Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield
Theoretical results of process of the wake field excitation in the dielectric waveguide by sequence of relativistic bunches in the presence of detuning between the bunch repetition frequency and the frequency of the fundamental wave waveguide. In this case the wake waves radiated by individual bunch...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. 2017-01-15T16:37:42Z 2017-01-15T16:37:42Z 2013 Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield / V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2013. — № 4. — С. 80-83. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 41.75.Jv, 41.75.Lx, 41.75.Ht, 96.50.Pw https://nasplib.isofts.kiev.ua/handle/123456789/111930 Theoretical results of process of the wake field excitation in the dielectric waveguide by sequence of relativistic bunches in the presence of detuning between the bunch repetition frequency and the frequency of the fundamental wave waveguide. In this case the wake waves radiated by individual bunches, there is the phase shift, and, beginning with some number of the bunch, the phase shift will be. The polarity of the field is changed and the subsequent bunches gain energy. Energy spectra of accelerated (decelerated) particles are obtained. Викладено результати теоретичних досліджень процесу збудження кільватерного поля в діелектричному хвилеводі послідовністю релятивістських згустків при наявності розстройки між частотою проходження згустків і частотою основної хвилі хвилеводу. У цьому випадку для кільватерних хвиль, випромінюваних окремими згустками, виникає набіг фази і, починаючи з деякого номера згустку, набіг фази складе π . Змінюється полярність поля і наступні згустки набирають енергію. Отримані енергетичні спектри прискорених (уповільнених) частинок. Изложены результаты теоретических исследований процесса возбуждения кильватерного поля в диэлектрическом волноводе последовательностью релятивистских сгустков при наличии расстройки между частотой следования сгустков и частотой основной волны волновода. В этом случае для кильватерных волн, излучаемых отдельными сгустками, возникает набег фазы и, начиная с некоторого номера сгустка, набег фазы составит π . Изменяется полярность поля и последующие сгустки набирают энергию. Получены энергетические спектры ускоренных (замедленных) частиц. This work was supported by the US Department of Energy/NNSA through the Global Initiatives for Proliferation Prevention (GIPP) Program in Partnership with the Science and Technology Center in Ukraine (Project ANL-T2-247-UA and STCU Agreement P522). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Новые методы ускорения заряженных частиц Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield Збудження кільватерного поля і прискорення електронів за наявності розстройки частоти слідування згустків і частоти основної власної кільватерної хвилі Возбуждение кильватерного поля и ускорение электронов при наличии расстройки частоты следования сгустков и частоты основной собственной кильватерной волны Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield |
| spellingShingle |
Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. Новые методы ускорения заряженных частиц |
| title_short |
Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield |
| title_full |
Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield |
| title_fullStr |
Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield |
| title_full_unstemmed |
Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield |
| title_sort |
wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield |
| author |
Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. |
| author_facet |
Balakirev, V.A. Onishchenko, I.N. Tolstoluzhsky, A.P. |
| topic |
Новые методы ускорения заряженных частиц |
| topic_facet |
Новые методы ускорения заряженных частиц |
| publishDate |
2013 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Збудження кільватерного поля і прискорення електронів за наявності розстройки частоти слідування згустків і частоти основної власної кільватерної хвилі Возбуждение кильватерного поля и ускорение электронов при наличии расстройки частоты следования сгустков и частоты основной собственной кильватерной волны |
| description |
Theoretical results of process of the wake field excitation in the dielectric waveguide by sequence of relativistic bunches in the presence of detuning between the bunch repetition frequency and the frequency of the fundamental wave waveguide. In this case the wake waves radiated by individual bunches, there is the phase shift, and, beginning with some number of the bunch, the phase shift will be. The polarity of the field is changed and the subsequent bunches gain energy. Energy spectra of accelerated (decelerated) particles are obtained.
Викладено результати теоретичних досліджень процесу збудження кільватерного поля в діелектричному хвилеводі послідовністю релятивістських згустків при наявності розстройки між частотою проходження згустків і частотою основної хвилі хвилеводу. У цьому випадку для кільватерних хвиль, випромінюваних окремими згустками, виникає набіг фази і, починаючи з деякого номера згустку, набіг фази складе π . Змінюється полярність поля і наступні згустки набирають енергію. Отримані енергетичні спектри прискорених (уповільнених) частинок.
