On dispersion relation of slow circularly polarized electromagnetic waves in plasmas
In the present communication, Hamilton equations for electrons interacting with slow circular polarized electromagnetic wave are solved in a self-consistent way. Basing on these solutions the interaction between the fast electrons and propagating circular wave is described kinetically, and the non-l...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1092322016-11-22T03:03:24Z On dispersion relation of slow circularly polarized electromagnetic waves in plasmas Gospodchikov, E.D. Suvorov, E.V. Фундаментальная физика плазмы In the present communication, Hamilton equations for electrons interacting with slow circular polarized electromagnetic wave are solved in a self-consistent way. Basing on these solutions the interaction between the fast electrons and propagating circular wave is described kinetically, and the non-linear dispersion relation is obtained. As a result, specific conditions for the slow wave propagation in a two component plasma are analyzed. Решаются уравнения Гамильтона для электронов, взаимодействующих с замедленной циркулярно поляризованной электромагнитной волной. На основе этих решений кинематически строится нелинейное дисперсионное соотношение. Обсуждаются специфические условия, при выполнении которых замедленная волна может распространяться в двухкомпонентной плазме. Розв’язується рівняння Гамільтона для електронів, що взаємодіють з уповільненою циркулярно поляризованою електромагнітною хвилею. На основі цих рішень кінематично будується нелінійне дисперсійне співвідношення. Обговорюються специфічні умови, при виконанні яких уповільнена хвиля може поширюватися в двокомпонентнiй плазмі. 2013 2013 Article On dispersion relation of slow circularly polarized electromagnetic waves in plasmas / E.D. Gospodchikov, E.V. Suvorov // Вопросы атомной науки и техники. — 2013. — № 1. — С. 87-89. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 41.20.Jb http://dspace.nbuv.gov.ua/handle/123456789/109232 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Фундаментальная физика плазмы Фундаментальная физика плазмы Gospodchikov, E.D. Suvorov, E.V. On dispersion relation of slow circularly polarized electromagnetic waves in plasmas Вопросы атомной науки и техники |
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In the present communication, Hamilton equations for electrons interacting with slow circular polarized electromagnetic wave are solved in a self-consistent way. Basing on these solutions the interaction between the fast electrons and propagating circular wave is described kinetically, and the non-linear dispersion relation is obtained. As a result, specific conditions for the slow wave propagation in a two component plasma are analyzed. |
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Article |
| author |
Gospodchikov, E.D. Suvorov, E.V. |
| author_facet |
Gospodchikov, E.D. Suvorov, E.V. |
| author_sort |
Gospodchikov, E.D. |
| title |
On dispersion relation of slow circularly polarized electromagnetic waves in plasmas |
| title_short |
On dispersion relation of slow circularly polarized electromagnetic waves in plasmas |
| title_full |
On dispersion relation of slow circularly polarized electromagnetic waves in plasmas |
| title_fullStr |
On dispersion relation of slow circularly polarized electromagnetic waves in plasmas |
| title_full_unstemmed |
On dispersion relation of slow circularly polarized electromagnetic waves in plasmas |
| title_sort |
on dispersion relation of slow circularly polarized electromagnetic waves in plasmas |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2013 |
| topic_facet |
Фундаментальная физика плазмы |
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http://dspace.nbuv.gov.ua/handle/123456789/109232 |
| citation_txt |
On dispersion relation of slow circularly polarized electromagnetic waves in plasmas / E.D. Gospodchikov, E.V. Suvorov // Вопросы атомной науки и техники. — 2013. — № 1. — С. 87-89. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT gospodchikoved ondispersionrelationofslowcircularlypolarizedelectromagneticwavesinplasmas AT suvorovev ondispersionrelationofslowcircularlypolarizedelectromagneticwavesinplasmas |
| first_indexed |
2025-07-07T22:44:16Z |
| last_indexed |
2025-07-07T22:44:16Z |
| _version_ |
1837029924206018560 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2013. №1(83) 87
ON DISPERSION RELATION OF SLOW CIRCULARLY POLARIZED
ELECTROMAGNETIC WAVES IN PLASMAS
E.D. Gospodchikov, E.V. Suvorov
Institute of Applied Physics of the Russian Academy of Sciences, N. Novgorod, Russia
E-mail: egos@appl.sci-nnov.ru
In the present communication, Hamilton equations for electrons interacting with slow circular polarized
electromagnetic wave are solved in a self-consistent way. Basing on these solutions the interaction between the fast
electrons and propagating circular wave is described kinetically, and the non-linear dispersion relation is obtained.
