Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator
The possibility of realization of resonator concept of dielectric wake field accelerator is studied. The requirements of implementation of this concept are obtained. Numerical simulations of wake field excitation in planar dielectric resonator by bunch sequence testify the obtained requirements ar...
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| Cite this: | Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator / N.I. Onishchenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2006. — № 2. — С. 73-75. — Бібліогр.: 5 назв. — англ. |
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Onishchenko, N.I. Sotnikov, G.V. 2015-03-20T20:08:02Z 2015-03-20T20:08:02Z 2006 Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator / N.I. Onishchenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2006. — № 2. — С. 73-75. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 41.60.-m https://nasplib.isofts.kiev.ua/handle/123456789/78770 The possibility of realization of resonator concept of dielectric wake field accelerator is studied. The requirements of implementation of this concept are obtained. Numerical simulations of wake field excitation in planar dielectric resonator by bunch sequence testify the obtained requirements are true. Исследована возможность реализации концепции ускорителя на кильватерных полях в диэлектрическом резонаторе. Аналитически определены условия когерентного сложения кильватерных полей последовательности электронных сгустков. Проведенное численное моделирование подтверждает условия когерентности. Досліджено можливість реалізації концепції прискорювача на кільватерних полях у діелектричному резонаторі. Аналітично визначені умови когерентного додавання кільватерних полів послідовності електронних згустків. Проведене чисельне моделювання підтверджує умови когерентності. This study was supported in parts by CRDF Grant # UP2-2569-KH-04 and Ukr. DFFD 02.07/325. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Линейные ускорители заряженных частиц Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator Когерентное сложение кильватерных полей последовательности электронных сгустков в плоском многомодовом диэлектрическом резонаторе Когерентне додавання кільватерних полів послідовності електронних згустків у плоскому многомодовому діелектричному резонаторі Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator |
| spellingShingle |
Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator Onishchenko, N.I. Sotnikov, G.V. Линейные ускорители заряженных частиц |
| title_short |
Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator |
| title_full |
Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator |
| title_fullStr |
Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator |
| title_full_unstemmed |
Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator |
| title_sort |
coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator |
| author |
Onishchenko, N.I. Sotnikov, G.V. |
| author_facet |
Onishchenko, N.I. Sotnikov, G.V. |
| topic |
Линейные ускорители заряженных частиц |
| topic_facet |
Линейные ускорители заряженных частиц |
| publishDate |
2006 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Когерентное сложение кильватерных полей последовательности электронных сгустков в плоском многомодовом диэлектрическом резонаторе Когерентне додавання кільватерних полів послідовності електронних згустків у плоскому многомодовому діелектричному резонаторі |
| description |
The possibility of realization of resonator concept of dielectric wake field accelerator is studied. The requirements
of implementation of this concept are obtained. Numerical simulations of wake field excitation in planar dielectric
resonator by bunch sequence testify the obtained requirements are true.
Исследована возможность реализации концепции ускорителя на кильватерных полях в диэлектрическом резонаторе. Аналитически определены условия когерентного сложения кильватерных полей последовательности электронных сгустков. Проведенное численное моделирование подтверждает условия когерентности.
Досліджено можливість реалізації концепції прискорювача на кільватерних полях у діелектричному
резонаторі. Аналітично визначені умови когерентного додавання кільватерних полів послідовності
електронних згустків. Проведене чисельне моделювання підтверджує умови когерентності.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/78770 |
| citation_txt |
Coherent summation of wake fields excited by an electron bunch sequence in a planar multimode dielectric resonator / N.I. Onishchenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2006. — № 2. — С. 73-75. — Бібліогр.: 5 назв. — англ. |
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| first_indexed |
2025-11-24T18:04:27Z |
| last_indexed |
2025-11-24T18:04:27Z |
| _version_ |
1850491276294619136 |
| fulltext |
COHERENT SUMMATION OF WAKE FIELDS EXCITED BY
AN ELECTRON BUNCH SEQUENCE IN A PLANAR MULTIMODE
DIELECTRIC RESONATOR
N.I. Onishchenko, G.V. Sotnikov
NSC KIPT, Kharkov, Ukraine
E-mail: sotnikov@kipt.kharkov.ua
The possibility of realization of resonator concept of dielectric wake field accelerator is studied. The require-
ments of implementation of this concept are obtained. Numerical simulations of wake field excitation in planar di-
electric resonator by bunch sequence testify the obtained requirements are true.
