Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma
The specific features of the superradiation processes excited by an ultra-short 1D monoenergetic bunch of charged particles, which moves through a plasma, are examined. When the size of the bunch is less then the wavelength of the excited plasma oscillations, the spontaneous bunch reshaping takes pl...
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Kirichok, A.V. Kuklin, V.M. Mischin, A.V. Pryimak, A.V. 2017-01-18T19:45:58Z 2017-01-18T19:45:58Z 2015 Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma / A.V. Kirichok, V.M. Kuklin, A.V. Mischin, A.V. Pryimak // Вопросы атомной науки и техники. — 2015. — № 4. — С. 255-257. — Бібліогр.: 16 назв. — англ. 1562-6016 PACS: 52.35.Qz; 52.40.Mj https://nasplib.isofts.kiev.ua/handle/123456789/112221 The specific features of the superradiation processes excited by an ultra-short 1D monoenergetic bunch of charged particles, which moves through a plasma, are examined. When the size of the bunch is less then the wavelength of the excited plasma oscillations, the spontaneous bunch reshaping takes place. As a result, the bunch excites a wakefield with maximum possible amplitude, which twice as much the amplitude of the wakefield driven by an extended bunch with the same number of particles. The domains of maximum amplitude keep in time their location in the laboratory frame of reference. Розглянуто особливості процесів супервипромінювання коротких одномірних моноенергетичних згустків заряджених частинок, що рухаються у плазмі. При розмірах згустка менших за довжину хвилі, що збуджується, в великій області позаду згустка формується кільватерний слід з максимально можливою амплітудою коливань. При цьому відбувається профілювання згустка, що забезпечує його ефективне випромінювання. Області максимуму амплітуди коливань локалізовані там, де вони були сформовані згустками частинок, що рухаються в лабораторній системі відліку. Рассмотрены особенности режимов сверхизлучения коротких одномерных моноэнергетических сгустков заряженных частиц, распространяющихся в плазме. При размерах сгустка меньше длины волны излучаемых колебаний происходит самопроизвольное профилирование сгустка, в результате которого в обширной области за сгустком формируется кильватерный след с максимально возможной амплитудой поля, в два раза превышающей предельно достижимые амплитуды поля в случае протяженных сгустков с тем же числом частиц. Области максимума поля, возбуждаемые движущимися сгустками частиц в лабораторной системе отсчета, локализованы в месте их образования. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы в плазменных средах Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma Моделювання процесів супервипромінювання ультракороткого згустка заряджених частинок, що рухається у плазмі Моделирование процессов сверхизлучения движущегося в плазме ультракороткого сгустка заряженных частиц Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma |
| spellingShingle |
Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma Kirichok, A.V. Kuklin, V.M. Mischin, A.V. Pryimak, A.V. Нелинейные процессы в плазменных средах |
| title_short |
Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma |
| title_full |
Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma |
| title_fullStr |
Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma |
| title_full_unstemmed |
Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma |
| title_sort |
modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma |
| author |
Kirichok, A.V. Kuklin, V.M. Mischin, A.V. Pryimak, A.V. |
| author_facet |
Kirichok, A.V. Kuklin, V.M. Mischin, A.V. Pryimak, A.V. |
| topic |
Нелинейные процессы в плазменных средах |
| topic_facet |
Нелинейные процессы в плазменных средах |
| publishDate |
2015 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Моделювання процесів супервипромінювання ультракороткого згустка заряджених частинок, що рухається у плазмі Моделирование процессов сверхизлучения движущегося в плазме ультракороткого сгустка заряженных частиц |
| description |
The specific features of the superradiation processes excited by an ultra-short 1D monoenergetic bunch of charged particles, which moves through a plasma, are examined. When the size of the bunch is less then the wavelength of the excited plasma oscillations, the spontaneous bunch reshaping takes place. As a result, the bunch excites a wakefield with maximum possible amplitude, which twice as much the amplitude of the wakefield driven by an extended bunch with the same number of particles. The domains of maximum amplitude keep in time their location in the laboratory frame of reference.
Розглянуто особливості процесів супервипромінювання коротких одномірних моноенергетичних згустків заряджених частинок, що рухаються у плазмі. При розмірах згустка менших за довжину хвилі, що збуджується, в великій області позаду згустка формується кільватерний слід з максимально можливою амплітудою коливань. При цьому відбувається профілювання згустка, що забезпечує його ефективне випромінювання. Області максимуму амплітуди коливань локалізовані там, де вони були сформовані згустками частинок, що рухаються в лабораторній системі відліку.
