Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder
Transition radiation of a relativistic electron bunch, which arises when it collides with the end face of a semi-infinite ideally conducting cylinder, is considered. An electron bunch moves along the axis of a semi-infinite cylinder. Expressions for the field strength of electromagnetic radiation in...
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irk-123456789-1956472023-12-06T12:08:57Z Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder Balakirev, V.A. Onishchenko, I.N. Novel and non-standard acceleration technologies Transition radiation of a relativistic electron bunch, which arises when it collides with the end face of a semi-infinite ideally conducting cylinder, is considered. An electron bunch moves along the axis of a semi-infinite cylinder. Expressions for the field strength of electromagnetic radiation in the wave zone are obtained. The influence of the guiding properties of an ideally conducting cylinder on the directional diagram of the transition radiation is investigated. Розглянуто перехідне випромінювання релятивістського електронного згустка, що виникає при його зіткненні з торцем напівнескінченного ідеально провідного циліндра. Електронний згусток рухається вздовж осі циліндра. Отримано вирази для напруженості поля електромагнітного випромінювання в хвильовій зоні. Досліджено вплив направляючих властивостей ідеально провідного циліндра на діаграму направленості перехідного випромінювання. Рассмотрено переходное излучение релятивистского электронного сгустка, возникающее при его столкновении с торцом полубесконечного идеально проводящего цилиндра. Электронный сгусток движется вдоль оси полубесконечного цилиндра. Получены выражения для напряженности поля электромагнитного излучения в волновой зоне. Исследовано влияние направляющих свойств идеально проводящего цилиндра на диаграмму направленности переходного излучения. 2021 Article Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder / V.A. Balakirev, I.N. Onishchenko // Problems of Atomic Science and Technology. — 2021. — № 6. — С. 103-106. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq DOI: https://doi.org/10.46813/2021-136-103 http://dspace.nbuv.gov.ua/handle/123456789/195647 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Novel and non-standard acceleration technologies Novel and non-standard acceleration technologies |
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Novel and non-standard acceleration technologies Novel and non-standard acceleration technologies Balakirev, V.A. Onishchenko, I.N. Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder Вопросы атомной науки и техники |
| description |
Transition radiation of a relativistic electron bunch, which arises when it collides with the end face of a semi-infinite ideally conducting cylinder, is considered. An electron bunch moves along the axis of a semi-infinite cylinder. Expressions for the field strength of electromagnetic radiation in the wave zone are obtained. The influence of the guiding properties of an ideally conducting cylinder on the directional diagram of the transition radiation is investigated. |
| format |
Article |
| author |
Balakirev, V.A. Onishchenko, I.N. |
| author_facet |
Balakirev, V.A. Onishchenko, I.N. |
| author_sort |
Balakirev, V.A. |
| title |
Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder |
| title_short |
Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder |
| title_full |
Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder |
| title_fullStr |
Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder |
| title_full_unstemmed |
Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder |
| title_sort |
transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2021 |
| topic_facet |
Novel and non-standard acceleration technologies |
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http://dspace.nbuv.gov.ua/handle/123456789/195647 |
| citation_txt |
Transition radiation of a relativistic electron bunch on a semi-infinite metal cylinder / V.A. Balakirev, I.N. Onishchenko // Problems of Atomic Science and Technology. — 2021. — № 6. — С. 103-106. — Бібліогр.: 13 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT balakirevva transitionradiationofarelativisticelectronbunchonasemiinfinitemetalcylinder AT onishchenkoin transitionradiationofarelativisticelectronbunchonasemiinfinitemetalcylinder |
| first_indexed |
2025-07-16T23:45:14Z |
| last_indexed |
2025-07-16T23:45:14Z |
| _version_ |
1837849132502351872 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2021. № 6(136) 103
https://doi.org/10.46813/2021-136-103
TRANSITION RADIATION OF A RELATIVISTIC ELECTRON BUNCH
ON A SEMI-INFINITE METAL CYLINDER
V.A. Balakirev, I.N. Onishchenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine
E-mail: onish@kipt.kharkov.ua
Transition radiation of a relativistic electron bunch, which arises when it collides with the end face of a semi-
infinite ideally conducting cylinder, is considered. An electron bunch moves along the axis of a semi-infinite cylin-
der. Expressions for the field strength of electromagnetic radiation in the wave zone are obtained. The influence of
the guiding properties of an ideally conducting cylinder on the directional diagram of the transition radiation is in-
vestigated.