Изложены результаты теоретических исследований процесса возбуждения кильватерного поля в диэлектрическом волноводе последовательностью релятивистских сгустков при наличии расстройки между частотой следования сгустков и частотой основной волны волновода. В этом случае для кильватерных волн, излучаемых отдельными сгустками, возникает набег фазы и, начиная с некоторого номера сгустка, набег фазы составит π . Изменяется полярность поля и последующие сгустки набирают энергию. Получены энергетические спектры ускоренных (замедленных) частиц.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111930 |
| citation_txt |
Wakefield excitation and electron acceleration at detuning bunch repetition frequency and frequency of eigen principal mode of wakefield / V.A. Balakirev, I.N. Onishchenko, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2013. — № 4. — С. 80-83. — Бібліогр.: 11 назв. — англ. |
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2025-11-26T00:17:51Z |
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ISSN 1562-6016. ВАНТ. 2013. №4(86) 80
WAKEFIELD EXCITATION AND ELECTRON ACCELERATION
AT DETUNING BUNCH REPETITION FREQUENCY AND FREQUENCY
OF EIGEN PRINCIPAL MODE OF WAKEFIELD
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
Theoretical results of process of the wake field excitation in the dielectric waveguide by sequence of relativistic
bunches in the presence of detuning between the bunch repetition frequency and the frequency of the fundamental
wave waveguide. In this case the wake waves radiated by individual bunches, there is the phase shift, and, beginning
with some number of the bunch, the phase shift will be. The polarity of the field is changed and the subsequent
bunches gain energy. Energy spectra of accelerated (decelerated) particles are obtained.
PACS: 41.75.Jv, 41.75.Lx, 41.75.Ht, 96.50.Pw
INTRODUCTION
Linear accelerators are an effective device for ob-
taining sequences of relativistic electron bunches of
high quality. Such sequences bunches can be used for
excitation of intense wake fields in dielectric
waveguide. Wake fields in turn can be used to acceler-
ate electrons (positrons). We note that various aspects of
the wake fields excitation by relativistic electron
bunches are considered in [1-8]. Development of accel-
erator based on wakefields excited by a long sequence
of bunches is encountered with the physical and techni-
cal difficulties associated with the injection of electron
bunches in the accelerating phase of the wake field. To
solve the problem of injection of accelerated particles
the scheme is proposed, in which a separate accelerator-
injector is not required. This scheme is based on input
of the detuning between the bunch repetition frequency
and frequency of the excited eigenmode of the dielectric
waveguide. In this case for the wakefield, excited by
individual bunches, a phase shift will originate. Begin-
ning from certain number of bunches, which is deter-
mined by the frequency detuning the phase shift will be
accumulated to π and, therefore, the polarity of wake-
field is reversed. Subsequent bunches will gain energy.
Thus, in this acceleration scheme, using the same se-
quence of bunches, one part of bunches moving in the
head of the sequence will lose its energy to excite wake-
field and another part of bunches beginning from a men-
tioned number, will be accelerated.
In this paper the results of theoretical investigations
of wake field excitation by a sequence of electron
bunches and accelerate bunches of the same sequence in
the dielectric waveguide in the presence of frequency
detuning are presented.
1. STATEMENT OF THE PROBLEM
Let’s consider an infinite dielectric waveguide,
which is a cylindrical metal tube of radius b , filled with
homogeneous dielectric of permittivity ε , having axial
vacuum channel of radius a . In the vacuum channel
along of the longitudinal axis z a periodic sequence of
N axisymmetric relativistic electron bunches with arbi-
trary spatial period L propagates. The longitudinal and
transverse density profile of bunches are arbitrary.
We will find the wakefields in a dielectric waveguide
excited by a finite sequence of relativistic electron
bunches. In the ultrarelativistic case, when relativistic
factor of the electron bunches 0 1γ >> , the motion of
electrons in the longitudinal and transverse directions is
strongly "frozen." Therefore, at the first step of the in-
vestigation the approximation of the “given motion” of
bunches.
