |
| title_sort |
axial symmetric surface waves in tubular magneto-active plasma column |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2006 |
| topic_facet |
Basic plasma physics |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/81798 |
| citation_txt |
Axial symmetric surface waves in tubular magneto-active plasma column / Yu.A. Akimov, V.P. Olefir, N.A. Azarenkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 121-123. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
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| first_indexed |
2025-11-26T23:30:45Z |
| last_indexed |
2025-11-26T23:30:45Z |
| _version_ |
1849897618152357888 |
| fulltext |
AXIAL SYMMETRIC SURFACE WAVES IN TUBULAR
MAGNETO-ACTIVE PLASMA COLUMN
Yu.A. Akimov, V.P. Olefir, N.A. Azarenkov
Karazin Kharkiv National University, 31 Kurchatov av., Kharkiv 61108, Ukraine,
e-mail: olefir@pht.univer.kharkov.ua
This paper is devoted to the dispersion properties of high-frequency axial symmetric potential surface waves propa-
gating in a cylindrical waveguide structure. The structure is supposed to consist of a radially non-uniform plasma layer,
partially filling a metal waveguide and immersed in an external axial magnetic field. The influence of the waveguide
structure parameters, as well as the magnetic field value, on frequency, phase and group velocities, resonance damping
of the surface waves is investigated both numerically and analytically.
PACS: 52.35.Fp
1. INTRODUCTION
To increase the plasma heating efficiency is one of the
important problems in discharge maintenance by traveling
surface waves (SWs) under low gas pressure [1, 2]. At
those conditions, the collision mechanism of SW power
transfer to a plasma becomes ineffective. It motivates the
study of collisionless methods of plasma heating. One of
them is resonant absorption of SWs that takes place in
those plasma regions where the wave frequency is close
to the upper hybrid one [3, 4]. The aim of this paper is to
study the plasma parameter and external magnetic field
influence on propagation and resonant damping of sym-
metric SWs in coaxial vacuum-plasma-vacuum-metal
structures.
2. TASK STATEMENT
Let us consider high-frequency axial symmetric poten-
tial SWs that propagate along a cylindrical waveguide
structure, which consists of a radially non-uniform plasma
layer partially filling a metal waveguide with a radius R .
The radial distribution of the plasma density is uniform,
0nn = , in the region dRrdR −<<+ 21 and varies
from 0n to zero in the narrow ( 1Rd << ) transition re-
gions dRrR +<< 11 and dRrR −<< 22 , where 1R
and 2R are the internal and external radiuses of the
plasma layer. An external steady magnetic field, 0H , is
supposed to be directed along the waveguide structure
axis. The plasma is considered to be a cold weakly colli-
sion medium with an effective electron collision fre-
quency ων << , whereω is the wave frequency.
The dispersion relation of the considered surface
waves has the following form
,
)(
)(
)(
)(
212322321
212322321
210
210
111311311
111311311
110
110
ZTkiTkZ
YTkiTkY
RK
RI
ZTkiTkZ
YTkiTkY
RK
RI
χπηχ
χπηχ
χ
χ
χπηχ
χπηχ
χ
χ
+−
+−
=
+−
+−
(1)
where
)(
)(
10
1
'
0
1
j
j
j RI
RI
Y
χ
χ
ε= ,
)(
)(
10
1
'
0
1
j
j
j RK
RK
Z
χ
χ
ε= ,
)(
)(
130
13
'
0
1 RkI
RkI
T = ,
)()()()(
)()()()(
2303030230
23
'
0303023
'
0
2 RkKRkIRkKRkI
RkKRkIRkKRkI
T
−
−
= ,
])[(
)(
1 22
2
1
e
e
i
i
ωνωω
νωΩ
ε
−+
+
−= ,
)(
1
2
3 νωω
Ω
ε
i
e
+
−= ,
( )
jrj drd 1
1 / −= εη ; 3k is the axial wavenumber; eΩ ,
0>eω are the electron plasma and cyclotron frequencies;
1331 /εεχ k= is the inverse depth of the SW penetra-
tion into the plasma; jr are the points inside the transient
regions, where the upper hybrid resonance 0)(1 =jrε
takes place, jj Rr ≈ ; 0I and 0K are the modified cylin-
drical Bessel and McDonald functions of the zero order.
3. RESULTS AND DISCUSSION
Dispersion relation (1) describes propagation of two
symmetric surface waves. The first wave propagates
along the internal interface of the non-uniform plasma
layer, whereas the second one does along the external
border. General solution of dispersion equation (1) at ar-
bitrary values of the waveguide parameters and external
magnetic field can be obtained numerically only (fig. 1).
Nevertheless, in some cases, analytical solution of equa-
tion (1) can be found.
