Features of coaxial waves in magnetized plasma cylinder
The resonance properties of plasma cylinder are carried out for nonzero axial wave vector. Waves with frequency greater then ion cyclotron frequency but less then electron frequency are considered. From Maxwell equations we get the system of two coupling differential equations of second order for th...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Features of coaxial waves in magnetized plasma cylinder / D.L. Grekov, V.I. Lapshin // Вопросы атомной науки и техники. — 2000. — № 3. — С. 42-44. — Бібліогр.: 2 назв. — англ. |
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| author_facet | Grekov, D.L. Lapshin, V.I. |
| citation_txt | Features of coaxial waves in magnetized plasma cylinder / D.L. Grekov, V.I. Lapshin // Вопросы атомной науки и техники. — 2000. — № 3. — С. 42-44. — Бібліогр.: 2 назв. — англ. |
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| description | The resonance properties of plasma cylinder are carried out for nonzero axial wave vector. Waves with frequency greater then ion cyclotron frequency but less then electron frequency are considered. From Maxwell equations we get the system of two coupling differential equations of second order for these waves. There exist two types of coaxial waves in the vacuum gap. One of them has zero axial electric field (TE-mode). This is the analog of azimuthal surface waves. Another one has zero axial magnetic field (TB-mode). As it is shown, in the broad range of plasma and device parameters the resonance frequencies of TB-mode are close to resonance frequencies of TEmode. This leads to effective coupling of these modes and energy transfer from one mode to another. The time of transfer is much shorter, then collisional absorption time. As an consequence, the wave fields change significantly in plasma and vacuum.
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Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 42-44 42
UDC 533.951
FEATURES OF COAXIAL WAVES
IN MAGNETIZED PLASMA CYLINDER
D.L.Grekov, V.I.Lapshin
Institute of Plasma Physics,
National Science Center "Kharkov Institute of Physics and Technology",
61108, Kharkov, Ukraine
The resonance properties of plasma cylinder are carried out for nonzero axial wave vector. Waves with
frequency greater then ion cyclotron frequency but less then electron frequency are considered. From Maxwell
equations we get the system of two coupling differential equations of second order for these waves. There exist two
types of coaxial waves in the vacuum gap. One of them has zero axial electric field (TE-mode). This is the analog of
azimuthal surface waves. Another one has zero axial magnetic field (TB-mode). As it is shown, in the broad range
of plasma and device parameters the resonance frequencies of TB-mode are close to resonance frequencies of TE-
mode. This leads to effective coupling of these modes and energy transfer from one mode to another. The time of
transfer is much shorter, then collisional absorption time. As an consequence, the wave fields change significantly in
plasma and vacuum.
The surface waves are applied in plasma sources
and plasma electronics. Also they are excited at RF
plasma in magnetic traps (so called coaxial modes) and
effect plasma periphery. These stimulate theoretical
investigations of surface waves. They often use the
model of homogeneous circular plasma cylinder with
axial magnetic field, separated from metal wall by
vacuum for surface waves studies. The axially
symmetric m = 0 surface waves with k || ≠ 0 ( k || is a
wave vector along confining magnetic field) were
analyzed in details e.g. in [1]. Since [2] axially
nonsymmetric ( m ≠ 0 , k || = 0 , E z = 0 ) surface
waves are intensively studied. Such waves is named as
azimuth surface waves. As for fast magnetosonic waves
the azimuth surface waves with m > 0 and with m < 0
propagates in different ways due to plasma gyrotropy.
Both for m = 0 case and for k || = 0 case one can
extract from Maxwell equations two independent
deferential equations of the second order. One of them
is for the Bz component. of wave and by analogy
to the theory of waveguides we shall call these waves as
TE coaxial modes. And another is for E z component
of wave. We shall call these waves as TB coaxial
modes. At the boundary plasma - the metal TB coaxial
modes does not exist. Moreover even in systems with
vacuum gap it was not considered as a rule. The real
plasma devices have final length along confining
magnetic field. Therefore it is worthwhile to take into
account k | | ≠ 0 in coaxial modes studies.