For the taken geometry of the dielectric waveguide
and electron bunches the wakefields will be excited as
E-type electromagnetic waves. The wakefield excitation
in a dielectric waveguide by a sequence of relativistic
electron bunches describes by inhomogeneous Maxwell
equations. We will solve the problem as follows. Firstly
we determine the wakefield of an infinitely thin ring
electron bunch with a longitudinal current density
0
0 0 0 0
0
( ) ( / / )
2ext
r rdj edN t z V z V
r
δ δ
π
−
= − − − , (1)
where 0r − is the radius of the ring, t − is the current
time, 0z − is the initial coordinate of the ring bunch,
0V − is the longitudinal velocity, z − is the longitudinal
coordinate, e is the electron charge, ( )xδ − is the delta
function, 0dN − is the number of particles in the electron
ring associated with the current density of the bunch at
the dielectric waveguide
0 0 0 0 0 0 0
0
1 ( , )2dN j z r r dr dz
eV
π= − , (2)
0 0 0( , )j z r is given initial current density electron bunch.
The resulting electromagnetic wake field of the bunch
of arbitrary axisymmetric form may be obtained by
summation (integration over initial longitudinal coordi-
nates 0z and initial radii 0r ) wake fields of the elemen-
tary ring bunches.
2. WAKE FIELD EXCITATION BY
SEQUENCE OF RELATIVISTIC BUNCHES
Wakefield of the periodic sequence of thin relativis-
tic ring electron bunches with period 0/T L V= is de-
termined by direct summation of individual bunch
wakefields. The expression for the longitudinal compo-
nent of the electric wakefield of a sequence of thin rela-
tivistic ring bunches (1) has the form
ISSN 1562-6016. ВАНТ. 2013. №4(86) 81
0 0 0 0
0
10 0 0
0
4 ( , ) ( )
( )
( )
cos( ( )) ( ),
n n n
zN n
n n n
N
n
m
edN F k a k b I r
E I r
aV D I a
mT mT
κ
κ
γ κ
ω τ θ τ
∞
⊥ ⊥
=
=
= ×
′
× − −
∑
∑
(3)
( ) /nD dD dω ω′ = at nω ω= , nω is the frequencies of
dielectric waveguide eigenmodes,
1 0 0 1( ) ( ) ( , ) ( ) ( , )D I a F a b I a F a bκεω κ κ κ κ κ κ
κ⊥ ⊥ ⊥ ⊥
⊥
= −
is dispersion equation, functions 0F and 1F are
0 0 0 0 0( , ) ( ) ( ) ( ) ( )n n n n n nF a b J a N b J b N aκ κ κ κ κ κ⊥ ⊥ ⊥ ⊥ ⊥ ⊥= − ,
1 1 0 0 1( , ) ( ) ( ) ( ) ( )n n n n n nF a b J a N b J b N aκ κ κ κ κ κ⊥ ⊥ ⊥ ⊥ ⊥ ⊥= − ,
2
2 2 0
0 02
0
( 1), ,n
n
V
V c
ωκ β ε β⊥ = − =
2
2
2 2
0 0
n
n V
ωκ
γ
= , ε − di-
electric constant, 0/t z Vτ = − .
Let’s consider the case when the bunch repetition
frequency 2 /r Tω π= is close to the fundamental mode
frequency 1ω of the dielectric waveguide, i.e. 1α << ,
1( ) /r rα ω ω ω= − . Then, for large amount of bunches
1N >> (in our case of the nonequidistant radial modes)
the excitation of the only fundamental mode 1n = will
be prevail. In such singlemode approximation the ex-
pression for the wakefield (3) takes the form
0 0 ( ) Re ,i
zm mE E I r e ωτκ −= Λ (4)
where 1 exp[ ( 1) ]
1 exp( )m
i m
i
− + Δ
Λ =
− Δ
, sin[( 1) / 2]
sin( / 2)m
m + Δ
Λ =
Δ
,
2παΔ = , 0E is the wakefield amplitude of separate
ring bunch, 1 1,κ κ ω ω≡ ≡ . Factor mΛ describes the
interference of the wakefield excited by separate
bunches and depends on the frequency detuning. In turn,
the function mΛ takes into account the change of am-
plitude with the number of bunch in the sequence. In the
absence of detuning 0Δ = we have 1m mΛ = + .
