Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma
Wakefield excitation by long sequence of short Gaussian bunches of relativistic electrons and electron bunch acceleration in excited wakefield is numerically simulated for the parameters of the experiments. It is shown that at change of the system parameters and shaping laws of sequence of bunches o...
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irk-123456789-1123612017-01-21T03:02:42Z Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma Levchuk, I.P. Maslov, V.I. Onishchenko, I.N. Новые и нестандартные ускорительные технологии Wakefield excitation by long sequence of short Gaussian bunches of relativistic electrons and electron bunch acceleration in excited wakefield is numerically simulated for the parameters of the experiments. It is shown that at change of the system parameters and shaping laws of sequence of bunches of relativistic electrons in the intervals of finite width the transformation ratio remains large. Числовим моделюванням досліджується для параметрів експериментів збудження кільватерного поля до-вгою послідовністю коротких гауссівських згустків релятивістських електронів і прискорення електронних згустків з малим зарядом у збудженому полі. Показано, що при зміні параметрів системи і законів профілювання послідовності згустків релятивістських електронів в інтервалах кінцевої ширини коефіцієнт трансформації залишається значним. Численным моделированием исследуется для параметров экспериментов возбуждение кильватерного по-ля длинной последовательностью коротких гауссовских сгустков релятивистских электронов и ускорение электронных сгустков с малым зарядом в возбужденном поле. Показано, что при изменении параметров системы и законов профилирования последовательности сгустков релятивистских электронов в интервалах конечной ширины коэффициент трансформации остается большим. 2015 Article Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma / I.P. Levchuk, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2015. — № 6. — С. 37-41. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 29.17.+w; 41.75.Lx; http://dspace.nbuv.gov.ua/handle/123456789/112361 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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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 Levchuk, I.P. Maslov, V.I. Onishchenko, I.N. Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma Вопросы атомной науки и техники |
| description |
Wakefield excitation by long sequence of short Gaussian bunches of relativistic electrons and electron bunch acceleration in excited wakefield is numerically simulated for the parameters of the experiments. It is shown that at change of the system parameters and shaping laws of sequence of bunches of relativistic electrons in the intervals of finite width the transformation ratio remains large. |
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Levchuk, I.P. Maslov, V.I. Onishchenko, I.N. |
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Levchuk, I.P. Maslov, V.I. Onishchenko, I.N. |
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Levchuk, I.P. |
| title |
Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma |
| title_short |
Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma |
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Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma |
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Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma |
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Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma |
| title_sort |
transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Новые и нестандартные ускорительные технологии |
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| citation_txt |
Transformation ratio at wakefield excitation by linearly shaped sequence of short relativistic electron bunches in plasma / I.P. Levchuk, V.I. Maslov, I.N. Onishchenko // Вопросы атомной науки и техники. — 2015. — № 6. — С. 37-41. — Бібліогр.: 8 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T03:48:16Z |
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2025-07-08T03:48:16Z |
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1837049051014496256 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №6(100) 37
TRANSFORMATION RATIO AT WAKEFIELD EXCITATION
BY LINEARLY SHAPED SEQUENCE OF SHORT RELATIVISTIC
ELECTRON BUNCHES IN PLASMA
I.P. Levchuk, V.I. Maslov, I.N. Onishchenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vmaslov@kipt.kharkov.ua
Wakefield excitation by long sequence of short Gaussian bunches of relativistic electrons and electron bunch ac-
celeration in excited wakefield is numerically simulated for the parameters of the experiments. It is shown that at
change of the system parameters and shaping laws of sequence of bunches of relativistic electrons in the intervals of
finite width the transformation ratio remains large.
PACS: 29.17.+w; 41.75.Lx;
INTRODUCTION
Transformation ratio is the important value in the
wakefield method of charged particle acceleration (see,
for example, [1 - 5]). It determines to what energy the
electrons can be accelerated by sequence of electron
bunches with fixed energy. The transformation ratio,
determined as the ratio TE=E2/E1 of wakefield E2, which
is excited in plasma by sequence of electron bunches to
the field E1, in which the electron bunch is decelerated,
is considered. The excitation of wakefield by long se-
quence of short Gaussian bunches of relativistic elec-
trons is investigated by numerical simulation, using the
code lcode [6], for the parameters of the experiments
[7, 8]. The distance between the bunches, equal to
wavelength plus the bunch width at half-maximum, and
the distance between bunches, equal to one and a half of
the wavelength, are selected. It is shown that not only in
these cases, but also at varying of the parameters within
a certain range the transformation ratio increases with
the number of bunches.