The paper presented concerns with analytical studies
of the excitation and propagation of this waves. We
consider a homogeneous plasma cylinder of radius
r a= with axial magnetic field B Oz0 | | . It is
separated from metal wall of r b= ( )a b< by
vacuum gap. The waves are exited by an azimuth
surface current
r r
j e j i m k z t= + −ϕ ϕ ω0 exp ( )| | of radius
r d= ( )a d b< < . We take into account presence of a
surface charge ρ ω= − i div j
r
/ . The RF field takes the
form
r r r r
B E B E r i m k z t, ~ , ( )exp ( )ϕ ω+ −| | . From Maxwell
equations we have
( )( )ε ω ε ε ε1
2
2
2
2
2 1
2 2
2
2
2
2
2
1 1
0−
−
+ − −
−
−
=|| || ||N
r
d
dr
r
d
dr
m
r
B
c
N B iN
r
d
dr
r
d
dr
m
r
Ez z z
(1)
( )[ ] [ ]iN
r
d
dr
r
d
dr
m
r
B N
r
d
dr
r
d
dr
m
r
E
c
N Ez z z|| || ||
−
+ − −
−
+ − − =ε ε ε ε
ω
ε ε ε2
2
2 1 1
2
2
2
2
2
2
2 3 1
2 2
2
21 1
0( )
Here εα are components of dielectric permeability
tensor of cold two-component (ions and electrons)
plasma. The collisional absorption of waves is taken
into account. The effective collision frequency
ν ω<< is included in εα . We carry out our research
for ω ω ωci ce< < ( ω αc is cyclotron frequency). The
solutions of (1) can be written as B r BI k rz m( ) ( )= ⊥ ,
E r E I k rz m( ) ( )= ⊥ (B, E are constants, I xm ( ) is
modified Bessel function). Then for N k c⊥ ⊥= / ω we
have
( )( ) ( )( )[ ] ( )N N N NTE TB⊥ || || ||= − + − ± − + − − − −
,
/
2
1
2
2
1 3 1
2
2
2
1 3 1
2
2
1 3 1
2 2
2
2
1 2
1
2
4
ε
ε ε ε ε ε ε ε ε ε ε ε ε (2)
43
The sign “+” meets TE coaxial modes and sign “-“ -
TB coaxial modes. The electro-magnetic field in
vacuum consist of two independent modes. One of them
is TE m mode with E z = 0 and another is TBm with
B z = 0 . In the space between plasma and surface
current they look like
B r AV k r k b j V k r k d k dz v v v v v( ) ( , ) ( , )= −
π
2 0 ,
E r CW k r k b
i
j W k r k d mNz v v v v( ) ( , ) ( , )= + ||
π
2 0 . (3)
Here W k r k b Y k b J k r Y k r J k bv v m v m v m v m v( , ) ( ) ( ) ( ) ( )= − ,
V k r k b Y k b J k r Y k r J k bv v m v m v m v m v( , ) ( ) ( ) ( ) ( )= ′ − ′ , Y xm ( ),
J xm ( ) - Bessel functions. Equating tangential
components of electric and magnetic fields at the
plasma - vacuum boundary we have a system of four
linear equations to obtain A,C,B and E. The right side of
this system is proportional to j0 . Putting to zero the real
part of determinant of this system we get the
eigenfrequencies that is resonance curves of coaxial
modes. We investigate resonance properties of coaxial
modes using Ω − N A
2 plane (here Ω = ω ω/ ci and
N c vA A= / , v A is Alfven velocity) (see Fig.1). To
number of parameters which characterize device and
remain fixed at resonance curve we shall refer
a b d B, , , 0 and k || . They are included in simulations as
b b cci0 = ω / , a b d b/ , / and N k c ci|| ||=0 / ω . Then
changing the plasma density we change NA
2.
There are two ways of coaxial modes absorption.
One of them is absorption of power, oscillating in the
vacuum gap by the metallic wall due to finite
conductivity of the wall. As the calculations show this
absorption is negligibly small both for TE m and for
TBm modes. Another one is collisional absorption of
the power oscillating in the plasma. We shall
characterize this absorption by γ = P Pab os/ . Here Pab
is the power absorbed in the plasma and Pos is the
power oscillating in the plasma. The value of γ changes
with frequency within the resonance range (see Fig.2).
This is due to different absorption value of TE mode
and TB mode. But this difference strongly depends on
plasma density as it is shown in Fig.3.
Now we turn to resonance properties of the coaxial
modes of magnetized plasma cylinder. In principle these
properties are well described by the relations
′
+
′
+
−
=
⊥
⊥
⊥ ⊥ ⊥ ||
V k b k a
V k b k a N
I k a
I k a
m
k a N N
v v
v v F
m F
m F F F
( , )
( , )
( )
( )
1 1
02
1
2
ε
ε
(4)
′
−
′
=⊥
⊥
⊥
W k a k b
W k a k b
N
I k a
I k a
v v
v v
S
m S
m S
( , )
( , )
( )
( )
0 .
