Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches
Results of the theoretical investigations of excitation process of wakefields in semi-infinite dielectric waveguides by finite sequence of nonresonance relativistic bunches are presented. Dependence of wakefield amplitude on length of matched section of dielectric waveguide are investigated. Proce...
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| Cite this: | Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches / V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2014. — № 3. — С. 102-106. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-801362015-04-13T03:01:57Z Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. Новые и нестандартные ускорительные технологии Results of the theoretical investigations of excitation process of wakefields in semi-infinite dielectric waveguides by finite sequence of nonresonance relativistic bunches are presented. Dependence of wakefield amplitude on length of matched section of dielectric waveguide are investigated. Process of wakefield excitation under coherent summing of traverse modes with equidistant spectrum of frequencies at semi-infinite plane dielectric waveguide is studied. It is shown that at this condition essential increasing of pulse wakefield amplitude takes place. Представлены результаты теоретических исследований процесса возбуждения кильватерных полей в полубесконечных диэлектрических волноводах конечной нерезонансной последовательностью релятивистских электронных сгустков. Исследованa зависимость кильватерного поля от длины согласованного отрезка диэлектрического волновода. Изучен процесс возбуждения кильватерных полей при когерентном сложении поперечных мод с эквидистантным частотным спектром в полубесконечном плоском диэлектрическом волноводе. Показано, что в этих условиях происходит значительное увеличение амплитуды импульсов кильватерного поля. Представлено результати теоретичних досліджень процесу збудження кільватерних полів у напівнескінченних діелектричних хвилеводах кінцевою нерезонансною послідовністю релятивістських електронних згустків. Досліджено залежність кільватерного поля від довжини узгодженого відрізка діелектричного хвилеводу. Вивчено процес збудження кільватерних полів при когерентному додаванні поперечних мод з еквідистантним частотним спектром у напівнескінченному плоскому діелектричному хвилеводі. Показано, що у цих умовах відбувається значне збільшення амплітуди імпульсів кільватерного поля. 2014 Article Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches / V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2014. — № 3. — С. 102-106. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 41.75.Jv, 41.75.Lx, 41.75.Ht, 96.50.Pw http://dspace.nbuv.gov.ua/handle/123456789/80136 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 Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches Вопросы атомной науки и техники |
| description |
Results of the theoretical investigations of excitation process of wakefields in semi-infinite dielectric waveguides
by finite sequence of nonresonance relativistic bunches are presented. Dependence of wakefield amplitude on length
of matched section of dielectric waveguide are investigated. Process of wakefield excitation under coherent summing
of traverse modes with equidistant spectrum of frequencies at semi-infinite plane dielectric waveguide is studied.
It is shown that at this condition essential increasing of pulse wakefield amplitude takes place. |
| format |
Article |
| author |
Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. |
| author_facet |
Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. |
| author_sort |
Balakirev, V.A. |
| title |
Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches |
| title_short |
Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches |
| title_full |
Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches |
| title_fullStr |
Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches |
| title_full_unstemmed |
Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches |
| title_sort |
excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2014 |
| topic_facet |
Новые и нестандартные ускорительные технологии |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80136 |
| citation_txt |
Excitation of wake fields in a semi-infinite dielectric waveguide by nonresonance sequence of bunches / V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2014. — № 3. — С. 102-106. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT balakirevva excitationofwakefieldsinasemiinfinitedielectricwaveguidebynonresonancesequenceofbunches AT onishchenkoin excitationofwakefieldsinasemiinfinitedielectricwaveguidebynonresonancesequenceofbunches AT tolstoluzhskyap excitationofwakefieldsinasemiinfinitedielectricwaveguidebynonresonancesequenceofbunches |
| first_indexed |
2025-07-06T04:03:32Z |
| last_indexed |
2025-07-06T04:03:32Z |
| _version_ |
1836868816903077888 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2014. №3(91) 102
EXCITATION OF WAKE FIELDS IN A SEMI-INFINITE DIELECTRIC
WAVEGUIDE BY NONRESONANCE SEQUENCE OF BUNCHES
V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: tolstoluzhsky@kipt.khakov.ua
Results of the theoretical investigations of excitation process of wakefields in semi-infinite dielectric waveguides
by finite sequence of nonresonance relativistic bunches are presented. Dependence of wakefield amplitude on length
of matched section of dielectric waveguide are investigated. Process of wakefield excitation under coherent sum-
ming of traverse modes with equidistant spectrum of frequencies at semi-infinite plane dielectric waveguide is stud-
ied. It is shown that at this condition essential increasing of pulse wakefield amplitude takes place.
