Plasma wakefield excitation providing homogeneous focusing of electron bunches
Wakefield plasma lens, which focuses all relativistic electron bunches of the sequence identically and uniformly, are investigated analytically and by numerical simulation. The necessary conditions of such lens operation are the followings: length of the bunches is ξb=q(λ/2), q=1, 2, …(λ is the plas...
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| Zitieren: | Plasma wakefield excitation providing homogeneous focusing of electron bunches / V.I. Maslov, I.N. Onishchenko, I.P. Yarovaya // Вопросы атомной науки и техники. — 2013. — № 1. — С. 134-136. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1092732016-11-23T03:02:43Z Plasma wakefield excitation providing homogeneous focusing of electron bunches Maslov, V.I. Onishchenko, I.N. Yarovaya, I.P. Плазменная электроника Wakefield plasma lens, which focuses all relativistic electron bunches of the sequence identically and uniformly, are investigated analytically and by numerical simulation. The necessary conditions of such lens operation are the followings: length of the bunches is ξb=q(λ/2), q=1, 2, …(λ is the plasma wavelength), the distance between bunches is Δξ=pλ, p=1, 2, … All bunches have the same charge and the 1-st bunch has a half of this charge. It is shown that with the exception of 1-st bunch all other bunches are occurred in zero longitudinal wakefield and in uniform along bunch length focusing radial wakefield. In the case of inhomogeneous longitudinal distribution of electron bunch density the middle of bunches are focused slower in comparison with their edges. Аналитически и численным моделированием исследуется кильватерная плазменная линза, в которой все релятивистские электронные сгустки последовательности фокусируются одинаково и однородно. При этом необходимо, чтобы сгустки имели длины, равные ξb=q(λ/2), q=1, 2, …, скважность между сгустками была равна Δξ=pλ, p=1, 2,…, заряд 1-го сгустка был в два раза меньше зарядов всех остальных сгустков. Показано, что только 1-й сгусток находится в конечном продольном электрическом кильватерном поле Ez≠0. Другие сгустки находятся в Ez=0. Радиальная кильватерная сила Fr в областях расположения сгустков постоянна вдоль сгустков. В случае неоднородного продольного распределения плотности электронных сгустков их середины фокусируются медленнее, чем фронты. Аналітично і чисельним моделюванням досліджується кільватерна плазмова лінза, в якій усі релятивістські електронні згустки послідовності фокусуються однаково і однорідно. При цьому необхідно, щоб згустки мали довжини, рівні ξb=q(λ/2), q=1, 2, ..., шпаруватість між згустками була рівною Δξ=pλ, p=1, 2,…, заряд 1-го згустка був в два рази менше зарядів усіх інших згустків. Показано, що тільки 1-й згусток знаходиться в кінцевому поздовжньому кільватерному електричному полі Ez≠0. Інші згустки знаходяться в Ez=0. Радіальна кільватерна сила Fr в областях розташування згустків постійна уздовж згустків. У разі неоднорідного поздовжнього розподілу щільності електронних згустків їх середини фокусуються повільніше, ніж фронти. 2013 2013 Article Plasma wakefield excitation providing homogeneous focusing of electron bunches / V.I. Maslov, I.N. Onishchenko, I.P. Yarovaya // Вопросы атомной науки и техники. — 2013. — № 1. — С. 134-136. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 29.17.+w; 41.75.Lx; http://dspace.nbuv.gov.ua/handle/123456789/109273 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Плазменная электроника Плазменная электроника |
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Плазменная электроника Плазменная электроника Maslov, V.I. Onishchenko, I.N. Yarovaya, I.P. Plasma wakefield excitation providing homogeneous focusing of electron bunches Вопросы атомной науки и техники |
| description |
