To the theory of plasma waves in pereodic plasma waveguides
Forbidden bands for plasma waves in periodical plasma waveguides are predicted and correctly studied. It was shown that the periodicity influence over frequencies far from the forbidden bands is neglectfully small even for the comparatively large corrugation depth. At the same time while frequency i...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2000 |
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| Мова: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Цитувати: | To the theory of plasma waves in pereodic plasma waveguides / G.I. Zaginaylov, V.I. Lapshin, I.V. Tkachenko // Вопросы атомной науки и техники. — 2000. — № 6. — С. 41-43. — Бібліогр.: 5 назв. — англ. |
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Zaginaylov, G.I. Lapshin, V.I. Tkachenko, I.V. 2015-03-18T15:37:55Z 2015-03-18T15:37:55Z 2000 To the theory of plasma waves in pereodic plasma waveguides / G.I. Zaginaylov, V.I. Lapshin, I.V. Tkachenko // Вопросы атомной науки и техники. — 2000. — № 6. — С. 41-43. — Бібліогр.: 5 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/78488 533.9: 621.315.81 Forbidden bands for plasma waves in periodical plasma waveguides are predicted and correctly studied. It was shown that the periodicity influence over frequencies far from the forbidden bands is neglectfully small even for the comparatively large corrugation depth. At the same time while frequency is approaching to any of forbidden bands periodicity influence over dispersion properties and field distribution increases. A group velocity decreases down to zero on the boundaries of forbidden bands. Plasma wave fields concentrates in small space domains and concentration power increases by approaching to the cut off frequencies. This is caused by higher space harmonics contribution which unlike electromagnetic waves in periodical vacuum waveguides have spatial nature and their calculation is of principle. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Magnetic confinement To the theory of plasma waves in pereodic plasma waveguides Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
To the theory of plasma waves in pereodic plasma waveguides |
| spellingShingle |
To the theory of plasma waves in pereodic plasma waveguides Zaginaylov, G.I. Lapshin, V.I. Tkachenko, I.V. Magnetic confinement |
| title_short |
To the theory of plasma waves in pereodic plasma waveguides |
| title_full |
To the theory of plasma waves in pereodic plasma waveguides |
| title_fullStr |
To the theory of plasma waves in pereodic plasma waveguides |
| title_full_unstemmed |
To the theory of plasma waves in pereodic plasma waveguides |
| title_sort |
to the theory of plasma waves in pereodic plasma waveguides |
| author |
Zaginaylov, G.I. Lapshin, V.I. Tkachenko, I.V. |
| author_facet |
Zaginaylov, G.I. Lapshin, V.I. Tkachenko, I.V. |
| topic |
Magnetic confinement |
| topic_facet |
Magnetic confinement |
| publishDate |
2000 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| description |
Forbidden bands for plasma waves in periodical plasma waveguides are predicted and correctly studied. It was shown that the periodicity influence over frequencies far from the forbidden bands is neglectfully small even for the comparatively large corrugation depth. At the same time while frequency is approaching to any of forbidden bands periodicity influence over dispersion properties and field distribution increases. A group velocity decreases down to zero on the boundaries of forbidden bands. Plasma wave fields concentrates in small space domains and concentration power increases by approaching to the cut off frequencies. This is caused by higher space harmonics contribution which unlike electromagnetic waves in periodical vacuum waveguides have spatial nature and their calculation is of principle.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/78488 |
| citation_txt |
To the theory of plasma waves in pereodic plasma waveguides / G.I. Zaginaylov, V.I. Lapshin, I.V. Tkachenko // Вопросы атомной науки и техники. — 2000. — № 6. — С. 41-43. — Бібліогр.: 5 назв. — англ. |
| work_keys_str_mv |
AT zaginaylovgi tothetheoryofplasmawavesinpereodicplasmawaveguides AT lapshinvi tothetheoryofplasmawavesinpereodicplasmawaveguides AT tkachenkoiv tothetheoryofplasmawavesinpereodicplasmawaveguides |
| first_indexed |
2025-11-27T00:21:34Z |
| last_indexed |
2025-11-27T00:21:34Z |
| _version_ |
1850788111153364992 |
| fulltext |
Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 41-43 41
UDC 533.9: 621.315.81
TO THE THEORY OF PLASMA WAVES IN PERIODIC PLASMA
WAVEGUIDES
Gennadiy I. Zaginaylov, Vladimir I. Lapshin*, and Ivan V. Tkachenko
Kharkov National University, Svobody Sq. 4, Kharkov, 61077, Ukraine
* Institute of plasma physics of the NSC KIPT
Akademicheskaya st. 1, 61108 Kharkov, Ukraine
Forbidden bands for plasma waves in periodical plasma waveguides are predicted and correctly studied. It was
shown that the periodicity influence over frequencies far from the forbidden bands is neglectfully small even for the
comparatively large corrugation depth. At the same time while frequency is approaching to any of forbidden bands
periodicity influence over dispersion properties and field distribution increases. A group velocity decreases down to
zero on the boundaries of forbidden bands. Plasma wave fields concentrates in small space domains and
concentration power increases by approaching to the cut off frequencies. This is caused by higher space harmonics
contribution which unlike electromagnetic waves in periodical vacuum waveguides have spatial nature and their
calculation is of principle.
