Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth
Analytical expressions for the dispersion equations of E-waves of flat and cylindrical sinusoidal rippled plasma waveguides with super conducting walls in a strong external magnetic field are received. Origin of forbidden bands, which are formed at crossings of the dispersion curves describing vario...
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nasplib_isofts_kiev_ua-123456789-1104922025-02-09T22:31:21Z Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth Особливості дисперсійних характеристик гофрованих плазмових хвилеводів з малою глибиною гофра Особенности дисперсионных характеристик гофрированных плазменных волноводов с малой глубиной гофра Lapshin, V.I. Tkachenko, V.I. Tkachenko, I.V. Basic plasma physics Analytical expressions for the dispersion equations of E-waves of flat and cylindrical sinusoidal rippled plasma waveguides with super conducting walls in a strong external magnetic field are received. Origin of forbidden bands, which are formed at crossings of the dispersion curves describing various own modes, is investigated. The width of formed forbidden bands and group speeds of own waves near to the forbidden bands are determined. Comparison of the theory with results of the previous researches in special cases gives good qualitative and quantitative conformity. Отримано аналітичні вирази для дисперсійних рівнянь Е-хвиль плоского і циліндричного плазмових хвилеводів з сінусоідально гофрованими ідеально провідними стінками в сильному зовнішньому магнітному полі. Досліджено утворення смуг непрозорості, які утворюються при перетинанні дисперсійних кривих, що характеризують різноманітні власні моди. Визначено ширину смуг непрозорості, що утворюються, і групові швидкості власних хвиль поблизу смуг непрозорості. Порівняння теорії з результатами попередніх досліджень в окремих випадках дає гарну якісну і кількісну відповідність. Получены аналитические выражения для дисперсионных уравнений Е-волн плоского и цилиндрического плазменных волноводов с синусоидально гофрированными идеально проводящими стенками в сильном внешнем магнитном поле. Исследовано образование полос непрозрачности, которые образуются при пересечении дисперсионных кривых, характеризующих различные собственные моды. Определены ширина образующихся полос непрозрачности и групповые скорости собственных волн вблизи полос непрозрачности. Сравнение теории с результатами предыдущих исследований в частных случаях дает хорошее качественное и количественное соответствие. 2003 Article Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth / V.I. Lapshin, V.I. Tkachenko, I.V. Tkachenko // Вопросы атомной науки и техники. — 2003. — № 1. — С. 89-91. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 52.35.Fp https://nasplib.isofts.kiev.ua/handle/123456789/110492 en Вопросы атомной науки и техники application/pdf 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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DSpace DC |
| language |
English |
| topic |
Basic plasma physics Basic plasma physics |
| spellingShingle |
Basic plasma physics Basic plasma physics Lapshin, V.I. Tkachenko, V.I. Tkachenko, I.V. Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth Вопросы атомной науки и техники |
| description |
Analytical expressions for the dispersion equations of E-waves of flat and cylindrical sinusoidal rippled plasma waveguides with super conducting walls in a strong external magnetic field are received. Origin of forbidden bands, which are formed at crossings of the dispersion curves describing various own modes, is investigated. The width of formed forbidden bands and group speeds of own waves near to the forbidden bands are determined. Comparison of the theory with results of the previous researches in special cases gives good qualitative and quantitative conformity. |
| format |
Article |
| author |
Lapshin, V.I. Tkachenko, V.I. Tkachenko, I.V. |
| author_facet |
Lapshin, V.I. Tkachenko, V.I. Tkachenko, I.V. |
| author_sort |
Lapshin, V.I. |
| title |
Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth |
| title_short |
Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth |
