Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma
Coupled surface extraordinarily polarized waves and bulk ordinarily polarized waves are proved to propagate along azimuthal direction in the cylindrical metal waveguide that is partially filled by cold magneto-active plasma in the frequency range above upper hybrid resonance. Their interaction and...
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| Date: | 2006 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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| Cite this: | Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma / O.I. Girka, I.O. Girka, V.O. Girka, I.V. Pavlenko // Вопросы атомной науки и техники. — 2006. — № 5. — С. 28-33. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859707509895331840 |
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| author | Girka, O.I. Girka, I.O. Girka, V.O. Pavlenko, I.V. |
| author_facet | Girka, O.I. Girka, I.O. Girka, V.O. Pavlenko, I.V. |
| citation_txt | Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma / O.I. Girka, I.O. Girka, V.O. Girka, I.V. Pavlenko // Вопросы атомной науки и техники. — 2006. — № 5. — С. 28-33. — Бібліогр.: 14 назв. — англ. |
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| description | Coupled surface extraordinarily polarized waves and bulk ordinarily polarized waves are proved to propagate
along azimuthal direction in the cylindrical metal waveguide that is partially filled by cold magneto-active plasma in
the frequency range above upper hybrid resonance. Their interaction and linear conversion is investigated under the
condition of an external magnetic field with both axial and azimuthal components application.
Показано, что связанные поверхностные необыкновенно поляризованные волны и объемные обыкновенно поляризованные волны могут распространяться в азимутальном направлении в цилиндрическом металлическом волноводе, который частично заполнен холодной магнитоактивной плазмой в частотном диапазоне выше верхнего гибридного резонанса. Их взаимодействие и линейная конверсия исследованы в условиях приложения внешнего магнитного поля с аксиальной, а также азимутальной составляющими.
Показано, що зв’язані поверхневі незвичайно поляризовані хвилі та об’ємні звичайно поляризовані хвилі
можуть поширюватись в азимутальному напрямку в циліндричному металевому хвилеводі, який частково
заповнено холодною магнітоактивною плазмою в частотному діапазоні вище верхнього гібридного
резонансу. Їхні взаємодія та лінійна конверсія досліджені за умов прикладення зовнішнього магнітного поля
з аксіальною, а також азимутальною складовими.
|
| first_indexed | 2025-12-01T03:58:29Z |
| format | Article |
| fulltext |
COUPLED HF AZIMUTHAL WAVES IN MAGNETOACTIVE WAVEG-
UIDE PARTIALLY FILLED BY CURRENT-CARRYING PLASMA
O.I. Girka, I.O. Girka, V.O. Girka, I.V. Pavlenko
Kharkiv National University, Svobody sq., 4, Kharkiv 61077, Ukraine
E-mail: girkai@univer.kharkov.ua
Coupled surface extraordinarily polarized waves and bulk ordinarily polarized waves are proved to propagate
along azimuthal direction in the cylindrical metal waveguide that is partially filled by cold magneto-active plasma in
the frequency range above upper hybrid resonance. Their interaction and linear conversion is investigated under the
condition of an external magnetic field with both axial and azimuthal components application.
PACS: 52.35.-g, 52.40. Fd
1. INTRODUCTION
Capabilities of using the ring-type charged particle
flows in sources of coherent radiation are intensively
studied [1-4] with the goal to increase an efficiency of
electronic devices utilized in the field of high frequency
engineering alongside with elaborating the devices
working on longitudinal charged particle beams (see, for
example, [5] and references therein). In paper [6], it is
proposed to use the ring-type flows for excitation of ex-
traordinarily polarized azimuthal surface waves (ASW).
For azimuthal electromagnetic oscillations, the depen-
dence of the wave fields on coordinates and time t is
chosen in the following way: f(r)ехр[i(mϑ − iωt)],
where ϑ is azimuthal angle. Thus usage of plasma fill-
ing of waveguides as contrasted to vacuum devices al-
lows continuous frequency controlling of electromag-
netic signal in definite frequency range. The dispersion
properties of the ASW propagating across an axial
steady magnetic field in the cylindrical metal waveg-
uide, partially-filled by plasma, are studied in [7]. As far
as the eigen modes are excited in waveguide structures
with a maximum increment, then research into the prop-
erties of eigen electromagnetic oscillations of magneto-
active plasma-filled waveguides is interesting from
practical point of view for plasma electronics.