1. Introduction
Periodic plasma-filled waveguide structures are
widely used in plasma microwave electronics for
development of efficient methods of microwave
generation and for charged particles acceleration. Also
they are used in plasma heating systems.
However, despite of a great number of practical
applications, dispersion properties of plasma-filled
periodic structures at frequencies below plasma one
have not been studed yet even qualitatively.
As it is well known, plasma filled periodic
waveguides support two sets of modes: electromagnetic
and plasma ones. In the simplest case of transversally
homogeneous plasma electromagnetic waves occure at
frequencies more then plasma one do not overlaping
with plasma waves which exist at frequencies below
plasma one. In contrast with electromagnetic modes,
which can be successfully analyzed by conventional
approaches, plasma modes form so-called “dense”
spectrum [1,2]. Such spectral behavior being quite
different from usual one can not be described on the
basis of conventional approaches which are widely used
for the analysis of electromagnetic modes in periodic
structures [3].
In this article the plasma “dense” spectrum is
investigated basing on a new approach [4,5] according
to which, the corresponding spectral problem is
formulated in the kind of some homogeneous integral
equation. It allowed us to correctly calculate the
dispersion curves for plasma modes revealing some new
features. Particularly, forbidden bands for plasma waves
in periodical plasma waveguides are predicted and
correctly studied. It was shown that the periodicity
influence at frequencies far from the forbidden bands is
neglectfully small even for the comparatively large
corrugation depth. At the same time while the mode
frequency is approaching to any of forbidden bands
periodicity influence on dispersion properties and field
distribution significantly increases. A group velocity
decreases down to zero on the boundaries of forbidden
bands (cut off frequencies). Meanwhile, plasma wave
fields concentrates in small space domains and the field
concentration increases at approaching to the cut off
frequencies. This is caused by higher space harmonics
contribution which unlike electromagnetic waves in
periodical vacuum waveguides have spatial nature and
their calculation is of principle.
Fig.1. Geometry of the problem
II. Formulation of the Problem
Let’s consider a planar waveguide with periodically
varying thickness loaded with cold homogenous
collisionless plasma with a longitudinally applied
infinite magnetic field (see, Fig.1). We suppose that all
perturbations are of TM – polarization
( ) ti
zyx eEHE ω−∝,, symmetric with respect of z-
axis: ( ) ( )zxEzxE zz ,, =− .
In this case Maxwell equations can be reduced to
only equation for the transverse magnetic field:
( ) 0,2
2
2
2
2
=
+
∂
∂+
∂
∂ zxHk
zx yεε (1)
42
In the region of plasma wave existence
01 2
2
!
ω
ω
ε p−= . So, making transition to a new
variable xx ~21 =ε , from (1) we come to the
telegrafist's equation
( ) 0,~~
~
2
2
2
2
2
=
+
∂
∂−
∂
∂ zxHk
zx y (2)
where ( ) ( )zxHzxH yy ,,~~ = .
Equation (2) is a hyperbolic type equation. Its
solution can be expressed with integrals of the unknown
function and its derivative on the z -axis:
( ) ( ) ×−−= ∫
+
−
)~(
2
1,~~ 22
~
~
0 xzkJzxH
xz
xz
y ζ
ζζ dg )(× (3)
where
0~
~
~
)(
=
∂
∂
=
x
y
x
H
g ζ
Making transition to the old variable we obtain
( ) ( )×= ∫
+
−
ζζ
ε
ε
gdzxH
xz
xz
y
2/1
2/12
1, (4)
( ) )( 22
0 xzkJ εζ −−×
Applying the boundary condition for the tangential
component of the electric field on the waveguide wall
( ) 0cossin =α+α = zXxzx EE , which is equivalent
to the following condition for ),( zxH y :
01)(
)(
=
∂
∂
−
∂
∂
′
= zXx
yy
x
H
z
H
zX
ε
(5)
we obtain the basic integral equation
( )( ) ( )( ) ( )( )
( )( ) ( )( )
( )
( )
( )
( ) ,0
)(
)(
]
[1
1
22
22
1
)(
)(
=
−−
−−
×
×−
−−′−′−×
×−+′++
∫
+
−
ζ
ϕζ
ϕζ
ϕ
ζϕζϕ
ϕϕϕ
ϕ
ϕ
g
zz
zzkJ
z
zzdkz
zzgzzzg
zz
zz
(6)
where ( ) ( )zXz 2/1εϕ = .
Approximate integral value in the last equation can
be derived by the spline functions. For this let’s divide
inergrating interval by the discrete sections and
introduce following values dz N
i
i 2= , where N - is
a number of discrete sections, and i - is a section
number.