Рассмотрены особенности режимов сверхизлучения коротких одномерных моноэнергетических сгустков заряженных частиц, распространяющихся в плазме. При размерах сгустка меньше длины волны излучаемых колебаний происходит самопроизвольное профилирование сгустка, в результате которого в обширной области за сгустком формируется кильватерный след с максимально возможной амплитудой поля, в два раза превышающей предельно достижимые амплитуды поля в случае протяженных сгустков с тем же числом частиц. Области максимума поля, возбуждаемые движущимися сгустками частиц в лабораторной системе отсчета, локализованы в месте их образования.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/112221 |
| citation_txt |
Modelling of superradiation processes driven by an ultra-short bunch of charged particles moving through a plasma / A.V. Kirichok, V.M. Kuklin, A.V. Mischin, A.V. Pryimak // Вопросы атомной науки и техники. — 2015. — № 4. — С. 255-257. — Бібліогр.: 16 назв. — англ. |
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ISSN 1562-6016. ВАНТ. 2015. №4(98) 255
MODELLING OF SUPERRADIATION PROCESSES
DRIVEN BY AN ULTRA-SHORT BUNCH OF CHARGED PARTICLES
MOVING THROUGH A PLASMA
A.V. Kirichok, V.M. Kuklin, A.V. Mischin, A.V. Pryimak
V.N. Karazin Kharkiv National University, Kharkov, Ukraine
The specific features of the superradiation processes excited by an ultra-short 1D monoenergetic bunch of
charged particles, which moves through a plasma, are examined. When the size of the bunch is less then the wave-
length of the excited plasma oscillations, the spontaneous bunch reshaping takes place. As a result, the bunch excites
a wakefield with maximum possible amplitude, which twice as much the amplitude of the wakefield driven by an
extended bunch with the same number of particles. The domains of maximum amplitude keep in time their location
in the laboratory frame of reference.
PACS: 52.35.Qz; 52.40.Mj
INTRODUCTION
The problem of obtaining the wakefields of highest
possible amplitude driven by an electron bunch moving
through a plasma had its origin from the analysis of the
possibility to use the high-energy and high-currentshort
electron beams – bunches, moving through a plasma, for
ion acceleration [1, 2].Despite the fact that the bunches,
which initial size is much smaller than the wavelength
of the radiation, are unstable in a one-dimensional and
three-dimensional cases (see, e.g. [3, 4]), at a certain
reshaping of the particles density and velocity profile,
one could hope for significant wakefield intensity, com-
parable with the wakefield, excited by particles, which
are bundled in a very small spatial domain [5 - 8].If the
goal is to ensure the stability of the bunch, it is prefera-
ble to use more extended beams, which further bunch-
ing allows to obtain the fields of high amplitude as in-
side the beam and behind it [9 - 12].
The wakefield behind the radiating particle moving
with the velocity v can be written in the particle refer-
ence frame as a cosine curve cos[ ] cos[ ( )]k k x vtζ = − .
In cold plasma, the group velocity of the Langmuir os-
cillations having a frequency ωpe is equal to zero. So,
after transfer to the laboratory frame of reference, the
field amplitude in any point will change with time as
cos[ ] cos[ ]pet kvtω = . It is important to note that this is
one and the same field, but in different frames of refer-
ence. If the velocities of each particle are different, the
cosine curves will have different values of the wave
number /pek vω= . The total wakefield represents an
interference pattern of the individual wakefields driven
by particles composing the bunch. When the relative
positions and velocities of particles are changed, the
interference pattern is changed too. Wherein in the la-
boratory frame, the total wakefield is a sum of oscilla-
tions with different phases. There are exist the spatial
and velocity configurations of particles, composing the
bunch, which allow to minimize a phase spread of the
wakefields driven by individual particles in separate,
generally speaking, quite lengthy domains in the labora-
tory system of reference. It was proposed [13] to use a
special profiling of the bunch over the velocity and den-
sity in order to achieve the wakefield of maximum am-
plitude, at least in some domains of the plasma space. It
is clear that in the bunch reference frame, the domain
with wakefield maximum amplitude moves in the oppo-
site direction with approximately the same velocity [14].
Below, we will show that such profiling of the bunch
inside its volume happens spontaneously. In addition, the
domain with wakefield maximum amplitude appears,
which in the bunch reference frame moves away from the
bunch the mean velocity of the bunch, i.e. keeps the fixed
position in the laboratory reference frame. Of particular
interest are the regimes, when the highest wakefield
amplitude can be achieved, that is found to be in the
case of short bunches, which longitudinal size is smaller
than the wavelength of the wakefield.
1. ASIMPLE MODEL OF SUPERRADIATION
PROCESS DRIVEN BY COMPACT MOVING
BUNCH OF CHARGED PARTICLES
It is important to note that if we consider the infinite
periodic consequence of particles, as is often done, and
then let the distance between them to infinity, the transi-
tion to the wake field of a single particle would be diffi-
cult. This is because of the field in the periodic system
is present both in front of and behind the individual par-
ticles, but as known a single charge, moving in plasma,
does not generate a radiation field in the direction of its
movement (see, for example [3]).
