Acceleration of charged particles by elliptic polarized waves of large amplitude
In this paper the results of investigations on the charged particle dynamics in the homogeneous constant magnetic field and in the field of a flat wave of arbitrary intensity are presented. When the magnetic field is absent the particle trajectory is defined. It is shown, that a particle can be effe...
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| Zitieren: | Acceleration of charged particles by elliptic polarized waves of large amplitude / A.V. Buts, V.A. Buts, V.V. Kuzmin // Вопросы атомной науки и техники. — 2004. — № 2. — С. 144-146. — англ. |
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irk-123456789-793712015-04-01T03:02:17Z Acceleration of charged particles by elliptic polarized waves of large amplitude Buts, A.V. Buts, V.A. Kuzmin, V.V. Динамика пучков In this paper the results of investigations on the charged particle dynamics in the homogeneous constant magnetic field and in the field of a flat wave of arbitrary intensity are presented. When the magnetic field is absent the particle trajectory is defined. It is shown, that a particle can be effectively drawn along the wave vector of the wave. Average speed of this drawing is proportional to the square of the wave force parameter. The trajectories of particles are significantly dependent on initial conditions. In the presence of constant external magnetic field the particles can be effectively accelerated by the wave field. The particle dynamics is always chaotic if the wave force parameter is enough large. This dynamic is characterized by the alternation. Викладені результаті дослідження динаміки заряджених часток у однорідному постійному магнітному полі та в полі плоскої хвилі довільної напруженості. За відсутності постійного магнітного поля вигляд траєкторії частки був визначений. Показано, що частка може ефективно захоплюватися полем зовнішньої хвилі. Середня швидкість захоплення пропорційна квадрату параметру сили хвилі. При наявності постійного магнітного поля. частки можуть ефективно прискорюватись полем хвилі. При великих напруженостях поля динаміка часток завжди хаотична. Динаміка характеризується перемежністю. Изложены результаты исследования динамики заряженных частиц в однородном постоянном магнитном поле и в поле плоской волны произвольной напряженности. В отсутствии магнитного поля определен вид траектории частиц. Показано, что частицы могут эффективно увлекаться полем внешней волны. Средняя скорость увлечения пропорциональна квадрату параметра силы волны. Вид траектории существенно зависит от начальных условий. При наличии постоянного магнитного поля частицы могут эффективно ускоряться полем волны. При больших напряженностях поля динамика частиц всегда хаотична. Динамика характеризуется перемежаемостью. 2004 Article Acceleration of charged particles by elliptic polarized waves of large amplitude / A.V. Buts, V.A. Buts, V.V. Kuzmin // Вопросы атомной науки и техники. — 2004. — № 2. — С. 144-146. — англ. 1562-6016 PACS: 29.17.+w http://dspace.nbuv.gov.ua/handle/123456789/79371 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Динамика пучков Динамика пучков |
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Динамика пучков Динамика пучков Buts, A.V. Buts, V.A. Kuzmin, V.V. Acceleration of charged particles by elliptic polarized waves of large amplitude Вопросы атомной науки и техники |
| description |
In this paper the results of investigations on the charged particle dynamics in the homogeneous constant magnetic field and in the field of a flat wave of arbitrary intensity are presented. When the magnetic field is absent the particle trajectory is defined. It is shown, that a particle can be effectively drawn along the wave vector of the wave.
