Dynamics of the charged particles in a field of intensive electromagnetic waves
The results of the investigations of the charged particles dynamics in field of intense electromagnetic waves and electromagnetic impulse are represented. It was shown, that the traditional scheme of the acceleration, such as inverse free electron laser in such fields are not efficiently. Accelerati...
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irk-123456789-1105252017-01-05T03:03:34Z Dynamics of the charged particles in a field of intensive electromagnetic waves Buts, V.A. Kuzmin, V.V. Plasma electronics The results of the investigations of the charged particles dynamics in field of intense electromagnetic waves and electromagnetic impulse are represented. It was shown, that the traditional scheme of the acceleration, such as inverse free electron laser in such fields are not efficiently. Accelerating particles by laser impulse allows to all accelerated particles to have absolutely identically trajectories. The limitation on maximum energy, which particles can received and which is caused by radiation friction are removed. Наведено результати досліджень динаміки заряджених часток у полі інтенсивних електромагнітних хвиль й електромагнітних імпульсів. Показано, що традиційна схема прискорення типу зверненого лазера на вільних електронах у таких полях мало ефективна. При прискоренні заряджених часток лазерним імпульсом всі прискорені частки, можуть рухатися абсолютно ідентично. Знято обмеження на гранично можливі значення енергії прискорених часток при лазерному прискоренні, які обумовлені радіаційним тертям. Изложены результаты исследований динамики заряженных частиц в поле интенсивных электромагнитных волн и электромагнитных импульсов. Показано, что традиционная схема ускорения типа обращенного лазера на свободных электронах в таких полях мало эффективна. При ускорении заряженных частиц лазерным импульсом все ускоряемые частицы могут двигаться абсолютно идентично. Сняты ограничения на предельно возможные значения энергии ускоряемых частиц при лазерном ускорении, которые обусловлены радиационным трением 2007 Article Dynamics of the charged particles in a field of intensive electromagnetic waves / V.A. Buts, V.V. Kuzmin // Вопросы атомной науки и техники. — 2007. — № 1. — С. 127-129. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.20.-j; 05.45.-a http://dspace.nbuv.gov.ua/handle/123456789/110525 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Plasma electronics Plasma electronics Buts, V.A. Kuzmin, V.V. Dynamics of the charged particles in a field of intensive electromagnetic waves Вопросы атомной науки и техники |
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The results of the investigations of the charged particles dynamics in field of intense electromagnetic waves and electromagnetic impulse are represented. It was shown, that the traditional scheme of the acceleration, such as inverse free electron laser in such fields are not efficiently. Accelerating particles by laser impulse allows to all accelerated particles to have absolutely identically trajectories. The limitation on maximum energy, which particles can received and which is caused by radiation friction are removed. |
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Article |
| author |
Buts, V.A. Kuzmin, V.V. |
| author_facet |
Buts, V.A. Kuzmin, V.V. |
| author_sort |
Buts, V.A. |
| title |
Dynamics of the charged particles in a field of intensive electromagnetic waves |
| title_short |
Dynamics of the charged particles in a field of intensive electromagnetic waves |
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Dynamics of the charged particles in a field of intensive electromagnetic waves |
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Dynamics of the charged particles in a field of intensive electromagnetic waves |
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Dynamics of the charged particles in a field of intensive electromagnetic waves |
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dynamics of the charged particles in a field of intensive electromagnetic waves |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2007 |
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Plasma electronics |
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http://dspace.nbuv.gov.ua/handle/123456789/110525 |
| citation_txt |
Dynamics of the charged particles in a field of intensive electromagnetic waves / V.A. Buts, V.V. Kuzmin // Вопросы атомной науки и техники. — 2007. — № 1. — С. 127-129. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT butsva dynamicsofthechargedparticlesinafieldofintensiveelectromagneticwaves AT kuzminvv dynamicsofthechargedparticlesinafieldofintensiveelectromagneticwaves |
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2025-07-08T00:42:18Z |
| last_indexed |
2025-07-08T00:42:18Z |
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1837037350373294080 |
| fulltext |
Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 127-129 127
DYNAMICS OF THE CHARGED PARTICLES IN A FIELD OF INTENSIVE
ELECTROMAGNETIC WAVES
V.A. Buts, V.V. Kuzmin
NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine,
e-mail: vbuts@kipt.kharkov.ua
The results of the investigations of the charged particles dynamics in field of intense electromagnetic waves and
electromagnetic impulse are represented. It was shown, that the traditional scheme of the acceleration, such as inverse
free electron laser in such fields are not efficiently. Accelerating particles by laser impulse allows to all accelerated par-
ticles to have absolutely identically trajectories. The limitation on maximum energy, which particles can received and
which is caused by radiation friction are removed.