Wakefield plasma lens, which focuses all relativistic electron bunches of the sequence identically and uniformly, are investigated analytically and by numerical simulation. The necessary conditions of such lens operation are the followings: length of the bunches is ξb=q(λ/2), q=1, 2, …(λ is the plasma wavelength), the distance between bunches is Δξ=pλ, p=1, 2, … All bunches have the same charge and the 1-st bunch has a half of this charge. It is shown that with the exception of 1-st bunch all other bunches are occurred in zero longitudinal wakefield and in uniform along bunch length focusing radial wakefield. In the case of inhomogeneous longitudinal distribution of electron bunch density the middle of bunches are focused slower in comparison with their edges. |
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Article |
| author |
Maslov, V.I. Onishchenko, I.N. Yarovaya, I.P. |
| author_facet |
Maslov, V.I. Onishchenko, I.N. Yarovaya, I.P. |
| author_sort |
Maslov, V.I. |
| title |
Plasma wakefield excitation providing homogeneous focusing of electron bunches |
| title_short |
Plasma wakefield excitation providing homogeneous focusing of electron bunches |
| title_full |
Plasma wakefield excitation providing homogeneous focusing of electron bunches |
| title_fullStr |
Plasma wakefield excitation providing homogeneous focusing of electron bunches |
| title_full_unstemmed |
Plasma wakefield excitation providing homogeneous focusing of electron bunches |
| title_sort |
plasma wakefield excitation providing homogeneous focusing of electron bunches |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2013 |
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Плазменная электроника |
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http://dspace.nbuv.gov.ua/handle/123456789/109273 |
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Plasma wakefield excitation providing homogeneous focusing of electron bunches / V.I. Maslov, I.N. Onishchenko, I.P. Yarovaya // Вопросы атомной науки и техники. — 2013. — № 1. — С. 134-136. — Бібліогр.: 4 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-07-07T22:50:37Z |
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| fulltext |
134 ISSN 1562-6016. ВАНТ. 2013. №1(83)
PLASMA WAKEFIELD EXCITATION PROVIDING HOMOGENEOUS
FOCUSING OF ELECTRON BUNCHES
V.I. Maslov1, I.N. Onishchenko1, I.P. Yarovaya2
1NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2V.N. Karazin Kharkov National University, Kharkov, Ukraine
Wakefield plasma lens, which focuses all relativistic electron bunches of the sequence identically and uniformly,
are investigated analytically and by numerical simulation. The necessary conditions of such lens operation are the
followings: length of the bunches is ξb=q(λ/2), q=1, 2, …(λ is the plasma wavelength), the distance between
bunches is ∆ξ=pλ, p=1, 2, … All bunches have the same charge and the 1-st bunch has a half of this charge. It is
shown that with the exception of 1-st bunch all other bunches are occurred in zero longitudinal wakefield and in
uniform along bunch length focusing radial wakefield. In the case of inhomogeneous longitudinal distribution of
electron bunch density the middle of bunches are focused slower in comparison with their edges.
PACS: 29.17.+w; 41.75.Lx;
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, is
inhomogeneous. In [3] the mechanism of focusing by
plasma wakefield, in which all bunches of sequence are
focused identically and uniformly, has been proposed
and numerically investigated. We analytically and
numerically investigate by 2.5D code lcode [4] the
longitudinal distribution of radial wake-force, excited
by sequence of lengthy electron bunches in
homogeneous plasma.
1. RESULTS OF SIMULATION
At first the wakefield excitation by resonant
sequence of bunches is considered. We find the
longitudinal Ez, radial Er fields and focusing force Fr in
the areas of location of bunches. For this purpose we
use the theory developed in [1]. For the bunch of
permanent density, the length of which equals ξb=λ/2,
we get that Ez and Er are proportional to ZII
(λ/2)(ξ) and
Z⊥
(λ/2)(ξ)
ZII
(λ/2)(ξ)=(2/k)sin(kξ), Z⊥
(λ/2)(ξ)=-(2/k)cos(kξ). (1)
Ez in the middle of 1st bunch equals
ZII.1
(λ/2)=(1/k)∫0
π/2 dx0 cos(kξ-x0)|kξ=π/2=(1/k). (2)
One can see it is, as well as observed, in 2 times less
than amplitude of the wakefield after 1st bunch.