Average speed of this drawing is proportional to the square of the wave force parameter. The trajectories of particles
are significantly dependent on initial conditions. In the presence of constant external magnetic field the particles can
be effectively accelerated by the wave field. The particle dynamics is always chaotic if the wave force parameter is
enough large. This dynamic is characterized by the alternation. |
| format |
Article |
| author |
Buts, A.V. Buts, V.A. Kuzmin, V.V. |
| author_facet |
Buts, A.V. Buts, V.A. Kuzmin, V.V. |
| author_sort |
Buts, A.V. |
| title |
Acceleration of charged particles by elliptic polarized waves of large amplitude |
| title_short |
Acceleration of charged particles by elliptic polarized waves of large amplitude |
| title_full |
Acceleration of charged particles by elliptic polarized waves of large amplitude |
| title_fullStr |
Acceleration of charged particles by elliptic polarized waves of large amplitude |
| title_full_unstemmed |
Acceleration of charged particles by elliptic polarized waves of large amplitude |
| title_sort |
acceleration of charged particles by elliptic polarized waves of large amplitude |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2004 |
| topic_facet |
Динамика пучков |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79371 |
| citation_txt |
Acceleration of charged particles by elliptic polarized waves of large amplitude / A.V. Buts, V.A. Buts, V.V. Kuzmin // Вопросы атомной науки и техники. — 2004. — № 2. — С. 144-146. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT butsav accelerationofchargedparticlesbyellipticpolarizedwavesoflargeamplitude AT butsva accelerationofchargedparticlesbyellipticpolarizedwavesoflargeamplitude AT kuzminvv accelerationofchargedparticlesbyellipticpolarizedwavesoflargeamplitude |
| first_indexed |
2025-07-06T03:26:32Z |
| last_indexed |
2025-07-06T03:26:32Z |
| _version_ |
1836866489176555520 |
| fulltext |
ACCELERATION OF CHARGED PARTICLES BY ELLIPTIC POLAR-
IZED WAVES OF LARGE AMPLITUDE
A.V. Buts, V.A. Buts, V.V. Kuzmin
National Scientific Center “Kharkov Institute of Physics & Technology”
61108 Akademicheskaya st.1, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
In this paper the results of investigations on the charged particle dynamics in the homogeneous constant magnet-
ic field and in the field of a flat wave of arbitrary intensity are presented. When the magnetic field is absent the par-
ticle trajectory is defined. It is shown, that a particle can be effectively drawn along the wave vector of the wave.
Average speed of this drawing is proportional to the square of the wave force parameter. The trajectories of particles
are significantly dependent on initial conditions. In the presence of constant external magnetic field the particles can
be effectively accelerated by the wave field. The particle dynamics is always chaotic if the wave force parameter is
enough large. This dynamic is characterized by the alternation.
PACS: 29.17.+w
1. INTRODUCTION
Interaction of charged particles (electrons) with
fields of electromagnetic waves underlies in the theory
of accelerators, amplifiers and generators. At present the
time dynamics of particles at such interaction in details
is investigated, when intensity of electromagnetic wave
fields is small enough. Under the value smallness we
understand a smallness of /eE mcω==E , which some-
times is named as a parameter of a wave force or param-
eter of nonlinearity. For 10 cm radiation this parameter
is of about unit when the intensity of an electric field of
a wave reaches 105V/cm. For laser radiation the (
410 −≈λ cm) intensity of a field should by higher than
1010V/cm. In majority of experiments this parameter is
small. In this case the significant exchange of energy
between particles and electromagnetic waves occurs for
times considerably large then the period, i.e. interaction
of particles with a field should be long. Therefore the
special role conditions of synchronism of the charged
particle with phases of electromagnetic waves play.
These conditions look like resonant conditions. As the
field intensity is sufficiently high that a particle gets a
velocity close to the velocity of light during the period
of an electromagnetic wave, i.e. the exchange of energy
may be very fast, as compared with resonant conditions
in this case lose the exclusiveness. We shall note the im-
portant feature of interaction of a flat transverse electro-
magnetic wave with particles in vacuum. If parameter
E is small, the particle in main makes transverse (con-
cerning a direction of wave distribution) oscillation. If
the parameter of nonlinearity is great (E >>1), the parti-
cle makes a basic movement along the direction of wave
distribution. In the same direction it gets also the maxi-
mum velocity, therefore, the period of a wave, which is
perceived by a particle, is essentially increased.
Some results of research on particle dynamics in in-
tense ( 1≥E ) electromagnetic waves are represented.
The basic models of interaction of the charged particles
with electromagnetic waves are formulated in section 2,
the equations describing these models, and also some
integrals are written out. Some features of the charged
particle dynamics in the field of a transverse electro-
magnetic wave with linear polarization in vacuum are
analyzed in section 3. The basic result of this section
consists in that the particles quickly scatter in the cross
direction. In section 4 the particle dynamics in the trans-
verse electromagnetic wave with elliptic polarization is
considered. In Section 5 the dynamics of particles in a
field of an electromagnetic wave with linear polariza-
tion at presence of a constant homogeneous magnetic
field are considered. In this case the dynamics of parti-
cles has alternated character. The average energy of the
particles grows. However this growth is not
monotonous, i.e. the particle in the casual moments of
time can gain or loss energy.
