Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field
For carrying out of the further numerical comparisons with experimental researches on breakdown of a mixture of gases by microwave radiation with a stoсhastic jumping phase the theoretical researches of wave dispersive properties of the coaxial waveguide are carried out. Electromagnetic modes of a c...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2015 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
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| Цитувати: | Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field / I.V. Karas, I.A. Zagrebelny // Вопросы атомной науки и техники. — 2015. — № 4. — С. 36-42. — Бібліогр.: 5 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860096756233011200 |
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| author | Karas, I.V. Zagrebelny, I.A. |
| author_facet | Karas, I.V. Zagrebelny, I.A. |
| citation_txt | Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field / I.V. Karas, I.A. Zagrebelny // Вопросы атомной науки и техники. — 2015. — № 4. — С. 36-42. — Бібліогр.: 5 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | For carrying out of the further numerical comparisons with experimental researches on breakdown of a mixture of gases by microwave radiation with a stoсhastic jumping phase the theoretical researches of wave dispersive properties of the coaxial waveguide are carried out. Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field are investigated. The existence of quasi-TEM modes in a finite-strength magnetic field is demonstrated. It is shown that, in the limits of infinitely strong and zero magnetic fields, this mode transforms into a true TEM mode.
Для проведення подальших кількісних порівнянь з експериментальними дослідженнями з пробою суміші газів за допомогою мікрохвильового випромінювання зі стохастичними стрибками фази проведені детальні теоретичні дослідження дисперсійних властивостей плазмового коаксіального хвилеводу. Досліджені електромагнітні хвилі, що поширюються в коаксіальному плазмовому хвилеводі, який вміщено в зовнішнє магнітне поле. Продемонстрована наявність режимів квазі ТЕМ-типу хвиль у магнітному полі скінченної величини. Показано, що в межах нескінченно сильного та нульового магнітних полів існують ТЕМ-типу хвилі.
Для проведения дальнейших численных сравнений с экспериментальными исследованиями по пробою смеси газов с помощью микроволнового излучения со стохастическими скачками фазы проведены детальные теоретические исследования дисперсионных свойств плазменного коаксиального волновода. Исследованы электромагнитные волны, распространяющиеся в коаксиальном плазменном волноводе, помещенном во внешнее магнитное поле. Продемонстрировано наличие режимов квази ТЕМ-типа волн в магнитном поле конечной величины. Показано, что в пределах бесконечно сильного и нулевого магнитных полей существуют ТЕМ-типа волны.
|
| first_indexed | 2025-12-07T17:27:00Z |
| format | Article |
| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №4(98) 36
ELECTROMAGNETIC MODES OF A COAXIAL PLASMA WAVEGUIDE
IN AN EXTERNAL MAGNETIC FIELD
I.V. Karas’, I.A. Zagrebelny
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: ira@kipt.kharkov.ua
For carrying out of the further numerical comparisons with experimental researches on breakdown of a mixture
of gases by microwave radiation with a stoсhastic jumping phase the theoretical researches of wave dispersive prop-
erties of the coaxial waveguide are carried out. Electromagnetic modes of a coaxial plasma waveguide in an external
magnetic field are investigated. The existence of quasi-TEM modes in a finite-strength magnetic field is demonstrat-
ed. It is shown that, in the limits of infinitely strong and zero magnetic fields, this mode transforms into a true TEM
mode.
PACS: 52.80.Pi, 52.65.-y, 52.65.Ff, 52.70. Ds, 52.70.Kz, 84.40Fe
INTRODUCTION
For carrying out of the further numerical compari-
sons with experimental researches on breakdown of a
mixture of gases by microwave radiation with a
stoсhastic jumping phase [1 - 3] the theoretical re-
searches of wave dispersive properties of the coaxial
plasma waveguide are carried out in detail.
1. ELECTROMAGNETIC MODES
OF A COAXIAL PLASMA WAVEGUIDE
IN AN EXTERNAL MAGNETIC FIELD
In the theoretical study examines the axisymmetric
waves that extended in the coaxial waveguide, which is
filled a plasma, along the waveguide axis is applied an
external constant magnetic field. Given the dependence
of the components of the electric and magnetic fields of
the coordinates and time has the form
)](exp[)(),( 3 tzkirAtxA ω−=
from Maxwell's
equations for the matter we get the following equation
system (compare with works [3, 4]):
rkHEk −=ϕ3 , zikHrE
rr
=
∂
∂
ϕ)(1
,
zEikrH
rr 3)(1
ε−=
∂
∂
ϕ
,
r
EikHEik z
r ∂
∂
=− ϕ3 ,
ϕϕ ε=−ε EkHikEik r 231 ,
,231 rHEkHikEik zrr ∂∂=ε−+ε ϕ (1)
c
k ω
= , 22
2
1 1
He
pe
ω−ω
ω
−=ε ,
)( 22
2
2
He
Hepe
ω−ωω
ωω
−=ε , 2
2
3 1
ω
ω
−=ε pe
, (2)
where ( ) 2/1
0
2 /4 ααπ=ω mnepe
is the electron Langmuir
frequency, )/(0 cmHeHe αα=ω is the electron Lar-
mor frequency.