Now we derive the fields into the 2nd resonant
bunch,
ZII.2
(ξ)(ξ)=(3/k)sin(kξ), Z⊥.2
(ξ)(ξ)=(2/k)[1-2cos(kξ)]. (3)
ZII.2
(ξ)(kξ) changes from ZII.2
(ξ)(x=2π)=0 to
ZII.2
(ξ)(kξ=2.5π)=ZII.2
(max)=(3/k) and then again
ZII.2
(ξ)(kξ=3π)=0. Thus Z⊥.2
(ξ)(kξ) changes from
Z⊥.2
(ξ)(x=2π)=-(2/k) to Z⊥.2
(ξ)(x=3π)=(6/k), reaching
zero in the 1st half of bunch, where cos(xa)=1/2,
xa=2π+π/3<2π+π/2. I.e. longer (in (π-xa)/xa=2 times)
part (back front) of bunch focuses in larger field Er than
1st front (more short) of bunch defocuses (in 3 times
less field Er). In resonant case 1st bunch only focuses,
and for other bunches (if length of bunch equals ξb=λ/2)
back fronts (more long) are in larger focusing field Er,
than (more short) 1st fronts are in defocusing Er. I.e. this
lens is inhomogeneous.
In [3] wakefield plasma lens has been numerically
simulated with homogeneous focusing force for the
sequence of bunches, lengths of which equal ξb=λ/2,
with the 1st bunch, charge of which in two times less
than charges of other bunches Q1=Qi/2, i=2, 3, …,
distance between bunches equals 1.5λ. From Fig. 1 one
can see that in this case dips of electron plasma density,
in which bunches are localized, wide and shallow, and
humps are narrow and high. From Fig. 1 one can see
that Ez=0 in the areas of location of bunches.
Fig. 1. Longitudinal distribution of density nb of
sequence of bunches (yellow), longitudinal wakefield Ez
(red), δne (grey) and coupling factor <Ez> of bunches
with Ez (black), excited by sequence of 10 bunches. t is
normalized on ωpe
From Fig. 2 one can see that in the areas of bunches
location Fr does not approximately depend on a
longitudinal coordinate.
Fig. 2. nb (yellow), radiuses rb (blue), Ez (red), wake
radial force Fr (dark blue), magnetic-field Hθ (pale
blue)
In the areas of location of bunches the radial field of
their volume charge is compensated by radial field,
appearing as a result of shift of several number of
plasma electrons from the areas of bunches location.
Ez=0 in the areas of bunches location, except 1st one
(see Fig. 2).
Thus screening of bunches takes place due to that
wide (≈λ) dips of plasma electron density δne<0 and
narrow (≈λ/2) humps of δne>0 appear (see Fig. 1). I.e.
ISSN 1562-6016. ВАНТ. 2013. №1(83) 135
plasma in the vicinity of bunches is positively charged.
Due to inertness of electrons screening of 1st bunch is
realized only in its end, where dip of density δne<0 is
not flat, as for 2nd bunch, but approximately three-
cornered.
For the bunch of permanent density and ξb=λ/2
longitudinal ZII
(λ/2)(ξ) and transversal Z⊥
(λ/2)(ξ) fields are
equal (1). Now we derive the wakefield into 2nd bunch
ZII.2
(ξ)(ξ)=(2/k)sin(kξ)+2∫0
ξ dξ0cos[k(ξ-ξ0)+3π]=0 (4)
As well as in numeral simulation, Ez=0 got in the areas
of bunches location.
Fig. 3. nb (yellow), rb (blue), Ez (red), Fr (dark blue)
Now we consider wakefield plasma lenses for the
sequence of bunches for three cases of their lengths at
the interbunch gap equal to ∆ξ=λ. The bunch-precursor
of half-charge is used. We connect it with next (1st)
bunch. 1st case: ξb=λ (Fig. 3), 2nd case:
ξb=1.5λ (Figs. 4, 5), 3rd case: ξb=2λ (Figs. 6–8). From
Figs. 3–5, and 7, 8 one can see that in the areas of
bunches location Ez=0, Fr≈const.