Influence of steady azimuthal magnetic field B0ϑ,
caused, for example, by electric current, which is carry-
ing by the plasma column, for the case of the waveguide
that is partially-filled by plasma, on low-frequency (LF)
ASW dispersion properties is studied in [8]. Availability
of an external steady magnetic field with two compo-
nents: axial and azimuthal (in cylindrical coordinate
system 0B
= В0z ze + В0ϑ ϑe ) does not allow presenting
the set of Maxwell equations for azimuthal electromag-
netic oscilaltions in the form of two independent subsys-
tems, which describe respectively, ASW with field com-
ponents Er , Eϑ , Bz and ordinarily polarized bulk az-
imuthal electromagnetic wave with field components
Ez , Br , Bϑ . LF ASW can resonantly interact with ordi-
narily polarized surface wave, if the dielectric layer,
which separates the plasma column from a metal wall of
the chamber, is broad enough and if the external axial
magnetic field is nonzero one [8]. Then the correction to
the eigen frequency of the ASW, caused by the az-
imuthal magnetic field, becomes proportional to the first
power of the B0ϑ .
In the present paper, the propagation of azimuthal
electromagnetic waves in magneto-active metal waveg-
uide, which is partially filled by plasma at presence of
an external magnetic field with both axial and azimuthal
components, is considered. The interaction of surface
azimuthal waves with ordinarily polarized azimuthal
bulk waves (АВW) is the object of the research in the
present paper. Here the dispersion relation for azimuthal
waves is derived. This relation allows taking into the ac-
count weak (as compared with the axial component) ex-
ternal azimuthal magnetic field and describing linear in-
teraction between ASW and ABW. In the paper, the
correction to eigen frequency of azimuthal waves
caused by an external azimuthal magnetic field is deter-
mined, and its influence on spatial distribution of elec-
tromagnetic fields of the waves is studied.
2. FORMULATION OF THE PROBLEM
Let's consider cylindrical metal waveguide with ide-
ally conductive wall and circular cross-section of radius
b. It is partially filled by cold magneto-active plasma
column of radius a with a radially inhomogeneous den-
sity profile. The plasma column is separated from the
chambers’ wall by dielectric layer with a permeability ε
d=1. Plasma is considered to be homogeneous along
axis z and azimuthal angle ϑ (in cylindrical
coordinates). Electrodynamical properties of plasma are
described by permeability tensor εij, obtained in the ap-
proach of cold magneto-active plasma [9,10]. It is as-
sumed, that an external constant magnetic field has two
components: axial B0z and azimuthal B0ϑ (r), respective-
ly. We restrict our consideration by the case of small ex-
ternal azimuthal magnetic fields, β≡ B0ϑ /B0z<<1, which
one corresponds to flowing of enough weak axial cur-
rents. The choice of such limiting case is explained, on
one hand, by a simplicity of its experimental realization,
and on the other hand, by the fact that it can be investi-
gated both numerically and analytically.
Neglecting the small addenda of the order above the
first one with respect to the small parameter β, from
Maxwell equations we derive the following set of differ-
ential equations for axial components of the ABW elec-
tric field Ez and ASW magnetic field Bz :
dr
dEr
dr
d
r
1 z + z2
2
2
o E
r
mk
− = M̂ ·Bz, (1)
2
x
z
z2
x
2
x
2
2
z
2
н k
EKB
krr
m
kr
m1
r
B
k
r
rr
1
=
∂
∂−+−
∂
∂
∂
∂ µ
. (2)
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2006. № 5.
Серия: Плазменная электроника и новые методы ускорения (5), с.28-33.28
The following nomenclature is applied here:
M̂ = − ikβ
dr
d
− ikβ 2
x
2
o
k
k
+
r
m
dr
d µ
, (3)
( ) ( )
−−−−−−=
dr
d
r
ikK 2
231
2
1
2
231
2
1
1
ˆ εεεεβεεεεβ
ε
−
βε2ε3 r
m
− ( )( )
dr
d 2
231
2
1 εεεεβ −−
+ ( ) ( )
−−−
dr
lnd 2
2
2
12
231
2
1
εεεεεεβ . (4)
The radial wave number of H-wave (that is ordinari-
ly polarized) kо and depth of penetration of E-wave (that
is extraordinarily polarized) kx
-1 in plasma are deter-
mined through the components of permeability tensor ε
ij of cold plasma [9,10] as follows:
ko
2=k2ε3, kx
2=k2ε1 (µ2 − 1), (5)
where ck=ω, µ=ε2/ε1. The fields of investigated waves
meet the following boundary conditions: the amplitudes
of the waves’ fields are finite values inside the waveg-
uide; the tangential components of electrical and mag-
netic fields of the waves are continuous at the plasma-
dielectric interface; the tangential components of the
waves’ electrical field are equal to zero on the internal
surface of the metal chamber.
