Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field
Regions of the most effective transformation of current wave into electromagnetic wave for the charged particle moving along the density gradient in magnetic field, which is parallel to the plasma density gradient, are found. Magnitude of transition radiation of the extraordinary wave is calculated....
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2010
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| Cite this: | Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field / I.O. Anisimov, Yu.N. Borokh // Вопросы атомной науки и техники. — 2010. — № 6. — С. 120-122. — Бібліогр.: 6 назв. — англ. |
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| author | Anisimov, I.O. Borokh, Yu.N. |
| author_facet | Anisimov, I.O. Borokh, Yu.N. |
| citation_txt | Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field / I.O. Anisimov, Yu.N. Borokh // Вопросы атомной науки и техники. — 2010. — № 6. — С. 120-122. — Бібліогр.: 6 назв. — англ. |
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| description | Regions of the most effective transformation of current wave into electromagnetic wave for the charged particle moving along the density gradient in magnetic field, which is parallel to the plasma density gradient, are found. Magnitude of transition radiation of the extraordinary wave is calculated.
Найдены области наиболее эффективной трансформации волны тока в электромагнитные волны для заряда, движущегося вдоль градиента концентрации слабонеоднородной холодной плазмы в магнитном поле, также направленном вдоль градиента концентрации плазмы. Найдена амплитуда переходного излучения необыкновенной волны.
Знайдені області найбільш ефективної трансформації хвилі струму в електромагнітні хвилі для заряду, що рухається вздовж градієнту концентрації слабконеоднорідної холодної плазми в магнітному полі, що також спрямоване вздовж градієнту концентрації плазми. Знайдена амплітуда перехідного випромінювання незвичайної хвилі.
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TRANSITION RADIATION OF THE CHARGED PARTICLE
IN THE INHOMOGENEOUS PLASMA WITH THE LONGITUDINAL
MAGNETIC FIELD
I.O. Anisimov1, Yu.N. Borokh2
Taras Shevchenko National University of Kiev, Radio Physics Faculty, Kiev, Ukraine
E-mail: 1ioa@univ.kiev.ua; 2y.borokh@gmail.com
Regions of the most effective transformation of current wave into electromagnetic wave for the charged particle
moving along the density gradient in magnetic field, which is parallel to the plasma density gradient, are found.
Magnitude of transition radiation of the extraordinary wave is calculated.
PACS: 52.35.Qz; 52.65.Rr
1. INTRODUCTION
Transition radiation in plasma attracts interest because of
its possible applications (usage of modulated electron beams
as radioemitters in ionosphere [1], transillumination of the
dense plasma barriers via electron beams [2], diagnostics of
the inhomogeneous plasma using transition radiation of
electron bunches [3] etc). But this fundamental problem was
not yet solved even for the simplest model of cold planarly-
stratified plasma with magnetic field parallel to its density
gradient [4]. In this work linear transformation of the
given current waves into electromagnetic waves for such
model was studied.
2. MODEL DESCRIPTION AND WAVE
EQUATION FOR VECTOR-POTENTIAL
Cold collisionless plasma is considered. Its density
gradient is constant and parallel to the z-axis. Charged
particle is moving along the density gradient. Magnetic
field parallel to the density gradient is applied to the
system, so plasma permittivity tensor has a form [5]
120 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2010. № 6.
Series: Plasma Physics (16), p. 120-122.
( ) ( )
( ) ( )
( )⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛ −
= ⊥
⊥
z
zzi
ziz
//00
0
0
ε
εα
αε
ε) , (1)
where
( )
( )
( )
2
2 2
2
2
2
1 ;
1
1 ;
.
1
p
c
II p
c p
c
z
z
z
ε
ε
α
⊥
Ω
= −
−Ω
= −Ω
Ω Ω
=
−Ω
(2)
Charged particle moves along z-axis with the velocity
v0. Ampere's circuital law with Maxwell's correction is
taken as a basic equation to find the field caused by this
particle:
1 4 ,DrotH j
c t c
π∂
= +
∂
r
r r (3)
here j is a current density
( ) ( ) ( )0 0zj e ev x y z v tδ δ δ= −
r r
,
(4)
and e is the particle charge.
Electric and magnetic field intensities can be expressed
in the terms of scalar and vector potentials
,
1 .