2. TEM-TYPE MODES OF A COAXIAL
PLASMA WAVEGUIDE WITHOUT
AN EXTERNAL MAGNETIC FIELD
First, consider the system of equations (1) in the ab-
sence of an external magnetic field, i.e. ωHe=0, and ac-
cordingly ε1=ε3, ε2=0.
03 =+ rkHEk ϕ ,
rEikHEik zr ∂∂=− ϕ3 ,
rHHikEik zr ∂∂=+ε ϕ 31 ,
031 =−ε ϕHikEik r , (3)
zikHrE
rr
=
∂
∂
ϕ )(1 ,
zEikrH
rr 1)(1
ε−=
∂
∂
ϕ
.
In this case, the system is divided into two systems
of equations unrelated to describe the E- and H-wave. In
the wave of E-type nonzero components of the electro-
magnetic field Ez, Er, Hφ and H-type wave have the
nonzero components Hz, Hr, Eφ. In a separate class you
need to make waves in the TEM are zero z-component
of the electric and magnetic fields.
We first look at the structure of TEM wave for
which, by definition, [5] Ez = 0 and Hz = 0. From the
equations (3) it follows:
rCE 1=ϕ , rCH 2=ϕ . (4)
For the existence of waves is necessary to satisfy the
boundary conditions, which in the case of coaxial
waveguide are of the form:
,0)()( == bEaE zz .0)()( == ϕϕ bEaE (5)
This means that the components of the field Eφ and
correspondly Hr=-k3/k·Eφ in TEM wave since no cannot
be a solution in the form of (2.4) satisfy the boundary
conditions (5). The nonzero components, defined by the
formulas:
r
CH 2=ϕ ,
r
C
k
kEr
2
1
3
ε
= . (6)
The frequency of the wave is TEM from the disper-
sion equation:
02
31
2 =−ε kk , (7)
when ω≤ ωpe TEM waves are absent, and when ω≥ωpe a
dispersion law has the form:
222
3
2
peck ω+=ω . (8)
The dispersion curves for different values ωpe shown
in Fig. 1. Curve 1 corresponds to the zero-density plas-
ma for its ε1 = ε3 = 1. In fact, the only wave coaxial vac-
uum waveguide and the frequency ω = k3c. Curve 2
corresponds to ωpe=1·109rad/s; 3 – ωpe = 2·109rad/s;
4 - ωpe= 2.5·109rad/s, 5 – ωpe= 5.6·109rad/s.
mailto:ira@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2015. №4(98) 37
Fig. 1. The dispersion curves for different values of ωpe
In experiments on the beam-plasma generator (BPG)
signal is fed into the coaxial plasma waveguide with the
main frequency 3·109 rad/s, which is marked by the hori-
zontal line in Fig. 1. With increasing k3thedispersion
curves 2, 3, 4, and 5 converge closer to a curve 1, ω = k3c.
Topography fields Hφ and Er is shown in Fig. 2. In
the middle are calculated for the case of the dispersion
curve 3 in Fig. 1 (ωpe = 2·109 rad/s) at the point
ω = 3·109 rad/s. The curves are normalized to the value
of the Hφ at r = a = 0.6 cm. It can be shown that the
graphics fields Hφ and Er in the vicinity of the intersec-
tion points corresponding-frequency signal with a
straight ω = k3c will be very close.
Fig. 2. Topography fields Hφ and Er for ωpe=2·109rad/s
From (6) and Fig. 2 shows that the field Hφ and Er
largest differ slightly for all radii and fall of hyperbole.