PACS: 41.75.Jv, 41.75.Lx, 41.75.Ht, 96.50.Pw
INTRODUCTION
A theoretical model of a semi-infinite dielectric
waveguide is adequate for a number of pilot schemes
for the acceleration of electrons by wakefields excited
by a sequence of relativistic electron bunches. The pres-
ence of the front boundary leads to a number of features
of the wakefields excitation in semi-infinite dielectric
waveguide [1, 2]. First of all, in such systems, there is a
“quenching” wave that restricts the number of bunches
which wakefields are coherently summed. In addition
there is a transition radiation bunches chain on the
boundary of semi-infinite dielectric waveguide.
An important problem in the excitation theory of
wakefields of electron bunches sequence in the
segments of the dielectric waveguide is the dependence
of the wakefield on the length of the segment. This rela-
tionship brightly reflects of the influence of the wave
quenching on the excitation of wakefield sequence
bunches in the waveguide segment.
This paper presents the results of theoretical investi-
gations of the wakefields excitation process, in total
case, non-resonant sequence of electron bunches in
semi-infinite dielectric waveguides. There is view
sequence of bunches, frequency of which is close to the
frequency of the excited wakefield.
Two schemes of semi-infinite waveguide are con-
sidered. In the first scheme, considered before in the 2nd
quarter, the sequence of bunches are injected from the
perfectly conducting diaphragm located at the face of
semi-infinite waveguide. In the second scheme, the se-
quence of bunches moves in the vacuum region of the
waveguide and then transverses the sharp boundary of
semi-infinite dielectric waveguide. Dependence of am-
plitude of a wakefield on length of the segment of die-
lectric waveguide is presented too. This dependence
brightly reflects influence of the quenching wave on the
process of wakefields excitation by a sequence of
bunches. To increase the accelerating gradient in dielec-
tric waveguide the effect of coherent addition of the
transverse modes of the excited wakefield can be used.
Most effectively process of addition takes place at the
frequency spectrum of the transverse modes which is
closed to the equidistant one. In this case the wakefield
looks like a sequence of narrow peaks of an opposite
polarity of large amplitude.
In the paper process of wakefield excitation in a
semi-infinite plane dielectric waveguide by sequence of
electron bunches is studied too. It is known that in such
slowing down structure the frequency spectrum of ex-
cited transverse wakefield modes is equidistant. Respec-
tively, as a result of the coherent addition of wakefield
transverse modes the resultant wakefield field looks like
sequence of narrow peaks. Influence of boundary of
semi-infinite plane dielectric waveguide (in the bunch
propagation direction) upon the process of excitation of
the large amplitude wakefield is considered.