H rotA
AE
c t
φ
=
∂
= − −∇
∂
rr
r
r (5)
Scalar potential vanishing calibration is used, so the
following equation for vector-potential is obtained:
j
ct
A
c
Arotrot
r
r)r πε 4
2
2
2 =
∂
∂
+ . (6)
3. SCALAR EQUATION FOR VECTOR-
POTENTIAL COMPONENT
Equation (6) should be expanded in a Fourier integral
by time and transversal coordinates for the further
solving. Current density can be written as a jmexp[i(ωt–
κ⊥y–κ//z)], κ//=ω/v0. Solution should be searched in a form
of
( ) ( ) ( ), exp .A r t A z i t yω κ⊥= −⎡⎣ ⎤⎦
r rr (7)
After such substitutions equation (7) can simply be
rewritten as a set of scalar equations. Since there are three
scalar equations and three unknown functions, equation
for one of the vector potential components is obtained
( ) ( ) ( ) ( )
4 2
4 2 04 2 ,x x
x l
d A d A
k z k z k z A k z
dz dz
′ ′ ′+ + = ′ (8)
where
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
4 / /
2 2
2 0 / / / /
2 2
0 0 / /
2 2 2 2 2
0
/ / / /
;
2 ;
;
4 exp .l m
k z z
k z k z z z z
k z k z
k z z z
k z i z j i z
c
ε
ε ε κ ε ε
κ ε
α κ ε ε κ
π κ κ α κ
⊥ ⊥ ⊥
⊥
⊥ ⊥ ⊥ ⊥
⊥
′ =
′ = − +⎡ ⎤⎣ ⎦
′ ⎡ ⎤= − ×⎣ ⎦
⎡ ⎤× − +⎣ ⎦
′ = −
(9)
4. METHOD OF GEOMETRICAL OPTICS
Equation (8) can be solved using method proposed in
[6] for analysis of the distributed reflection. Characteristic
length of the plasma inhomogeneity is the large
parameter. Then solution of (8) can be given in a form
( ) ( ) ( )exp[ ],xA z A z i zϕ= (10)
mailto:ioa@univ.kiev.ua
mailto:y.borokh@gmail.com
http://multitran.ru/c/m.exe?t=401671_1_2
http://multitran.ru/c/m.exe?t=401671_1_2
where A is an amplitude, and φ is an eikonal function. I-th
derivative of A(z) has i-th order of vanishing and i-th
derivative of φ(z) has (i-1)-th order of vanishing.
After substitution of (10) to (8) and neglecting of the
higher orders of vanishing summands general solution of
the homogeneous equation (8) may be found:
( ) [ ] ( )
( )
[ ] ( )
( )
0
0
0
0
1 1 1
1 1
2 2 2
2 2
exp exp
exp
exp exp
exp ,
z
z
z
z
z
z
z
z
A z I A i K z dz
A i K z dz
I A i K z dz
A i K z dz
+
−
+
−
⎛ ⎡ ⎤
′ ′⎜= +⎢ ⎥
⎜ ⎢ ⎥⎣ ⎦⎝
⎞⎡ ⎤
′ ′ ⎟+ − +⎢ ⎥
⎟⎢ ⎥⎣ ⎦ ⎠
⎛ ⎡ ⎤
′ ′⎜+ +⎢ ⎥
⎜ ⎢ ⎥⎣ ⎦⎝
⎞⎡ ⎤
′ ′ ⎟+ −⎢ ⎥
⎟⎢ ⎥⎣ ⎦ ⎠
∫
∫
∫
∫
(11)
121
where
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( )
( )
( )
2
2 2 0 4
1,2
4
4 1,2 2 1,2
1,2
1,22 4 1,2
4
;
2
6
.
2 4
k z k z k z k z
K z
k z
k z K z k z K z
I dz
K zk z k z K z
⎛ ⎞
⎜ ⎟
⎝ ⎠
⎛ ⎞
⎜ ⎟
⎝ ⎠
′ ′ ′ ′± −
=
′
′′ ′−
=
′ ′−∫
)
(12)
5. METHOD OF THE CONSTANTS
VARIATION AND MUTUAL
TRANSFORMATION OF WAVES
To obtain the general solution of inhomogeneous equation
(8) dependencies on z-coordinate should be implemented
for amplitudes A1,2
± and then constants’ variation method
should be used. As a result we obtain a set of four
equations for A1,2
± amplitudes and their derivatives. From
this set derivatives of amplitudes can be expressed.
Expressions for the derivatives contain summands with
, where ( 1 2exp i K K⎡ ⎤± ±⎣ ⎦
% %
( ) ( )
0
1,2 1,2
z
z
K z K z′ dz= ∫%
⎤
⎦
′ , (13)
that describe mutual transformation of electromagnetic
waves, and summands with , that
describe transformation of the current wave into
electromagnetic waves, i.e. transition radiation. To
estimate the amplitude of this radiation, these summands
should be integrated. Integrals have a form
( )1exp zi K zκ⎡ ± −⎣
%
( )( ) ( )
( ) ( )
0
1,2 1,2 1,2
4 1,2
2
exp z y
z
ev z
A i z K z I d
ck z K z
π κ κ α
κ
+∞
±
−∞
⎡ ⎤= − ± −⎣ ⎦ ′∫ %m z . (14)
6. INTEGRATING
Integrals (14) can be taken using residue method for
poles’ vicinities and stationary phase method for the
vicinities of Cherenkov resonant points. Analysis of (14)
shows that only one Ωp
2=1 pole is of a mathematical
interest. For this pole two qualitatively different situations
exists: Ωc
2<1 and Ωc
2>1.