3. ELECTROMAGNETIC MODES
OF A COAXIAL PLASMA WAVEGUIDE
WITHOUT AN EXTERNAL MAGNETIC
FIELD
Consider what else besides the TEM waves at the
absence in waveguide of external-magnetic field. In this
case the set equations (3) break up into two independent
subsystems. The equation for the components of field
Hr, Eφ,
Hz. 0)()(1 2
31
2 =−ε+
∂
∂
∂
∂
rr HkkrH
rrr
,
0)()(1 2
31
2 =−ε+
∂
∂
∂
∂
ϕϕ EkkrE
rrr
. (9)
Equations (9) in the case 02
31
2 ≠−ε kk are the
equations for cylindrical (Bessel) functions of the first
order Z1 (λr). Their decision in the case of a two-
connected domain (coaxial waveguide) is the combina-
tion of Bessel functions first and second kinds:
( ) ( ))()()()( 1111 rNrNBrJrJCE λ+λ+λ+λ=ϕ , (10)
where 2
31
2 kk −ε=λ . Here C, B is arbitrary coeffi-
cients that are the boundary conditions. By the Bessel
functions added someone complex-conjugate of the
(horizontal bar over the function denotes complex con-
jugation) to condition real fields was performed for
complex eigen values λ. Substitution Eφ, recorded thus
in the boundary conditions (5) and taking into account
that )()( 11 rJrJ λ=λ that allows us to obtain the dis-
persion-equation:
1 1
1 1
( ) ( )
0.
( ) ( )
J a N a
J b N b
λ λ
λ λ
= (11)
Fig. 3 shows the solution of the dispersion equation
for the following parameters of waveguide and plasma:
a = 0.6 cm, b = 2.25 cm, ωpe= 5.64·109 rad/s. The pa-
rameters taken for the real apparatus used in the exper-
iment [2]. Type curves did not change with decreasing
of plasma density and accordingly ωpe, under big in-
creasing of the plasma density the curves slightly lift up
the ω.
Fig. 3. The dispersion curves for the case
ωpe = 5.64·109 rad/s. The parameters of the waveguide:
a = 0.6 cm, b = 2.25 cm. The numbers 1, 2, 3 are
marked different radial waves
For the given geometrical dimensions of the wave-
guide and the plasma density is not solution at frequen-
cies below 6.034·1010 rad/s. For comparison, Fig. 4
shows the curves for the same parameters of the plasma
but in the case where the outer radius of the waveguide
b = 38.5 cm. At this case the lower curve in Fig. 4 at
k3 = 0.013 cm-1 corresponds to the frequency of the sig-
nal received from the BPG equal to ω = 3⋅109 rad/s.
Fig. 4. Dispersion curves for the case ωpe= 5.64·109 rad/s.
Waveguide parameters: a= 0.6 cm, b = 38.5 cm
The following shows the radial distribution of com-
ponents of the field for three dispersion curves shown in
Fig. 5 at ω = 2⋅1011 rad/s and respective k3. We see that
ISSN 1562-6016. ВАНТ. 2015. №4(98) 38
for the first curve Fig. 5 the field distributions Hr, Eφ
corresponds to one extreme point (maximum). For sec-
ond curve is the two extreme points, for the third curve
is three extreme points. The distribution of the field Hz
for the first curve has no extreme points, for the second
curve is one extreme point, for the third curve is the two
extreme points.
Fig. 5. Topography fields Hr, Eφ, Hz.
The numbers 1, 2, 3 marked radial modes
The equations for components of Er, Hφ, Ez fields.
0)()(1 2
31
2 =−ε+
∂
∂
∂
∂
rr EkkrE
rrr
,
0)()(1 2
31
2 =−ε+
∂
∂
∂
∂
ϕϕ HkkrH
rrr
. (12)
Just as in the previous case, the solution of these
equations is a combination of Bessel functions:
( ) ( ))()()()( 1111 rNrNBrJrJCH λ+λ+λ+λ=ϕ
. (13)
Substituting into the equation of the system (3):
zEikrH
rr 1)(1
ε−=
∂
∂
ϕ .
Obtain an expression for Ez:
( ) ( ))()( 00 rNBrJCEz λ+λ= , (14)
and after the satisfaction of the boundary conditions (5),
we obtain the dispersion equation:
0)()()()( 0000 =λλ−λλ aNbJbNaJ . (15)
The dispersion curves of the second triple of fields
shown in Figs. 6, 7 shows the dispersion curves for the
E-type wave (solid line) and H-type wave (dashed line).
On Fig. 8 shows the radial distribution of the compo-
nents of the field, three dispersion curves shown in
Fig. 6 at ω = 2⋅1011 rad/s and related k3.