1. WAKEFIELD OF NONRESONANCE
SEQUENCE OF ELECTRON BUNCHES
IN SEMI-INFINITE CIRCULAR
DIELECTRIC WAVEGUIDE
Let's consider the following model of semi-infinite
dielectric waveguide. The semi-infinite dielectric wave-
guide occupies region 0z∞ > ≥ . The butt 0z = is
short-circuited by perfectly conducting diaphragm. In
the axial region there is a vacuum channel. In the wave-
guide from conducting diaphragm the sequence of rela-
tivistic electron bunches with initial value of a relativ-
istic factor 0 1γ >> is injected. Longitudinal and trans-
verse profiles of bunches density have rectangular pro-
files with sharp boundaries. Let's neglect influence of
the narrow vacuum channel on electrodynamics of sys-
tem and for simplification of calculations we will con-
sider dielectric filling continuous. The solution of the
electrodynamic problem by method of Fourier transform
gives the following expression of the wakefield excited
by nonresonance sequence of bunches in a semi-infinite
dielectric waveguide
,ch tr
z z zE E E= +
1
( ) ( )1 0
02
0 1 1
( / ) ( / )8 ,
( )
N
ch s sn b n
z n gn
j nb n b n n
J r b J r bQE Z Z
br t J
λ λ
ε ω λ λ
− ∞
= =
= − ∑∑ (1)
where Q is the bunch charge, br is the radius of the
bunches, bt is the duration of a single bunch,
2 2
0/n n b c vω λ ε − −= − are frequencies of radial modes,
ε is dielectric constant, nλ are roots of Bessel function
0 ( )J x , b is waveguide radius,
0 0
( )
0 0
sin ( / ) sin (1 / ) / ,
( ) ( ),
sin ( / ) sin ( / ),
( ),
n n g g
s g g b
gn
n n b
g b
t z v sT z v v v
v t sT z v t t sT
Z
t z v sT t z v t sT
z v t t sT
ω ω
ω ω
− − − −
− − −=
− − − − − −
− −
ISSN 1562-6016. ВАНТ. 2014. №3(91) 103
0 0 0
( )
0 0 0
0
sin ( / ), ( ) ( ),
sin ( / ) sin ( / ),
( ),
n b
s
n n n b
b
t z v sT v t sT z v t t sT
Z t z v sT t z v t sT
z v t t sT
ω
ω ω
− − − − −
= − − − − − −
− −
0v is the velocity of bunches, 2
0/grv c v ε= is the group
velocity of wakefield waves,
1
1 0
2 2
0 1 1
( / ) ( / )8 ,
( )
N
tr n b n
z ns
j nb n n
J r b J r bQE S
b t J
λ λ
ε λ λ
− ∞
= =
= ∑∑ (2)
{0 0
0
2 2
1 2 2
1
2 2
0 0 1 2 2
1
( / ) ( / )
( 1) ( ) ( )
( / ) ( ) ( 1) ( ) ( ) ,
bt
ns pr
m m m
s s m s
m
m m m
s s s m s
m
S dt t z v sT t z v sT
r r J y
t z v sT J y r r J y
θ θ
θ
∞
=
∞
−
=
= − − − − − ×
× − − +
+ − − + − +
∫
∑
∑
/ ,prv c ε= 2 2 ,s sy α τ ς= − / ,n nk cα ε⊥=
,s t sTτ = − / ,z cς ε= ( ) / ( ).s s sβ τ ς τ ς= − +
1 0 0(1 / ) / (1 / ),s sr c v c vβ ε ε= − +
2 0 0(1 / ) / (1 / ) .s sr c v c vβ ε ε= + −
The term (1) describes Cherenkov field of bunches
sequence with the quenching waves, the term (2) de-
scribes the transition radiation of the bunches chain,
caused by existence of perfectly conducting butt 0z = .
In expressions for fields (1), (2) external summing on
index s corresponds to summing on bunches, and
summing on index n corresponds to summing on radial
modes of the dielectric waveguide.
Let's consider now excitation of a wakefield in semi-
infinite dielectric waveguide by periodic sequence from
N infinitely thin bunches. We will be restricted to the
most interesting case of existence of detuning between
the frequency of bunches repetition and frequency of the
excited wave. Numerical calculations were performed
for different numbers of bunches in chain and value of
the relative detuning. The pictures of the wakefield ex-
citation field in semi-infinite dielectric waveguide de-
pends on number of bunches in chain. Fig. 1 shows the
spatial structure of the excited field for 22 bunches It
can be seen in semi-infinite waveguide as the distance
from the conductive diaphragm two pulses of field, the
following one after another are formed.