Wakefield amplitude increases linearly with the number
of bunch. Linear growth is due to the coherent summa-
tion of wakefields of separate bunches. Dependence
mΛ on the number of bunch at small detuning is pre-
sented in Fig. 1. At the beginning, linear growth of am-
plitude with increasing number of the bunch takes place.
Then the amplitude growth diminishes and for bunch
with number 1 1 /m π+ = Δ amplitude reaches the
maximum value 1 / sin( / 2) 2 /mΛ = Δ ≈ Δ . Further
the amplitude begins to decrease and for bunch with
number 2 1 2 /m π+ = Δ becames zero, and th picture is
periodically repeated. Bunches with numbers up to
/m π= Δ increases wakefield due to summation and,
therefore, lose its energy. On the other hand the region
of slowdown of amplitude wakefield for bunches with
number 2 / /mπ πΔ > > Δ contrary take away energy
from the wakefield spend its energy on corresponding
bunches acceleration.
Now we consider the change of the bunch energy
with the bunch number m . It is described by the equa-
tion
sin[ ( 1 / 2)] , 0,1,2,...
2sin( / 2)
md m m N
d
γ ε
ζ
Δ +
= − =
Δ
, (5)
where 2
0/ , / ez L eE L m cζ ε= = , mγ is the relativistic
factor. At the exact synchronism condition of the
bunches with the fundamental wave (i.e. detuning is
absent 0Δ = ) we have instead of (5) the equation
/ ( 1 / 2)md d mγ ζ ε= − + . The first bunch ( 0m = ) ex-
periences the deceleration field, which is equal / 2ε− ,
the second one 3 / 2ε− , the third one 5 / 2ε− , etc. In the
presence of frequency detuning in result of phase shift
the polarity of the wakefield will be changed. The proc-
ess of deceleration will be replaced by process of
bunches acceleration.
The graph of function sin[ ( 1 / 2)]mF m= − Δ + ,
which determines the force sign in the right side of the
equation (5), is shown in Fig. 1. Bunches with numbers
/ 1 ( 1)mπ Δ > > Δ << are decelerated, i.e. / 0sd dγ ζ < .
Accordingly, the wakefield amplitude is increased. In
turn, bunches with numbers 2 / /mπ πΔ > > Δ gain
the energy. The amplitude of the wakefield decreases.
Bunch with number max 3 / 2mγ π= Δ has maximum of
acceleration rate max / 1 / 1d dγ ζ = Δ >> .
Fig. 1. mΛ (red line) and mF (blue line) as functions
of bunch number. Frequency detuning α=0.05
Fig. 2 shows the dependence of the longitudinal
component of the electric field on the axis upon τ for
sequence of 10 infinitely thin ring bunches for the ra-
dius of bunches 0,5br
a
= , 1,44a = cm, radius of the
waveguide 4,325b = cm, the permittivity 2,1ε = ,
bunch energy 4,5bunchE = MeV in the absence of fre-
quency detuning α=0 and in its presence α=0.1. For
given parameters of the waveguide and the energy of
the electron bunches the frequency of the excited fun-
damental radial mode is 1 2,804f = GHz. In the reso-
nant case (without detuning) Fig. 2 shows a coherent
(linear) growth of wakefield with bunch number with its
jumps that corresponds to adding excited field of the
each next bunch. If detuning α = 0.1 the growth of
wakefield after 5-th bunch is changed by its decrease up
to zero on 10-th bunch.