Fig. 6. The dispersion curves of the relation (15)
for the case ωpe = 5.64·109 rad/s. Parameters
of waveguide are: a = 0.6 cm, b = 2.25 cm
Fig. 7. Comparison of the dispersion curves for different
triples fields. The dispersion relation (11) corresponds
dashed line and the dispersion relation (15)
corresponds solid line
We see that for the first curve of Fig. 6 the distribu-
tion of Er, Hφ fields have not extreme points, for the
second curve has one extreme point, for the third curve
have the two extreme points. Allocation of Ez field for
the first curve corresponds to one extremal point (max-
imum). The second curve have the two extreme points,
the third curve have the three extreme points.
Fig. 8. Topography of Er, Hφ, Ez fields.
The numbers 1, 2, 3, marked radial modes
corresponding dispersion curves in Fig. 6
ISSN 1562-6016. ВАНТ. 2015. №4(98) 39
4. ELECTROMAGNETIC MODES
OF A COAXIAL PLASMA WAVEGUIDE
IN AN EXTERNAL MAGNETIC FIELD
Let's go back to the original system of equations (1)
and is considered to be at arbitrary values of the mag-
netic field and plasma density. In this case, the inde-
pendent E-and H-type waves does not exist. Transform-
ing the system of equations, we find the relationship
between Hφ and Eφ, as well as between Hr and Er:
,0)()(1
1
32
3
1
3
2
31
2
=
ε
εε
+
ε
ε−ε
+
∂
∂
∂
∂
ϕϕϕ EkkHkkrH
rrr
0)()(1
1
2
3
2
3
1
2
2
2
12 =
ε
ε
−−
ε
ε+ε
+
∂
∂
∂
∂
ϕϕϕ HkkEkkrE
rrr
. (16)
Set equations for Hr and Er:
0)(1
=++
∂
∂
∂
∂
rrr sEqHrH
rrr
, (17)
0)(1
=++
∂
∂
∂
∂
rrr tHpErE
rrr
,
where r
rrr ∂
∂
∂
∂
=∆
1~
, )( 2
31
2 kkq −ε= , 23ε= kks ,
])([
1
32
33
1
2
22
ε
ε
−ε+
ε
ε
= kkp , ])1([
1
2
3
1
3
2
3
3
ε
ε
−
ε
ε
−ε= kk
k
kt .
This system of equations (17) is equivalent to two
fourth-order equations for Hr and Er:
( ) ( ) 0~~2 =−+∆++∆ rrr HstqpHqpH , (18)
( ) ( ) 0~~2 =−+∆++∆ rrr EstqpEqpE . (19)
From these equations it is seen that the radial com-
ponents of the electric and magnetic fields are described
by the same equation. The solution of the equation for
Hr and Er is the first order Bessel function Z1(λr). It
should be noted that ∆~ is not radial component of the
transverse Laplacian operator, and a solution of both
equations (18) is the first order Bessel function Z1(λr).
Substituting Z1 (λr) in equation (18) we obtain an ex-
pression for the determination of λ:
( ) 0)(24 =−+λ+−λ stqpqp , (20)
from its we find the values of λ2:
stpqpq
+
−
±
+
=λ
4
)(
2
2
2
2,1 . (21)
As can be seen from (21) generally have 4-root of
equation (20):
stpqpq
+
−
+
+
=λ
4
)(
2
2
1
,
stpqpq
+
−
−
+
=λ
4
)(
2
2
2
,
stpqpq
+
−
+
+
−=λ−=λ
4
)(
2
2
13
,
stpqpq
+
−
−
+
−=λ−=λ
4
)(
2
2
24
.
The properties of the Bessel functions )()( 11 rZrZ λ−=λ−
it should be that of Bessel functions for λ1 and λ3, as
well as for λ2 and λ4 are linearly dependent. Therefore,
the solution of the first equation (18) will be expressed
by λ1 and λ2. In addition, as Hr is a real value, it will be
limited to a combination of:
( ) ( ))()()()( 21211111 rZrZBrZrZCHr λ+λ+λ+λ= . (22)
Here C and B are the arbitrary coefficients, which
are determined from the boundary conditions. Substitut-
ing Hr in the form (22) with the expression (21) for λ1
and λ2 and take into account the properties of Bessel
functions )()( 11 rZrZ λ=λ as well as rules for their dif-
ferentiation )()( 1
/
0 rZrZ λλ−=λ , and obtain expressions
for
rH∆~ и rH2~
∆ :
λλ+λλ−
λλ+λλ−=∆ )()()()(~
21
2
221
2
211
2
111
2
1 rZrZBrZrZСHr ,
λλ+λλ+
λλ+λλ=∆ )()()()(~
21
4
221
4
211
4
111
4
1
2 rZrZBrZrZСHr .
