Homogeneous focusing of electron bunch sequence by plasma wakefield
The mechanism of focusing of relativistic electron bunches by plasma wakefield, in which all bunches of sequence are focused identically and uniformly, has been proposed and investigated. In this scheme of focusing it is necessary that length of each bunch should be equal to a half of wavelength, th...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1087382016-11-16T03:02:29Z Homogeneous focusing of electron bunch sequence by plasma wakefield Lotov, K.V. Maslov, V.I. Onishchenko, I.N. Svistun, O.M. Новые и нестандартные ускорительные технологии The mechanism of focusing of relativistic electron bunches by plasma wakefield, in which all bunches of sequence are focused identically and uniformly, has been proposed and investigated. In this scheme of focusing it is necessary that length of each bunch should be equal to a half of wavelength, the charge of first bunch equals half of the charge of the others bunches and the distance between bunches equals 1.5 of wavelength. It is shown that in this case only 1-st bunch is in the finite longitudinal electrical wakefield Ez. Other bunches are in zero longitudinal electrical wakefield Ez= 0. The focusing radial electrical field in regions, occupied by bunches, is constant along each bunch. Предложен и исследован механизм фокусировки релятивистских электронных сгустков плазменными кильватерными полями, в которых все сгустки последовательности фокусируются одинаково и однородно. В этой схеме фокусировки необходимо, чтобы длина каждого сгустка была равна половине длины кильватерной волны, заряд первого сгустка был в 2 раза меньше заряда каждого из остальных сгустков, а расстояние между ними составляло 1,5 длины волны. Показано, что в этом случае только на 1-й сгусток действует продольное поле Ez. Остальные сгустки находятся в нулевом продольном кильватерном поле Ez = 0. Фокусирующее радиальное поле в областях, занятых сгустками, постоянно по длине сгустков. Запропонований і досліджений механізм фокусування релятивістських електронних згустків плазмовими кільватерними полями, у яких усі згустки послідовності фокусуються однаково та однорідно. В цій схемі фокусування необхідно, щоб довжина кожного згустка дорівнювала половині довжини кільватерної хвилі, заряд першого згустку був у 2 рази менше заряда кожного із решти згустків, а відстань між ними була рівною 1,5 довжини хвилі. Показано, що в цьому випадку лише на 1-й згусток діє повздовжнє електричне поле Ez. Останні згустки знаходяться у нульовому повздовжньому кільватерному полі Ez = 0. Фокусуюче радіальне електричне поле в зонах, що заняті згустками, постійне по довжині згустків. 2012 Article Homogeneous focusing of electron bunch sequence by plasma wakefield / K.V. Lotov, V.I. Maslov, I.N. Onishchenko, O.M. Svistun // Вопросы атомной науки и техники. — 2012. — № 3. — С. 159-163. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 29.17.+w; 41.75.Lx; http://dspace.nbuv.gov.ua/handle/123456789/108738 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Новые и нестандартные ускорительные технологии Новые и нестандартные ускорительные технологии |
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Новые и нестандартные ускорительные технологии Новые и нестандартные ускорительные технологии Lotov, K.V. Maslov, V.I. Onishchenko, I.N. Svistun, O.M. Homogeneous focusing of electron bunch sequence by plasma wakefield Вопросы атомной науки и техники |
| description |