| title_full |
Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth |
| title_fullStr |
Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth |
| title_full_unstemmed |
Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth |
| title_sort |
peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2003 |
| topic_facet |
Basic plasma physics |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/110492 |
| citation_txt |
Peculiarities of dispersion characteristics of sinusoidally rippled plasma waveguides with small ripple depth / V.I. Lapshin, V.I. Tkachenko, I.V. Tkachenko // Вопросы атомной науки и техники. — 2003. — № 1. — С. 89-91. — Бібліогр.: 9 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| first_indexed |
2025-12-01T10:42:24Z |
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2025-12-01T10:42:24Z |
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| fulltext |
PECULIARITIES OF DISPERSION CHARACTERISTICS OF
SINUSOIDALLY RIPPLED PLASMA WAVEGUIDES
WITH SMALL RIPPLE DEPTH
Vladimir I. Lapshin, Victor I. Tkachenko and Ivan V. Tkachenko
NSC KIPT,
Akademicheskaya st. 1, Kharkov, 61108, Ukraine,E-mail tkachenko@kipt.kharkov.ua
Analytical expressions for the dispersion equations of E-waves of flat and cylindrical sinusoidal rippled plasma
waveguides with super conducting walls in a strong external magnetic field are received. Origin of forbidden bands,
which are formed at crossings of the dispersion curves describing various own modes, is investigated. The width of
formed forbidden bands and group speeds of own waves near to the forbidden bands are determined. Comparison of the
theory with results of the previous researches in special cases gives good qualitative and quantitative conformity.
PACS: 52.35.Fp
For the first time use of plasma filling in periodical
waveguides was offered in work [1] Further development idea
of application of plasma filling and sinusoidal ripple has
received in works [2-4]. In work [2] dispersion characteristics
of axial-symmetric Е-waves are investigated, dependence of
forbidden band on ripple depth in a vicinity and far from a
point of crossing of dispersion curves is determined. It is
shown, that the forbidden band width is proportional to ripple
depth. In works [3,4] excitation of rippled plasma structures
by electron beams is investigated.
However in connection with the recent theoretical
description of a Trivelpiece-Gould waves spectrum in periodic
waveguides and introduction in a terminology of concept of a
dense spectrum [5] mentioned above results become unfair.
The matter is that in periodic wave guides instead of the
determined dispersion curves the dot set having fractal
properties [6,7] is organized. Thus the dot set represents the
monotonous function having some constant value at equality
last rational number (a devil ladder) [6,8].
In the present work the analysis of dispersion
characteristics of electromagnetic E-waves of flat and
cylindrical waveguides is carried out in view of finite, but
satisfying Rayleigh hypothesis [9], ripple depth.
Let's receive the dispersion equation of a plasma
waveguide with sinusoidally rippled walls. Dependence
of distance between walls of a waveguide on longitudinal
coordinate )(zX we shall choose as
( )zkRzX 00 sin1)( −= , where
D
k π2
0 = , 1α -
small parameter (
0R
R∆=α ), R∆ - ripple depth,
∞∞− z , 0R , D , - average radius and the spatial
period of a waveguide accordingly. Tensor of dielectric
permeability ikε is considered to be
=
II
ik
ε
ε
00
010
001
, where
2
2
1
ω
ω
ε p
II −= .
Similarly [9] we count, that the wave guide is filled
with homogeneous plasma and placed in a strong
longitudinal magnetic field zeH
↑ ↑0 . We assume also,
that ions of plasma are infinitely heavy and their
contribution plasma fluctuations is neglected.
By virtue of spatial periodicity of structure on z axis
we shall present required fields as infinite sums of partial
waves (spatial harmonics)
Analytical expressions for the dispersion equations of
symmetric E-waves of flat and cylindrical plasma
waveguides with sinusoidally rippled super conducting
walls in a strong magnetic field with small ripple depth
can be received from infinite Hill’s determinant [9].