After the substitution of which in the first equation
(18) we get:
( )
( )
4 4 2 2 2
1 1 1 2 1 2 1 2 1 1 1
2 2 2 2 2
1 2 2 1 2 1 2 1 1 1 2
4 4 22 2
1 1 2 2 1 11 1 1 2 1
22 2 2 2
2 2 1 21 2 1 1 2 1 1
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) 0.
Ñ Z r B Z r Ñ Z r
B Z r ÑZ r BZ r
Ñ Z r B Z r Ñ Z r
B Z r ÑZ r BZ r
λ λ λ λ λ λ λ λ
λ λ λ λ λ λ λ λ
λ λ λ λ λ λ λ λ
λ λ λ λ λ λ λ λ
+ − + −
− + + + +
+ + − + −
− + + + =
Obviously, the first part of the expression-containing
sponding part with λ equal to zero and after bringing the
similar items have:
( )
4 4 2 2 2
1 1 1 2 1 2 1 2 1 1 1
2 2 2 2 2
1 2 2 1 2 1 2 1 1 1 2
( ) ( ) ( )
( ) ( ) ( ) 0.
Ñ Z r B Z r Ñ Z r
B Z r ÑZ r BZ r
λ λ λ λ λ λ λ λ
λ λ λ λ λ λ λ λ
+ − + −
− + + + =
The second part of the expression that contains the
complex conjugate λ the solvable only if factors both
at constant C and at constant B intercept to zero, that is:
[ ] 0)()()( 11
2
2
2
111
2
1
2
2
2
111
4
1 =λλλ+λλλ+λ−λλ rСZrZСrZС ,
[ ] 0)()()( 21
2
2
2
121
2
2
2
2
2
121
4
2 =λλλ+λλλ+λ−λλ rBZrZBrZB .
And these conditions lead us to the fact that the complex
conjugate parts 1λ and 2λ must satisfy the equations:
[ ] 02
2
2
1
2
1
2
2
2
1
4
1 =λλ+λλ+λ−λ , (23)
[ ] 02
2
2
1
2
2
2
2
2
1
4
2 =λλ+λλ+λ−λ .
Which, in turn, means that:
2
2
2
1
2
1 λλ=λ , (24)
2
2
2
1
2
2 λλ=λ .
From this we can conclude that we are entitled to
write the expression for the field in the form:
1 1 1 2( ) ( ).rH CZ r BZ rλ λ= +
In our case, we will conduct a review of coaxial
waveguide with the conductive walls, which means so-
lution will be put through a Bessel function and Neu-
mann function, i.e.:
)()()()( 212111212111 rNBrNBrJCrJCHr λ+λ+λ+λ= . (25)
Expressing Eφ and Ez by Hr written as (25), we ob-
tain:
ISSN 1562-6016. ВАНТ. 2015. №4(98) 40
( ),)()(
2
1
11
3
∑
=
ϕ λ+λ
−
=
s
ssss rNBrJC
k
kE
0 0
3 3
( ( ) ( )) ,z s s s s s
s
iE C J r B N r
k
λ λ
ε
−
= Λ +∑
where
2
3
2,12
2,1
12,1
2,1 ελ+
λ−
ελ
=Λ
k
k
p
t . (26)
Satisfying the boundary conditions (5), we obtain
the dispersion equation:
1 0 1 2 0 2 1 0 1 2 0 2
1 0 1 2 0 2 1 0 1 2 0 2
1 1 1 2 1 1 1 2
1 1 1 2 1 1 1 2
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
J a J a N a N a
J b J b N b N b
J a J a N a N a
J b J b N b N b
λ λ λ λ
λ λ λ λ
λ λ λ λ
λ λ λ λ
Λ Λ Λ Λ
Λ Λ Λ Λ
=
(27)
Below are shown the solution of the dispersion equa-
tion for the characteristic frequency with parameters
corresponding to the experimental conditions. It should
be noted that in some region on the plane (ω, k3) square
Eigen values λ (expression (21)) becomes complex. The
boundaries of this region are determined by the condi-
tion that 04)( 2 <+− stpq , after substitution of val-
ues q, p, s, t, is given by:
( ) ( )+ω+ω⋅ω−+ωω 2222
3
222
3
24 224 pHH ckck
2 4 4
3 0.H k cω+ <
The solution of this inequality leads to a condition
on
ω: 12 ω<ω<ω ,
( )
+ω
ω−ωω+ω±ω+ω
=ω 22
3
2
2122
3
222422
32,1 4
22
ck
ck
ck
H
ppHpHp . (28)
In this area, the Eigen values λ are complex conjugate,
i.e. 21 λ=λ . Fig. 9 shows the dependence of the fre-
quency ω vs. the longitudinal wave number kz = k3 when
the electron-cyclotron frequency ωHe = 1.4·1010rad/s. In
Figs. 9 and 10 a complex λ2 is limited lobe.