The mechanism of focusing of relativistic electron bunches by plasma wakefield, in which all bunches of sequence are focused identically and uniformly, has been proposed and investigated. In this scheme of focusing it is necessary that length of each bunch should be equal to a half of wavelength, the charge of first bunch equals half of the charge of the others bunches and the distance between bunches equals 1.5 of wavelength. It is shown that in this case only 1-st bunch is in the finite longitudinal electrical wakefield Ez. Other bunches are in zero longitudinal electrical wakefield Ez= 0. The focusing radial electrical field in regions, occupied by bunches, is constant along each bunch. |
| format |
Article |
| author |
Lotov, K.V. Maslov, V.I. Onishchenko, I.N. Svistun, O.M. |
| author_facet |
Lotov, K.V. Maslov, V.I. Onishchenko, I.N. Svistun, O.M. |
| author_sort |
Lotov, K.V. |
| title |
Homogeneous focusing of electron bunch sequence by plasma wakefield |
| title_short |
Homogeneous focusing of electron bunch sequence by plasma wakefield |
| title_full |
Homogeneous focusing of electron bunch sequence by plasma wakefield |
| title_fullStr |
Homogeneous focusing of electron bunch sequence by plasma wakefield |
| title_full_unstemmed |
Homogeneous focusing of electron bunch sequence by plasma wakefield |
| title_sort |
homogeneous focusing of electron bunch sequence by plasma wakefield |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2012 |
| topic_facet |
Новые и нестандартные ускорительные технологии |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/108738 |
| citation_txt |
Homogeneous focusing of electron bunch sequence by plasma wakefield / K.V. Lotov, V.I. Maslov, I.N. Onishchenko, O.M. Svistun // Вопросы атомной науки и техники. — 2012. — № 3. — С. 159-163. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2012. №3(79) 159
HOMOGENEOUS FOCUSING OF ELECTRON BUNCH SEQUENCE
BY PLASMA WAKEFIELD
K.V. Lotov1, V.I. Maslov, I.N. Onishchenko, O.M. Svistun2
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
1Budker Institute of Nuclear Physics, Novosibirsk, Russia;
2Karazin Kharkov National University, Kharkov, Ukraine
E-mail: vmaslov@kipt.kharkov.ua
The mechanism of focusing of relativistic electron bunches by plasma wakefield, in which all bunches of se-
quence are focused identically and uniformly, has been proposed and investigated. In this scheme of focusing it is
necessary that length of each bunch should be equal to a half of wavelength, the charge of first bunch equals half of
the charge of the others bunches and the distance between bunches equals 1.5 of wavelength. It is shown that in this
case only 1-st bunch is in the finite longitudinal electrical wakefield Ez. Other bunches are in zero longitudinal elec-
trical wakefield Ez= 0. The focusing radial electrical field in regions, occupied by bunches, is constant along each
bunch.
PACS: 29.17.+w; 41.75.Lx;
1. INTRODUCTION
The focusing of relativistic electron bunches by
wakefield, excited in plasma, is very interesting and
important (see, for example, [1-2]). The focusing of
bunches by wakefield, excited in plasma by resonant
sequence of relativistic electron bunches (repetition
frequency of the bunches ωm coincides with the plasma
frequency ωm=ωp), is inhomogeneous. In this paper the
specific mechanism of focusing by plasma wakefield, in
which all bunches of sequence are focused identically
and uniformly, has been proposed and numerically in-
vestigated by 2.5D code LCODE [3]. We numerically
simulate the self-consistent dynamics of lengthy elec-
tron bunches in homogeneous plasma. In simulation we
use the hydrodynamic description of plasma. In other
words the plasma is considered to be cold electron liq-
uid, and bunches are aggregate of macroparticles.
2. RESULTS OF SIMULATION
We investigate focusing of sequence of electron
bunches, length of each bunch equals half of wave-
length, because in this case the wakefield has maximal
value at increase of bunch length of the same density.