These equations turn out from (1) ignoring the
interaction of TE and TB coaxial modes in the plasma.
As we can see in Fig.1, the difference between the
eigenfrequencies of TB coaxial mode and TE coaxial
mode approaches to zero with density increase. This easy
to understand analyzing Eqs.(4). With N A
2 growth N ⊥
2 is
increased both for TE coaxial mode and TB coaxial
mode. Therefore zero of the first equation in (4) are
determined by zero of a numerator of the first addend and
the zero of the second equation are determined by zero of
a denominator of the first addend of this equation. The
roots of these functions are very close. That is why it is
interesting to investigate excitation of coaxial modes in
a range of such "collective" resonance. In this range we
can consider the plasma cylinder as the two resonators
system. Neglecting the coupling of resonators we get the
eigenfrequencies ωT E 1 and ωT B 1 from (4). Taking into
account the coupling of coaxial modes we have another
pare of resonance frequencies ωT E 2 and ωT B 2 . The
differences between these pares are rather small (Fig.4) but
very important from the point of view of theory of
oscillations.
Let us consider the system of two coupling contours and
put x T E and x T B as the generalized coordinates of
contours. Then the potential and kinetic energy of the
system are
U x x x x x xT E T B T E T E T B T B( , ) = + +α α α11
2
12 22
22 ,
T x x x xT E T E T B T B= + +β β β11
2
12 22
22& & & & .
The partial resonance frequencies are defined as
ϖ α βT E = 11 11/ and ϖ α βT B = 22 22/ . The
resonance frequencies of system can be obtained from
( ) ( )β β β ω α β α β α β ω α α α11 22 12
2 4
11 22 22 11 12 12
2
11 22 12
22 0− − + − + − =
Consequently, having partial resonance frequencies and
resonance frequencies of coupling system, we can
calculate the coupling coefficients γ β β β1
2
12
2
11 22= / ( )
(inductive) and γ α α α2
2
12
2
11 22= / ( ) (capacitive). They
characterize the swap time between the resonators
t s ≈ = + −π ϖγ γ γ γ γ γ/ ( ),
2
1
2
2
2
1 2 (Fig.5). As we
can see from Fig.5 the time of power transfer from one
mode to another is the order of few periods.
Now we define AT E - amplitude of TE mode with
frequency ωT E , BT E - amplitude of TB mode with
frequency ωT E , AT B - amplitude of TE mode with
frequency ωT B , BT B - amplitude of TB mode with
frequency ωT B , and coefficients of amplitudes
distribution k A BT E T E1 = / and k A BT B T B2 = / . These
coefficients are connected to the contours tie-up
coefficient σ like
k k1
11
22
2
2
11
22
21 1
2
1 1
2
=
− +
=
+ +β
β
σ
σ
β
β
σ
σ
, ,
where σ γ ϖ ϖ ϖ ϖ= −2 2 2
T E T B T B T E/ . In the case of
weak coupling or great difference between partial
frequencies of resonators , σ → → → ∞0 01 2, ,k k
and excitation of resonator (TE mode) does not
influence another resonator (TB mode). But this is not
our case (Fig.6). Even if the external currents is
managed so as to excite only TE mode, then in the swap
44
time t s RF power transfers to the TB mode (and then
back).
CONCLUSION
- strong coupling of fast (TEm) and slow (TBm) coaxial
modes determined
- this coupling does not depend on wave damping value
- dependencies of coupling on plasma density and
longitudinal wave number established
- absorption of coaxial modes in the wall is negligible
- collisional absorption of these modes in plasma is
rather weak.
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0
2 0 0
2 5 0
3 0 0
3 5 0
4 0 0
2
A
N
a / b = 0 . 8
a / b = 0 . 9
ω / ω
c i
C O A X I A L M O D E S
, - T E
, - T B
Fig.1. Resonance curves of TE and TB modes,
calculated with use of system (1).