3. SPATIAL DISTRIBUTION OF THE
WAVES’ FIELDS
We derive their solutions by the method of variation
of constants in the following form:
Ez = (A1− A2 ∫
r
a
xo drMr ψϕ ˆ~ ) ϕo+ o~ϕ A2 ∫
r
нo drMr
0
ˆψϕ ,(6)
Bz = (A2− A1 ( )∫
r
a xx
ox
~,W
drK̂~
ψψ
ϕψ
) xψ + x
~ψ A1 ( )∫
r
0 xx
ox
~,W
drK̂
ψψ
ϕψ
, (7)
where ψx(r) is the solution of the equation (2) with the
zero right hand side, which is finite one at r = 0, x
~ψ (r)
is the solution of the equation (2) with the zero right
hand side, which is linearly independent from the solu-
tion ψx(r). The Wronskian of the functions ψx(r) and x
~ψ
(r) is equal:
W (ψx, x
~ψ ) = ψx d x
~ψ /dr - x
~ψ d xψ /dr ∝ - kx
2/r, (8)
where ϕo and o~ϕ are the linearly independent from each
other solutions of the equation (1) with the zero right
hand side. The solution ϕo(r) is finite one at the axis
(r=0), and o~ϕ (r) is a singular solution at the axis.
Wronskian of the functions ϕo(r) and o~ϕ (r) is inversely
proportional to r. Solutions (6) and (7) for the fields of
azimuthal waves contain only two constants of integrat-
ing А1 and А2, because two other constants are deter-
mined from the boundary conditions, that fields Ez and
Bz are finite values at the waveguide axis (r=0).
Other components of the wave fields are expressed
through Ez and Bz as follows:
Er= 2
xk
1
−+ z
2
o
z
z Eki
dr
dBkB
r
km µββ , (9)
Bϑ=
dr
dE
k
i z , Br= kr
mEz ,
Eϑ= 2
xk
1 ( )
+++ z
2
x
2
o
z
z Ekk
dr
dBikB
r
kmi βµ
. (10)
Note, that, as it should be for ordinarily polarized waves
[10], the properties of АВW don’t depend on the value
of an external axial magnetic field Вoz and, as it is
shown in [8], ordinarily polarized ASW can’t propagate
in the metal waveguide, which is entirely filled by cold
plasma.
The radial dependence of the waves’ fields in dielec-
tric (a < r < b) region can be expressed through Bessel
cylindrical function of the first kind Jn(ξ) and Neumann
function Nn(ξ):
Ez (d) (r, ϑ) = C1 [Jm(κr) Nm (κb) − Jm (κb) Nm (κr)]
exp(imϑ − iω t), (11)
Bz (d) (r, ϑ) = C2 Pm (κr) exp (imϑ − iωt), (12)
Pm (κr) =Jm (κr) N'm (κb) − J'm (κb) Nm (κr). (13)
Here κ2=εdk2, C1,2 – are normalization factors, the prime
denotes the derivative with respect to the argument. The
expressions (11) and (13) are constructed so that tangen-
tial components of the waves’ electric field are equal to
zero on the metal wall interface.
4. DISPERSION RELATION
Application of boundary conditions which consist in
continuity of tangential components of electrical and
magnetic fields of the considered waves Eϑ, Ez, Bϑ and
Bz on plasma - dielectric interface (r=a) allows to de-
duce the following dispersion relation:
DoDx=D (β). (14)
Here Do=0 is dispersion relation of АВW in the absence
of an external steady azimuthal magnetic field,
Do=ϕo
1
o
dr
d −
ϕ
−
κ
1 ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )aNbJbNaJ
aNbJbNaJ
mmmm
mmmm
κκκκ
κκκκ
′−′
− , (15)
Dx=0 is dispersion relation of АSW in the absence of an
external constant azimuthal magnetic field,
Dx =− 2
x
2
k
k
+
a
m
dr
d1 x
x
µψ
ψ −
− κ
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )aNbJbNaJ
aNbJbNaJ
mmmm
mmmm
κκκκ
κκκκ
′−′
′′−′′
, (16)
D(β) is coupling coefficient which describes interaction
between АВW and АSW, its value is of the second or-
der of smallness on β‚
D (β) =
2
o
dr
d −
ϕ
a
1 ∫
a
0
xo drM̂r ψϕ ( )
2
x
2
x
xx
2
k
~,Wk
ψ
ψψ
( )∫
a
0 xx
ox
~,W
drK̂
ψψ
ϕψ
− iκ β
+
x
o
2
x
2
o
k
k1
ψ
ϕ
. (17)
Let's note, that all values included in expressions (15) -
(17), which are the functions of radius r and are not
written under the sign of integral, have been computed
in the point r=a.