1. ANALYTICAL INVESTIGATION
OF WAKEFIELD EXCITATION IN PLASMA
WITH LARGE TRANSFORMATION RATIO
BY SEQUENCE OF BUNCHES WITH
LINEAR GROWTH OF CHARGE
First we analytically compare the wakefield, excited
by resonance (ωm=ωpe, ωm is the repetition frequency of
bunches, ωpe is the electron plasma frequency) sequence
of rectangular (uniform) bunches with the ratio of the
successive charges, equal to 1:3:5: … with 1.5ωm=ωpe.
At using of shaped bunches through 1.5λ the decel-
erating field is equal to Esl=E1/2, and the accelerating
field is equal to Eac=NE1. N is the number of bunches.
I.e. after 2nd bunch Eac=2E1, and after 3rd bunch
Eac=3E1.
In the case of resonant shaped bunches through λ
the decelerating wakefield is approximately equal to the
accelerating wakefield and accelerating wakefield after
2nd bunch is equal to Eac=4E1, after 3rd bunch it is
equal to Eac=9E1, i.e. it should be Eac=N2E1. Indeed
WN-WN-1 = ηεN,
εN = 2πenbcENc/ωp, ENc = EN-1+βδEN β≈1/2.
Then after 1st bunch one can derive
ε1 = πenb1cδE1/ωp, W1 = ηε1, δE1 = E1, W1 = E1
2/4π
E1
2/4π = ηπenb1cE1/ωp, E1 = (2π)2ηenb1c/ωp.
After 2nd bunch one can derive
W2-W1 = ηε2, ε2 = 2πe3nb1cENc/ωp, ENc = E1+δE2/2.
(E1+δE2)2/4π-E1
2/4π =
= δE2(E1+δE2/2)/2π = η6πenb1c(E1+δE2/2)/ωp
δE2 = 3η(2π)2enb1c/ωp = 3E1.
Hence after 2nd bunch
E2 = E1+δE2 = 4E1.
After N-th bunch
WN-WN-1 = ηεN, εN = 2πenbcENc/ωp, ENc = EN-1+δEN/2.
EN
2/4π-EN-1
2/4π = (EN-1+δEN)2/4π-EN-1
2/4π =
= δEN(E1+δEN/2)/2π = η2πe(2N-1)nb1c(EN-1+δEN/2)/ωp,
δEN = η(2π)2e(2N-1)nb1c/ωp = (2N-1)E1.
Then
E3 = E2+δE3 = 4E1+5E1 = 9E1.
EN = EN-1+δEN∼1+3+…+(2N-1)
1+3+...+(2N-1) = 1+2+...+2N-2(1+2+...+N).
Because
1+2+...+N = N(N+1)/2,
we have
1+3+…+(2N-1) = 2N(2N+1)/2-2N(N+1)/2 = N2.
EN = N2E1.
Then after 300 bunches the accelerating fields in
resonant shaped case through λ EN
(res) and in shaped
case through 1.5λ EN
(sh) are different in
EN
(res)/EN
(sh)=N2E1/NE1=N=300
times.
In the case of shaped bunches through 1.5λ the de-
celerating wakefield is in 300 times smaller than the
accelerating wakefield and in (300)2≈105 times smaller
than the wakefield in resonant case.