These expressions are represented by following equations:
( ) ( )
( ) ( )
+
+×
×′′=
=
−
−
−−
−
2
0
2
02
0
0000
2
000
11
n
nII
n
II
nn
n
n
kkkh
R
RZRZ
RZRZ
κ
ε
κ
ε
κκκκλ
κκ
(1)
where 0nkhkn +=
−= 2
2
2
nIIn k
c
ωεκ ,
( ) ( ){ }xJxxZ 0);cos(= ; { }2; ααλ = ; ( )
dx
dZxZ =′ , the
first member in braces corresponds to the flat rippled
waveguide, the second - to the cylindrical one.
We’ll investigate the dispersion characteristics of such
waveguides using the equation (1). All calculations we shall
carry out for a flat rippled waveguide. The received results, in
view of the mentioned above replacement, may be easily
transformed for the description of rippled cylindrical plasma
waveguides.
DISPERSION OF ELECTROMAGNETIC Е-
WAVES ( 22
pωω 〉 〉 )
Let's determine the frequency displacement caused by
the presence of small ripple in a case, when the basic own
mode ( 0=n ) is crossed with a counter own mode
ln −= , 0l . From the equation (1) for a flat rippled
waveguide we have:
( ) ( )
( ) ( )
+
+×
×=
=
−
−
−−
−
2
0
2
0
0
00000
2
000
11
sinsin
coscos
l
l
ll
l
l
kkhk
RRR
RR
κκ
κκκκα
κκ
(2)
Problems of Atomic Science and Technology. 2003. № 1. Series: Plasma Physics (9). P. 89-91 89
In points of crossing of dispersion curves
2
0lkh = ,
decomposing the equation (2) in a line in a vicinity of
points ωω ∆+0 and klk
∆+
2
0 ,
〈 〈
−
=∆∆ 12, 2
0
0
00 k
lkh
k
k
ω
ω
, we shall receive the following
equation connecting the amendments to frequency ω∆
and wave number k∆ :
( ) ( ) 2
2
2
0
2
0
2
2
2
2
0
2
0
2
0
2222
2
1
2
1
+
+
=∆−∆
k
kl
гр
Rkl
Rkl
kv
σ
σ
ωαω (3)
where:
2
1
2
2
0
2
0
2
00
2
1
2
−
+±=
kk
гр
RklRkl
cv
σσ
- group
velocities of own modes of a smooth гладкого waveguide
at 0→α .
From the equation (3) follows, that in a point of
crossing
2
0lkh = of a zero mode ( 0=n ) with the
counter mode ln −= extending in an opposite direction,
dispersion curves on a plane ω∆ , k∆ ( ω∆ -an axis of
abscissas, - k∆ an axis of ordinates) are split. The
received curves are described by the equation of a
hyperbole with the beginning of coordinates in a point (
0ω ,
2
0lk
) and with the imaginary axis parallel to an axis
of wave numbers h .
The distance between tops of hyperbolas (width of a
band e∆ Ω ) is determined by expression:
1
2
2
0
2
0
22
0
2
0
0 2
1
22
1
2
−
+
+=
=∆≡∆ Ω
k
l
e
RklRkl
σ
ωα
ω
The relative width of forbidden bands
le
l α
ω
η ∝∆ Ω=
0
and with increasing of mode number decreases.
Hyperbolas asymptotes are the straight lines
ω∆±=∆ грvk which correspond to the equations of
tangents to the appropriate dispersion curve of a smooth
wave guide with 0→α .
Proceeding from above-stated it is possible to draw a
conclusion, that in a vicinity of points of crossing
2
0lkh =
of dispersion curves of rippled waveguide forbidden
bands with width e∆ Ω , i.e. frequency intervals in which
the equation (3) has no valid solutions relative to k∆ .
Value of group velocity in a vicinity of
2
0lkh =
follows from (3) and may be presented as:
ω∆
∆±= kvw гргр
2 (4)
From expression (4) follows that in a vicinity of a
point
2
0lkh = group velocities may decrease to zero.
DISPERSION OF PLASMA Е-WAVES
( 22
pωω )
Let's consider dispersion properties of plasma E-
waves of a flat rippled waveguide.