We derive the solution of the equation (14) in the
_______________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2006. № 5.
Серия: Плазменная электроника и новые методы ускорения (5), с.28-33.29
following form: ω=ωx+∆ωx (|∆ωx| << ωx) and ω=ωo+∆
ωo (|∆ωo| <<ωo), where ωx and ωo are eigen frequencies
of the АSW and АВW, respectively, calculated in the
absence of an external azimuthal magnetic field. One
can found from equation (14), that the correction ∆ωx to
the eigen frequency of АSW, which is caused by exter-
nal steady azimuthal magnetic field, is as follows:
∆ωx = D (β) ( ) 1
xo x
DD −
=∂∂ ωωω ∝B0ϑ
2. (18)
The correction ∆ωo to eigen frequency of АВW can be
written in the following form:
∆ωo = D (β) ( ) 1
ox o
DD −
=∂∂ ωωω ∝B0ϑ
2. (19)
Under the definite conditions dispersion curves of the
АВW and АSW intersect: ωx =ωo. Under these condi-
tions, the influence of the field В0ϑ on the interaction be-
tween the АВW and АSW increases, and the correction
to their eigen frequency becomes linear with respect to
an external steady azimuthal magnetic field (namely,
proportional to the parameter β):
(∆ωx)2=(∆ωo)2=D(β) ( ) 1
x x
D −
=∂∂ ωωω ( ) 1
o o
D −
=∂∂ ωωω .(20)
5. ANALYTICAL SOLUTION OF THE
DISPERSION EQUATION
We undertake the further consideration for the case
of homogeneous radial density profile. Such limitation
of the consideration can be justified by the following
reasons. First, this model describes the case of semicon-
ductor plasma. Second, the radial non-uniformity of the
plasma density can be neglected, if plasma density
varies weakly in radial layer which thickness is of the
order of the depth of the waves’ field penetration into
the plasma. Thus, for example, in the case of gas dis-
charge maintenance due to propagation of surface
waves, the uniformity of produced plasma density is
provided just at distances about penetration-depth of the
operating wave into the plasma.
Let's assume also, that the plasma density value is
high enough (Ωe
2>>ωe
2, where Ωe and ωe are Langmuir
and electron cyclotron frequencies, respectively). Such
inequality is always true for n-semiconductor plasma,
but it can be realized also in laboratory gas plasma in
the case of utilization of not too strong steady axial
magnetic field. Such limiting case introduces the great-
est interest for plasma technologies [11], since utiliza-
tion of a strong magnetic field results in essential rising
in price of a unit volume of produced plasma. Let's
mark also, that the dispersion properties of АSW in
magnetized, Ωe
2<ωe
2, cylindrical plasma waveguides are
studied in [12].
In the homogeneous plasma chamber АВW exist in
frequency range ω > Ωe. Since the inequality ko
2 > 0 is
satisfied under this condition, then one can consider the
value ko (5) as a radial wave number of the АВW. In the
case of radial homogeneous plasma the equations (1)
and (2) for Ez and Bz have the form of inhomogeneous
Bessel equations. The solutions of the respective homo-
geneous Bessel equations, which are restricted on the
waveguide axis, can be expressed through cylindrical
functions of imaginary and real arguments, accordingly:
ψx=Im (kx r), ϕo=Jm (kor). (21)
The solutions, which are linearly independent from the
indicated above solutions of equations (1) and (2) with
the zero right hand sides, are functions of McDonald
and Neumann functions:
( )rkK~
xmx =ψ , ( )rkN~
omo =ϕ . (22)
The solutions (21) and (22) are applicable, if ko
2 > 0 and
kx
2 > 0. In its turn, just the condition kx
2 > 0 determines
the frequency ranges of ASW existence [7]. Let’s write
them down:
e
12
e
2
e
2
ie )( ωωωΩΩω <<+ − ,
|ωe | < ω < 2
e
2
e25,0 Ωω + − 0.5|ωe |, (23)
here Ωi is ion plasma frequency. Analyzing the ranges
(23), one can see that they are low enough, so it is natu-
ral to refer to them as to the low frequency (LF) ranges.
The minimum frequency is the lower hybrid frequency
and the maximum frequency is the cut-off frequency.
Besides that, the condition kx
2 > 0 is valid also in the
narrow frequency range that is little bit higher than the
upper hybrid frequency:
2
e
2
ee
2
e
2
e 25,05,0 ΩωωωΩω ++<<+ . (24)
It is natural to refer to this frequency range as to high
frequency (HF) range.
The analysis of value ko
2 testifies that it has opposite
signs in the ranges (23) and (24): namely, ko
2 < 0 in the
LF ranges (23), and it becomes positive, ko
2 > 0, in the
HF range (24). Thus, the resonant linear interplay be-
tween ordinarily polarized bulk waves and АSW can
take place only in the HF range (24).
Let's assume, that the steady azimuthal magnetic
field Вoϑ depends on radius r linearly: Вoϑ (r)= Вoϑ(a)
r/a. Such choice of radial dependence for the magnetic
field is explained by the following reasons. If the az-
imuthal magnetic field is produced by an axial current
of completely ionized plasma, which conductivity is de-
termined by pair Coulomb collisions, then it means that
the temperature profile of electrons is homogeneous
one. (Let’s note also, that the participation of electrons
in an axial motion, which is connected with the current,
does not result in Doppler shift of frequency due to zero
value of the axial wave number.) It is necessary to point
out, that in this case it is possible to calculate the integrals
in the expression (17) for D(β) in the explicit form.