3.1 LONG SURFACE WAVES
Firstly, we consider long symmetric SWs, when
1, 213 <<RRk χ . For the waves propagating at the inter-
nal border of the plasma layer, for an arbitrary ratio of the
radiuses 1R and 2R , one can write
.ln
42
,ln
2
1
1
2
4
1
4
3
1
1
1
2
2
1
2
3222
R
RRk
R
R
RRk
ee
πη
ω
ν
ω
γ
Ωωω
−=
−+=
(2)
The presented expressions demonstrate that these waves
are backward. Their frequency grows with an increase of
the external magnetic field (fig. 1a) and weakly depends
on the ratio of the external and internal radiuses of the
plasma. It does not depend on value of the metal
waveguide radius also (fig. 2a).
Fig.1. Influence of the external magnetic field on disper-
sion of the SWs propagating at the internal (a) and exter-
nal (b) interfaces of the cylindrical plasma layer, in the
case of 12 / RR =2 and 1/ RR =2.1. The curves 1-3 corre-
spond to ee Ωω / =0; 0.3; 0.5
The frequency, ω , of the long SWs propagating at the
external boundary of the plasma layer is described by
.ln)(
2 2
2
1
2
2
2
3
2
2
R
RRRke −=
Ωω (3)
It is necessary to mark, the condition 121 <<Rχ holds for
weak magnetic fields only, 22
ee Ωω << . At that, the waves
propagating at the external boundary are forward, in con-
trast to the waves propagating at the internal boundary of
the layer (2). The frequency of these waves slightly de-
pends on the magnetic field (fig. 1b) and is determined by
the plasma layer width, as well as by the metal radius. As
the metal comes to the plasma, the wave frequency de-
creases (fig. 2b). In the limit RR =2 , the SWs do not
exist at the external boundary of the plasma. In the ab-
sence of the metal, expressions for the frequency and
damping rate for the waves at the external boundary be-
come
.
ln22
,
ln
)(
2 23
3
2
2
1
2
2
4
4
2
23
2
1
2
2
2
3
2
2
RkR
RR
Rk
RRk ee −
−=
−
=
ω
Ωπη
ω
ν
ω
γΩω (4)
Thus, the efficiency of resonant damping is proportional
to the plasma layer width.
3.2 SHORT SURFACE WAVES
Now we consider short symmetric SWs, when
1)(,, 1211113 >>− RRRRk χχ . For the waves propagat-
ing at the internal border of the plasma layer,
.
)(
)(
)(2
,
2
11
2
31222
222
222
422
13
22
2
k
Rk
eee
e
ee
ee
ee
ηπ
ωΩΩ
ωω
ωΩω
ωωΩ
ω
ν
ω
γ
ωΩω
−
−
+
−
−
=
+
+
=
(5)
Fig.2. Influence of the metal on dispersion of the SWs
propagating at the internal (a) and external (b) interfaces
of the cylindrical plasma layer, in the case of 12 / RR =2
and ee Ωω / =0.3. The curves 1-3 correspond to
1/ RR =2.1; 2.3; 5.0
In this region, SWs at the internal border of the plasma
layer are backward with small group velocity and strong
dependence on the external magnetic field (fig. 1a). They
do not depend on the metal, as the long waves (fig. 2a).
Below, we study the influence of the metal waveguide
on properties of the short SWs propagating at the external
plasma boundary. Their frequency and damping rate look
like
[ ].)(
)(
)(
)(2
],)(4
)()(2[
2
1
2332222
222
222
422
2
2
2
3
44
222
2
2
3
22
RRkcthk
RRk
RRk
eee
e
ee
ee
ee
eee
−
−
−
+
+
−
−
=
−++
+−−−−=
πη
ωΩΩ
ωω
ωΩω
ωωΩ
ω
ν
ω
γ
Ωω
ωΩωω
(6)
The resonant item in damping rate (6) increases as the
metal comes close to the plasma, whereas the frequency
decreases (fig. 2b). In the case, when the plasma bounds
with the metal, the SWs do not propagate. In the metal
waveguide absence, we obtain
.
)(
)(
)(2
,
2
11
2
32222
222
222
422
23
22
2
k
Rk
eee
e
ee
ee
ee
πη
ωΩΩ
ωω
ωΩω
ωωΩ
ω
ν
ω
γ
ωΩω
−
−
+
−
−
=
−
+
=
(7)
Thus, the frequency of SWs at the external border of the
plasma layer increases from eω at 123 <<Rk up to the
value 2/)( 22
ee ωΩ + at 123 >>Rk (fig. 1b).