PACS: 41.75.Lx, 41.85.Ja, 41.69.Bq
INTRODUCTION
The effects of transition and diffraction radiation of
relativistic electron bunches [1 - 4] can underlie the pro-
cesses of excitation of pulsed electromagnetic radiation
in a wide frequency range [5 - 10]. In [10] the process
of excitation of an infinite ideally conducting cylinder
by a relativistic electron bunch was investigated. There-
at the situation was considered when an electron bunch
collides with a cylinder perpendicular to its cylindrical
surface. Below we will consider a semi-infinite ideally
conducting cylinder, which is excited by a relativistic
electron bunch when it collides with the end face of the
cylinder. It is in this way the ultra-wideband antennas
were excited by a relativistic electron beam [7 - 9]. Ex-
pressions for the intensity of electromagnetic radiation
in the wave zone are obtained. The directional diagram
of transition radiation and, first of all, the influence of
the guiding properties of a cylindrical conductor on its
formation is investigated.
1. STATEMENT OF THE PROBLEM.
BASIC EQUATIONS
We consider a semi-infinite ideally conducting circu-
lar cylinder of radius a. A relativistic electron bunch
moves in vacuum along the cylinder axis and collides
with its face end. For simplicity, we will consider an infi-
nitely thin tubular electron bunch with a current density
0 0( ) /1
2
z
b b
r r t z v
j e Q T
r t t
, (1)
where
ze is the unit vector in the longitudinal direction,
Q is the charge of the bunch,
0r is the radius of the
bunch,
0r a ,
0v ,
bt are velocity and duration of the
bunch,
0( )r r is the delta function. The dimensionless
function ( )T describes the longitudinal density profile
of the bunch and satisfies the normalization condition
( ) 1T d
,
where ( ) ( )T T is symmetrical function.
Let us represent the excited electromagnetic field
and the current density in the form of Fourier integrals
over frequencies. So, for example, for a magnetic field
we have
( , , ) ( , ) ,i tH r z t H r z e d
(2)
where
1
( , ) ( , , )
2
i tH r z H r z t e dt
is Fourier amplitude of the magnetic field.
The system of Maxwell's equations for the compo-
nents of the Fourier amplitudes of the electromagnetic
field has the form
0
r zE E
ik H
z r
, (3a)
0
0
( )1 2
lik z
z
b
r rQ
r H ik E T e
r r r ct r
, (3b)
0 r
H
ik E
z
, (3c)
where
0 /k c ,
0/lk v ,
00
0
1
2
i t
b
t
T T e dt
t
.