As a result, specific conditions for the slow wave propagation in a two component plasma are analyzed.
PACS: 41.20.Jb
INTRODUCTION
Finding the dispersion relation for electromagnetic
waves in homogeneous media is basic fundamental
problem of wave propagation. In cold isotropic plasma
within a linear theory, the dispersion relation for
transverse electromagnetic waves with frequency higher
than electron plasma frequency results in the fact that
only fast electromagnetic waves with cph >υ can
propagate [1]. Some time ago the question was under
discussion if there a possibility to arrange circularly
polarized slow waves due to the trapping of some supra-
thermal electron fraction into the wave field (see e.g.
[2,3]). In such a situation slowing-down of waves is
provided due to electron trapping by a finite amplitude
electromagnetic wave. In the present communication we
analyze specific conditions for the existence of
circularly polarized slow waves in a plasma with two
electron components. The nonlinear dispersion relation
for such waves is obtained self-consistently with taking
into account Maxwell equations and motion equations
for electrons. The treatment is performed both in
hydrodynamics approximation and kinetically basing on
the solutions of Hamilton equations for electrons
interacting with slow circular polarized electromagnetic
wave. Problems of the formation of such waves and of
electrons with two fractions are out of the scope of
present communication.
1. SELF-CONSISTENT STATIC SHEARED
MAGNETIC FIELD
In the investigation of slow waves it may be
convenient to shift to the reference frame moving with
the phase velocity, where the plane wave is presented as
purely static magnetic configuration. In particular,
circularly polarized in the laboratory frame wave
corresponds to sheared magnetic field
kzBB
kzBB
y
x
sin
cos
=
=
. (1)
To provide self-consistency of such magnetic
configuration it is necessary to have corresponding
electron current in which every electron perform the
motion allowed by magnetic field (1). For the sake of
simplicity we shall consider that the current is produced
by electrons with constant longitudinal velocities
constVz = (the ion motion is neglected).
For the electron with constVz = it is necessary to
satisfy the following set of equations:
⎪
⎪
⎩
⎪⎪
⎨
⎧
−=
=
=
tkVVV
tkVVV
BVBV
zzBy
zzBx
xyyx
cos
sin
ω
ω
&
& , (2)
where mceBB /=ω . From Eqs. (2) for 0≠zV it
follows that mckBeV /
rr
−=⊥ , and this relation does not
depend on the longitudinal velocity. Electrons with
0=zV (“trapped” electrons in the laboratory frame of
reference), can have arbitrary 0⊥V
r
which is parallel to
the magnetic field B
r
in the corresponding z =const
plane. Transverse current of electrons with 0≠zV
(“untrapped” electrons) is
( ) ( ) mckBNeVNej /11 2
rrr
αα −=−−= ⊥⊥ , (3)
where N is the electron density, α is a fraction of
“trapped” electrons. From the static Maxwell equation
j
c
Brot
rr π4
= ,
we obtain following condition
( )
B
PP kV
cc
k
ω
ωαωα 0
2
2
2
2
2 1 ⊥=−+ , (4)
with mNeP /4 22 πω = . The condition that Eq.(4) possess
real solutions for k is
P
B
c
V
ω
ω
α
α 2
1
0 >
−
⊥ , which impose
the following limitation toα
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+> ⊥
⊥
1
4
12 2
2
2
2
0
22
0
22
B
P
P
B
c
V
V
c
ω
ω
ω
ωα ; (5)
in the case 1<<α this corresponds to the inequality
0
2
⊥
>
V
c
P
B
ω
ωα . (6)
Taking into account the evident inequality cV <⊥0 ,
one can obtain overestimated limitation to α
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+> 1
4
12 2
2
2
2
B
P
P
B
ω
ω
ω
ωα , (7)
which does not contain 0⊥V .
Eq. (4) for a fixed plasma density impose the relation
between field amplitude, fraction of “trapped” electrons
88 ISSN 1562-6016. ВАНТ. 2013. №1(83)
and their transverse velocity, which may be considered
as nonlinear dispersion relation. This relation can be
modified with taking into account that longitudinal
current formed by ions with the density N moving with
the velocity phυ− and by “untrapped” electrons with the
density αN must be zero. This means that in the
laboratory reference frame “untrapped” electrons must
move also in z-direction relative to immovable ions to
compensate z-component of “trapped” electron current.
The mean velocity of “untrapped” electrons in z-
direction >< zV is defined by the relation
><−=− zph V/1 υα .
It should be noted that Eq. (4) has two real
solutions for k, which means that in the laboratory
reference frame there are two slow nonlinear circularly
polarizes waves with the equal phase velocities.