PACS: 41.60.-m
1. INTRODUCTION
Historically, the most of studies on the dielectric
waveguides used both for Cherenkov radiation sources
and for the dielectric wake field accelerators (DWFA) is
carried out for cylindrical structures. Intensity of electric
fields in DWFA can be considerably increased due to
coherent summation of fields of many traversal harmon-
ics of a field [1]. For this purpose it is necessary to pro-
vide the equal spacing of resonant eigen-frequencies of
the structure. In rectangular structures [2] such mode to
realize is essentially easier. The equal spacing of reso-
nant frequencies is especially important at summation of
fields from regular sequence of bunches.
In the dielectric waveguide excited wake field is re-
moved from structure with group velocity. In finite
length waveguide on building up of the wake field max-
imum the part of bunches of sequence works only [3].
The using of resonator eliminates effect of removing of
wake field [4,5]. Under optimum conditions, excitation
of dielectric resonator by a sequence of bunch is similar
to the excitation of an optical resonator by a mode-
locked laser equipped with an “optical switch”. At the
moment when the short pulse of radiation reflected from
the output of the resonator appears at the input, the opti-
cal switch injects the next impulse into the resonator.
Below we shall find optimum regimes of excitation
of the dielectric resonator by sequence of bunches and
we investigate the time dynamics of excited fields.
2. WAKE FIELD OF BUNCH TRAIN IN DI-
ELECTRIC RESONATOR
Let the rectangular metal resonator have a transverse
size a ( / 2 / 2a x a− Ј Ј ) and b ( / 2 / 2b y b− Ј Ј ), its
length equal to L . The resonator is filled with a homo-
geneous dielectric with permittivity ε . Along the axis
of the resonator there is a drift channel, the transverse
dimensions which are small if compared with those of
the resonator. We will suppose that monoenergetic, thin
electron bunches are injected into the input of the res-
onator 0=z and then move with a constant velocity 0v
along the axis. The distribution of current density of a
single bunch has the form of:
( ) ( ) ( )0 0 0 0
0 0 0 0
/
[ ( ) ( / )] / ,
z b i
i i
j Q x x y y t t z v
t t t t L v v
δ δ δ
θ θ
= − − − −
− − − −ґ
where bQ is the charge of a bunch, 0it is the time of the
i-th bunch injection into the resonator, 0 0,x y are trans-
verse coordinates of a bunch, ( )tθ is the Heaviside
function, ( )tδ is the Dirac function.
Having solved the wave equation and taking into ac-
count the vanishing of the tangential components of
electric field on the metal walls of the resonator, we ob-
tain the expression for the longitudinal electric field:
100
0 0 0 2 2
1 , 1
0
2 2
0
0 02
0
2
0
0 0 0
2
02
0
( , , , ) cos( )
sin ( ) 1 sin ( )
( ) ( 1) sin ( / )
1 sin ( /
bN
l
z mn l
i m n mnl l
l
mn
mnl i l l i
mnl
l mn
i mnl i
mnl
l l i
E E G x y x y k z
ct t t t
v
t t t t L v
c t t L v
v
δ ω
ω ω
ω ω ω ω
ω ε
ωθ ω
ω
ω ω
ε
Ґ
= =
=
=
−
м щцй жп− − − −ґ н ъчзк
п л и шо ы
й
− − − − − −ґ к
л
цж
− − −чз
и ш
е е
] }0 0 0) ( / ) ,it t L vθ − −
where: 0 0 10016 /bE Q v abLπ ε ω= − ; /lk l Lπ= ,
0l lk vω = , 2 2 2 2 2( ) /
xm ynmnl lc kω κ κ ε= + + , /xm m aκ π= ,
/yn n bκ π= , the function lδ is equal 1 if 0l = and is
equal 2 if 0l № ; bN is the number of bunches in a se-
quence; 0( , ) sin[ ( / 2)]m xmG x x x aκ= + ґ
0 0sin[ ( / 2)]sin[ ( / 2)]sin[ ( / 2)]xm yn ynx a y a y aκ κ κ+ + + .