PACS: 52.20.-j; 05.45.-a
1. INTRODUCTION
The acceleration of the charged particles by a field of
laser radiation in vacuum represents tempting prospect
[1,2]. One of the most interesting and perspective scheme
of acceleration is the scheme of the inverted free electron
laser (IFEL). However, as we shall see below, with in-
crease of intensity of laser radiation in the scheme IFEL
the stochastic instability develops. This scheme ceases to
be effective. We shall show that the acceleration charged
particles by laser pulse can be enough effective. At laser
acceleration the particles are moving with acceleration.
They radiate. This radiation can restrict maximal energy,
which particles can receive at laser acceleration [3]. Be-
low we shall show that this restriction can be deleted.
Except it we shall show that forces of friction can pro-
mote laser acceleration.
2. THE SCHEMENS OF THE INVERSE
FREE ELECTRON LASER
Acceleration in scheme IFEL occurs by field of a
combinational electromagnetic wave which is formed as a
result of beat of two laser waves. Accelerated particles are
in Cherenkov resonance with this beat-wave. The dimen-
sionless amplitude of a beat-wave is proportional to prod-
uct of amplitudes of the waves forming this wave. In ex-
isting schemes it is supposed, that amplitudes of laser
waves in scheme IFEL are small. The amplitude of a beat-
wave is smaller. Therefore there is a natural desire to in-
crease these amplitudes. However, as we shall see below,
such increase leads to development of stochastic instabil-
ity. Presence of this instability does scheme IFEL practi-
cally disabled. Below we shall define values of ampli-
tudes of laser fields which else are admissible in scheme
IFEL.
Let's consider dynamics of the charged particles in a
field of several electromagnetic waves. Expressions for
electric and magnetic fields of these waves can be pre-
sented in such kind:
n
n
E E= ∑
r r
, n
n
H H= ∑
r r
,
Re( )ni
n nE a e ψ= ⋅
r r , [ ]n n n
n
cH k E
ω
=
rr r
, (1)
where n n nk r tψ ω= −
r r , 0/a eE mcn n ω≡
rr .
The equations of movement in fields (1) look like:
[ ]dP eeE vH
dt c
= +
r
r rr . (2)
These equations are convenient rewrite in dimensionless
variables:
PP
mc
→ ,
0
n
n
ω
ω
ω
= , dPP
dτ
≡
r
r& , 0tτ ω≡ , vr
c
=
r
r& ,
n
n
eE
E
mcω
≡
r
r
, n
n
k c
k
ω
≡
r
r
, 0r r
c
ω
≡
r r .
( ) ( )n n n n n
n n
P E k r k rEω= − +∑ ∑
r rr rr r& & & (3)
n n
n
P Eγ ω
γ
= ∑
r
r
& ,
where: ( )Re ni
n nE a e ψ= ⋅
r r
n n nk rψ ω τ≡ −
r r .
For the further analysis it is convenient to introduce
the auxiliary characteristic of a particle, which we in the
further shall name partial energy of a particle. This energy
satisfies to the following equation:
( )n n nrEγ ω=
rr&& . (4)
The equations (3) (4) have following integral:
Re( )ni n
n n
n n n
kP i a e Cψ γ
ω
− ⋅ ⋅ − =∑ ∑
r
rr r . (5)
Generally, the equations (3) and (4), together with in-
tegral (5), can be studied only by numerical methods. For
receiving of analytical results we shall consider, that the
parameter ar each of waves acting on a particle is small.
In this case all characteristics (its energy, an impulse, co-
ordinate, velocity) can be presented in the form of the
sum slowly varying and quickly varying component:
P P P= +
r r r%
n n nγ γ γ= + % .
In this case it is possible to receive following expres-
sions and the equations which connect fast and slow vari-
ables:
n
n
n n
k
P Cγ
ω
= +∑
r
r
,
Re( ) /ni
n n n n
n n
P i a e kψ γ ω= ⋅ ⋅ +∑ ∑
rr r%
% (6)
Re( )ni
n n n n nv E v a e ψγ ω ω= ⋅ ⋅ = ⋅ ⋅ ⋅
rr r r&% ,
n n nv Eγ ω= ⋅ ⋅
rr
& Re( )ni
n ne ψγ = Γ% ,
mailto:vbuts@kipt.kharkov.ua
128
where /n n n ni v aω ψΓ = − ⋅
r r
& .