The first pulse of a wakefield is caused by the Che-
renkov radiation of bunches and propagates with bunch-
es velocity. The second pulse is caused by existence of
boundary and propagates with group velocity which in
dielectric waveguide always is less than velocity of
bunches. As for the first pulse, in case of the chosen
value of detuning the phase of wakefield changes on π
after the tenth bunch. The Cherenkov wakefield in vi-
cinity of the tenth bunch reaches the maximum value.
After the twentieth bunch the phase of a field changes
on 2π . Thus wakefield is zero. Last 21 and the 22nd
bunches excite in region between field pulses the mono-
chromatic wake wave having relatively low level. With
increase of number of bunches in chain amplitude of
monochromatic wake wave increases and for 30 bunch-
es reaches the maximum value. This case is illustrated
by Fig. 2,a. It can be seen that amplitude of monochro-
matic wave wake behind the first pulse is equal in accu-
racy to amplitude of the first field. As a whole in region
0 g gv t z z v t= picture of wakefield excitation is the
same, as in infinite dielectric waveguide.
As always 0 gv v , the second pulse will lag behind
constantly from chain of bunches and pulse wakefield
connected with chain. The distance between pulses will
increase and, respectively, length of a monochromatic
wave wake behind the first pulse will increase also. In
Figs. 1,a; 2,а full fields which include both a Cherenkov
field, and transition radiation are shown. Pictures of the
transition radiation are shown in Figs. 1,b; 2,b. The
pulse of the transition radiation basically is in region
0 gz z and level of this pulse is rather small. In
region gz z pulse of the transition radiation extends
to the plane
a b
Fig. 1. Spatial structure of wakefields at the moment
of time / 100t T = , system length is 102L λ= , number
of bunches is 22N = , value of detuning is 0.05α = ,
full wakefield (a); transition radiation (b)
a b
Fig. 2. Spatial structure of full wakefields at the moment
of time / 100t T = , system length is 102L λ= , number
of bunches is 30N = , value of detuning is 0.05α = ,
full wakefield (a); transition radiation (b)
An important characteristic of the process of wake-
fields excitation in dielectric waveguide segment by
sequence of relativistic electron bunches is dependence
of the wakefield amplitude on the length of the segment
of the dielectric slow-wave structure. Input end
completely is short-circuited perfectly conducting
diaphragm. The output end is assumed fully match with
external RF line (the reflected wave from the output end
is absent). Figs. 3,a,b shows the dependence of the am-
plitude of the Cherenkov wakefield at the output end of
the segment of the dielectric waveguide length
/ 4L λ= and / 2L λ= . The number of bunches is
10N = . The field is a non-sinusoidal oscillations with a
constant amplitude. Such form of the field is caused by
the fact that in the dielectric waveguide of length
0/ ( / 1)gL v vλ= − , as a result of quenching waves (ef-
fect of group velocity), a bunch does not have time to
excite the whole period of wakefield oscillation. So in
the process of the sequence of bunches propagation in
such a short waveguide Cherenkov pulses of separate
bunches do not overlap each other. The coherent addi-
tion of Cherenkov fields of single bunches is absent.
Therefore at an output end we observe pulses of Che-
renkov radiation of single bunches. In Figs. 3,a; 3,b it is
ISSN 1562-6016. ВАНТ. 2014. №3(91) 104
shown that on every period of oscillations there is a pe-
riod during which the Cherenkov field is absent. This
period corresponds to an interval of time (coordinate)
between this bunch and hind front of Cherenkov radia-
tion of a previous bunch. In Fig. 3,c dependence on time
of full field at output end of waveguide is presented. It
is well seen that the transition radiation significantly
perturbs oscillations. Besides decreasing ring appears,
which has been completely caused by the transition ra-
diation. Similar dependences of fields on time are pre-
sented in Figs. 4,a,b for segment of a dielectric wave-
guide by length 2L λ= . In this case in the region
adjoined to output end pulses of Cherenkov radiation of
three bunches have time to be overlapped. The transi-
tion radiation (see Fig. 4,b), as well as the previous case,
strongly perturbs Cherenkov oscillations.