When all particles left the resonator ( 0 0/Nbt t L v> +
), the field of the space charge (second expression in
square brackets) disappears, and the expression for the
longitudinal electric field is the sum of traveling for-
ward and backward eigen-waves of the dielectric res-
onator:
[
100
0 0 0 2 2
1 , 1
0
2
0
0 0 0
( , , , ) cos( )
sin ( ) ( 1) sin ( / ) .
bN
l
z mn l
i m n mnl l
l
lmn
mnl i mnl i
mnl
E E G x y x y k z
t t t t L v
δ ω
ω ω
ω ω ω
ω
Ґ
= =
=
=
−
щ− − − − −ґ ы
е е
We note, that for the condition
mnl lω ω=
the relevant items in the sum (2),(3) become dominant.
This condition is nothing but the condition of
Cherenkov radiation in a slowing medium. This is why
these resonant items can be treated as a Cherenkov field
___________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2006. № 2.
Series: Nuclear Physics Investigations (46), p.73-75.
73
accumulated in the dielectric resonator. The rest of the
field is the field of the transition radiation from both
boundaries.
Because of the discreteness of the longitudinal wave
numbers of the oscillation spectrum, the resonant condi-
tion for the optiomum sizes of the resonator, permittivi-
ty, and energy of the bunch can be fulfilled only approx-
imately and only for a finite number of harmonics. Tak-
ing into account that we want to implement a multimode
condition of DWFA, it makes sense to find the relation
between the above quantities from the resonant require-
ment.
We now find the relation for a 2-d dielectric res-
onator (size in the y-direction is greater than the size in
x-direction). Let the resonant condition be fulfilled for
harmonic 1m = , l N= . Then for coherent summation
of fields of all harmonics m k= , l kN= from expres-
sion (3) we obtain
2
0 0 01, /L Na v cβ ε β= − = .
Conditions (5) and (6) are the basis of the resonator
concept of the dielectric wake field accelerator. In such
a resonator there will occur a multimode regime of field
excitation and a coherent summation of fields from the
injected bunches.
3. NUMERICAL SIMULATIONS OF WAKE
FIELDS BUILD UP
The wakefield from a sequence of bunches of finite
size is obtained by integration of expression (2) over the
time of injection 0it and the locations 0x of the point
bunches.
We choose the following parameters for numerical
calculations: dielectric permittivity 2.1ε = ; frequency
of bunch repetition 2.85f = GHz; energy of the bunch-
es is 4MeV; transverse size chosen according to condi-
tion (6) 5,045a = cm and the length of the resonator,
chosen according to requirement (5), 15.677L = cm
(tenth longitudinal harmonic is chosen as resonant,
10N = ); the length of bunch 1.7bL = cm; the height of
bunch 0.1ba a= ; 2 / 0.32bQ b = nC/cm.
In Fig.1 the time dynamics of the wakefield ( 0x = )
at the input of the dielectric resonator after injection of a
single bunch is presented, the bunch having left the res-
onator at time 1.75t ≈ ns. The wake field at the input of
the resonator (as well as at any spatial point inside the
resonator) has a nonregular behavior in time (see the left
graph), even though the peaks of the wake field follow
with a period approximately equal to the period of the
basic resonant mode.
The longitudinal distribution of the wakefield also
has a nonregular character. The envelope of the field is
nonuniform along the length of the resonator. Its struc-
ture varies with time; the maximum is periodically dis-
placed from one wall of the resonator to another. The ir-
regularity of the wake field is related to the excitation
by a single bunch of set of nonresonant longitudinal har-
monics. These nonresonant harmonics can be attributed
to the transition radiation [3], which has a wide spec-
trum. This is verified by the spectral density of the lon-
gitudinal electric field given in right part of Fig.1. In
this graph the peaks of spectral density corresponding to
the excited longitudinal harmonics of the field are clear-
ly visible.