Let's divide the space into three regions:
0 0, 0;I r r z
0 , 0;II a r r z
, .III r a z
The problem is reduced to solving the inhomogene-
ous system of Maxwell's equations (3) with the follow-
ing boundary conditions. On the current surface
0r r ,
the longitudinal component of the electric field is con-
tinuous, and the magnetic field experiences a leap
0r r , ( 0) ( 0)II I
z zE z E z ,
0
2
( 0) ( 0) lik zII I
b
Q
H z H z T e
ct r
. (4)
On the boundary r a in the left half-space 0z ,
the tangential components of the electromagnetic field
are continuous
r a , ( 0) ( 0)III II
z zE z E z ,
( 0) ( 0)III IIH z H z . (5)
In the right half-space 0z on the surface of a per-
fectly conductive cylinder, the longitudinal component
of the electric field vanishes
r a , ( 0) 0III
zE z . (6)
And, finally, at the end of a semi-infinite cylinder,
the radial component of the electric field also vanishes
0a r , ( 0, ) ( 0, ) 0I II
r rE z r E z r . (7)
ISSN 1562-6016. ВАНТ. 2021. № 6(136) 104
2. METHOD OF SOLUTION. MAIN RESULTS
In the system under consideration, the electromag-
netic field disappears at infinity, so the electromagnetic
field can be sought in the form of a superposition of
waves propagating in the longitudinal direction
1 0( , ) ( ) ( )I ikz
zE r z e k J vr e dk
, (8a)
1
0 1 1( , ) ( ) ( )I ikzH r z ik e k v J vr e dk
, (8b)
(1)
2 0 3 0( , ) ( ) ( ) ( ) ( )II ikz
zE r z e k J vr e k H vr e dk
, (8c)
1 (1)
0 2 1 3 1( , ) ( ) ( )II ikzH r z ik v e J vr e H vr e dk
,(8d)
(1)
4 0( , ) ( ) ( )III ikz
zE r z e k H vr e dk
, (8e)
1 (1)
0 4 1( , ) ( ) ( )III ikzH r z ik e k v H vr e dk
, (8f)
2 2
0v k k ,
1 4
( )... ( )e k e k are the sought amplitudes of
the fields in the corresponding regions, (1)( ), ( )n nJ vr H vr
are the Bessel and Hankel functions, 0.1n .
Substituting fields (8) into boundary conditions (4) -
(6), we obtain a system of paired integral equations,
which, using the Wiener-Paley-Rappoport lemma, can
be reduced to the following system of functional equa-
tions
(1)
1 2 0 0 3 0 0[ ( ) ( )] ( ) ( ) ( ) ( )e k e k J vr e k H vr k , (9a)
(1)0
1 2 1 0 3 1 0{[ ( ) ( )] ( ) ( ) ( )} ( )
k
i e k e k J vr e k H vr k
v
0
2 1
2 ( )b l
Q
T
ct r i k k
, (9b)
(1)
2 0 3 4 0 0( ) ( ) [ ( ) ( )] ( ) ( )e k J va e k e k H vr k , (9c)
(1)0
2 1 3 4 1 0{ ( ) ( ) [ ( ) ( )] ( )} ( )
k
i e k J va e k e k H vr k
v
,
(1)
4 0 0( ) ( ) ( )e k H vr k , (9d)
where ( )k , ( )k , ( )k , ( )k are analytic func-
tions in the lower half-plane of the complex variable k,
and ( )k is the analitic function in the upper one,
Im 0.lk
From the boundary condition at the face end of the
cylinder (7) it follows that the following amplitudes are
even
( ) ( ), 1,2,3e k e k . (10)
It can be shown that the system of functional equa-
tions (9), taking into account relations (10), is equiva-
lent to the system of Hilbert boundary value problems
on the real axis Im 0k
( ) ( ) 0,k k (11a)
2 2
0
2
( ) ( ) ,
( )b l
Q k
k k T
ct r i k k
(11b)
( ) ( ) ( ) ( )k k k k , (12a)
( ) ( ) ( )[ ( ) ( )]k k H k k k , (12b)
0
1 0 0 0( )[ ( ) ( )] ( )[ ( ) ( )]
k
i J vr k k J vr k k
v
0
1 0
0
( )[ ( ) ( )] ( )[ ( ) ( )]
ka
i J va k k J va k k
r v
0
0 02 (1) 2 2
00 0
2 ( ) ( ) 2
( )
( ) ( )
l
b l
k kk k Q
T J vr
ct rv r H va i k k
,
where ( ) ( ), ( ) ( )k k k k etc,
(1)
0 1
(1)
0
( )
( )
( )
ik H va
H k
v H va
.