- 1 , 2 - 0 , 8 - 0 , 4 0 , 0 0 , 4 0 , 8
0 , 0 0
0 , 0 1
0 , 0 2
0 , 0 3
γ
γ
T E
γ
T B
γ
( ω − ω
a v
) / ( ω
T B
− ω
T E
)
Fig.2. Dependencies of γ = P Pab os/ on frequency for
TE, TB modes and total. Here ω ωT E T B, - resonance
frequencies of TE , TB modes, ω ω ωav T E T B= +( ) / 2 .
- 0 , 8 - 0 , 6 - 0 , 4 - 0 , 2 0 , 0 0 , 2 0 , 4 0 , 6 0 , 8
0 , 0 2 1
0 , 0 2 4
0 , 0 2 7
0 , 0 3 0
γ
( ω − ω
a v
) / ( ω
T B
− ω
T E
)
N
A
= 2 5
N
A
= 3 5
N
A
= 5 0
N
A
= 7 7
Fig.3. Dependencies of γ = P Pab os/ on frequency for
different density values. Here ω ωT E T B, - resonance
frequencies of TE , TB modes, ω ω ωav T E T B= +( ) / 2 .
2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0
0 , 0
0 , 2
0 , 4
0 , 6
0 , 8
1 , 0
1 , 2
1 , 4
∆ ω T E
∆ ω T B
N
l l
= . 2
N
A
Fig.4. Dependencies of difference ∆ω between
resonance frequencies, calculated for coupling and
isolated TE and TB modes, on plasma density.
2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0
0 , 0 5
0 , 0 6
0 , 0 7
0 , 0 8
0 , 0 9
0 , 1 0
0 , 1 1
0 , 1 2
0 , 1 3
i n d u c t i v e
c a p a c i t i v e
N
l l
= . 2
N
A
Fig.5. Dependence of the coupling of TE and TB
coaxial modes on plasma density.
2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0
0
5
1 0
1 5
2 0
2 5
3 0
N
l l
= . 2
N
A
Fig.6. Dependence of the tie-up of TE and TB coaxial
modes on plasma density
References
1. A. N. Kondratenko. Surface and volume waves in bounded plasma. Moscow: ”Energoatomizdat”, 1985, 208 p.
2.V.A. Girka, I.A. Girka, A.N. Kondratenko, V.I. Tkachenko. Azimuthal surface waves in magnetized plasma
waveguides // Sov. J. of Communications Tec. & Electronics. 1988, v.3, N 5, p.1031-1035.
|
| id | nasplib_isofts_kiev_ua-123456789-82365 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:28:07Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Grekov, D.L. Lapshin, V.I. 2015-05-29T05:59:12Z 2015-05-29T05:59:12Z 2000 Features of coaxial waves in magnetized plasma cylinder / D.L. Grekov, V.I. Lapshin // Вопросы атомной науки и техники. — 2000. — № 3. — С. 42-44. — Бібліогр.: 2 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/82365 533.951 The resonance properties of plasma cylinder are carried out for nonzero axial wave vector. Waves with frequency greater then ion cyclotron frequency but less then electron frequency are considered. From Maxwell equations we get the system of two coupling differential equations of second order for these waves. There exist two types of coaxial waves in the vacuum gap. One of them has zero axial electric field (TE-mode). This is the analog of azimuthal surface waves. Another one has zero axial magnetic field (TB-mode). As it is shown, in the broad range of plasma and device parameters the resonance frequencies of TB-mode are close to resonance frequencies of TEmode. This leads to effective coupling of these modes and energy transfer from one mode to another. The time of transfer is much shorter, then collisional absorption time. As an consequence, the wave fields change significantly in plasma and vacuum. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Мagnetic Confinement Features of coaxial waves in magnetized plasma cylinder Article published earlier |
| spellingShingle | Features of coaxial waves in magnetized plasma cylinder Grekov, D.L. Lapshin, V.I. Мagnetic Confinement |
| title | Features of coaxial waves in magnetized plasma cylinder |
| title_full | Features of coaxial waves in magnetized plasma cylinder |
| title_fullStr | Features of coaxial waves in magnetized plasma cylinder |
| title_full_unstemmed | Features of coaxial waves in magnetized plasma cylinder |
| title_short | Features of coaxial waves in magnetized plasma cylinder |
| title_sort | features of coaxial waves in magnetized plasma cylinder |
| topic | Мagnetic Confinement |
| topic_facet | Мagnetic Confinement |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82365 |
| work_keys_str_mv | AT grekovdl featuresofcoaxialwavesinmagnetizedplasmacylinder AT lapshinvi featuresofcoaxialwavesinmagnetizedplasmacylinder |