The equations for fast variables can be integrated. The
equations for slow variables will get a following form:
( ) ( )
,
1 Re Rem ni i
n m n
m n
P k i a e a eψ ψ
γ
= ⋅ ⋅ ⋅ ∑
rr& r r ,
( )
( )
,
,
1 Re( ) Re( )
1 [cos / 2
2
cos / 2 ]
m ni i
m n n
m n
n n m m n
m n
m n
i a e a e
a a
ψ ψγ ω
γ
ω ψ ψ π
γ
ψ ψ π
= ⋅ ⋅ ⋅ =
= ⋅ ⋅ + + +
+ − +
∑
∑
r r
&
r r (7)
The equations (7) are equivalent to the equation of a
nonlinear pendulum (a mathematical pendulum) on which
external periodic force acts. Really, let’s consider exam-
ple, when there are available only two waves. Then if we
introduce new variable 1 2θ ψ ψ≡ − , the equations (7) can
be rewritten in the form:
( )1 cosd a F
d
γ
θ τ
τ γ
= ⋅Ω + , ( )d v
dt
θ
χ γ= − Ω = ∆
r r , (8)
where: 1 2k kχ ≡ −
r rr , 1 2ω ωΩ ≡ − , 1 2a a a= ⋅
r r , ( )F τ - peri-
odic function.
We supposed that / vχΩ ≅ .
At small changes of energy, the system (8) can be rewrite
in the form:
( )
0 0
cosa F
γ
θ θ τ
γ γ
∂∆ ⋅Ω
= + ∂
&& . (9)
We fulfilled series of numerical researches of the equa-
tions (3). The dynamics of particles in the most interesting
configuration of fields which is represented with a field of
two electromagnetic waves, which are moving towards
each other, was investigated. Such configuration meets in
the schemes of acceleration IFEL. The basic results of
these numerical researches consist in the following:
– If amplitudes of waves are small ( 1ar and 2ar less than
0.1) qualitatively dynamics of particles are similar to dy-
namics of a mathematical pendulum.
– When amplitudes of waves become greater 0.1, the
dynamics of some particles, namely those particles which
are located in a vicinity separatrix of mathematical pendu-
lum, becomes chaotic. And the more the amplitude of
waves, the more of particles joins to chaotic dynamics.
– Only those particles, which appear in zero phases of a
beat-wave, do not participate in chaotic dynamics. They
are located in islands of stability. However with increase
amplitude of the waves the number of such particles be-
comes less.
For an illustration of formulated above results on
Fig.1 characteristic dependence of a longitudinal impulse
of a particle from time is presented. From this figure ir-
regular dynamics of movement of a particle is visible. The
same irregularity is proved by the statistical analysis:
spectra of movement is wide (Fig. 2), correlation function
quickly falls down, Lyapunov's parameters are positive.
The received numerical results are in the good qualita-
tive consent with the analysis of dynamics of particles on
the basis of the equation (9).
0 200 400 600 800 1000
0.98
0.59
0.21
0.17
0.56
Pz n
T n
Fig. 1. Dependence of a longitudinal impulse of a particle
on time
0 1.4 2.8 4.2 5.6 7 8.4 9.8 11.2 12.6 14
1 .10 6
1 .10 4
0.01
1
100
sn
Ω n
Fig. 2. A spectrum of movement of a particle at 0.5a =
3. INTERACTION OF A PARTICLE
WITH AN IMPULSE
OF AN ELECTROMAGNETIC WAVE
One of interesting schemes of acceleration is the
scheme in which movement of the charged particle occurs
in a field of an impulse of the flat running electromagnetic
wave. This impulse characterized by vector potential.
( ) ( )A A k r Aτ ψ= − ⋅ ≡
rr r rr . Such scheme was considered. In
this case it is possible to receive analytical expressions,
describing dynamics of the charged particles [4].
In case of interaction of particles with an impulse of a
running electromagnetic wave, the particles are dragging
by the wave along wave-vector. Under it’s, the longitudi-
nal impulse oscillate, but don’t change his sign. The lon-
gitudinal coordinate is defined by integral from non-
negative function. Character of interaction of particles
with a wave does not depend on their initial position since
front of an impulse consistently run over on all particles
and they appear in identical entry conditions concerning a
phase of a wave. And in a field of a high-frequency im-
pulse having circular polarization, a longitudinal impulse
of particles repeats the form of the impulse envelope.