Solutions of the boundary problems (11a), (11b) are
found trivially
( ) ( ) 0k k ,
0
2 1
( )
2 ( )b l
Q
k T
ct r i k k
,
0
2 1
( )
2 ( )b l
Q
k T
ct r i k k
.
Using the Sokhotsky formulas [11], the system of
boundary problems (12) can reduce to the following
singular equation for the function
ˆ ˆ ˆ2 ( ) ( ) ( ) ( ) ( ) ( )I k L k LH k k H k L k
0 0
2 2
0
( )4
( ) ( )
l
b l
J vr kQ
T
ct a J va i k k
, (13)
where
0 1
0
( )
( )
( )
ik J va
I k
v J va
,
1 ( )ˆ ( ) .
f k
Lf k dk
i k k
(14)
Integral (14) should be understood in the sense of
the principal value. It is not possible to find a general
solution to this integral equation. However, in the limit-
ing case of a conducting cylinder of small radius
0 1k a , the solution can be found approximately. In
this limiting case, the term in the integral equation (13),
which proportional
0( ) ~I k k a can be neglected. Then
the integral equation is simplified and takes the form
2 2
4ˆ ˆ( ) ( ) ( ) ( ) .
( )
l
b l
kQ
LH k k H k L k T
ct a i k k
(15)
After replacement
( ) ( ) ( )U k H k k
the simplified integral equation (15), in turn, is reduced
to the boundary value problem [12]
2 2
2
( )
l
b l
kQ
U H T
ct a i k k
,
the solution of which can be found explicitly
0
2
0
( )4
( )
(2 )
l
b
X k k kkQ
k T
ct a k i
0
2 2( )( )( )lC
k k
dk
X k k k k k
, (16)
where ( )X k is solution of homogeneous conjugation
problem
( ) ( ) ( )X k G k X k ,
(1)
0
(1)
1
( )
( )
( )
H va
G k
iH va
.
The solution to this boundary value problem has the
form [11]
1 ln ( )
( ) exp
2
C
G k
X k dk
i k k
,
1
( )
( )
X k
X k
. (17)
ISSN 1562-6016. ВАНТ. 2021. № 6(136) 105
The contour C runs along the real axis and bypasses
the singularity k k from below. The integral included
in (16) is easily calculated by closing the integration
contour into the lower half-plane. As a result, we obtain
0 0
0
( )(2 ( )
( )
2 ( ) ( )
l
b l l
k k k kQ X k
k T
сt ak i k k X k
.
Relation (9d), taking into account the obtained ex-
pression for the function ( )k , makes it possible to de-
termine the amplitude
4 ( )e k for the region r a and,
accordingly, the expression for the magnetic field in this
region
02
2 ( )
lIII
b l
k kQT
H
ct a X k
(1)
1
(1)
00
( ) ( )
( )( )
ikz
l
X k H vr e dk
H vak k k k
. (18)
We are primarily interested in electromagnetic radia-
tion in the wave zone
0 1k R , R is the distance from
the origin of coordinates to the point of observation.