0 2 4 6 8
0.70
0.75
0.80
0.85
0.90
0.95
1.00
a
1, 2, 3
k3R1
ω/Ωe
0 2 4 6 8
0.3
0.4
0.5
0.6
0.7 3
2
1
b
k3R1
ω/Ωe
0 2 4 6 8
0.70
0.75
0.80
0.85
0.90
0.95
1.00
3
2
1
k3R1
ω/Ωe
a
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
3
2
1
k3R1
ω/Ωe
b
3.3 PLASMA THICKNESS INFLUENCE
Influence of the ratio of the external and internal radi-
uses of the plasma layer, 12 / RR , on properties of the
SWs is investigated also. It is shown both numerically
(fig. 3a) and analytically (2) that an increase of the pa-
rameter 12 / RR results in a decrease of the phase velocity
of the waves propagating at the internal boundary of the
layer. The phase velocity dependence for the waves at the
external boundary of the layer on its width is more com-
plicated. So, if the vacuum gap is wide enough, an in-
crease of the layer thickness results in a growth of the SW
phase velocity. However, at a fixed value of the metal
radius, an increase of 2R results in a decrease of the vac-
uum gap width. In this case, a decrease of the SW phase
velocity by the metal coming to the plasma appears more
essential, than its growth owing to the increase of the
plasma layer width. Their relation for the case of a narrow
waveguide is described by (3), and for the waveguide of a
finite size is presented in fig. 3b (curve 4).
CONCLUSIONS
In this paper, the dispersion properties and damping
rates of the high-frequency axial symmetric potential sur-
face waves propagating in the cylindrical metal
waveguide partially filled with the radially non-uniform
plasma immersed to the external steady axial magnetic
field have been studied. It has been shown that the group
velocity of the waves propagating at the internal and ex-
ternal interfaces of the cylindrical layer, have opposite
signs. It has been obtained that the frequency of SWs,
which can propagate at the internal plasma boundary, is
greater than the frequency of SWs propagating at the ex-
ternal boundary. An increase in the external magnetic
field has been shown to cause a growth of the wave fre-
quency, whereas the area of axial wavenumbers, at which
the waves exist, decreases.
Fig.3. Influence of the plasma layer thickness on the dis-
persion of SWs propagating at the internal (a) and exter-
nal (b) boundaries of the cylindrical plasma layer, in the
case of 1/ RR =2.1 and ee Ωω / =0.3. The curves 1-4 cor-
respond to 12 / RR =1.1; 1.3; 1.5; 2.0
REFERENCES
1. A. Shivarova, Kh. Tarnev // Plasma Sources Sci.
Technol. 2001, v. 10, p. 260.
2. H. Schlüter, A. Shivarova, Kh. Tarnev // Plasma
Sources Sci. Technol. 2001, v. 10, p. 267.
3. K. N. Stepanov // Sov. Phys. Tech. Phys. 1965, v. 35,
p. 1002.
4. Yu. A. Romanov // Sov. Phys. JETP. 1964, v. 47, p.
2119.
АКСИАЛЬНО-СИМЕТРИЧНЫЕ ПОВЕРХНОСТНЫЕ ВОЛНЫ В ТРУБЧАТОМ СТОЛБЕ
МАГНИТОАКТИВНОЙ ПЛАЗМЫ
Ю.А. Акимов, В.П. Олефир, Н.А. Азаренков
В данной статье изучены дисперсионные свойства высокочастотных аксиально-симметричных потенциаль-
ных поверхностных волн, распространяющихся в цилиндрической волноводной структуре, состоящей из ради-
ально неоднородного пламенного слоя, частично заполняющего металлический волновод и помещенного во
внешнее аксиальное магнитное поле. Численно и аналитически исследуется влияние параметров волноводной
структуры и внешнего магнитного поля на частоты, фазовые и групповые скорости поверхностных волн, а так-
же на декременты их затухания.
АКСИАЛЬНО-СИМЕТРИЧНІ ПОВЕРХНЕВІ ХВИЛІ У ТРУБЧАТОМУ СТОВПІ
МАГНІТОАКТИВНОЇ ПЛАЗМИ
Ю.О. Акімов, В.П. Олефір, М.О. Азарєнков
В роботі вивчено дисперсійні властивості високочастотних аксиально-симетричних потенціальних поверх-
невих хвиль, що розповсюджуються в циліндричній хвилеводній структурі, яка містить радіально неоднорідний
плазмовий шар, що частково заповнює металевий хвилевод і знаходиться у зовнішньому аксіальному магнітно-
му полі. Досліджено вплив параметрів хвилеводної структури та зовнішнього магнітного поля на частоти, фа-
зові та групові швидкості поверхневих хвиль, а також на декременти їхнього загасання.
0 2 4 6 8
0.75
0.80
0.85
0.90
0.95
1.00
4 3
2
1
k3R1
ω/Ωe
a
0 2 4 6 8
0.3
0.4
0.5
0.6
0.7
4
3
2
1
k3R1
ω/Ωe
b
|