The conditions of resonant linear interaction be-
tween ordinarily polarized ASW and extraordinarily po-
larized АSW can be provided only in the case of a wide
dielectric layer between plasma and metal wall of the
waveguide (when a lot of half-wavelengths can be locat-
ed along the radial coordinate in this layer) for the elec-
tromagnetic oscillations propagating along the direction
of electrons’ rotation on Larmor orbits [8].
In the case of a narrow dielectric layer (∆=(b−
a)/a<1) and homogeneous plasma, the dispersion rela-
tions for АВW (15) and АSW (16) in an axial magnetic
field are essentially simplified:
Do=
( )
( )akJk
akJ
omo
om
′ + (b− a) (1− 0.5∆), (25)
Dx =− 2
x
2
k
k ( )
( )
+
′
a
m
akI
akIk
xm
xmx µ
−
− ∆ [κ 2a2− m2+0.5∆ (3m2− κ 2a2)] /a. (26)
30
Analysis of the dispersion relation (26) Dx=0 for
АSW in an axial magnetic field (Вoϑ=0) demonstrates,
that in narrow waveguides (a << |m|δ, where δ=с/Ωe is
the skin-depth) HF АSW do not propagate [9]. An ana-
lytical estimation of the waveguide radius, in which one
HF АSW can exist in absence of the dielectric layer, can
be determined by the following inequality: a > |m/ωe|c.
The availability of a narrow dielectric gap leads to
little decreasing of the HF ASW eigen frequency. In
wide waveguides (a >> |m|δ) its value can be approxi-
mately determined as follows:
ωx ≈ ( ) 2
e
2
ef
2
e k1 ωΩ ++ − mωeεd∆ (1− kef
2/εd). (27)
Effective wave number kef = | m|δ/a is utilized here.
If the applied magnetic field and the sizes of the
waveguide are not too large, so that the following in-
equality takes place:
22
s,m
22
e
2 mjca −<ω , (28)
then the eigen bulk ordinarily polarized azimuthal
waves do not propagate in the HF range (24). Here j|m|,s
is the magnitude of the s-th root of the cylindrical
Bessel function of the first kind of the |m |-th order, i.e.
J|m|(j|m|, s) =0.
With increasing of the waveguide radius and/or the
magnetic field value, the condition of coincidence of
eigen frequencies АSW and АВW can be realized. This
condition follows from the inequality (28), in which the
sign of inequality is replaced by the sign of equality. In
this case eigen frequency of АSW takes the form ω=ω
x+∆ωx, where ωx is given by the equation (27), and the
correction ∆ωx (18) to ASW frequency is equal to:
∆ωx ≈ β|m|1.5Z −2.5j|m|,s
−2Ωe, (29)
where Z=Ωe/|ωe| > 1. One can draw a conclusion from
the analysis of expression (29), that the resonant influ-
ence of В0ϑ on the ASW spectra strengthens at increas-
ing an axial constant magnetic field, and also at decreas-
ing the plasma density, azimuthal wave number and ra-
dius of the waveguide chamber. The correction (29) to
the wave frequency appears to be small as compared
with the main addend (27) even if В0ϑ is not a small, be-
cause the smallness of this correction is provided with
small multiplicands Z−2.5j|m|,s
−2. At the same time, it is
necessary to note, that for definite values of plasma pa-
rameters, namely, if j|m|,s
2 − m2 < a2ωe
2/c2 < j|m|,s+1
2 − m2,
i.e. in the intervals between two neighboring points of
intersection for dispersion curves of ordinarily and ex-
traordinarily polarized modes, the derivative dϕo/da lo-
cated in the denominators in equations (15) and (17) be-
comes equal to zero. It means, that for such values of
parameters, the influence of the steady azimuthal mag-
netic field on the dispersion properties of azimuthal
waves becomes even more than that nearby the points of
intersections for dispersion curves of ordinarily and ex-
traordinarily polarized modes.
6. NUMERICAL RESEARCH
OF DISPERSION PROPERTIES
In the Fig.1 the fine structure of spectral interaction
between АSW and the first, the second and the third ra-
dial modes of АВW, which propagate with azimuthal
number m =−1, is shown. We have introduced the di-
mensionless variable y: y4=ω2/ωe
2−1−Z2. The level y=0
corresponds to lower boundary of HF frequency range
(24). The higher boundary (dashed line) of HF frequen-
cy range (24) corresponds to the level y ≈ 1.261. The
abscissa axis is represented in a logarithmic scale. At
construction of dispersion curves in the Fig.1, the fol-
lowing values of the waveguide parameters are chosen:
β=0.1, Z=3, ∆=0.
The behavior of dispersion curves indicated by num-
bers 6 and 7 differs weakly from that of the curves indi-
cated by numbers 4 and 5. That is why we do not
demonstrate in Fig.1 fine structure of spectral interac-
tion between АSW and higher radial modes of АВW.