Fig. 4. nb (yellow), Ez (red), <Ez> (black)
Fig. 5. nb (yellow), rb (blue), Ez (red), Fr (dark blue)
Fig. 6. Spatial distribution of δne in the wakefield
Fig. 7. nb (yellow), Ez (red), <Ez> (black)
Fig. 8. nb (yellow), rb (blue), Ez (red), Fr (dark blue)
Now we consider the wake plasma lens for the
bunches of identical charge with lengths equal to λ/2
and the interbunch gap equal to λ (Figs. 9–11).
Fig. 9. nb (yellow), Ez (red), <Ez>
(black)
Fig. 10. nb (yellow) и δne (grey)
Fig. 11. nb (yellow), rb (blue), Fr (dark blue)
If all bunches are identical and they are placed
through 1.5λ, all bunches are focused identically, but
inhomogeneously along bunches (see Figs. 10, 11). Odd
number bunches are decelerated, and the even number
bunches are accelerated (see Fig. 9).
We consider the wake plasma lens for the bunches of
identical charges with lengths and interbunch gap equal
to λ (Figs. 12, 13). Both longitudinal and radial fields
between bunches are equal to zero, and the fields in
every bunch are equal to
ZII1
(ξ)(ξ)=(1/k)sin(kξ), Z⊥1
(ξ)(ξ)=(2/k)sin2(kξ/2). (5)
The radial field has one sign along bunch. Therefore
bunches are focused, however on fronts of bunches
Fr=0. The longitudinal field is oscillated between E0 in
the forehand of bunch and -E0 in the back-end of bunch.
Therefore the front half of bunch is decelerated, and the
back is accelerated.
Fig. 12. nb (yellow), Ez (red), <Ez> (black)
Fig. 13. nb (yellow), rb (blue), Ez (red), Fr (dark blue)
The results of numeral simulation for the case of
length of bunches, equal to 2λ, are presented in Fig. 14.
We consider the case of electron density distribution
along bunch according to sin2(kξ) (Figs. 15, 16).
Bunches are placed through 1.5λ. The charge of 1st
bunch is less than charges of other bunches in 2 times.
From Fig. 16 one can see that 1st front of bunch is
accelerated, and the back is decelerated.
Fig. 14. nb (yellow), rb (blue), Ez (red), Fr (dark blue)
136 ISSN 1562-6016. ВАНТ. 2013. №1(83)
Fig. 15. Er (red), Fr (dark blue), Hθ (pale blue), excited
by sequence of 10 inhomogeneous bunches
From Fig. 15 one can see that fronts of bunches are
focused stronger than their centers.
If the electrons of lengthy (ξb=λ/2) bunch are
distributed according to sin2(kξ), 0<ξ0<λ/2, then after
1st bunch we get
ZII
(sin2)(ξ)=(4/3k)sin(kξ). (6)
Fig. 16. nb of 10 inhomogeneous bunches (yellow), ne
(grey), Ez (red)
Now we derive the radial field
Z⊥
(sin2)(ξ)=(4/3k)cos(kξ) (7)
We obtain the field in the center of 1st bunch
ZII.1
(sin2)= ∫0
λ/4 dξ0cos[k(ξ-ξ0)]sin2(kξ0)|kξ=π/2=(2/3k). (8)
One can see that, as well as observed, the field in the
center of 1st bunch is in 2 times less than amplitude of
the wake field after the 1st bunch.
For the fields into the 2nd bunch we derive
ZII.2
(sin2)(ξ)=(2/3k)sin(2kξ). (9)
As well as observed, into the 2nd bunch period is in 2
times shorter and amplitude is in 2 times less than after
the 1st bunch.
Z⊥.2
(sin2)(ξ)=-(4/3k)+(2/3k)sin2(kξ). (10)
At the change into the 2nd bunch 3π<kξ<4π on the
edges (at kξ=3π and kξ=4π) of 2nd bunch focusing is
the strongest (the force is equal to -(4/3k)) and in the
middle of the 2nd bunch (at x=3π+π/2), focusing is the
weakest (force is in 2 times less -(2/3k)).