2. ELECTRON MOTION IN CIRCULARLY
POLARIZED ELECTROMAGNETIC WAVE
Above we present the simplest demonstration of the
existing of nonlinear slow circularly polarized waves in
a plasma with two cold electron fractions (“trapped” and
“untrapped”), in which every “trapped” electron is
moving with constant velocity along its own rectilinear
trajectory, while “untrapped electrons are moving with
constant velocity along spiral trajectories. In such a
wave the “trapped” fraction can possess arbitrary spread
over transverse velocities, and “untrapped” fraction
allows spread over longitudinal velocities.
Now again having in mind investigation of slow
nonlinear circularly polarized waves in a more general
case, we consider arbitrary electron motion (non-
relativistic) in the electromagnetic wave defined by
vector-potential:
( ) ( )
tkz
cEAcEA yx
ωφ
φ
ω
φ
ω
−=
== sin;cos 00
, (8)
From the translation symmetry transverse momentum
conservation follows:
constPA
c
eVm ==− ⊥⊥
rrr
. (9)
The longitudinal motion is governed by the equation:
)cossin(0 φφ
ω yx PPk
m
eEzm −=&& , (10)
or
( )0
0 sin φφ
ω
−−= ⊥kP
m
eEzm && , (11)
where P⊥ and φ0 are introduced by
00 sin;cos φφ ⊥⊥ −=−= PPPP yx . Note also the following
condition
00 φφ
ω
=
⊥
⊥ −= A
cE
PP
rr
. (12)
By substitution 0
~ φω −−= tkzzk one can obtain the first
integral of equation (11):
constWzkP
m
eEzm
z ==− ⊥
~cos
2
~
0
2
ω
&
, (13)
from which it follows
( )
k
t
k
EPtzz z
0,,~ φω
τ −−−= ⊥ , (14)
where z~ – takes the form of inverse elliptical function
∫
⊥−
=−
zkP
m
eEW
zdmt
z
~cos
~
2 0
ω
τ , (15)
with the constant τ characterizing the initial phase of
electron oscillation.
3. DISPERSION RELATION
The plane wave (8) can propagate in the plasma
without support from external sources if the following
equations are satisfied:
( )
⎪⎩
⎪
⎨
⎧
=
=− ⊥
0
42
0
2
zj
j
c
Akk
rr π
. (16)
The transverse current can be calculate as
( ) ( )( )∫ −−= ⊥⊥ ..., dtzzfVetzj δ
rr
, (17)
where f is the – electron distribution function, )(tz is
the solution of motion equation; integration is
performed over the set of constants characterizing
electron motion. In our case such these constants are
τφ ,,, 0⊥PEz . Actually the electron motion must be
characterized by 6 constant, but two constants arising
from initial transverse coordinates can be omitted.
When the distribution function depends on zE and ⊥P ,
Eq. (17) takes the form:
( ) ( )
( )0
0
, ,
1 , ,
2 z z
j z t e V z t
z z t f W P d d dE dP
k k
ϕωδ ϕ τ
π
⊥ ⊥
⊥ ⊥
= − ×
⎛ ⎞× − − −⎜ ⎟
⎝ ⎠
∫
rr
%
(18)
where transverse velocity ⊥V
r
is also dependent on
⊥PEz , . From Eqs. (9) and (12) it follows:
( ) ( ) ( )0
0
1, φωφ A
mcE
PA
cm
eP
m
A
cm
etzV
rrrrr
⊥−=+= . (19)
Integrating over variable 0φ and using normalization
condition
( )( ) ( ) 0
1 , ... ... ,
2 2z
kz z t f W P d d fd const Nδ ϕ
π π⊥− = = =∫ ∫
one can obtain
( ) ( )
( )( ) ( )
2
0
,
, , , .
2z z z
ej z t NA
mc
e kA kz t W P P f W P d dE dP
mcE
ϕ
ω ϕ τ τ
π
⊥
⊥ ⊥ ⊥ ⊥
=− +
+ − −∫
rr
r
%
(20)
Substituting (20) into (16) results in the nonlinear
dispersion relation:
( ) ( )∫∫ ⊥⊥⊥⊥=+− dPdEPWFPPWФ
mcE
e
cc
kk zzz
p ,,4
0
2
2
2
0
2 ωπω
, (21)
where F is the distribution function normalized as:
NdPFdWz =∫ ⊥ , (22)
ISSN 1562-6016. ВАНТ. 2013. №1(83) 89
and
∫
∫
−
⊥
−
⊥
⎟
⎠
⎞
⎜
⎝
⎛ +
⎟
⎠
⎞
⎜
⎝
⎛ +
=
ζζ
ω
ζζ
ω
ζ
dP
m
eEW
dP
m
eEW
Ф
z
z
2/1
0
2/1
0
cos
coscos
(23)
integration limits in (23) are determined by zero points
of expression ωζ mPeEWz /cos0 ⊥+ .