2. NUMERICAL SIMULATION
OF WAKEFIELD EXCITATION IN PLASMA
WITH LARGE TRANSFORMATION RATIO
BY SEQUENCE OF BUNCHES WITH
LINEAR GROWTH OF CHARGE
Numerical simulation has been performed using 2d3v-
code lcode [6]. Parameters: nres=1011cm-3 is the resonant
plasma density which corresponds to ratio
ωpe=ωm=2π·2.8·109, relativistic factor of bunches equals
γb=5, have been selected for numerical simulation. ωm is
the repetition frequency of bunches, ωpe= (4πnrese2/me)1/2
is the electron plasma frequency. The density of bunches
8 3
bn 6 10 ñm−= × is distributed in the transverse direction
approximately according to Gaussian distribution,
ISSN 1562-6016. ВАНТ. 2015. №6(100) 38
r 0.5cmσ = , 10.55cmλ = is the wavelength, ξ=Vbt-z, Vb is
the velocity of bunches. Time is normalized on ωpe
-1, dis-
tance – on c/ωpe, density − on nres, current Ib – on
Icr=πmc3/4e, fields – on (4πnresc2me)1/2.
We show that in the case of often used shaped
bunches-uniform-cylinders with the ratio of charges,
equal to 1:3:5:... and with the distance between bunches,
equal to one and half of the wavelength δξ=1.5λ, with a
width of bunches, equal to ξb=λ/2, at of plasma wake-
field excitation the problem of the formation of the ac-
celerated bunch is solved easily. In this case, the last
bunch, if its charge is small in comparison with the
charge of previous bunch, becomes accelerated bunch
(Figs. 1-3).
Fig. 1. Current of bunches with ξb=λ/2, δξ=1.5λ,
q1:q2:q3: … =1:3:5: … at linear growth and sharp
decrease at back front
Fig. 2. Longitudinal distribution of density nb of sequence
of bunches and of longitudinal wakefield Ez in the case
of sequence of bunches with the ratio of the successive
charges, equal to 1:3:5:…, at ξb=λ/2, Ib=0.2·10-3
The wakefield excitation by sequence of N bunches
with the ratio of the charges of the successive bunches,
equal to 1:3:5: ..., the distance between bunches, equal to
the sum of the excited wavelength λ and bunch width at
half maximum ξb is considered. One can see (Fig. 4), that
bunches on part of the sequence, on which the charges of
bunches increases, get into small decelerating wakefield.
Thus the large transformation ratio achieves, which can
be determined as the ratio of the maximum accelerating
wakefield Ema to the maximum decelerating wakefield
Emd in the area of bunch TR=Ema/Emd. Also one can see
that on the back front of the sequence, where the charges
of bunches decrease, the bunches automatically get into
large accelerating wakefield.
Fig. 3. Perturbation of longitudinal momenta δpz
of bunches
When bunches are formed due to trapping of elec-
trons with linearly increasing charge density by electric
field on each successive identical time interval and due
to their bunching and when on half periods of time elec-
trons are lost, then the ratio of charges of consecutive
bunches is equal to 1:5:9: ... If the distance between the
bunches is equal to the sum of the excited wavelength
and of the width of the bunch at half-maximum, then
(Fig. 5) the amplitude of the beatings of decelerating
wakefield, in which bunches get, is more. However, the
transformation ratio remains large. Again at the back front
of the sequence, in which the charges of the bunches de-
crease, bunches get into large accelerating field.
In the case of a sequence of 45 bunches with the ra-
tio of charges of the successive bunches, equal 1:5:9: ...,
excited wakefield has the shape, shown in Fig. 6. In the
case of a sequence of 100 bunches with the ratio of
charges of the successive bunches, equal 1:3:5: ..., excit-
ed wakefield has the shape, shown in Fig. 7.
If the ratio of the charges of successive bunches is
equal to 1:2:3: ... [2] and the distance between bunches
is equal to the sum of excited wavelength and of width
of the bunch at half-maximum, and the charge of bunch-
es distributed in the longitudinal direction according to
the Gaussian distribution, the excited wakefield has the
shape, shown in Fig. 8. One can see that at the back
front of the sequence, in which the charges of the
bunches decrease, bunches get automatically into large
accelerating wakefield.
If the distance between the bunches, the charge of
which is distributed in the longitudinal direction accord-
ing to Gauss distribution, is equal to one and half of the
wavelength, the length of the bunch at the base equals to
the wavelength and the charge ratio of successive
bunches is equal to 1:3:5: ... the excited wakefield has
the shape, shown in Fig. 9. One can see that decelerating
wakefield is small and approximately the same for the
majority of the bunch electrons. The bunches are in fairly
homogeneous close focusing wakefields (see Fig. 9) due
to the fact that the bunches get into the dips of the plas-
ma electron density (Fig. 10).