Fig. 9. The frequency ω of the longitudinal wave num-
ber k3 in case ωpe = 5.64·109 rad/s. The numbers repre-
sent the different branches of the plasma waves.
For each of them a specific value λ: λ1 = 1.86;
λ2 = 3.85; λ3 = 5.7; λ4 = 7.6; λ5 = 9.5
The numbers (1) - (5) represent the different branch-
es of high-frequency plasma waves. High-frequency
plasma waves under given parameters of the waveguide
and plasma are very close to the hybrid-frequency
22
Hepe ω+ω
and, for ease of analysis, are shown in a
separate chart (Fig. 11). High-frequency and low-
frequency TEM-type waves tend to ω = k3c, which is
highlighted in the chart the dashed blue line. Numerals
(6) and (7) on the graph designated two first high-
frequency electromagnetic waves. It can be seen that the
branches of plasma waves with the increase in the value
of k3 is committed to ωpe = 5.64·109 rad/s, but do not
cross it. For ease of analysis Fig. 10 shows some disper-
sion curves, but on a linear scale, as opposed to Fig. 9,
where the scale is logarithmic ω.
Fig. 10. The dependence of ω vs. k3 at case
ωpe = 5.64·109 rad/s in the linear scale
Fig. 11. The dependence ωvs. k3for case
ωpe = 5.64·109 rad/s in the linear scale
Fig. 11 shows a part of Fig. 10 is an enlarged axially ω.
It shows that near hybrid-frequency 22
Hepe ω+ω are
presented two branches of the high-frequency plasma
waves. Here topography fields for this case. The expres-
sions for the fields:
,))()(( 11∑ λ+λ=
s
ssssr rNBrJCH
),)()(( 11
3
∑ λ+λ
−
=ϕ
s
ssss rNBrJC
k
kE
,))()(( 00
33
∑ λ+λΛ
ε
−
=
s
sssssz rNBrJC
k
iE
,))()(( 11
2
∑ λ+λ⋅
−λ
=
s
ssss
s
r rNBrJC
s
qE
,))()(( 112
3
2
22
3
1∑ λ+λ
ε
−
−λε
=ϕ
s
ssss
s rNBrJC
k
k
s
q
k
kH
,))()(( 00
3
∑ λ+λ
λ−
=
s
ssss
s
z rNBrJC
k
iH
ISSN 1562-6016. ВАНТ. 2015. №4(98) 41
0 0
3 3
( ( ) ( )).z s s s s s
s
iE C J r B N r
k
λ λ
ε
−
= Λ +∑
Fig. 12 shows the topography golf first radial
threads plasma waves, (λ1 = 1.86) and the waveguide
parameters and plasma: a = 0.6 cm, b = 2.25 cm,
ωpe = 5.64·109 rad/s, ωHe = 1.4·1010 rad/s. For conven-
ience field analysis are normalized to the maximum
value of Hr.
Fig. 12. Topography fields for the case λ1= 1.86:
curve 1 in Fig. 9 at ω = 4.6∙109 rad/s, k3 = 2.86.
Parameters of the magnetic field and plasma density
correspond ωpe= 5.64·109 rad/s, ωHe = 1.4·1010 rad/s
Fig. 13. Topography of low frequency fields for the ca-
seterm TEM-wave at the point ω = 9∙109 rad/s,
k3 = 0.339. The parameters of the magnetic field
and plasma density correspond ωpe = 5.64·109 rad/s,
ωHe = 1.4·1010 rad/s
This point corresponds to the phase velocity
υph=2.65⋅1010 cm / s, which is close to the light velocity
c. As can be seen from figure 13 there are the dominant
components of the field Er, Hϕ and Hz. The Umov-
Poynting vector in the z direction for this wave is great
because it is determined by Er⋅Hϕ, Hr⋅Eϕ , the first of
which is large.