a b
c
Fig. 3. Dependences Cherenkov wakefields on time
at output end of the dielectric waveguide at length
/ 4L λ= (a); / 2L λ= (b); full wakefields
/ 2L λ= (c). Number of bunches is 10N =
Further growth of field amplitude doesn't come due
to removal of the field of previous bunches from wave
guide with group velocity. Thus amplitude of the wake-
field reaches stationary level. Value of maximum level
of the wakefield amplitude depends on number of pulses
of separate bunches which are able to overlapped on the
waveguide length. After exiting of the last bunch the
field amplitude falls down to null value in time which
equals to the time rise of amplitude.
a b
Fig. 4. Dependences wakefields on time at output end
of the dielectric waveguide at length 2L λ= .
Cherenkov wakefield (a). Full wakefield (b).
Number of bunches is 10N =
In Fig. 5,a it is presented dependence of amplitude
of the longitudinal component Cherenkov wakefield
from length of the dielectric waveguide. It can be seen
that with growth of the waveguide length the amplitude
of wakefield increases. In more detail dependence of
wakefield amplitude on time in case of the fixed maxi-
mum of the dielectric waveguide lengths is shown in
Fig. 5,b. As seen from this figure, field amplitude in-
creases abruptly with growth of the wave guide length
and reaches stationary value at maximum length of the
waveguide, as it was stated above, due to removal of
field of the previous bunches from wave guide.
a b
c
Fig. 5. Dependence of Cherenkov wakefield amplitude
on waveguide length at exit (a); dependence Cherenkov
wakefield on time at exit of the dielectric waveguide at
the fixed length (b); dependence of average value of the
module wakefield ( )zE t on waveguide length at output (c)
In Fig. 5,c dependence of average value of the mod-
ule of longitudinal component of electric field ( )zE t of
wakefield on length of dielectric waveguide is present-
ed. Averaging was performed on realization length. It is
seen that with increasing of waveguide length amplitude
of wakefield grows practically under the linear law.
Such regularity is explained by that with waveguide
lengthening as a result of lag of quenching wave in-
creases the number of the bunches which fields coher-
ently are added.
In all previous cases the semi-infinite dielectric
waveguide which input end is short-circuited perfectly
conducting diaphragm was considered. Important con-
clusion of the theory of wakefields excitation by elec-
tron bunches is formation in such semi-infinite slow
wave structure of quenching wave of Cherenkov radia-
tion. This wave plays a basic role in processes of wake-
fields excitation both for single electron bunches, and
for sequences of bunches. Below we will consider a
semi-infinite dielectric waveguide of other geometry
[3]. In perfectly conducting metal pipe the region 0z
is filled with dielectric with the vacuum channel at vi-
cinity of axial region. In region 0z it is vacuum.
From vacuum region into dielectric the relativistic elec-
tron bunch flies. Let's consider excitation of wakefield
for considered geometry of semi-infinite dielectric
waveguide. Let's solve the electrodynamics problem by
method of expansion of field on Bessel functions with
respect to radial coordinate and Fourier transform with
respect to time. As a result for a longitudinal component
of an electric wakefield in the region filled with dielec-
tric 0z , we will obtain the following expression
ch bord
m m mE E E= + , 0
1
( / ),z m N
n
E E J R Bλ
∞
=
= ∑
0 0
2 2
1
( )4 ,
( )
ch chn
m n
n
J rQE i I
b J
λ
ε λ
=
2
0 0
2 2
1
4 ( ) ,
( )
bord bordn n
m n
n
Q J rE i I
b J
λ λ
λ
=
0 0
2 2
( / )0
2 2
2
1 ,
2
i t t z vch l
n
n l
k kdI e
k k
ωεω
π ω
∞
− − −
=∞
−
=
−∫
0 2( )
1
2 2 2 2
12 1 2
1 1 ,
2 ( )
ni t t ik z
bord n l
n
nn n n l
k kd eI
kk k k k
ω εω
π ω ε ε
∞ − − +
=∞
+
= − + −
∫
ISSN 1562-6016. ВАНТ. 2014. №3(91) 105
where 0/ ,lk vω= 2 2 2
1 0 / ,n nk k bλ= −
2 2 2
2 0 /n nk k bε λ= − are longitudinal wave numbers in
vacuum and dielectric respectively, 0 /k cω= . Integral
ch
nI describes wake Cherenkov in boundless dielectric
and integral bord
nI takes into account the presence of a
boundary 0z = in semi-infinite dielectric waveguide.