CONCLUSIONS
It has been shown that all bunches of the sequence
can be focused identically and uniformly under the
conditions: all bunches lengths are equal to qλ/2, q=1,
2, 3, .., the distance between them equals pλ, p=1, 2, 3,
.., the charge of 1-st bunch equals a half of the charges
of the other bunches. It has been 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 loss the energy for excitation of wakefield, which
amplitude is constant along the sequence. Radial wake
force Fr in regions, occupied by bunches, is constant
along bunches length.
In the case of inhomogeneous longitudinal
distribution of electron bunch density the middle of
bunches are focused slower than edges.
If all bunches are identical and they are placed over
1.5λ, then bunches are focused identically, but not
uniformly along bunch length.
REFERENCES
1. Ya. Fainberg, M. Ayzatskiy, V. Balakirev, et al.
Focusing of Relativistic Electron Bunches at the
Wakefield Excitation in Plasma // Proceedings PAC’97.
12-16 May, 1997 Vancouver, Canada, v. II, p. 651-653.
2. G. Hairapetian, P. Davis, C.E. Clyaton, C. Joshi,
C. Pellegrini. Transverse dynamics of a short,
relativistic electron bunch in a plasma lens // Phys.
Plasmas. 1995, v. 2, p. 2555-2561.
3. K. V.Lotov, V.I. Maslov, I.N. Onishchenko,
O.M. Svistun. Homogeneous focusing of electron
bunch sequence by plasma wakefield // Problems of
Atomic Science and Technology. 2012, №3 (79), p. 159-
163.
4. K.V. Lotov. Simulation of ultrarelativistic beam
dynamics in plasma wake-field accelerator // Phys.
Plasmas. 1998, v. 5, № 3, p. 785-791.
Article received 20.09.12
ВОЗБУЖДЕНИЕ КИЛЬВАТЕРНОГО ПОЛЯ В ПЛАЗМЕ, ОБЛАДАЮЩЕГО ОДНОРОДНОЙ
ФОКУСИРОВКОЙ ЭЛЕКТРОННЫХ СГУСТКОВ
В.И. Маслов, И.Н. Онищенко, И.П. Яровая
Аналитически и численным моделированием исследуется кильватерная плазменная линза, в которой все релятивистские
электронные сгустки последовательности фокусируются одинаково и однородно. При этом необходимо, чтобы сгустки
имели длины, равные ξb=q(λ/2), q=1, 2, …, скважность между сгустками была равна ∆ξ=pλ, p=1, 2,…, заряд 1-го сгустка был
в два раза меньше зарядов всех остальных сгустков. Показано, что только 1-й сгусток находится в конечном продольном
электрическом кильватерном поле Ez≠0. Другие сгустки находятся в Ez=0. Радиальная кильватерная сила Fr в областях
расположения сгустков постоянна вдоль сгустков. В случае неоднородного продольного распределения плотности
электронных сгустков их середины фокусируются медленнее, чем фронты.
ЗБУДЖЕННЯ КІЛЬВАТЕРНОГО ПОЛЯ В ПЛАЗМІ, ЯКЕ ОДНОРІДНО ФОКУСУЄ ЕЛЕКТРОННІ ЗГУСТКИ
В.І. Маслов, І.М. Онищенко, І.П. Ярова
Аналітично і чисельним моделюванням досліджується кільватерна плазмова лінза, в якій усі релятивістські електронні
згустки послідовності фокусуються однаково і однорідно. При цьому необхідно, щоб згустки мали довжини, рівні ξb=q(λ/2),
q=1, 2, ..., шпаруватість між згустками була рівною ∆ξ=pλ, p=1, 2,…, заряд 1-го згустка був в два рази менше зарядів усіх
інших згустків. Показано, що тільки 1-й згусток знаходиться в кінцевому поздовжньому кільватерному електричному полі
Ez≠0. Інші згустки знаходяться в Ez=0. Радіальна кільватерна сила Fr в областях розташування згустків постійна уздовж
згустків. У разі неоднорідного поздовжнього розподілу щільності електронних згустків їх середини фокусуються повільніше,
ніж фронти.
|