Let's note only the following common features of these
curves. In process of increasing of radial number of the
mode, first, splitting of the frequencies in the transfor-
mation point of the waves becomes less and less pro-
nounced. Second, the range of values of kef between two
adjacent points of the waves’ transformation becomes
more and more narrow. Third, at decreasing of kef the
ASW frequency value becomes more closed to the lower
boundary of the frequency range (24), and value of di-
mensionless parameter y turns to zero point, respectively.
0,1
0,0
0,4
0,8
1,2
0,04 0,2
7
6
5
4
3
2
1
y
k
ef
Fig.1. Fine structure of spectral interaction be-
tween АSW and first three radial modes of АВW in
absence of the dielectric layer. m =−1, Z=3, β=0,1
0,1
0,0
0,4
0,8
1,2
5
4
3
2
1
0,04 0,2
y
k
ef
Fig.2. The same as in Fig. 1, but in presence of
very thin vacuum layer, ∆=0,03, εd=1
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2006. № 5.
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32
0,1
0,0
0,4
0,8
1,2
5 3
2
1
y
0,05 0,2
k
ef
Fig.3. The same, as in a Fig.2, but in the case of
more wide vacuum layer: ∆=0,1
0,1
0,0
0,4
0,8
1,2
5 3
2
1
0,05 0,2
y
k
ef
Fig.4. The same, as in the Fig.2, but in presence of
the dielectric layer: ∆=0,0333, εd=3
0,1
0,0
0,4
0,8
1,2
5 3
2
1
0,04 0,2
y
k
ef
Fig.5. The same, as in Fig.2, but in the case of
more weak constant axial magnetic field: Z=4
Let's point out, that the interaction between АSW
and АВW results in appearance of forbidden bands in
the frequency spectrum (non-passing bands of a signal),
that is representative for problems on wave propagation
in mediums with a periodic spatial non-uniformity (see,
for example, [13]). The appearance of forbidden bands
in the frequency spectrum on the graph of these waves
frequency dependence on kef is accompanied by appear-
ance of ranges of forbidden values of the effective wave
number, which in fact is dimensionless inverse radius of
the plasma cylinder. It is explained by the fact that for
these ranges of plasma radius values АВW are not eigen
modes anymore. That is why at approaching to these
forbidden ranges of kef the correction to frequency of
АSW (these waves are coupled with АВW) sharply in-
creases (being calculated in the first approximation from
the equation (18)). The width of these forbidden ranges
of kef is larger for smaller radii of plasma a/δ.
The fine structure of spectral interaction between
АSW and two radial modes of AВW in presence of very
thin vacuum (εd=1) layer: ∆=0.03, and for the same val-
ues of other parameters of the waveguide is shown in
Fig.2. As contrasted to Fig.1 value of ASW frequency
has notably decreased.
The changes which take place at further increase of
the vacuum layer width up to ∆=0.1 under the same val-
ues of other parameters of the waveguide are shown in
Fig.3. As contrasted to the curves, which were intro-
duced in Fig.2, the branch of the dispersion curve indicat-
ed by number 4 has vanished completely; further decreas-
ing of ASW frequency takes place; and the range of kef ,
within which the waves can exist, moves to the right.
In Fig.4 as contrasted to Fig.3 width of the dielectric
layer is chosen three times less: ∆=0.1/3, and the dielec-
tric permeability is chosen three times larger: εd=3. In ac-
cordance with conceptions of General Physics, three lay-
ers with the same value of permeability exert the same in-
fluence as one layer with the tripled permeability value.
In Fig.5 as contrasted to Fig.2 steady axial magnetic
field is reduced a little: Z=4. This reduction is accompa-
nied by the narrowing of frequency range (24) and its
approaching to Langmuir frequency. As the result the
ABW dispersion curves move in the direction of smaller
kef as contrasted to Fig.2.
Let's estimate an opportunity of experimental obser-
vation of the studied phenomenon. For example, for the
concentration n=1011 cm−3 the Langmuir frequency is
equal to Ωe ≈ 1.8×1010 sec−1. To get the chosen ratio be-
tween Langmuir and electron cyclotron frequencies
(Z=3) the axial magnetic field of moderate value B0z≈
483 G is necessary. In accordance with the condition
(28), minimum radius of plasma is necessary to meet the
resonant conditions for the wave with azimuthal mode
number m = −1. In this case it appears, that
68.131jca 2
1,1
22
ce
2 ≈−=ω , whence it is possible to eval-
uate the radius of plasma: a ≈12 cm. As it was noted
above, АSW propagate only in a wide waveguides (ra-
dius of plasma a > Z|m|δ). For the waveguide with ra-
dius a ≈ 12 cm this inequality holds true, since skin-
depth is δ ≈1.7 cm. To get the steady azimuthal magnet-
ic field that is ten times weaker than the axial one, it is
necessary to supply the axial current I≈3 KA in plasma.