The integral describes wake Cherenkov in boundless
dielectric, and the integral considers boundary existence
at a semi-infinite dielectric waveguide. Wakefield in
boundless dielectric waveguide we find, using the theo-
rem of residues
0 0
0 0 0 02 2
1
( / )4 ( / ) cos ( / ),
( )
ch
n n
n
J r bQE t t z v t t z v
b J
λ
θ ω
ε λ
= − − − −
0
2 2
0 / 1
n
n
v
b v c
λω
ε
=
−
are frequencies of dielectric wave-
guide eigenwaves.
The analysis of integral by the approximate method
proposed by E.L. Burstein and G.V. Voskresensky [2],
shows that in case of realization of the condition
0( ) 0gv t t z− it is necessary to take into considera-
tion a pole in the integrand expression bord
nI . Residues in
poles give a field of quenching wave
0 0
0 0 02 2
1
( / )4 ( / ) cos ( / ).
( )
qn
n g n
n
J r bQE t t z v t t z v
b J
λ θ ω
ε λ
= − − − − −
The field of quenching wave in accuracy coincides
on value with Cherenkov field, but has an opposite sign.
Summing of these fields shows that the Cherenkov
wakefield exists in region 0 0 0( ) ( )gv t t z v t t− − .
2. MULTIMODE REGIME WAKEFIELD
SEMI-INFINITE PLANE DIELECTRIC
WAVEGUIDE
For increasing of the field transformation ratio in di-
electric waveguide may be used the effect of the coher-
ent addition of radial (transverse) harmonic of an excit-
ed wakefield. Most effectively process of addition takes
case under frequency spectrum of transverse harmonic
as much as possible the close to the equidistant. In this
case the wakefield looks like sequence of narrow peaks
of an opposite polarity of large amplitude. Let's consider
a semi-infinite waveguide, having form of two parallel
ideally conducting planes, distance between which is
L . The waveguide is completely filled with uniform
dielectric with dielectric permittivity. The input end face
is short-circuited by perfectly conductive transverse
wall. In volume of waveguide from transverse wall the
sequence of bunches with period T is injected. Let's
chose the following model transverse profile of bunches
0
0
0
0 0
cos , ,
( ) 2
0, / 2 , / 2.
b b
b
b b
x x x x
R x x
L x x x x L
π
≥ ≥ − =
≥ ≥ − ≥ ≥ −
In the longitudinal direction bunches are infinitely
thin. Expression for the wakefield in semi-infinite plane
dielectric waveguide has form
,ch tr
z z zE E E= +
3 1
0
0 0
2 ( )
( ) ( ) cos ( ),
N
ch
z n
s n нечетн
n
g
QE R x
L
z z zt sT t sT t sT
v v v
π
ε
θ θ ω
− ∞
= =
= ×
× − − − − − − −
∑ ∑
(3)
{
1
0
0
2 2
1 2 2
1
2 2
0 0 1 2 2
1
8 ( ) ( / ) ( / )
( 1) ( ) ( )
( / ) ( ) ( 1) ( ) ( ) , (4)
N
tr
z n pr
s n нечетн
m m m
s s m s
m
m m m
s s s m s
m
QE R x t z v sT t z v sT
L
r r J y
t z v sT J y r r J y
π θ θ
ε
θ
− ∞
= =
∞
=
∞
−
=
= − − − − − ×
× − − +
+ − − + − +
∑ ∑
∑
∑
where ch
zE is field of Cherenkov radiation, tr
zE is field
of the transition radiation, Q is charge of bunch,
2
2 2( ) cos( )cos( ) / ( )
4n n b n n bR x k x k x k xπ
⊥ ⊥ ⊥= − /nk n Lπ⊥ =
are transverse wave numbers, 2 2
0 0/ / 1n nv L v cω π ε= −
are equidistant frequencies of transverse modes,
/prv c ε= is velocity of harbinger propagation.