19,5 20,2 20,9 21,6 22,3
-3
-2
-1
0
1
2
3 E z , kV/cm
t, ns
2,9 14,3 25,6 37,0 48,4 59,8
0,0
0,1
0,2
0,3
Ez, kV/cm/GHz
1,71 2,28 2,85 3,42 3,99
0,0
0,1
0,2
0,3
fm, GHz
Fig.1. Time dependence of longitudinal electric field
at the input of the dielectric resonator and its spec-
tral density after injection of a single bunch
In Fig.2 the axial distributions of wakefield build up
by injecting a sequence of 101 bunches into resonator is
presented: the top figure corresponds to the time
1.738t = ns (the first 5 bunches are injected into the res-
onator) and the down figure corresponds to the time
35.639t = ns (when the last bunch of the sequence is
injected into the resonator).
I
0 10 20 30 40 50
-3
-2
-1
0
1
2
3
z, cm
Ez, kV/cm
0 10 20 30 40 50
-80
-60
-40
-20
0
20
40
60
80 E
z
, kV/cm
z, cm
Fig.2. Axial distributions of wake field in the centre
of resonator ( 0x = ) when injecting a train of
bunches. Dashed lines show the location of bunches
At the initial stage, the amplitude of the field grows
from the head of a sequence of the bunches to the loca-
tion of the group wave front, excited by the first bunch,
and then decreases backwards to the input of the res-
onator. The wakefield in the resonator before the first
bunch leaves is qualitatively and quantitatively the same
as the field in a semi-infinite waveguide. The shape of
the wake field impulses and their duration approximate-
74
ly repeats the shape and duration of bunches. At later
times, after all bunches have been injected into the res-
onator, a nearly homogeneous distribution of field am-
plitude is formed.
I
50,00 50,71 51,42 52,13 52,84 53,55
-80
-40
0
40
80
t, ns
Ez, kV/cm
2,9 14,3 25,6 37,0 48,4 59,8
0
10
20
30
Ez, kV/cm/GHz
1,71 2,28 2,85 3,42 3,99
0
10
20
30
f
m
, GHz
Fig.3. Time dependence (left) of the longitudinal
electric field and its spectral density (right) in the di-
electric resonator after injection of 101 bunches
The case in the semi-infinite waveguide in contrast
to the resonator with regard to the increasing input to
output distribution of field amplitude [3] is formed. The
number of bunches participating in the build up to maxi-
mum amplitude is much lower, compared with the res-
onator. For a dielectric waveguide having the same
length, transverse size and permittivity as the resonator,
this number of bunches is equal 6. By comparison of the
top and down graphs of Fig.2 we conclude that all
bunches of the sequence equally contribute to the for-
mation of the longitudinal electric field amplitude in
resonator: i.e., it is possible to excite a wakefield in the
resonator with the amplitude considerably exceeding the
amplitude of the field in the semi-infinite waveguide.
The regularity of the oscillations is maintained.
We now show the mode-locking from the excitation
of the dielectric resonator by a sequence of electron
bunches. For this purpose we shall compare the tempo-
ral dynamics of the longitudinal electric field at a fixed
point of the resonator when the field is excited by a sin-
gle bunch and by a train of 101 bunches. In Fig.3 (top
part), the time dependence of the longitudinal electric
field at the input end of the resonator is presented. The
last bunch has left the resonator at time 38t ≈ ns. The
wakefield at the input of the resonator has a pattern of a
sequence of rectangular pulses with period equal to the
period of the bunch repetition rate. The amplitude of the
pulses varies weakly with time. At the down side of
Fig.3 the spectrum of the longitudinal electric field is
presented. It is seen that wakefield contains only odd
resonant frequencies. Non-resonant harmonics con-
tribute very little to the amplitude of the wakefield.