Fig.1. Density of bunches nb (blue), their averaged
radius <rb> (red), longitudinal wakefield Ez (black)
and radial wake force Fr (green)
We use the charge of first bunch, equal in two times
smaller than charge of each next bunch
Q1=Q/2, Qi=Q, i=2, 3, 4 ...,
and distance between bunches equals 1.5λ
ξi+1-ξi=1.5λ,
λ is the wakefield wavelength. Then for three bunches
the distribution of bunch density nb, their averaged ra-
dius <rb>, excited longitudinal wakefield Ez and radial
wake force Fr are of the form, shown in Fig.1. One can
see that all bunches (which are in Ez = 0) with the ex-
ception of first one (which are in Ez ≠ 0) do not change
by energy with longitudinal wakefield. Then wakefield
after i-th bunch is the same as before it (with the excep-
tion of first bunch). But the bunches are focused be-
cause radial wake force is not equal zero Fr ≠ 0. This
wakefield lens has following qualities:
1) radial wake force Fr does not approximately depend
on coordinate in regions, occupied by bunches (with
the exception of first bunch), Fr ≈ const, i.e. lengthy
bunches are focused identically;
2) only first bunch is slowed down;
3) identical focusing force effects on all bunches (with
the exception of first bunch);
4) longitudinal wakefield equal zero Ez = 0 in regions,
occupied by bunches (with the exception of first
bunch).
Fig.2. Plasma electron density ne (black) in wakefield,
longitudinal wakefield Ez (green), density of bunches nb
(blue) and <Ez>=∫dr r Eznb/∫dr rnb (red) coupling rate
of bunch with wakefield Ez
Such ideal focusing is realized due to formation of
flat holes of plasma electron density ne in regions, oc-
cupied by bunches (Fig.2), which neutralize charges of
bunches and focuse them. In region, occupied by first
bunch (see Fig.2), the formed hole of plasma electron
density ne is not flat. One can see from Fig.2 that wave
of plasma electron density ne is nonsinusoidal: wide
ISSN 1562-6016. ВАНТ. 2012. №3(79) 160
holes of ne (with the exception of region, occupied by
first bunch) are interchanged by narrow humps of ne.
The bunches are located in the regions of the holes of ne.
In Fig.1 one can see that the usual for linear case
connection of radial and longitudinal wakefields Er ∼
∂rEz is not true.
In Fig.3 one can see that on wavelength (regions a
and b) the positive and negative perturbations of Ez
compensate each other, hence on the last part (region c)
of wave period (λ/2), where bunches are localized, lon-
gitudinal wakefield Ez can become zero (it is observed).
Radial wake force Fr does not equal zero Fr ≠ 0 in re-
gions, occupied by bunches.
Fig.3. Longitudinal wakefield Ez (black) and radial
wake force Fr (green) on one wave period
Fig.4. The density of ten bunches
Fig.5. Change of longitudinal momentum of bunches Pz
at wakefield excitation. N is the number of wavelengths
These results are right and for larger number of
bunches. Let us consider ten bunches (Fig.4). Again the
charge of first bunch is two times smaller than charge of
each next bunch Q1=Q/2, Qi=Q, i=2, 3, 4 ... and dis-
tance between bunches equals ξi+1-ξi=1.5λ. Then the
spatial distributions of plasma electron density ne, den-
sity of bunches nb, their averaged radius <rb>, excited
longitudinal wakefield Ez, radial wake force Fr, cou-
pling rate of bunch with longitudinal wakefield <Ez>
and longitudinal momentum of bunches Pz have views,
shown in Figs.4-8. From Fig.6 one can see that <Ez> of
only 1-st bunch does not equal zero and therefore only
1-st bunch is slowed down (see Fig.5).
Fig.6. The density of bunches nb (blue), longitudinal
wakefield Ez (black) and <Ez> (red) coupling rate of
bunch with wakefield Ez
Fig.7. The density of bunches nb (dark blue), their aver-
aged radius <rb> (red), longitudinal wakefield Ez
(black), radial wake force Fr (green)
Fig.8. The density of bunches nb (black) and their
averaged radius <rb> (red) after focusing
Again one can (see Fig.7) that only 1-st bunch is in fi-
nite Ez≠0. Other bunches are in zero longitudinal electri-
cal wakefield Ez = 0. Radial wake force Fr in regions,
occupied by bunches, is finite (see Fig.7). In Fig.8 one
can see that radius of bunches (and their density nb2 on r,
equal initial radius of bunches) decreases due to focusing.