Fig. 14. Topography of low frequency fields for the case
TEM-type wave at the point ω = 14∙109 rad/s,
k3 = 2.294. The parameters of the magnetic field
and plasma density correspond ωpe = 5.64·109 rad/s,
ωHe = 1.4·1010 rad/s
This point corresponds to the phase velocity
υph=6.1⋅109 cm / s, which is less than the light velocity
c. As can be seen from Fig. 14 are dominant field com-
ponent Hz, Hφ, Hr, Er. The Umov-Poynting vector in the
direction of z for this wave is not small, but significant-
ly less than the point from Fig. 13.
Fig. 15. Topography of fields for high-frequency TEM-
type wave at the point ω = 2∙1010 rad/s, k3 = 0.613. The
parameters of the magnetic field and plasma density
correspond ωpe = 5.64·109 rad/s, ωHe = 1.4·1010 rad/s
ISSN 1562-6016. ВАНТ. 2015. №4(98) 42
This point corresponds to the phase velocity
υph=3.26⋅1010 cm/s, which is greater than the light ve-
locity c. As can be seen from Fig. 15 are the dominant
components of the field Er, Hφ and Hz. The Umov-
Poynting vector in the z direction for this wave is great
because it is determined by Er⋅Hϕ, Hr⋅Eϕ the first of
which is large.
CONCLUSIONS
For carrying out of the further numerical compari-
sons with experimental researches on breakdown of a
mixture of gases by microwave radiation with a
stoсhastic jumping phase the theoretical researches of
wave dispersive properties of the created coaxial wave-
guide are carried out. Electromagnetic modes of a coax-
ial plasma waveguide in an external magnetic field are
investigated. The existence of quasi TEM-type modes in
a finite-strength magnetic field is demonstrated. It is
shown that this mode transforms into the true
TEM-mode under the limits of infinitely strong and ze-
ros magnetic fields.
REFERENCES
1. V.I. Karas`, Ya.B. Fainberg, A.F. Alisov,
A.M. Artamoshkin, R. Bingham, I.V. Gavrilenko,
V.D. Levchenko, M. Lontano, V.I. Mirny, I.F.
Potapenko and A.N. Starostin. Interaction of Micro-
wave Radiation Undergoing Stochastic Phase Jumps
with Plasmas or Gases // Plasma Phys. Rep. 2005,
v. 31, № 5, p. 748-760.
2. V.I. Karas`, A.F. Alisov, A.M. Artamoshkin,
S.A. Berdin, V.I. Golota, A.M. Yegorov,
A.G. Zagorodny, I.A. Zagrebelny, V.I. Zasenko,
I.V. Karas’, I.F. Potapenko, and A.N. Starostin. Low
Pressure Discharge Induced by Microwave Radia-
tion with a Stochastically Jumping Phase // Plasma
Phys. Rep. 2010, v. 36, № 8, p.736-749.
3. I.A. Zagrebelny, P.I. Markov, V.O. Podobinsky.
About breakdown in a coaxial waveguide of atomic
gas of low pressure by a microwave radiation with a
stoсhastic jumping phase // Problems of Atomic Sci.
and Technol. Ser. “Plasma Electronics and New Ac-
celeration Methods” (6). 2008, v. 4(56), p. 195-198.
4. I.N. Kartashov, M.V. Kuzelev. Quasi-TEM Elec-
tromagnetic Modes of a Plasma Waveguide with a
No simply Connected Cross Section in an External
Magnetic Field // Plasma Phys. Rep. 2014, v. 40,
№ 12, p. 965-974.
5. L.D. Landau, E.M. Lifshits. Electrodynamics of con-
tinuous media. Physical kinetics. Moscow: “Nauka”,
1982, 620 p.
Article received 20.05.2015
ВОЛНЫ В КОАКСИАЛЬНОМ ПЛАЗМЕННОМ ВОЛНОВОДЕ В МАГНИТНОМ ПОЛЕ
И.В. Карась, И.А. Загребельный
Для проведения дальнейших численных сравнений с экспериментальными исследованиями по пробою
смеси газов с помощью микроволнового излучения со стохастическими скачками фазы проведены деталь-
ные теоретические исследования дисперсионных свойств плазменного коаксиального волновода. Исследо-
ваны электромагнитные волны, распространяющиеся в коаксиальном плазменном волноводе, помещенном
во внешнее магнитное поле. Продемонстрировано наличие режимов квази ТЕМ-типа волн в магнитном поле
конечной величины. Показано, что в пределах бесконечно сильного и нулевого магнитных полей существу-
ют ТЕМ-типа волны.