Fig.2. Dispersion diagrams for the first three plasma
modes of the sinusoidal rippled plasma waveguide
( ) ( )
( ) ( )
⋅+=
⋅−=
N
d
zziP
N
d
zziM
2
2
ϕ
ϕ
(7)
),(iM )(iP - are the numbers of the first and the last
discrete sections correspondingly.
( )( ) ( )( ) ( )( )
( )( ) ( ) ( )( )
( )( ) ( )( ) ( )( )
( ) ( )
( )( ) ( )( ) ( )( )
( ) ( )[
( )
( )
( ) ( )] 0,
,
2
2
1,
2
2
1,
2
1
1
11
1
11
1
1
=+
++
+
+⋅−×
×
−+
+
+
−⋅−
×
−−
+′−×
×−+′++
++
+=
++
+
∑
−
kk
iMk
kk
iPiP
iP
iMiM
iM
zgzzG
zgzzG
N
d
zzgkzgzzG
zzz
zzgkzgzzG
zzz
z
zzgzzzg
iP
ϕ
ϕ
ϕ
ϕ
ϕ
ϕϕϕ
(8)
where ( )ζ,zG is an integral kernel. Since the
argument of the Bessel function is imaginary all over
the inergrating interval we will make transition to the
modified Bessel function
43
( ) ( )( ) ( )
( )
( )
( )( ) ( )
( )
( )22
22
1
22
22
1
)(
)(
][
)(
)(
][,
ζϕ
ζϕ
ϕζϕ
ϕζ
ϕζ
ϕζϕζ
−−
−−
×
×−−′⋅−=
=
−−
−−
×
×−−′⋅−=
zz
zzkI
zzzk
zz
zzkJ
zzzkzG
(9)
Computer simulations for the equation (8) can be
carried out.
In the case when ∞→c the integral equation (6)
coincides with the equation obtained in [4] based on
expansion of fields in spatial harmonic series. Full
mathematical identity of these approaches under the
condition 1)( ≤ϕ ′ z was shown also in [5].
Now we consider the case of electrostatic waves
( ∞=c ). In this case the integral equation (6)
transforms into the functional equation:
( ) ( )( ) ( )( )
( ) ( )( ) ( )( ) 01
1
=′−−Ψ−+
+′++Ψ
zzzzike
zzzzike
z
z
ϕϕϕ
ϕϕϕ
(10)
where the new function ( ) ( ) zik
z
zezEz −=Ψ ,0 has
been introduced, zk is a wavenumber of perturbations.
It was shown that increasing of exactness in
numerical calculations of equation (10) does not lead to
sufficient modification of numerical results obtained in
[5].
III. Numerical Results
Numerical calculations of (13) at 0=n have been
performed by expansion of )(zθ into the series of the
spline functions and into Fourier series. The results
were identical for the large number of terms (~100)
taken into account. Fig. 2 displays the dispersion curves
for the first three plasma modes in the case of
sinusoidally rippled waveguide:
))cos(1()( 00 zkxzX α+= with parameters
1.0=α , 67.30 =k cm 1− , 4.10 =x cm.
At these parameters plasma modes have four
allowed bands:
Where ( )( )( ) 22
00 1/1/
−
± ±+= απωω kxqpq ,
.4,3,2,1=q
Below 4+ω the equation (10) has no solution. Thus,
it seems plasma modes in periodic plasma-filled
structures have lower cut off frequency, which is
determined mostly by the ripple height what has a clear
physical meaning: at low frequencies a number of radial
modes is very large and they are located very closely to
each other. So, reflections from rippled walls of
waveguide, which are provided by them become very
efficient, blocking the propagation of any mode.
However, we can not state this for sure since the lower
cut off frequency usually lies out of region of validity of
equation (10). The latter, in our case, is defined by the
relation [5]: 12/1
00 ≤εα xk .
References
[1] Lou W.R., Y. Carmel, T.M. Antonsen, Jr.,
W.W.Destler, and V.I.Granatstein, Phys. Rev. Lett.,
1991, vol. 67, p. 2481.
[2] Ogura K., Ali M.M., Minami K. et al., J. Phys. Soc.
Japan, vol.61, p.4022.
[3] See, for example, Swell J.A. et al., Phys. Fluids,
1985, vol. 28, p.2882.
[4] Zaginaylov G.I., Rozhkov A.A., and Raguin J.-Y.,
Phys. Rev. E, 1999, vol. 60, р. 7391.
[5] Verbitskii I.L. and Zaginaylov G.I., IEEE Trans.
Plasma Sci., 1999, vol. 27, p.1101.
433
2211
,
,,
+−+
−+−+
<<<
<<<<<<
ωωωω
ωωωωωωωω p
Gennadiy I. Zaginaylov, Vladimir I. Lapshin*, and Ivan V. Tkachenko
Kharkov National University, Svobody Sq. 4, Kharkov, 61077, Ukraine
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