We considered bunches with close radius and
length. If the bunches are represented «needles», i.e. if
their lengths are larger than radius, then plateau-
electron-holes (Fig.10), Fr longitudinal distribution in
regions, occupied by bunches (Fig.9), and focusing
(Fig.11) are more ideal.
ISSN 1562-6016. ВАНТ. 2012. №3(79) 161
Fig.9. The density of needle bunches nb (blue), their
averaged radius <rb> (red), longitudinal wakefield Ez
(black) and radial wake force Fr (pink)
Fig.10. Plasma electron density ne in wakefield, excited
by needle bunches
Fig.11. The density nb of ten bunches before and after
focusing
Fig.12. Longitudinal wakefield Ez (black), density
of bunches nb (blue) and <Ez> (red) coupling rate
of bunch with wakefield Ez
For radial wake force Fr enhancement let us consider
following shaped sequence of bunches. The charges of
first n = 3 shaped bunches increase along sequence ac-
cording to: 2k-1, k≤n (Fig.12,14). The charges of next
bunches equal 2n, k>n. Then the distribution of plasma
electron density ne, density of bunches nb, their aver-
aged radius <rb>, excited longitudinal wakefield Ez,
radial wake force Fr, coupling rate of bunch with longi-
tudinal wakefield <Ez> and longitudinal momentum of
bunches Pz have views, shown in Figs.12-14.
Fig.13. Change of longitudinal momentum of bunches
Pz at wakefield excitation. N is the number
of wavelengths
Fig.14. The density nb (blue) of shaped bunches, their
averaged radius <rb> (red), longitudinal wakefield Ez
(black) and radial wake force Fr (green)
From Fig.12 one can see that the coupling rates <E>
of only first three bunches n = 3 with longitudinal
wakefield Ez do not equal zero and therefore only first
three bunches are slowed down (see Fig.13).
Fig.15. Plasma electron density ne (black), longitudinal
wakefield Ez (green), density nb (blue) of two bunches and
<Ez> (red) coupling rate of bunch with wakefield Ez
ISSN 1562-6016. ВАНТ. 2012. №3(79) 162
From Fig.14 one can see that radial wake force Fr
does not depend approximately on coordinate in re-
gions, occupied by bunches k>n=3, Fr ≈ const, i.e. the
lengthy bunches are focused approximately identically.
Fig.16. The distribution of plasma electron density ne
(grey) in wakefield
Fig.17. The density of bunches nb (blue), their averaged
radius <rb> (red) and radial wake force Fr (black)
It is for ideal case, when bunches are represented by
cylinders of constant density in longitudinal direction,
i.e. nb=const. The bunch electrons are distributed in
radial direction according to Gauss distribution. How-
ever the bunch density nb distribution is inhomogeneous
in longitudinal direction. Because Gauss distribution
and cosine distribution are close, we consider the cosine
electron density distribution along bunch (Figs.15-17).
From Fig.15 one can see that in this case of inhomoge-
neous longitudinal distribution of electron bunch den-
sity the 1-st front of 2-nd bunch is accelerated and back
front is decelerated. I.e. the 2-nd bunch is in Ez ≠ 0.
Fig.18. The density nb (blue) of inhomogeneous needle
bunches, their averaged radius <rb> (red), longitudinal
wakefield Ez (black) and radial wake force Fr (green)
The distribution of radial wake force Fr in the case
of inhomogeneous nb distribution along needle bunch is
shown in Fig.18. From Fig.17 and Fig.18 one can see
that the middle of bunch is focused more slower than
fronts.
Fig.19. Plasma electron density ne (black) in wakefield
and density of bunches nb (blue)
Fig.20. The density of bunches nb (dark blue),
their averaged radius <rb> (red) and radial wake force
Fr (black)
If all bunches are identical and they are distributed
through 1.5λ, then all bunches are focused identically, but
inhomogeneously along each bunch (Figs.19, 20).