2. THE BASIC EQUATIONS AND INTE-
GRALS
Let us consider the movement of the charged parti-
cle in the field of a flat electromagnetic wave with any
polarization. Components of electric and magnetic
fields of such a wave can be presented as
Re( )0
ie ψ=Ε Ε , [ ]1Re
0k
=
Η kE , (1)
where krtψ ω≡ − , 0 0Eα=Ε r
; }{ , ,i zx yα α α α=r
– a
vector of polarization of a wave; /0k cω= ; ω , k –
frequency and a wave vector of a wave. We shall enter
the following dimensionless variables:
/1 mc=p p , /1 0k=k k , tτ ω= , 1 0k=r r ,
/0e mcω= EE , /1 c=v v , / /1 c kcph phυ υ ω= = .
In these variables the equation of movement gets a form
(the index "1" is omitted)
( ) ( ){ }Re 1 +d ie
d
ψ
τ
≡ = −
pp kv k v& E E . (2)
It is convenient to add to equations (2) the equation
which defines the particle energy and may be obtained
from system (2):
( )Re ie ψγ = v& E , (3)
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.144-146. 144
where 21 pγ = + is the dimensionless energy of a par-
ticle (measured in terms of 2mc ).
Equations (2) and (3) have the integrals:
Re 0 0
0Re const
ii e
i
i e
ψγ γ
ψ
− + = − +
+ = =
p k p k
C
E
E
, (4)
Here the index "0" designates initial variables. Further,
without restriction of a generality, we shall consider,
that the wave travels along the axis z , i.e. {0,0, }k=k .
3. DYNAMICS OF PARTICLES AT INTER-
ACTION WITH THE WAVE OF LINEAR PO-
LARIZATION
If the particle moves in the field of only one wave
then the dynamics of its movement may be expressed by
analytical expressions. So, for values of the pulses and
values of its energy, one can obtain such expressions
(the solving of system of equations (2)):
( )
( ) ( )
( )
0 0
2 2
0
0
0 0
sin sin ,
,
2
,
x x x
x x
z z
z z
p p
p p
p p
p p
ψ ψ
γ ψ
γ γ
= + −
−
= ±
= ± −
&
E
(5)
where 0 /x eE mcω≡E ; The upper sign (+) in the ex-
pressions for γ and zp corresponds to the case 1k = .
The inferior sign (–) corresponds to the case 1k = − . Let
us look at the particles which originally were in rest:
From formulas (5) it follows that the value of a trans-
verse pulse essentially depends on the initial position of
a particle concerning a wave, i.e. from 0ψ . Really, even
if originally the particle has no transverse velocity (
0xp = ), depending on of an initial phase 0ψ , the maxi-
mal values of the module of a transverse pulse vary
from xE up to 2 xE . Besides, in an initial phase, the size
of an average transverse pulse varies too. The average
pulse is equal to 0, for particles, which are located in
phases nπ . Such particles are not displaced in a trans-
verse direction concerning the initial position. The aver-
age value of a longitudinal pulse, which the particles get
during interaction with a wave, reaches the value 2 / 4xE
. For particles which are located in phases
0 ( 1/ 2)nψ π= + , the maximal values of a transverse
pulse on the module reach the value 2 xE . And the aver-
age value of the transverse pulse for such particles is not
equal to 0 and also reaches value xE .