1) Ωc
2>1:
For A2
+ amplitude only Ωp
2=1 pole makes contribution
to the integral:
( )
( )
( )
( )( ) }
2
22
0
2 22 2
2
2
2
1
exp .
p
pz
p p
z p
ev dA
c d K
i z K
απ κ κ
κ
+ ⊥
Ω =
⎧ Ω⎪= − ×⎨
Ω Ω⎪⎩
⎡ ⎤× − + Ω⎣ ⎦
%
(15)
For A2
− amplitude also vicinities of the Cherenkov
resonant point make contributions to the integral.
Cherenkov resonant points correspond to the roots of
equation
6 4 2 0,p p pa b c dΩ + Ω + Ω + = (16)
where
( )
( ) ( )
4
0
2
2 2 2
02
0 2
2
2 2 2 2 2
0 2
2 2
2 2 2 0
0 0 2
2 2
4 2 2 2 2 2 2 20
0 0 02
;
1
2 2
;
1
22
1
2 ;
1
22 .
1
c
y z
c
c
p z z y
c
c
y
c
c
y y y y y
c
ka
k
b k
c k
kk k
kd k k k
κ κ
κ κ κ
κ
κ κ κ κ κ
=
Ω −
− −
=
−Ω
⎛ ⎞−Ω
=Ω − + + +⎜ ⎟−Ω⎝ ⎠
⎛ ⎞Ω
+ − +⎜ ⎟−Ω⎝ ⎠
⎛ ⎞Ω
= − + − − + −⎜ ⎟−Ω⎝ ⎠
(17)
After substitution Wp=Ωp
2+b/3a roots of the equation
(16) can be found:
1
2,3
;
3,
2 2
p
p
W
W i
α β
α β α β
= +⎡
⎢
+ −⎢ = − ±⎢⎣
(18)
where
3
3
3 2
,
2
,
2
.
3 2
q Q
q Q
p qQ
α
β
= − +
= − −
⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
(19)
We are interested only in the points that satisfy the
following conditions:
( ) ( )
2
2 2 2
4 2
;
2 0
p
p z pk kκ
⎧Ω ∈⎪
⎨
′ ′ .Ω − Ω ≥⎪⎩
(20)
Then amplitude can be written as
( )
( )
( )
( )( ) }
( ) ( )
( ) ( )
2
22
0
2 22 2
2
2
2
1
2
2 2
2
22
2
exp
exp exp 4
2 ,
p
pz
p p
z p
pl l
l
pl
pl
ev dA
c d K
i z K
iS i
f
S
απ κ κ
κ
π δ
π
− ⊥
Ω =
⎧ Ω⎪= ×⎨
Ω Ω⎪⎩
⎡ ⎤× − − Ω +⎣ ⎦
⎡ ⎤+ Ω ×⎡ ⎤⎣ ⎦⎣ ⎦
× Ω
′′ Ω
∑
% (21)
122
where
( ) ( )( )
( )
( )
( ) ( )
[ ]
2 2
2
0
2
4 2
;
sgn ;
2
exp .
p z
p
z y
S z K z
S
ev z
2f I
ck z K z
κ
δ
π κ κ α
Ω = − −
′′= Ω
=
′
%
−
(22)
2. Transformation of the current wave into
extraordinary wave occurs in the vicinities of local plasma
resonance point and Cherenkov resonance point.
3. In the case of current wave transformation into
ordinary wave the local plasma resonance point is a
branch point, and calculation of transition radiation needs
more detailed analysis.
REFERENCES
2) Ωc
2<1:
1. M. Starodubtsev, C. Krafft, P. Thevenet, A. Kostrov//
Physics of Plasmas. 1999, v. 6, N5, p. 1427-1434.
For the amplitudes A1
+ only pole Ωp
2=1 makes
contribution to the integral, so
2. I.O. Anisimov, K.I. Lyubich // Journal of Plasma
Physics. 2001, v. 66, p. 157-165.