ISSN 1562-6016. ВАНТ. 2013. №4(86) 82
Fig. 2. Dependence of the longitudinal component Ez
of wakefield excited by a sequence of infinitely thin ring
bunches upon τ at absence of frequency detuning α=0
(blue line) and its presence α=0.1 (red line)
Wakefield of the bunch of arbitrary form and, there-
fore, wakefield of the sequence of such bunches can be
found by summation of wakefields of elementary thin
ring bunches (3). For this in formula (2) we must set the
current density of a sequence of bunches at waveguide
entrance. Let’s consider wakefield of bunches sequence
with rectangular longitudinal and transverse profiles.
The expression for the longitudinal component of the
electric field excited by a sequence of bunches has the
view
1 0
02
1 0
4 2 ( ) ( , ) ( ) ( ),
( )
b n b n n n
zN n Nn
n n b n n n
Q I r a F k a k bE I r Z
a r D I a
κ κ κ τ
κ ω κ
∞
⊥ ⊥
=
=
′∑
0
( ) ( ) ( ),
N
Nn mn
m
Z Z mTτ τ θ τ
=
= −∑ (6)
sin ( );
1( ) sin ( ) sin ( );
( 1) ,
n b
mn n n b
n b
b
mT mT t mT
Z mT mT t
t
m T mT t
ω τ τ
τ ω τ ω τ
ω
τ
− + > >⎧
⎪= − − − −⎨
⎪ + > > +⎩
bt is the duration of the rectangular bunch, bQ is its
charge, br is the radius of the bunch, ( )mTθ τ − is the
unit Heaviside function.
Let’s consider the wakefield between the bunches.
In this case for first mode, performing the summation in
accordance with the formula (6), we find
1 1 1
1
2( ) sin( / 2) Re[exp( ( / 2)) ]N b b N
b
Z t i t
t
τ ω ω τ
ω
= − Λ . (7)
Factor 1
1
sin( / 2)
/ 2
b
b
t
t
ω
ω
describes the interference of
wakefield excited by head and back fronts of each sepa-
rate rectangular bunches, and the factor NΛ takes into
account the interference of wakefields excited by a se-
quence of bunches.
Fig. 3 presents the dependence of the longitudinal
component of the electric field upon τ for sequence of
10 bunches with rectangular longitudinal and transverse
profiles with duration 1 / 3b btτ ω π= = in the absence of
frequency detuning α=0 and in its presence α=0.1. The
other parameters are the same as in Fig. 2, and corre-
spond to the experimental facility of the accelerator "Al-
maz-2". It is seen that the behavior of the field differs
little from the case of infinitely short bunches (see
Fig. 2). Jumps of the field becomes smoother and
maximum amplitude of the beating is slightly less. So
bunches of duration 1 / 3b btτ ω π= = can be considered
as infinitely short ones. Region of growth of wakefield
amplitude corresponds to the deceleration of electron
bunches (bunches give up energy to wakefield). When
wakefield amplitude decreases the acceleration of the
electron bunches takes place
Fig. 3. Dependence of the longitudinal component Ez
of wakefield excited by a sequence of rectangular
bunches of duration 1 / 3b btτ ω π= = upon τ at absence
of frequency detuning α=0 (blue line)
and its presence α=0.1 (red line)
3. ENERGY SPECTRA OF BUNCH
PARTICLES
The energy spectrum of particles in the single-mode
approximation (the fundamental mode) for the case of
thin circular bunches (red line) and the length of the
rectangular bunches (blue line) is shown in the graphics
of Fig. 4. Interaction length has been set at six wave-
lengths of the fundamental mode a wake field. Detuning
is α=0.002. The values of other parameters listed above.
Changing of the frequency of the exciting electromag-
netic field, depending on the energy of the bunch is not
taken into account.
Fig. 4. Energy spectra particles sequences of bunches
for infinitely thin ring bunches (red line) and rectangu-
lar (blue line) at detuning α=0.002, N=500
For infinitely thin bunches energy spectrum has two
narrow peaks corresponding to acceleration and decel-
eration of bunches. For the chosen parameters the maxi-
mum increase of the relativistic factor is 0 0.9γΔ = or
410 keV on the initial value 4.5 MeV. For bunches of
finite width is a significant broadening of the energy
spectrum of particles. Instead of narrow peaks have
weakly expressed maxima. The maximum of the parti-
cle acceleration has been the increment 612 keV. The
reason for the broadening of the energy spectrum is the
gradient of accelerating (decelerating) electric field in-
side the bunches, which leads to the gradient energy
and, consequently, to a broadening of the energy spec-
trum.