ХВИЛІ В КОАКСІАЛЬНОМУ ПЛАЗМОВОМУ ХВИЛЕВОДІ В МАГНІТНОМУ ПОЛІ
І.В. Карась, І.А. Загребельний
Для проведення подальших кількісних порівнянь з експериментальними дослідженнями з пробою суміші
газів за допомогою мікрохвильового випромінювання зі стохастичними стрибками фази проведені детальні
теоретичні дослідження дисперсійних властивостей плазмового коаксіального хвилеводу. Досліджені елект-
ромагнітні хвилі, що поширюються в коаксіальному плазмовому хвилеводі, який вміщено в зовнішнє магні-
тне поле. Продемонстрована наявність режимів квазі ТЕМ-типу хвиль у магнітному полі скінченної величи-
ни. Показано, що в межах нескінченно сильного та нульового магнітних полів існують ТЕМ-типу хвилі.
ELECTROMAGNETIC MODES OF A COAXIAL PLASMA WAVEGUIDE IN AN EXTERNAL MAGNETIC FIELD
I.V. Karas’, I.A. Zagrebelny
PACS: 52.80.Pi, 52.65.-y, 52.65.Ff, 52.70. Ds, 52.70.Kz, 84.40Fe
Introduction
Волны в коаксиальном плазменном волноводе в магнитном поле
И.В. Карась, и.а. загребельный
ХВИлІ в коаксІальномУ плазмоВОмУ ХвИлЕводІ в магнІтномУ полІ
І.В. Карась, І.а. загребельний
|
| id | nasplib_isofts_kiev_ua-123456789-112238 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:27:00Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Karas, I.V. Zagrebelny, I.A. 2017-01-18T20:02:06Z 2017-01-18T20:02:06Z 2015 Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field / I.V. Karas, I.A. Zagrebelny // Вопросы атомной науки и техники. — 2015. — № 4. — С. 36-42. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.80.Pi, 52.65.-y, 52.65.Ff, 52.70. Ds, 52.70.Kz, 84.40Fe https://nasplib.isofts.kiev.ua/handle/123456789/112238 For carrying out of the further numerical comparisons with experimental researches on breakdown of a mixture of gases by microwave radiation with a stoсhastic jumping phase the theoretical researches of wave dispersive properties of the coaxial waveguide are carried out. Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field are investigated. The existence of quasi-TEM modes in a finite-strength magnetic field is demonstrated. It is shown that, in the limits of infinitely strong and zero magnetic fields, this mode transforms into a true TEM mode. Для проведення подальших кількісних порівнянь з експериментальними дослідженнями з пробою суміші газів за допомогою мікрохвильового випромінювання зі стохастичними стрибками фази проведені детальні теоретичні дослідження дисперсійних властивостей плазмового коаксіального хвилеводу. Досліджені електромагнітні хвилі, що поширюються в коаксіальному плазмовому хвилеводі, який вміщено в зовнішнє магнітне поле. Продемонстрована наявність режимів квазі ТЕМ-типу хвиль у магнітному полі скінченної величини. Показано, що в межах нескінченно сильного та нульового магнітних полів існують ТЕМ-типу хвилі. Для проведения дальнейших численных сравнений с экспериментальными исследованиями по пробою смеси газов с помощью микроволнового излучения со стохастическими скачками фазы проведены детальные теоретические исследования дисперсионных свойств плазменного коаксиального волновода. Исследованы электромагнитные волны, распространяющиеся в коаксиальном плазменном волноводе, помещенном во внешнее магнитное поле. Продемонстрировано наличие режимов квази ТЕМ-типа волн в магнитном поле конечной величины. Показано, что в пределах бесконечно сильного и нулевого магнитных полей существуют ТЕМ-типа волны. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нерелятивистская электроника Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field Хвилі в коаксіальному плазмовому хвилеводі в магнітному полі Волны в коаксиальном плазменном волноводе в магнитном поле Article published earlier |
| spellingShingle | Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field Karas, I.V. Zagrebelny, I.A. Нерелятивистская электроника |
| title | Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field |
| title_alt | Хвилі в коаксіальному плазмовому хвилеводі в магнітному полі Волны в коаксиальном плазменном волноводе в магнитном поле |
| title_full | Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field |
| title_fullStr | Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field |
| title_full_unstemmed | Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field |
| title_short | Electromagnetic modes of a coaxial plasma waveguide in an external magnetic field |
| title_sort | electromagnetic modes of a coaxial plasma waveguide in an external magnetic field |
| topic | Нерелятивистская электроника |
| topic_facet | Нерелятивистская электроника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112238 |
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