One can show that the field distribution for positron
bunches are identical to electron bunches and they are
focused identically to electron bunches.
CONCLUSIONS
It has been shown that all bunches of sequence can
be focused identically and uniformly. For this it is nec-
essary that bunches have lengths, equal λ/2, the charge
of 1-st bunch equals half of the charges of the other
bunches, the distance between them equals 1,5 λ. It is
shown that only 1-st bunch is in finite Ez ≠ 0. Other
bunches are in zero longitudinal electrical wakefield
Ez = 0. Hence the 1-st bunch interchange by energy with
wakefield. The subsequent bunches do not interchange
by energy with wakefield and the amplitude of wake-
field does not change along sequence. Radial wake
force Fr in regions, occupied by bunches, is approxi-
mately constant along bunches.
In the case of inhomogeneous longitudinal distribu-
tion of electron bunch density the middle of bunches are
focused more slower than fronts.
ISSN 1562-6016. ВАНТ. 2012. №3(79) 163
If all bunches are identical and they are distributed
through 1,5λ, then all bunches are focused identically,
but inhomogeneously along each bunch.
REFERENCES
1. Ya. Fainberg, M. Ayzatskiy, V. Balakirev, et al.
Focusing of Relativistic Electron Bunches at the
Wakefield Excitation in Plasma // Proceedings
PAC’97. 1997, v.II, р.651-653.
2. H. Suky, et al. // Proc. of PAC. New York, 1999.
3. K.V. Lotov. Simulation of ultrarelativistic beam
dynamics in plasma wake-field accelerator // Phys.
Plasmas. 1998, v.5, №3, р.785-791.
Статья поступила в редакцию 07.11.2011 г.
ОДНОРОДНАЯ ФОКУСИРОВКА ПОСЛЕДОВАТЕЛЬНОСТИ ЭЛЕКТРОННЫХ СГУСТКОВ
КИЛЬВАТЕРНЫМ ПОЛЕМ В ПЛАЗМЕ
К.В. Лотов, В.И. Маслов, И.Н. Онищенко, Е.Н. Свистун
Предложен и исследован механизм фокусировки релятивистских электронных сгустков плазменными
кильватерными полями, в которых все сгустки последовательности фокусируются одинаково и однородно.
В этой схеме фокусировки необходимо, чтобы длина каждого сгустка была равна половине длины кильва-
терной волны, заряд первого сгустка был в 2 раза меньше заряда каждого из остальных сгустков, а расстоя-
ние между ними составляло 1,5 длины волны. Показано, что в этом случае только на 1-й сгусток действует
продольное поле Ez. Остальные сгустки находятся в нулевом продольном кильватерном поле Ez = 0. Фоку-
сирующее радиальное поле в областях, занятых сгустками, постоянно по длине сгустков.
ОДНОРІДНЕ ФОКУСУВАННЯ ПОСЛІДОВНОСТІ ЕЛЕКТРОННИХ ЗГУСТКІВ КІЛЬВАТЕРНИМ
ПОЛЕМ У ПЛАЗМІ
К.В. Лотов, В.І. Маслов, І.М. Онiщенко, О.М. Свистун
Запропонований і досліджений механізм фокусування релятивістських електронних згустків плазмовими
кільватерними полями, у яких усі згустки послідовності фокусуються однаково та однорідно. В цій схемі
фокусування необхідно, щоб довжина кожного згустка дорівнювала половині довжини кільватерної хвилі,
заряд першого згустку був у 2 рази менше заряда кожного із решти згустків, а відстань між ними була рів-
ною 1,5 довжини хвилі. Показано, що в цьому випадку лише на 1-й згусток діє повздовжнє електричне поле
Ez. Останні згустки знаходяться у нульовому повздовжньому кільватерному полі Ez = 0. Фокусуюче радіаль-
не електричне поле в зонах, що заняті згустками, постійне по довжині згустків.
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