0 20 40 60 80
0
0.45
0.9
1.35
1.8
2.25
2.7
3.15
3.6
4.05
4.5
A
Pzn
Tn
0 20 40 60 80 100
4
3.2
2.4
1.6
0.8
0
0.8
1.6
2.4
3.2
4
B
Py n
T n
Fig 1. Pulses of particles taking place in an initial
phase nπ , at 3x =E . A) a longitudinal pulse,
B) a transverse pulse
0 20 40 60 80
0
1.8
3.6
5.4
7.2
9
10.8
12.6
14.4
16.2
18
A
Pzn
T n
0 20 40 60 80 1000
0.6
1.2
1.8
2.4
3
3.6
4.2
4.8
5.4
6
B
Pyn
Tn
Fig 2. Pulses of particles taking place in an initial
phase 0 ( 1/ 2)nψ π= + , at 3x =E . A) a longitudinal
pulse, B) a transverse pulse
The average size of a longitudinal pulse of such parti-
cles reaches size 2
xE . Thus, all particles, except for those
which are located in initial phases nπ , will run up
quickly in the transverse direction. Clearly, that such a
scheme of laser acceleration of the charged particles
will be not effective.
4. INTERACTION OF PARTICLES WITH
THE WAVE OF CIRCULAR POLARIZA-
TION
As is shown above, if a charged particle is accelerat-
ed by the wave with linear polarization, the particles run
up in the transverse direction, and for the purpose of ac-
celeration it is necessary to find conditions at which
their run up will be limited. One of the simplest way of
restriction of run up, apparently, may be acceleration of
particles by a wave with circular polarization. Under
this condition we can expect, that the trajectory of run-
ning up particles will be a spiral trajectory. Below we
shall show that indeed it takes place. Really the dynam-
ics of particles in the elliptic polarized wave may be ex-
pressed by analytical formulas which can be obtained
from equation (2):
( )
( )
( ) ( )
( )
sin sin ,0 0
cos cos ,0 0
2 2 2 2
0 0 ,0 2
,0 0
p px xx
p py yy
p p p px y x yp pz z
p pz z
ψ ψ
ψ ψ
γ ψ
γ γ
= + −
= + −
+ − +
= ±
= ± −
&
E
E
(6)
Despite of simplicity of these formulas, they de-
scribe dynamics in a rather complex manner. It is
caused by that decisions (6) are submitted in an implicit
form since a phase ψ itself is a function of a pulse.
For a more detailed investigation of particle dynam-
ics in a wave with elliptic polarization the system of
equations (2) was investigated by numerical methods. In
figures the most typical examples of this dynamics are
given.
As follows from the analytical research and the nu-
merical analysis the main feature of movement of a par-
ticle in a wave with elliptic polarization consists in that
its trajectory represents a spiral with an wave axis di-
rected along the wave propagation and with the radiusE
in space of pulses (see figure 3).
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.144-146.145
Thus, acceleration of particles by a field elliptic po-
larization may remove a part of the difficulties connect-
ed with the run up of electrons in the transverse direc-
tion that is characteristic for the scheme of interaction
with a linearly polarized wave.
0 60 120 180 240 300
4
3.2
2.4
1.6
0.8
0
0.8
1.6
2.4
3.2
4
Pxn
T n
0 60 120 180 240 300
6
5.4
4.8
4.2
3.6
3
2.4
1.8
1.2
0.6
0
Py n
T n
0 60 120 180 240 300
4
3.2
2.4
1.6
0.8
0
0.8
1.6
2.4
3.2
4
Pxn
Tn
0 60 120 180 240 300
6
5.4
4.8
4.2
3.6
3
2.4
1.8
1.2
0.6
0
Py n
Tn
0 60 120 180 240 300
0
2
4
6
8
10
12
14
16
18
20
Pzn
Tn
Fig.3. Dynamics of a particle at interaction with a
wave of circular polarization, at 3=E
Really, in the space of transverse pulses the trajecto-
ry of a particle describes by circle. The equation of this
circle can be presented as:
( ) ( ) 22 2
x x y yP A P A− + − = E (7)
where 2
0 0cosx x xA P ψ= + E , 2
0 0siny y yA P ψ= + E
It is visible, that if initial particle pulses (pulses of com-
ing particles in area of interactions with a wave) will be
large than 2E ( 2 2
0 0,x x y yP P> > > >E E ), all particles will
get an average transverse pulse which differs insignifi-
cantly from the initial particle pulse. In this case accel-
erated particles practically will not be run up since their
average pulses will not depend on the arrangement of
particles with respect to the wave phase 0ψ .