a b
Fig. 6. Spatial structure of multimode wakefields at the
moment of time / 60t T = (a); fragment of Fig. 6,a (b)
The spatial structure of wakefield excited by se-
quence of electron bunches in a multimode regime in
the plane dielectric waveguide was calculated by nu-
merical methods. Calculations were executed in case of
the following parameter values: system length is
60L λ= , number of bunches is 20N = , number of
transverse modes is mod 21N = (increasing of modes
number didn't lead to notable change of results), fre-
quency of the transverse fundamental mode is
2.8025=f GHz, transverse size of waveguide is
5.12=xL cm, transverse size of a bunches is
/ 0.04b xx L = , dielectric permittivity 2.1ε = . In
Fig. 6,a the spatial structure of a wakefield in a multi-
mode regime is showed.
The qualitative picture of the wakefield in multi-
mode regime is close to the similar picture of the single
mode regime. In the beginning there is the linear growth
of amplitude caused by the coherent addition of fields of
all twenty bunches.
Then there is region of the wakefield with constant
amplitude. And, at last, on output butt amplitude de-
creases practically to zero. Similarity of behavior of
wakefield in multimode and single-mode approximation
is explained by that for considered system group veloci-
ties of all radial harmonic identical and are equal
2
0/gv c v ε= . Therefore group fronts of all harmonics
coinside. In Fig. 6,b the increased fragment of wakefield
oscillations from region of constant value of amplitude
is shown. It is seen that oscillations look like of se-
quence of narrow pulses with opposite polarity. Thus
ISSN 1562-6016. ВАНТ. 2014. №3(91) 106
amplitude of narrow pulses amplified, approximately,
by 5 times in comparison with the single-mode approx-
imation (see Fig. 5,c).
CONCLUSIONS
Thus, in the paper process of excitation of wake-
fields in a semi-infinite dielectric waveguide by se-
quence of relativistic electron bunches in the presence
of detuning between the repetition frequency of bunches
and frequency of the excited wave is considered. It is
shown that in a semi-infinite dielectric waveguide two
pulses, the following one after another are formed. The
first pulse of a wakefield propagates with the velocity of
bunches. The second pulse propagates with group veloc-
ity and is caused by existence of boundary. Between
pulses of the field there is monochromatic wake wave of
the constant amplitude, which value depends on number
of bunches in sequence. In the first pulse of wakefield
may be realized regime of auto acceleration of the elec-
tron bunches located in the region of wakefield decreas-
ing.
The alternative scheme of semi-infinite dielectric
waveguide which is the metal pipe, which half-space is
filled with dielectric is considered. Exact expressions
for Cherenkov radiation with taking into account
quenching wave for wakefield are found.
Dependence of a wakefield on time at output end of
the dielectric waveguide segment for different lengths
of waveguide is investigated. Also dependence of aver-
age value of the electric field module on waveguide
length is obtained. It is shown that with increasing of
waveguide length the average wakefield grows, approx-
imately, under the linear law. Process of wakefields
excitation in semi-infinite plane dielectric waveguide by
sequence of electron bunches in the multiwave regime is
considered. It is shown that the coherent addition of
transverse modes with the equidistant frequency spec-
trum leads to excitation of a wakefield in the form of
sequence of narrow pulses of field with opposite polari-
ty. Thus there is the strong increasing of pulses ampli-
tude. It is studied influence of waveguide input bounda-
ry which lead to first of all to excitation of quenching
waves, on process of excitation of wakefield in multi-
mode the regime.