The displacement of frequency caused by presence of
small ripple depth at crossing of the basic mode ( 0=n )
with a counter own mode ln −= in points
2
0lkh = we
shall determine from the equation:
( ) ( )
+
+
=∆−∆
2
0
2
0
22
2
2
0
2
0
2
2
22222
21
21
4
kRl
kRl
l
kv
k
k
l
гр
σ
σ
ωαω (5)
where:
0
2
3
0 2
lk
v
p
IIгр ω
ωε±= - group velocities of the
own modes with 0→α .
The equation (5) as well as in case of an
electromagnetic E-wave describes splitting dispersion
curves and formation of the forbidden bands. The relative
width of a forbidden band is determined by expression:
+
+
=
∆ Ω
=
2
0
2
0
22
2
0
2
0
2
0 21
21
4
kRl
kRl
l
k
k
le
l
σ
σ
α
ω
η (6)
From this expression follows, that the width of the
forbidden band decreases like
l
l 1α with growth of
number l of a mode.
Value of group velocity in a vicinity of the forbidden
band is determined by expression (4) in which it is
necessary to use
0
2
3
0 2
lk
v
p
IIгр ω
ωε±= . As well as in case
of electromagnetic E-waves, group velocity for wave
numbers 2
0lkh = may decrease to zero.
In conclusion it is necessary to note the following.
The received expressions are fair for small ripple
depths - 12
0
22 〈 〈Rnκα . Here follows the requirement on a
waveguide ripple depth:
90
( )
1
12
2
2
2
2
2 〈 〈
+
k
k
λ
π
α
At small ripple depth the quantity of modes for which
consideration is fair, may be big enough and thus,
decomposition of fields in infinite numbers is justified.
CONCLUSIONS
For the first time from infinite Hill’s determinant
analytical expressions for the dispersion equations of
symmetric Е-waves of flat and cylindrical plasma
waveguides with sinusoidally rippled super conducting
walls in a strong external magnetic field are received. It is
shown, that the dispersion equation of a cylindrical
plasma rippled waveguide may be received from the
dispersion equation of a flat rippled plasma waveguide if
in the last to replace ( )0Rtg nκ on
( )
( )00
01
RJ
RJ
n
n
κ
κ
and to
replace α on
2
α
.
Formation of forbidden bands for electromagnetic (
22
pωω ) and plasma ( 22
pωω ) waves of E-type
which are formed at the crossing of basic mode dispersion
curve with various own waves in flat and cylindrical
rippled plasma waveguides is analytically investigated. It
is shown, that near to any point of such crossing
dispersion curves are split also their behaviour is
described by the equation of a hyperbola-type in which
the imaginary axis is parallel to an axis of wave numbers,
and the distance between tops of a hyperbole determines
width of a forbidden band.
The width of l -th forbidden band is determined. It
appeared to be proportional lα at modes crossing with
identical radial wave numbers and proportional α at
crossing of the basic and first longitudinal modes
distinguished by radial wave numbers.
Values of group velocities of own waves in the
vicinity of the forbidden band are calculated. It is shown,
that near to points of crossing of own modes both with
identical and with various wave numbers group velocity is
small in comparison with light velocity and in the point of
crossing decreases to zero. The received theoretical
results in limiting cases will well be coordinated to results
of other authors.
REFERENCES
1. A.O. Ostrovskij, V.V. Ognivenko. The Dispersion of a
axial-symmetric plasma waveguide with sinusoidally ripled
super conducting walls in a strong magnetic field.//
Radiotehnika i Elektronica. 1979, v. 24, №12, p. 2470-
2474.
2. A.O. Ostrovskij. The dispersion and a picture of a field of a
axial-symmetric E-wave in a cylindrical waveguide with
sinusoidally ripled super conducting walls.// VANT(2),
1980, p. 25-28.