5. DYNAMICS OF PARTICLES IN FIELD
OF THE WAVE IN PRESENCE OF THE
СONSTANT HOMOGENEOUS MAGNETIC
FIELD
In the presence of a constant magnetic field 0H , hav-
ing components { }00, ,0yH=0H , the equation of move-
ment (2) will become:
( ) ( ){ } [ ]Re 1 + id e
d
ψ
τ
= − ⋅ ⋅ + 0
p k v k v vBE E , (8)
where 0 0 /e mcω=B H
The system of equations (8) was analyzed by numerical
methods. Characteristic results of calculations are given
in figure 4.
0 500 1000 1500
50.58
39.76
28.93
18.11
7.29
3.54
14.36
25.19
36.01
46.83
Pz n
Tn
0 400 800 1200 1600 2000 60 48 36 24 12 0
12 24 36 48 60
Py n
T n
Fig.4. Dynamics of particles in a wave field with lin-
ear polarization in the presence of a constant homo-
geneous magnetic field. 3=E , 1=0B
A feature of movement of a particle in a wave, follow-
ing from the numerical analysis in the presence of a
constant homogeneous magnetic field, consists in that
the dynamics has alternated character. Under these con-
ditions the particle receives the average energy which is
lower than it can receive as a result of diffusion in space
of energy. Really, all conditions for stochastic accelera-
tion of charged particles in this case are satisfied. As it
is known, under this condition, the particle should get
the energy under the law ~γ τ∆ ⋅E . Numerical calcu-
lations show, that there are intervals of time at which
the particle gets the energy considerably more quickly,
however, on a whole, for rather large times its energy is
lower than it follows from the law of diffusion. Discus-
sion of this fact is contained in other our paper submit-
ted at this conference.
УСКОРЕНИЕ ЗАРЯЖЕННЫХ ЧАСТИЦ ЭЛЛИПТИЧЕСКИ ПОЛЯРИЗОВАННЫМИ
ВОЛНАМИ БОЛЬШОЙ АМПЛИТУДЫ
А.В. Буц, В.А. Буц, В.В. Кузьмин
Изложены результаты исследования динамики заряженных частиц в однородном постоянном магнитном
поле и в поле плоской волны произвольной напряженности. В отсутствии магнитного поля определен вид
траектории частиц. Показано, что частицы могут эффективно увлекаться полем внешней волны. Средняя
скорость увлечения пропорциональна квадрату параметра силы волны. Вид траектории существенно зави-
сит от начальных условий. При наличии постоянного магнитного поля частицы могут эффективно ускорять-
ся полем волны. При больших напряженностях поля динамика частиц всегда хаотична. Динамика характе-
ризуется перемежаемостью.
ПРИСКОРЕННЯ ЗАРЯДЖЕНИХ ЧАСТОК ЄЛЕПТИЧНО ПОЛЯРІЗОВАНИМИ
ХВИЛЯМИ ВЕЛИКОЇ АМПЛІТУДИ
О.В.Буц, В.О. Буц, В.В. Кузьмін
Викладені результаті дослідження динаміки заряджених часток у однорідному постійному магнітному
полі та в полі плоскої хвилі довільної напруженості. За відсутності постійного магнітного поля вигляд
траєкторії частки був визначений. Показано, що частка може ефективно захоплюватися полем зовнішньої
хвилі. Середня швидкість захоплення пропорційна квадрату параметру сили хвилі. При наявності
постійного магнітного поля. частки можуть ефективно прискорюватись полем хвилі. При великих
напруженостях поля динаміка часток завжди хаотична. Динаміка характеризується перемежністю.
146
A.V. Buts, V.A. Buts, V.V. Kuzmin
УСКОРЕНИЕ ЗАРЯЖЕННЫХ ЧАСТИЦ ЭЛЛИПТИЧЕСКИ ПОЛЯРИЗОВАННЫМИ
ВОЛНАМИ БОЛЬШОЙ АМПЛИТУДЫ
А.В. Буц, В.А. Буц, В.В. Кузьмин
О.В.Буц, В.О. Буц, В.В. Кузьмін
|