Let present a charge density of an electron, moving
with the velocity 0v > as follows
( ) ( )e v t x s e sρ δ δ ξ= − ⋅ − ⋅ + − = − ⋅ − .
Here ( )xδ is the Dirac delta function.
Then, we use the Poisson equation / 4D x πρ∂ ∂ = ,
which after the Fourier transformation takes the form
( , ) ( , ) 4 ( , )ik k E k kε ω ω πρ ω− ⋅ = . (1)
The inverse Fourier transform of Eq. (1) gives
0
0 0
exp{ } [ ( ) ( )]
( ) exp{ }.
k
i ik dk k k E k
kk ik
k
ξ ε
ε ξ
ξ
∞
−∞
− − ⋅ ⋅ ⋅ ⋅ =
∂ ∂Ε
= −
∂ ∂
∫
(2)
After substituting the charge density into Eq.(1) and
taking into account Eq.(2) one can obtain the electric
strength of the wakefield behind the charge, moving in
positive direction
0
1
0
0
4 [ ( ) / | ]
( ) exp{ ( )}
kE e k k k
s ik s
π ε
θ ξ ξ
−= − ⋅ ⋅∂ ∂ ×
× − ⋅ −
, (3)
where ( )xθ is the Heaviside function.
The wakefield, produced by a single particle, can be
considered as result of its spontaneous radiation. The
ISSN 1562-6016. ВАНТ. 2015. №4(98) 256
spontaneous fields, generated by individual particles
composing the bunch (under condition of their uniform
spatial distribution and in absence of other synchroniza-
tion mechanisms) differ from one another by the phase,
which, in general, can be randomly distributed, at least,
at the initial moment. In other words, the spontaneous
emission of n uniformly distributed non-phase-locked
emitters is non-coherent. The intensity of the spontane-
ous radiation is proportional to the number of emitters,
i.e. n∝ .On each particle acts its own field and the field
of the particles, which are ahead of it.
The field energy of the spontaneous radiation inanen
closed volume growths with time linearly. In the open
system (bunch), the field energy growth is restricted by
energy flow outside the bunch. However, after a certain
time, the grouping of emitters may result in the situation
when the main impact on the particles will make the
emission produced by the grouped particles. If the num-
ber of particles in the group is s , the intensity of the
field, produced by this group is proportional to s2. When
s n∝ , the field of the grouped particles will dominate
in the process of self-modulation and wakefield for-
mation behind the bunch. The growth of this coherent
emission, which by now takes an appearance of the
stimulated emission, happens exponentially. The part of
coherent component in the total emission of the bunch
increases, i.e. the phases of many individual emitters
differ only slightly from each other. The change in the
field energy per unit of time is proportional in this case
to the squared number of synchronized oscillators. Note,
that such synchronization takes place under the action of
the emitted wave and is governed by it.
The total wakefield can be obtained after summation
over all particles of the bunch
2( ) cos [ 2 ( )] ( )
N
f g
N α α α α
α
ξ π ξ ξ ξ ξΕ = − − Θ −∑ . (4)
The Eq. (4) should be supplemented with equation
of particle motion
d
d
ξ ν
τ
= , ( )d
d
ν ξ
τ
= Ε , (5)
where 0 02 ( )K z V tπ ξ = − , 0 0( ) / 2 LK V Vν π γ= − ,
2 2
0 /L ee K M mγ = , 1(1 )g ν −= + ∆ ⋅ , 0 02 /L K Vπ γ∆ = ,
L tτ γ= , and M is the total number of particles in the
unit cross-section of the bunch, 2
0 / 2 e LeK E mπ γΕ = , Е
is the electric field strength, fα is a statistical weight of
the modelling particle, e , em are the electron charge and
mass, 0 0/peK Vω= is the wakefield wave-number The
Eqs. (4), (5) describe the nonlinear dynamics of a short
one-dimensional electron bunch, which moves through
a dense plasma in its rest frame. The conditions of ap-
plicability of the model and its comparison with more
full description [15] were discussed in [3].
2. SIMULATION RESULTS
The program, which implements a mathematical
model of the problem, was created using the technology
JCUDA. JCUDA provides execution of Java-program
code on the GPU (graphics processing unit (GPU)) writ-
ten in the C programming language with inserts of code
specific to technology CUDA (a brief description of this
technology can be found in [16]).
It follows from the results of calculation, that for
bunches, which longitudinal dimension is less than but
still comparable with the wavelength of the produced
wakefield, the phenomenon of bunch self-profiling ap-
pears. The wakefield amplitude E in this case reaches
in some domain behind the bunch the values, which are
significantly higher than the unit.