If the distance between the bunches, the charge of
which is distributed in the longitudinal direction accord-
ing to Gauss distribution, is equal to two and half of the
wavelength, the length of the bunch at the base equals to
the wavelength and the charge ratio of successive
bunches is equal to 1:3:5: ... the excited wakefield has
the shape, shown in Fig. 11. One can see that decelerat-
ing wakefield is also small and approximately the same
for the majority of the bunch electrons.
In the case of linear shaping of charges of bunch se-
quence along the sequence as well as along each bunch
with use of bunch-precursor the large transformation
ratio is achieved (Fig. 12)
TR≈2πNξb/λ.
One can see that the wakefield amplitude and the
transformation ratio increase with increasing of number
of bunches, exciting wakefield.
Now we consider the range of change parameters in
which the transformation ratio remains large. Numerical
simulation shows that TR and the accelerating wakefield
are large in the case of parameter changes
2πV0/ωm=λ+ξb in range 2πV0/ωm=λ+ξb±ξb/2. However,
in these cases the decelerating wakefield and their spa-
tial distributions are different (see Figs. 4, 13, 14).
ISSN 1562-6016. ВАНТ. 2015. №6(100) 39
Fig. 4. Longitudinal distribution of density nb of sequence of bunches and of longitudinal wakefield Ez
in the case of sequence of bunches with the ratio of charges of the successive bunches, equal to 1:3:5: … with
the distance between the Gaussian bunches, equal to the sum of excited wavelength and of width of the bunch
at half-maximum at ξb=λ/6, Ib=0.7·10-3
Fig. 5. Longitudinal distribution of density nb of sequence of bunches and of longitudinal wakefield Ez
in the case of sequence of 17 bunches with the ratio of charges of the successive bunches, equal to 1:5:9: … with
the distance between the the Gaussian bunches, equal to the sum of excited wavelength and of width
of the bunch at half-maximum at ξb=λ/6, Ib=0.7·10-3
Fig. 6. Longitudinal distribution of density nb of sequence of bunches and of longitudinal wakefield Ez
in the case of sequence of 45 bunches with the ratio of charges of the successive bunches, equal to 1:5:9: … with
the distance between the Gaussian bunches, equal to the sum of excited wavelength and of width of the bunch
at half-maximum at ξb=λ/6, Ib=0.23·10-3
Fig. 7. Longitudinal distribution of density nb of sequence of the Gaussian bunches and of longitudinal wakefield
Ez in the case of sequence of 100 bunches with the ratio of charges of the successive bunches, equal to 1:3:5: …
at ξb=λ/6, Ib=0.18·10-3
ISSN 1562-6016. ВАНТ. 2015. №6(100) 40
Fig. 8. Longitudinal distribution of density nb of sequence of bunches and of longitudinal wakefield Ez in the case of se-
quence of bunches with the ratio of charges of the successive bunches, equal to 1:2:3: … with the distance between the
bunches, equal to the sum of excited wavelength and of width of the bunch at half-maximum at ξb=λ/6, Ib=0.7·10-3
Fig. 9. Longitudinal distribution of density nb of sequence of bunches, of longitudinal wakefield Ez and of radial
wake force Fr in the case of sequence of the Gaussian bunches with the ratio of charges of the successive bunch-
es, equal to 1:3:5: … with the distance between the bunches, equal to one and half of the wavelength,
the length of the bunch at the base equals to the wavelength at ξb=λ/2, Ib=0.7·10-3
Fig. 10. Longitudinal distribution of density nb of sequence of bunches and of density of plasma electrons ne in the case
of sequence of bunches with the ratio of charges of the successive bunches, equal to 1:3:5: … with the distance between