Fig. 5 illustrates the above, which shows the change
of the relativistic factor of the first particle bunch (lilac
line) and last (blue line). Red curve corresponds to an
ISSN 1562-6016. ВАНТ. 2013. №4(86) 83
infinitely thin bunch. It is seen that for all bunches, ex-
cept bunches near the front and rear edges of the chain,
where the electric field is zero. There is a significant
difference in the increase in the energy of the first and
the last particles in each bunch.
Fig. 5. Dependences of energy first particle (magenta
line), last particle (blue line) of rectangular bunches
upon bunch number. Red curve corresponds case of
infinitely thin ring bunches. Detuning is α=0.002,
N=500
Thus, in this paper we have shown that the introduc-
tion of small detuning between the frequency of the
fundamental eigen mode excited in dielectric waveguide
and bunch repetition frequency allows to excite wake-
field efficiently and on the other hand, to solve the prob-
lem of injection of relativistic electron bunches in the
accelerating phase of the wakefield. In this acceleration
scheme the bunches of the head of the sequence, excite
the wakefield and thus they are decelerated. For
bunches with large numbers, in result of the phase shift,
polarity of the wakefield is reversed, and these bunches
are occurred in the acceleration process.
ACKNOWLEDGEMENTS
This work was supported by the US Department of
Energy/NNSA through the Global Initiatives for Prolif-
eration Prevention (GIPP) Program in Partnership with
the Science and Technology Center in Ukraine (Project
ANL-T2-247-UA and STCU Agreement P522).
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Article received 26.04.2013.
ВОЗБУЖДЕНИЕ КИЛЬВАТЕРНОГО ПОЛЯ И УСКОРЕНИЕ ЭЛЕКТРОНОВ ПРИ НАЛИЧИИ
РАССТРОЙКИ ЧАСТОТЫ СЛЕДОВАНИЯ СГУСТКОВ И ЧАСТОТЫ ОСНОВНОЙ
СОБСТВЕННОЙ КИЛЬВАТЕРНОЙ ВОЛНЫ
В.А. Балакирев, И.Н. Онищенко, А.П. Толстолужский
Изложены результаты теоретических исследований процесса возбуждения кильватерного поля в диэлек-
трическом волноводе последовательностью релятивистских сгустков при наличии расстройки между часто-
той следования сгустков и частотой основной волны волновода. В этом случае для кильватерных волн, из-
лучаемых отдельными сгустками, возникает набег фазы и, начиная с некоторого номера сгустка, набег фазы
составит π . Изменяется полярность поля и последующие сгустки набирают энергию. Получены энергетиче-
ские спектры ускоренных (замедленных) частиц.
ЗБУДЖЕННЯ КІЛЬВАТЕРНОГО ПОЛЯ І ПРИСКОРЕННЯ ЕЛЕКТРОНІВ ЗА НАЯВНОСТІ
РОЗСТРОЙКИ ЧАСТОТИ СЛІДУВАННЯ ЗГУСТКІВ І ЧАСТОТИ ОСНОВНОЇ ВЛАСНОЇ
КІЛЬВАТЕРНОЇ ХВИЛІ
В.А. Балакірєв, І.М. Оніщенко, О.П. Толстолужський
Викладено результати теоретичних досліджень процесу збудження кільватерного поля в діелектричному
хвилеводі послідовністю релятивістських згустків при наявності розстройки між частотою проходження
згустків і частотою основної хвилі хвилеводу. У цьому випадку для кільватерних хвиль, випромінюваних
окремими згустками, виникає набіг фази і, починаючи з деякого номера згустку, набіг фази складе π . Змі-
нюється полярність поля і наступні згустки набирають енергію. Отримані енергетичні спектри прискорених
(уповільнених) частинок.
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