Then the splitting of the wave frequency nearby the in-
tersection point for the dispersion curves of АSW and
first radial mode of АВW is equal to 1.8×107 seс−1,
though minimum spacing interval between the curves
indicated by numbers 2 and 3 in the Fig.2 along ordinate
makes only (y2 − y1) ≈ 0.087.
CONCLUSIONS
In the present paper, the linear interaction of extraor-
dinarily polarized azimuthal surface waves and ordinari-
ly polarized azimuthal bulk waves is studied for the case
of their propagation in metal waveguides which are par-
tially-filled by plasma with an axial current. This phe-
nomenon can be experimentally observed in wide plas-
ma waveguides for the waves propagating with negative
values of azimuthal mode number if utilized steady axi-
al magnetic field is not small, so that |ωe|~j|m|,sc/a. The
property of surface waves to propagate only in one di-
rection across the utilized steady magnetic field along
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2006. № 5.
Серия: Плазменная электроника и новые методы ускорения (5), с.28-33.33
the plasma – metal interface is well-known [9,15]. They
are called as unidirectional waves and are rather useful
in such plasma - filled radio devices, in which it is nec-
essary to provide for the absence of reflected signal.
Influence of the following plasma waveguide param-
eters: width of the dielectric layer, permeability of the
layer, value of the axial steady magnetic field, - on the
fine structure of spectral interaction between АSW and
the radial modes of АВW is studied numerically (see
Fig.2-5). The effect of the steady azimuthal magnetic
field B0ϑ on the spatial distribution of the waves’ fields
is determined (see equations (6) and (7)) by the aid of
perturbation theory with taking into account the adden-
da of the first order of smallness, which are proportional
to B0ϑ.
Far from the conditions of the waves’ linear resonant
interaction, the correction to eigen frequency of АSW,
caused by the external azimuthal magnetic field, is pro-
portional to the square of B0ϑ: ∆ωx ∝ (B0ϑ)2 (whereas the
corresponding frequency correction, introduced by
steady axial magnetic field, is proportional to the first
power of В0z). In other words, transversal component of
the external magnetic field (with respect to the direction
of the wave propagation) affects on the dispersion prop-
erties of АSW stronger, than the longitudinal compo-
nent. Such dependence of eigen frequency on the direc-
tion of an external magnetic field corresponds to the
conclusions, which were obtained in [14] for the case of
surface MHD waves propagation along flat plasma –
metal interface.
It is shown that nearby the points, in which the dis-
persion curves of АSW and АВW cross (see the equa-
tion (28)), the resonant linear interaction between ex-
traordinarily polarized surface and ordinarily polarized
bulk azimuthal waves takes place. Convenient analytical
expression (29) for the resonant correction to the eigen
frequency caused by a steady azimuthal magnetic field
is derived. In this case the correction to the eigen fre-
quency appears to be the value of the first order of
smallness: ∆ωx ∝ B0ϑ.
However, intersection points for the dispersion
curves of АSW and АВW do not indicate those condi-
tions under which a steady azimuthal magnetic field B0ϑ
exerts the strongest influence on the ASW dispersion
properties. Let's remind that under the conditions which
are determined by these points, the integer number of
АВW half-wavelengths compose the plasma radius.
Therefore, just as it happens in the case of linear inter-
action between АSW and АBW, the boundary condi-
tions can be satisfied due to a small frequency shift be-
cause of its simultaneous strong influencing on the spa-
tial distribution of the both interacting modes.
As the result of bulk nature of ABW fields’ spatial
distribution, for the definite sets of values of the waveg-
uide parameters the conditions are realized, under which
an odd number of quarters of forced ABW wavelength
compose the plasma radius. These conditions are ob-
served approximately in the middle between intersection
points for dispersion curves of ABW radial modes and
АSW. In this case to meet the boundary conditions the
frequency correction would be so large, that the corre-
sponding increasing of the wavelength allows withdraw-
ing a superfluous quarter of the wavelength. This cir-
cumstance distinguishes the linear interaction between
АSW and АВW qualitatively from the interaction be-
tween extraordinarily polarized and ordinarily polarized
surface azimuthal waves.
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V.I. Tkachenko // Soviet Journal of Communica-
tions Technology and Electronics. 1988, v.33, №8,
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8. V.O. Girka, I.O. Girka // Soviet Journal of Communi-
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9. A. Kondratenko. Surface and bulk waves in re-
stricted plasma. M.: “Energoatomizdat”, 1985.
10. A.F. Аlexandrov, L.S. Bogdankevich,
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cations, ed. by Ferreira C.M. and Moisan M. 1993,
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v.43, №12, p.1424.
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// Wave theory. M.: Nauka, 1979.