the bunches, equal to one and half of the wavelength, the length of the bunch at the base equals to the wavelength
Fig. 11. Longitudinal distribution of density nb of sequence of the Gaussian bunches and of longitudinal
wakefield Ez in the case of sequence of bunches with the ratio of charges of the successive bunches,
equal to 1:3:5: … with the distance between the bunches, equal to two and half of the wavelength, the length
of the bunch at the base equals to the wavelength at ξb=λ/2, Ib=0.7·10-3
Fig. 12. Longitudinal distribution of density nb of sequence of short bunches and of longitudinal wakefield Ez
in the case of linear shaping of charges of bunch sequence along the sequence as well as along each bunch
with use of bunch-precursor at ξb=λ/20, Ib=1.0·10-3
ISSN 1562-6016. ВАНТ. 2015. №6(100) 41
Fig. 13. Longitudinal distribution of density nb
of sequence of the Gaussian bunches and of longitudi-
nal wakefield Ez in the case of sequence of bunches with
the ratio of charges of the successive bunches, equal
to 1:3:5: … with the distance between the bunches,
equalλ+ξb+ξb/2 at ξb=λ/6, Ib=0.7·10-3
Fig. 14. Longitudinal distribution of density nb
of sequence of the Gaussian bunches and of longitudi-
nal wakefield Ez in the case of sequence of bunches with
the ratio of charges of the successive bunches, equal
to 1:3:5: … with the distance between the bunches,
equal λ+ξb-ξb/2 at ξb=λ/6, Ib=0.7·10-3
CONCLUSIONS
The transformation ratio has been investigated by the
numerical simulation at wakefield excitation in plasma
by sequence of Gaussian short relativistic electron
bunches with linearly increasing charges. It has been
shown that the transformation ratio increases with the
increasing of number of bunches when the distance be-
tween bunches is equal to the sum of excited wavelength
and of width of the bunch at half-maximum and when
the distance between the bunches is equal to the one and
half of the wavelength.
REFERENCES
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Article received 20.10.2015
КОЭФФИЦИЕНТ ТРАНСФОРМАЦИИ ПРИ ВОЗБУЖДЕНИИ КИЛЬВАТЕРНОГО ПОЛЯ
ЛИНЕЙНО ПРОФИЛИРОВАННОЙ ПОСЛЕДОВАТЕЛЬНОСТЬЮ КОРОТКИХ
РЕЛЯТИВИСТСКИХ ЭЛЕКТРОННЫХ СГУСТКОВ В ПЛАЗМЕ
И.П. Левчук, В.И. Маслов, И.Н. Онищенко
Численным моделированием исследуется для параметров экспериментов возбуждение кильватерного по-
ля длинной последовательностью коротких гауссовских сгустков релятивистских электронов и ускорение
электронных сгустков с малым зарядом в возбужденном поле. Показано, что при изменении параметров си-
стемы и законов профилирования последовательности сгустков релятивистских электронов в интервалах
конечной ширины коэффициент трансформации остается большим.
КОЕФІЦІЄНТ ТРАНСФОРМАЦІЇ ПРИ ЗБУДЖЕННІ КІЛЬВАТЕРНОГО ПОЛЯ
ЛІНІЙНО ПРОФІЛЬОВАНОЮ ПОСЛІДОВНІСТЮ КОРОТКИХ РЕЛЯТИВІСТСЬКИХ
ЕЛЕКТРОННИХ ЗГУСТКІВ У ПЛАЗМІ
І.П. Левчук, В.І. Маслов, І.М. Оніщенко
Числовим моделюванням досліджується для параметрів експериментів збудження кільватерного поля до-
вгою послідовністю коротких гауссівських згустків релятивістських електронів і прискорення електронних
згустків з малим зарядом у збудженому полі. Показано, що при зміні параметрів системи і законів профілю-
вання послідовності згустків релятивістських електронів в інтервалах кінцевої ширини коефіцієнт трансфор-
мації залишається значним.
INTRODUCTION
1. ANALYTICAL INVESTIGATION OF WAKEFIELD EXCITATION IN PLASMA WITH LARGE TRANSFORMATION RATIO BY SEQUENCE OF BUNCHES WITH LINEAR GROWTH OF CHARGE
2. NUMERICAL SIMULATION OF WAKEFIELD EXCITATION IN PLASMA WITH LARGE TRANSFORMATION RATIO BY SEQUENCE OF BUNCHES WITH LINEAR GROWTH OF CHARGE
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