Electrons may be specified as belonging to three
groups. The first group includes trapped electrons with
Cz WWP
m
cE
<<− ⊥ω
0 ( ⊥≈ P
m
cEWC ω
065.0 ), for these
electrons 10 <Φ< and kVz /ω>=< ; the second group
includes trapped electrons with ⊥<< P
m
cEWW zC ω
0 , for
these electrons 01 <Φ<− and kVz /ω>=< ; the third
group includes untrapped electrons with zWP
m
cE
<⊥ω
0
for this electrons 01 <Φ<− and kVz /ω>≠< .
In Figure the dependence of Φ on ⊥PcEmWz 0/ω is
shown.
For fast waves (with ck >/ω ) the expression (23)
after some cumbersome calculations can be simplified
and for waves of low enough amplitude the nonlinear
dispersion relation (21) can be transformed into well
known linear dispersion relation for transverse waves in
the isotropic plasma.
Weight function Ф dependence on normalized zW
Condition for existence of self-consistent non-linear
slow waves in plasma may be derived from Eqs. (21),
(23) (see also Figure). Such waves can exist, if there is
sufficiently large fraction of trapped electrons with high
enough ⊥P . This condition may be written in the form:
( ) ( ) 0,, 0 >⎟
⎠
⎞
⎜
⎝
⎛ −∫∫ ⊥⊥⊥⊥ dPdWPWFeEPPWФ zzz ω
. (24)
So for existence of such a wave two necessary
conditions must be satisfied: for the group of trapped
particles the condition ω/0eEP >⊥ must be fulfilled;
more strictly the trapped particle density should satisfy
the condition
ω/0NeEPN trtr >⊥ . (25)
If one take into account the conservation of
transverse momentum in the process of wave switch-on,
and assume that before switching-on the wave field
electrons had Maxwellian distribution, the condition of
existence of nonlinear slow waves takes the form
03
E
e
m T >
ωυ . (26)
This is a rather strong limitation, and more simple
realization of slow waves may be achieved in the
plasma with two electron fractions, e.g., in the presence
of electron beam in the direction of wave propagation
with large enough transverse energy.
In conclusion, we demonstrate the simple way of
constructing nonlinear slow circularly polarized waves
in the plasma, which allows evident modification to the
relativistic case.
REFERENCES
1. T. Stix. The Theory of Plasma Waves Mc Graw Hill.
New York: 1962, M.: «Atomizdat», 1965,
2. Ya.N. Istomin,V.I. Karpman. // JETP Lett. 1972,
№ 15, p. 143.
3. A.I. Matveev // Plasma Physics Report. 2009, v. 35,
№ 4, p. 315 .
Article received 14.09.12
О ДИСПЕРСИОННОМ СООТНОШЕНИИ ДЛЯ ЦИРКУЛЯРНО ПОЛЯРИЗОВАННЫХ
ЗАМЕДЛЕННЫХ ВОЛН В ПЛАЗМЕ
Е.Д. Господчиков, Е.В. Суворов
Решаются уравнения Гамильтона для электронов, взаимодействующих с замедленной циркулярно
поляризованной электромагнитной волной. На основе этих решений кинематически строится нелинейное
дисперсионное соотношение. Обсуждаются специфические условия, при выполнении которых замедленная
волна может распространяться в двухкомпонентной плазме.
ПРО ДИСПЕРСІЙНІ СПІВВІДНОШЕННЯ ДЛЯ ЦИРКУЛЯРНО ПОЛЯРИЗОВАНОГО
УПОВІЛЬНЕННЯ ХВИЛЬ У ПЛАЗМІ
Є.Д. Господчиков, Є.В. Суворов
Розв’язується рівняння Гамільтона для електронів, що взаємодіють з уповільненою циркулярно
поляризованою електромагнітною хвилею. На основі цих рішень кінематично будується нелінійне
дисперсійне співвідношення. Обговорюються специфічні умови, при виконанні яких уповільнена хвиля
може поширюватися в двокомпонентнiй плазмі.
- 1 1 2 3 4
- 0.4
- 0.2
0.2
0.4
0.6
0.8
1.0 Φ
⊥ePEWm z 0/ω
|