ACKNOWLEDGMENTS
This work was supported by the US Department of
Energy/NNSA through the Global Initiatives for Prolif-
eration Preventation (GIPP) Program in Partnership
with the Science and Technology Center in Ukraine
(Project ANL-T2-247-UA and STCU Agreement P522).
REFERENCES
1. E.L. Burstein and G.V. Voskresensky. Linear elec-
tron acceleration with intensive beams. M.: «Atomiz-
dat», 1970.
2. 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 // JETPh. 2001, v. 93, №1, p. 33-42.
3. T.Yu. Alekhina and A.V. Tyukhtin. Cherenkov-
transition radiation in a waveguide with a dielectric-
vacuum boundary // Phys. Rev. Special Topics –
Accelerators and Beams. 2012, v. 15, p. 091302.
4. V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky.
Wakefield excitation and electron accelerationat de-
tuning bunch repetition frequency and frequency of
eigen principal mode of wakefield // PAST. Series
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tions” (86). 2013, №4, p. 80-83.
Article received 20.03.2014
ВОЗБУЖДЕНИЕ КИЛЬВАТЕРНЫХ ПОЛЕЙ НЕРЕЗОНАНСНОЙ ПОСЛЕДОВАТЕЛЬНОСТЬЮ
СГУСТКОВ В ПОЛУБЕСКОНЕЧНОМ ДИЭЛЕКТРИЧЕСКОМ ВОЛНОВОДЕ
В.А. Балакирев, И.Н. Онищенко, А.П. Толстолужский
Представлены результаты теоретических исследований процесса возбуждения кильватерных полей в по-
лубесконечных диэлектрических волноводах конечной нерезонансной последовательностью релятивистских
электронных сгустков. Исследованa зависимость кильватерного поля от длины согласованного отрезка ди-
электрического волновода. Изучен процесс возбуждения кильватерных полей при когерентном сложении
поперечных мод с эквидистантным частотным спектром в полубесконечном плоском диэлектрическом вол-
новоде. Показано, что в этих условиях происходит значительное увеличение амплитуды импульсов кильва-
терного поля.
ЗБУДЖЕННЯ КІЛЬВАТЕРНИХ ПОЛІВ НЕРЕЗОНАНСНОЮ ПОСЛІДОВНІСТЮ ЗГУСТКІВ
У НАПІВНЕСКІНЧЕННОМУ ДІЕЛЕКТРИЧНОМУ ХВИЛЕВОДІ
В.А. Балакірєв, І.М. Онищенко, О.П. Толстолужський
Представлено результати теоретичних досліджень процесу збудження кільватерних полів у напівнескін-
ченних діелектричних хвилеводах кінцевою нерезонансною послідовністю релятивістських електронних
згустків. Досліджено залежність кільватерного поля від довжини узгодженого відрізка діелектричного хви-
леводу. Вивчено процес збудження кільватерних полів при когерентному додаванні поперечних мод з екві-
дистантним частотним спектром у напівнескінченному плоскому діелектричному хвилеводі. Показано, що у
цих умовах відбувається значне збільшення амплітуди імпульсів кільватерного поля.
1. wakefield of nonresonance sequence of electron bunches in semi-INFINITE CIRCULAR dielecTric waveguide
2. MULTIMODE REGIME wakefield semi-INFINITE pLANE dielectric waveguide
ВОЗБУЖДЕНИЕ КИЛЬВАТЕРНЫХ ПОЛЕЙ НЕРЕЗОНАНСНОЙ ПОСЛЕДОВАТЕЛЬНОСТЬЮ СГУСТКОВ В ПОЛУБЕСКОНЕЧНОМ ДИЭЛЕКТРИЧЕСКОМ ВОЛНОВОДЕ
ЗБУДЖЕННЯ КІЛЬВАТЕРНИХ ПОЛІВ НЕРЕЗОНАНСНОЮ ПОСЛІДОВНІСТЮ ЗГУСТКІВ У НАПІВНЕСКІНЧЕННОМУ ДІЕЛЕКТРИЧНОМУ ХВИЛЕВОДІ
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