3. N.E. Belov, N.I. Karbushev, A.A. Ruhadze,
S.J.Udovichenko. To the theory of relativistic karsionotron
in conditions of the big spatial charge.// Fizika plazmy. (4)
1983. v.9, p. 785-790.
4. V.I. Kurilko, V.I. Kucherov, A.O. Ostrovskij, Yu.A. Tkach,
To the theory of stability of a relativistic electronic beam in
a rippled cylindrical waveguide.// Journal of Technical
Physics. 1979, v.49, № 12, p. 2569-2575.
5. W.R. Lou, Y. Carmel, T.M. Antonsen et. al. New modes in
a Plasma with Periodic Boundaries: The Origin of the
Dense Spectrum. // Phys. Rev. Lett., 1991, v. 67, No 18, p.
2481-2484.
6. A.M. Ignatov Trivelpiece-Gould waves in a corrugated
plasma slab. // Phys. Rev. E., 1995, v. 51, p. 1391-1399.
7. V.I. Lapshin, G.I. Zaginajlov, V.I. Tkachenko,
I.V.Tkachenko Fractal properties of periodical plasma
waveguides.// VANT, Issue: Nuclear-physical
investigations (38) 2001, №3, p. 137.
8. V.I. Lapshin, G.I. Zaginaylov, I.V. Tkachenko To the
theory of plasma waves in periodic plasma waveguides.//
VANT, № 6, 2000, p. 41-43.
9. V.A. Balakirev, N.I. Karbushev, A.O. Ostrovskij,
Yu.A.Tkach. Theory of Cherenkov amplifiers and
generators on relativistic beams. Kiev: Naukova Dumka,
1993, p. 208.
ОСОБЛИВОСТІ ДИСПЕРСІЙНИХ ХАРАКТЕРИСТИК ГОФРОВАНИХ ПЛАЗМОВИХ ХВИЛЕВОДІВ
З МАЛОЮ ГЛИБИНОЮ ГОФРА
В.І. Лапшин, В.І . Ткаченко, І .В. Ткаченко
Отримано аналітичні вирази для дисперсійних рівнянь Е-хвиль плоского і циліндричного плазмових
хвилеводів з сінусоідально гофрованими ідеально провідними стінками в сильному зовнішньому магнітному
полі. Досліджено утворення смуг непрозорості, які утворюються при перетинанні дисперсійних кривих, що
характеризують різноманітні власні моди. Визначено ширину смуг непрозорості, що утворюються, і групові
швидкості власних хвиль поблизу смуг непрозорості. Порівняння теорії з результатами попередніх досліджень
в окремих випадках дає гарну якісну і кількісну відповідність.
ОСОБЕННОСТИ ДИСПЕРСИОННЫХ ХАРАКТЕРИСТИК ГОФРИРОВАННЫХ ПЛАЗМЕННЫХ
ВОЛНОВОДОВ С МАЛОЙ ГЛУБИНОЙ ГОФРА
В.И. Лапшин, В.И. Ткаченко, И.В. Ткаченко
Получены аналитические выражения для дисперсионных уравнений Е-волн плоского и цилиндрического
плазменных волноводов с синусоидально гофрированными идеально проводящими стенками в сильном
внешнем магнитном поле. Исследовано образование полос непрозрачности, которые образуются при
пересечении дисперсионных кривых, характеризующих различные собственные моды. Определены ширина
91
образующихся полос непрозрачности и групповые скорости собственных волн вблизи полос непрозрачности.
Сравнение теории с результатами предыдущих исследований в частных случаях дает хорошее качественное и
количественное соответствие.
92
NSC KIPT,
Akademicheskaya st. 1, Kharkov, 61108, Ukraine,E-mail tkachenko@kipt.kharkov.ua
Conclusions
В.І. Лапшин, В.І. Ткаченко, І.В. Ткаченко
В.И. Лапшин, В.И. Ткаченко, И.В. Ткаченко
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