( )
( )
( )
( )( ) }
2
22
0
1 22 2
1
2
1
1
exp .
p
pz
p p
z p
ev dA
c d K
i z K
απ κ κ
κ
+ ⊥
Ω =
⎧ Ω⎪= − ×⎨
Ω Ω⎪⎩
⎡ ⎤× − + Ω⎣ ⎦
(23) 3. I.O. Anisimov, S.M. Levitsky, D.B. Palets,
L.I. Romanyuk // Problems of atomic science and
technology. Ser. “Plasma Electronics and New
Acceleration Methods” (2). 2000, N1, p. 243-247.
4. V.L. Ginzburg, V.N. Tsytovich. Transition radiation
and transition dispersion (some matters of theory). M.:
“Nauka”, 1984 (in Russian).
For A1
− amplitude contribution from Cherenkov
resonant point is the same as in previous case.
5. V.L. Ginzburg. Propagation of electromagnetic waves
in plasma. M.: “Nauka”, 1967 (In Russian).
7. DISCUSSION
1. For charged particle moving along the plasma
density gradient in the magnetic field, parallel to the
density gradient, transition radiation is emitted both
forward and backward relatively to the particle motion
direction, and current wave is transformed into ordinary
and extraordinary waves.
6. M.I. Rabinovich, D.S. Trubetskov. Introduction to the
theory of oscillations and waves. M.: “Nauka”, 1984 (in
Russian).
Article received 13.09.10
ПЕРЕХОДНОЕ ИЗЛУЧЕНИЕ ЗАРЯЖЕННОЙ ЧАСТИЦЫ В НЕОДНОРОДНОЙ ПЛАЗМЕ
С ПРОДОЛЬНЫМ МАГНИТНЫМ ПОЛЕМ
И.А. Анисимов, Ю.Н. Борох
Найдены области наиболее эффективной трансформации волны тока в электромагнитные волны для заряда,
движущегося вдоль градиента концентрации слабонеоднородной холодной плазмы в магнитном поле, также
направленном вдоль градиента концентрации плазмы. Найдена амплитуда переходного излучения
необыкновенной волны.
ПЕРЕХІДНЕ ВИПРОМІНЮВАННЯ ЗАРЯДЖЕНОЇ ЧАСТИНКИ В НЕОДНОРІДНІЙ ПЛАЗМІ
З ПОВЗДОВЖНІМ МАГНІТНИМ ПОЛЕМ
І.О. Анісімов, Ю.Н. Борох
Знайдені області найбільш ефективної трансформації хвилі струму в електромагнітні хвилі для заряду, що
рухається вздовж градієнту концентрації слабконеоднорідної холодної плазми в магнітному полі, що також
спрямоване вздовж градієнту концентрації плазми. Знайдена амплітуда перехідного випромінювання
незвичайної хвилі.
|
| id | nasplib_isofts_kiev_ua-123456789-17477 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-02T12:03:18Z |
| publishDate | 2010 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Anisimov, I.O. Borokh, Yu.N. 2011-02-26T21:59:58Z 2011-02-26T21:59:58Z 2010 Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field / I.O. Anisimov, Yu.N. Borokh // Вопросы атомной науки и техники. — 2010. — № 6. — С. 120-122. — Бібліогр.: 6 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17477 Regions of the most effective transformation of current wave into electromagnetic wave for the charged particle moving along the density gradient in magnetic field, which is parallel to the plasma density gradient, are found. Magnitude of transition radiation of the extraordinary wave is calculated. Найдены области наиболее эффективной трансформации волны тока в электромагнитные волны для заряда, движущегося вдоль градиента концентрации слабонеоднородной холодной плазмы в магнитном поле, также направленном вдоль градиента концентрации плазмы. Найдена амплитуда переходного излучения необыкновенной волны. Знайдені області найбільш ефективної трансформації хвилі струму в електромагнітні хвилі для заряду, що рухається вздовж градієнту концентрації слабконеоднорідної холодної плазми в магнітному полі, що також спрямоване вздовж градієнту концентрації плазми. Знайдена амплітуда перехідного випромінювання незвичайної хвилі. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Плазменная электроника Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field Переходное излучение заряженной частицы в неоднородной плазме с продольным магнитным полем Перехідне випромінювання зарядженої частинки в неоднорідній плазмі з повздовжнім магнітним полем Article published earlier |
| spellingShingle | Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field Anisimov, I.O. Borokh, Yu.N. Плазменная электроника |
| title | Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field |
| title_alt | Переходное излучение заряженной частицы в неоднородной плазме с продольным магнитным полем Перехідне випромінювання зарядженої частинки в неоднорідній плазмі з повздовжнім магнітним полем |
| title_full | Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field |
| title_fullStr | Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field |
| title_full_unstemmed | Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field |
| title_short | Transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field |
| title_sort | transition radiation of the charged particle in the inhomogeneous plasma with the longitudinal magnetic field |
| topic | Плазменная электроника |
| topic_facet | Плазменная электроника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/17477 |
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