0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0
0
2 0
4 0
6 0
8 0
1 0 0
P z
τ
Fig. 3. Dependence of a longitudinal impulse on time
( 0 50ψ = )
129
As an example of an opportunity of acceleration of a
bunch of the charged particles we shall consider a bunch
which has initial energy 0 10γ = . Let on such bunch the
transversal electromagnetic field with components de-
fined by expression:
2
0 0/ exp[ ( ) ] cosxA Aψ β ψ ψ ψ∂ ∂ = − − ⋅ acts. ( 0 3A = ,
0.01β = ). In Fig. 3 it is visible, that on distance 0.4 m.
energy of particles reaches value 100γ ≈ . Besides, all
particles have practically identical trajectories. Such laser
pulse is convenient for acceleration.
4. ROLE OF FORCES OF FRICTION
AT LASER ACCELERATION
The authors of work [3], considering acceleration of
electrons by laser radiation, have equated force of radiat-
ing friction to accelerating forces (forces of high-
frequency pressure). As a result they have found that in a
field of laser radiation the electrons cannot get energy
greater, than 200 MeV ( ~ 1 kλ µ ).
In the present section we shall show, that forces of
friction, including forces of radiating friction, can pro-
mote transfers of energy from an external laser field to
accelerated particles. Besides restriction on the maximal
size of energy in 200 eV in common case can be de-
leted (see [5] too).
For the description of a role of friction we use equa-
tions (3) in which we should to add force of friction into
right part. In the beginning we shall consider model in
which we shall not concretize the nature of these forces:
( ) ( ){ }Re (1 v) (v ) expdp k a k a i v
d
µ
τ
= − ⋅ + ⋅ Ψ −
r
r rr r r r r . (10)
From (10) it is possible to receive a following equation:
( ) 2{ Re exp( ) }d p k i a i v kv
dt
γ µ − + ⋅ ⋅ Ψ = − −
r rr r r . (11)
If friction is absent ( )0µ = , expression in braces
represents integral of the equation (10). We shall con-
sider, that ( ),0,0a a=
r ; ( )1,0,0α =
r , ( )0,0,1k =
r
. In this
case the equation (10) can be simplified essentially.
Let’s make such replacement:
( ),0 0sin sinx x xp a p a ρ= ⋅ Ψ + − ⋅ Ψ + ,
( ) ( )2 2
,0 ,0 0/ 2 / 2z x z x zp p I p p I ρ= + − + . (12)
Here xρ and zρ new dependent variables, and «0»
are designated initial values of variables. From system
(11) it is easy to find the following equations for defini-
tion xρ and zρ :
( )[ ]/ sinx xI aρ µ ρ′ = − + ⋅ Ψ
( ) ( ) ( )2 2/ / 2 /z z x xI p I p Iρ µ ρ γ′ = − + ⋅ ⋅ − . (13)
The analysis of these equations shows that if 1Iγ ⋅ > ,
then the appearing friction will lead to acceleration. If the
opposite inequality takes place, then particles will be
brake by friction. We will have the same results if friction
forces are forces of radiating friction. It is important, that
acceleration can be unlimited. Numerical researches com-
pletely confirm the received results.
REFERENCES
1. B.M Bolotovskij, A.V. Serov. Features of move-
ment of particles in an electromagnetic wave // Physics-
Uspekhi. 2003, v. 173, N 6, p. 667-678.
2. V.A. Buts, A.V Buts. Dynamics of the charged parti-
cles in a field of an intense tranversal electromagnetic
wave // JETPh. 1996, v. 110, N 3(9), p. 818-831.
3. N.B. Baranova, M.O. Skalli, B.J. Zeldovich. Accel-
eration of the charged particles laser beams // JETPh.
1994. v. 105, N 3, p. 469-486.
4. V.A. Buts, V.V. Kuzmin. Dynamics of particles in
fields of the big intensity // Successes of modern radio
electronics. 2005, 11, p. 5-20.
5. V.A. Buts. Peculiarities of particles and field dynam-
ics at critical intensity of electromagnetic waves (part 1) //
Problems of Atomic Science and Technology. 2005, N 1,
p. 119-121.
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