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СВЯЗАННЫЕ ВЧ-АЗИМУТАЛЬНЫЕ ВОЛНЫ В МАГНИТОАКТИВНОМ ВОЛНОВОДЕ,
ЧАСТИЧНО ЗАПОЛНЕННОМ ТОКОВЕДУЩЕЙ ПЛАЗМОЙ
А.И. Гирка, И.А. Гирка, В.А. Гирка, И.В. Павленко
Показано, что связанные поверхностные необыкновенно поляризованные волны и объемные обыкновен-
но поляризованные волны могут распространяться в азимутальном направлении в цилиндрическом металли-
ческом волноводе, который частично заполнен холодной магнитоактивной плазмой в частотном диапазоне
выше верхнего гибридного резонанса. Их взаимодействие и линейная конверсия исследованы в условиях
приложения внешнего магнитного поля с аксиальной, а также азимутальной составляющими.
ЗВ’ЯЗАНІ ВЧ-АЗИМУТАЛЬНІ ХВИЛІ В МАГНІТОАКТИВНОМУ ХВИЛЕВОДІ, ЯКИЙ ЧАСТКОВО
34
ЗАПОВНЕНО СТРУМОВЕДУЧОЮ ПЛАЗМОЮ
О.І. Гірка, І.О. Гірка, В.О. Гірка, І.В. Павленко
Показано, що зв’язані поверхневі незвичайно поляризовані хвилі та об’ємні звичайно поляризовані хвилі
можуть поширюватись в азимутальному напрямку в циліндричному металевому хвилеводі, який частково
заповнено холодною магнітоактивною плазмою в частотному діапазоні вище верхнього гібридного
резонансу. Їхні взаємодія та лінійна конверсія досліджені за умов прикладення зовнішнього магнітного поля
з аксіальною, а також азимутальною складовими.
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2006. № 5.
Серия: Плазменная электроника и новые методы ускорения (5), с.28-33.35
5. Analytical solution of the
dispersion equation
Conclusions
References
|
| id | nasplib_isofts_kiev_ua-123456789-80438 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T03:58:29Z |
| publishDate | 2006 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Girka, O.I. Girka, I.O. Girka, V.O. Pavlenko, I.V. 2015-04-17T19:27:24Z 2015-04-17T19:27:24Z 2006 Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma / O.I. Girka, I.O. Girka, V.O. Girka, I.V. Pavlenko // Вопросы атомной науки и техники. — 2006. — № 5. — С. 28-33. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 52.35.-g, 52.40. Fd https://nasplib.isofts.kiev.ua/handle/123456789/80438 Coupled surface extraordinarily polarized waves and bulk ordinarily polarized waves are proved to propagate along azimuthal direction in the cylindrical metal waveguide that is partially filled by cold magneto-active plasma in the frequency range above upper hybrid resonance. Their interaction and linear conversion is investigated under the condition of an external magnetic field with both axial and azimuthal components application. Показано, что связанные поверхностные необыкновенно поляризованные волны и объемные обыкновенно поляризованные волны могут распространяться в азимутальном направлении в цилиндрическом металлическом волноводе, который частично заполнен холодной магнитоактивной плазмой в частотном диапазоне выше верхнего гибридного резонанса. Их взаимодействие и линейная конверсия исследованы в условиях приложения внешнего магнитного поля с аксиальной, а также азимутальной составляющими. Показано, що зв’язані поверхневі незвичайно поляризовані хвилі та об’ємні звичайно поляризовані хвилі можуть поширюватись в азимутальному напрямку в циліндричному металевому хвилеводі, який частково заповнено холодною магнітоактивною плазмою в частотному діапазоні вище верхнього гібридного резонансу. Їхні взаємодія та лінійна конверсія досліджені за умов прикладення зовнішнього магнітного поля з аксіальною, а також азимутальною складовими. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Релятивистская и нерелятивистская плазменная СВЧ-электроника Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma Связанные ВЧ-азимутальные волны в магнитоактивном волноводе, частично заполненном токоведущей плазмой Зв’язані ВЧ-азимутальні хвилі в магнітоактивному хвилеводі, який частково заповнено струмоведучою плазмою Article published earlier |
| spellingShingle | Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma Girka, O.I. Girka, I.O. Girka, V.O. Pavlenko, I.V. Релятивистская и нерелятивистская плазменная СВЧ-электроника |
| title | Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma |
| title_alt | Связанные ВЧ-азимутальные волны в магнитоактивном волноводе, частично заполненном токоведущей плазмой Зв’язані ВЧ-азимутальні хвилі в магнітоактивному хвилеводі, який частково заповнено струмоведучою плазмою |
| title_full | Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma |
| title_fullStr | Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma |
| title_full_unstemmed | Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma |
| title_short | Coupled HF azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma |
| title_sort | coupled hf azimuthal waves in magnetoactive waveguide partially filled by current-carrying plasma |
| topic | Релятивистская и нерелятивистская плазменная СВЧ-электроника |
| topic_facet | Релятивистская и нерелятивистская плазменная СВЧ-электроника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80438 |
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