For bunches which length exceeds few wave lengths
of the emitted waves, the wakefield amplitude is less or
of the order of unity [3]. If all particles, which compose
the bunch, are gathered into a point, than the amplitude
of the wakefield in the considered representation (4)-(5)
reaches the maximum value of 2.
At first, consider what happens to the bunch. It was
found that it slows down as a whole, forming at this
time the characteristic triangle distributions in the con-
figuration and velocity spaces (see Figs. 1, 2).
Fig. 1. The spatial distri-
bution of bunch particles:
1τ = (a); 2τ = (b);
3τ = (c)
Fig. 2. The velocity
distribution of bunch
particles: 1τ = (a);
2τ = (b); 3τ = (c)
Namely the formation of triangular distributions
similar to the initial profiling of the bunch leads to inter-
ference of wakefields driven by particles in a certain
spatial domain, which moves away from the bunch in its
rest frame with a mean velocity
0 0 1
2 L
K V td
d t
ξ
τ πγ
< >= =
∆
(6)
The maximum field domain moves with the velocity
of the bunch in an opposite direction (Fig. 3). Note that
the beam is on the right side of the figure. In the labora-
tory reference frame the domain with the maximum
field remains motionless. Some distortion of the wake-
field profile associated with a slight change of the bunch
configuration during this time interval. This model de-
scribes qualitatively the dynamics of the bunch and
emission produced by it, but allows to detect the effect
of bunch self-profiling.
As a result of such self-profiling, the wakefield am-
plitude reaches the practically maximum possible value
in the spatial domain, which dimension is more than an
order of magnitude greater than the initial size of the
bunch.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 257
Fig. 3. The wakefield profile ( )E ξ in the bunch rest
frame of reference at the moments τ=2 (a); τ=3 (b)
CONCLUSIONS
It is considered the model of one-dimensional ultra-
short bunch of charged particles, moving through plas-
ma. At the initial time, all particles are evenly distribut-
ed over the length L and have the same velocity. The
case was examined, when the bunch length is signifi-
cantly less then the wakefield wave length. As a result
of the evolution of a short bunch, the profiling of its
shape occurs and the amplitude of the wakefield in a
fairly extended domain behind the bunch reaches the
maximum possible value for a given number of particles
of the bunch, which is twice as much the value of the
wakefield generated by an extended bunch. The domain
with maximum wakefield amplitude keep in time its
location in the region of its formation in the bunch rest
frame. The dimension of this domain is of the order of
several tens of wavelengths.
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Article received 12.05.2015
МОДЕЛИРОВАНИЕ ПРОЦЕССОВ СВЕРХИЗЛУЧЕНИЯ ДВИЖУЩЕГОСЯ В ПЛАЗМЕ УЛЬТРАКОРОТКОГО
СГУСТКА ЗАРЯЖЕННЫХ ЧАСТИЦ
А.В. Киричок, В.М. Куклин, А.В. Мишин, А.В. Приймак
Рассмотрены особенности режимов сверхизлучения коротких одномерных моноэнергетических сгустков заряженных
частиц, распространяющихся в плазме. При размерах сгустка меньше длины волны излучаемых колебаний происходит
самопроизвольное профилирование сгустка, в результате которого в обширной области за сгустком формируется кильва-
терный след с максимально возможной амплитудой поля, в два раза превышающей предельно достижимые амплитуды
поля в случае протяженных сгустков с тем же числом частиц. Области максимума поля, возбуждаемые движущимися
сгустками частиц в лабораторной системе отсчета, локализованы в месте их образования.
МОДЕЛЮВАННЯ ПРОЦЕСІВ СУПЕРВИПРОМІНЮВАННЯ УЛЬТРАКОРОТКОГО ЗГУСТКА ЗАРЯДЖЕНИХ
ЧАСТИНОК, ЩО РУХАЄТЬСЯ У ПЛАЗМІ
О.В. Кірічок, В.М. Куклін, О.В. Мішин, О.В. Приймак
Розглянуто особливості процесів супервипромінювання коротких одномірних моноенергетичних згустків заряджених
частинок, що рухаються у плазмі. При розмірах згустка менших за довжину хвилі, що збуджується, в великій області поза-
ду згустка формується кільватерний слід з максимально можливою амплітудою коливань. При цьому відбувається профі-
лювання згустка, що забезпечує його ефективне випромінювання. Області максимуму амплітуди коливань локалізовані
там, де вони були сформовані згустками частинок, що рухаються в лабораторній системі відліку.
http://en.wikipedia.org/wiki/CUDA
INTRODUCTION
1. ASIMPLE MODEL OF SUPERRADIATION PROCESS DRIVEN BY COMPACT MOVING BUNCH OF CHARGED PARTICLES
2. SIMULATION RESULTS
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