We now compare this spectral density with the spec-
tral density of the field, excited by a single bunch (see
Fig.1). It is seen that a train of bunches regularizes the
wake field, suppresses non-resonant longitudinal har-
monics and strengthens the resonant harmonics of a
field. In other words, the regular sequence of bunches
realizes mode-locking of the dielectric resonator.
This study was supported in parts by CRDF Grant
# UP2-2569-KH-04 and Ukr. DFFD 02.07/325.
REFERENCES
1. T.B. Zhang, J.L. Hirshfield, T.C. Marshal,
B. Hafizi. Stimulated dielectric wake-field accelera-
tor // Phys. Rev. 1997, v.E 56, p.4647-4655.
2. T.C. Marshal, C. Wang, J.L. Hirshfield. Fem-tosec-
ond planar electron beam source for micron-scale
dielectric wake field accelerator // Phys. Rev. 2001,
STAB 4, 121301, p.1-7.
3. V.A. Balakirev, I.N. Onishchenko, D.Yu. Sidorenko,
G.V. Sotnikov. Excitation of a wake field by a rela-
tivistic electron bunch in a semi-infinite dielectric
waveguide // Zh. Eksp. Teor. Fiz. 2001, v.93, p.33-
42.
4. T.C. Marshall, J-.M. Fang, J.L. Hirshfield,
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field resonator accelerator. AIP Conf. Proc. 2001,
№569, p.316.
5. V.A. Balakirev, I.N. Onishchenko, D.Yu. Sidorenko,
G.V. Sotnikov. Charged Particles Accelerated by
Wake Fields in a Dielectric Resonator with Excit-
ing Electron Bunch Channel // Pisma v Zh. Tekh.
Fiz. 2003, 29, p.39-45 (in Russian).
КОГЕРЕНТНОЕ СЛОЖЕНИЕ КИЛЬВАТЕРНЫХ ПОЛЕЙ ПОСЛЕДОВАТЕЛЬНОСТИ ЭЛЕКТРОН-
НЫХ СГУСТКОВ В ПЛОСКОМ МНОГОМОДОВОМ ДИЭЛЕКТРИЧЕСКОМ РЕЗОНАТОРЕ
Н.И. Онищенко, Г.В. Сотников
Исследована возможность реализации концепции ускорителя на кильватерных полях в диэлектрическом
резонаторе. Аналитически определены условия когерентного сложения кильватерных полей последователь-
ности электронных сгустков. Проведенное численное моделирование подтверждает условия когерентности.
КОГЕРЕНТНЕ ДОДАВАННЯ КІЛЬВАТЕРНИХ ПОЛІВ ПОСЛІДОВНОСТІ ЕЛЕКТРОННИХ
ЗГУСТКІВ У ПЛОСКОМУ МНОГОМОДОВОМУ ДІЕЛЕКТРИЧНОМУ РЕЗОНАТОРІ
М.І. Оніщенко, Г.В. Сотніков
Досліджено можливість реалізації концепції прискорювача на кільватерних полях у діелектричному
резонаторі. Аналітично визначені умови когерентного додавання кільватерних полів послідовності
електронних згустків. Проведене чисельне моделювання підтверджує умови когерентності.
___________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2006. № 2.
Series: Nuclear Physics Investigations (46), p.73-75.
73
КОГЕРЕНТНОЕ СЛОЖЕНИЕ КИЛЬВАТЕРНЫХ ПОЛЕЙ ПОСЛЕДОВАТЕЛЬНОСТИ ЭЛЕКТРОННЫХ СГУСТКОВ В ПЛОСКОМ МНОГОМОДОВОМ ДИЭЛЕКТРИЧЕСКОМ РЕЗОНАТОРЕ
КОГЕРЕНТНЕ ДОДАВАННЯ КІЛЬВАТЕРНИХ ПОЛІВ ПОСЛІДОВНОСТІ ЕЛЕКТРОННИХ ЗГУСТКІВ У ПЛоСКОМУ МНОГОМОДОВОМУ ДІЕЛЕКТРИЧНОМУ РЕЗОНАТОРІ
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