Using the asymptotic representation of the Hankel func-
tion for large values of the argument 1vr , after
passing to a spherical coordinate system
cos , sinz R r R ( is polar angle) and replac-
ing the integration variable
0 cosk k w instead of the
integral representation (18), we obtain
0 0 /4
0
( )2 2
sin2 ( )
lIII i
b l
k k kQ
H i T e
ct a k RX k
0 cos( )0
(1)
0 0
( cos ) sin
1 cos( sin )
i
ik R w
i
X k w w
e dw
wH k w
. (19)
The integration contour runs along a straight line
from i to , then along a segment 0w and
again along a straight line from 0 to i . In the wave
zone, integral (19) can be estimated by the saddle-point
method. The saddle point w lies within the real axis
line segment. As a result, we obtain the following ex-
pression for the radiation field
0 0
0
( ) sin
(1 cos ) 1 cos
lIII
b l
k k kQ
H T
ct k
0
0( cos )
( )
ik R
l
X k e
RX k
, (20)
0 0 /v c . The function ( )X k can be calculated ex-
actly. However, in the considered quasistatic approxi-
mation
0
1k a , we can use the asymptotic representa-
tion of this function [13]
0
0
0 0
ln
1
( )
2( )
ln
2 ( )
i
i
k a
X k
ik k a
a k k k
,
1.78 is Euler's constant. Accordingly, the expres-
sion for the magnetic field of radiation (20) takes the
form
0
1
cos
ln
sin
2
III
b
Q
H T
ict
k a
0
0
sin sin
1 cos 1 cos
ik R
e
R
, (21)
where
0
00 0 0
22
ln ln
(1 ) k ak a
,
0 is a relativistic factor. The first term in formula (21)
describes the transition radiation arising from the colli-
sion of an electron bunch with the end face of a semi-
infinite conducting cylinder. The second term has a sin-
gularity for the angle 0 . The radiation field along
the cylindrical conductor goes to infinity. This effect is
explained by the guiding properties of a perfectly con-
ducting cylinder [13]. The feature of the field is
integrable. The total energy flow remains finite. An
analysis of the radiation pattern shows that the maxi-
mum of the radiation field in the direction characteristic
of transition radiation
01/ is never formed. This is
due to the strong distortion of the radiation field in this
direction by radiation focused along the surface of the
cylindrical conductor. Turning to infinity towards
0 , the amplitude of the radiation field decreases
monotonically with increasing polar angle . Thus, a
thin semi-infinite cylinder is actually an antenna of a
traveling wave excited by a relativistic electron bunch.
Directly from the form of the second term in square
brackets in formula (21) it follows that the antenna radia-
tion excites a current induced on the surface of a perfectly
conducting cylinder, which propagates along the cylinder
at the speed of light. Moreover, this current is in
antiphase with respect to the current of the electron
bunch.
A single bunch excites an electromagnetic field with
a continuous frequency spectrum, and the shape of the
radiation pulse is determined by the Fourier integral (2),
which is conveniently represented in the form
*
0
III III i t III i tH H e H e d
. (22)
Taking into account the expression (21), relation
(22) takes the form
2 2
20 0
cos sin
( ) 2
4
III
b
L t t
Q F
H T d
Rct
L
,(23)
where
0
sin
( )
(1 cos )(1 cos )
F
, 0ln
,
0
0
1
ln
2sin
2
L
, 0
0
2 c
a
,
R
t t
c
.
For small polar angles 1 under the condition
0 0
0 0
ln ~ ln
4
btL
expression (23) is essentially simplified and takes the
form
2
0
( )
/III
b
Q F
H t R c
Rct
. (24)
Function
ISSN 1562-6016. ВАНТ. 2021. № 6(136) 106
0
00
0
ln
/ cos /
ln
t R c T t R c d
describes the shape of the transition radiation pulse. For
not too small angles
0
0
0
1
ln ~ ln lnbt
or
0 0
1
bt
we have
/
/
b
t R c
t R c T
t
,
that is, the electromagnetic pulse exactly repeats the
shape of the electron bunch. Otherwise, the shape of the
electromagnetic pulse will depend, albeit weakly, on the
polar angle and will slightly differ from the shape of
the electron bunch.
CONCLUSIONS
Transition radiation of a relativistic electron bunch,
which arises when it collides with the end face of a
semi-infinite perfectly conductive cylinder, is consid-
ered. The electron bunch moves along the cylinder axis.
Expressions for the intensity of the magnetic field of
radiation in the wave zone are obtained. It is shown that
this expression contains two terms. The first term de-
scribes the actual transition electromagnetic radiation of
the electron bunch. Its characteristics substantially de-
pend on the energy (relativistic factor) of the electron
bunch. The second term describes the radiation of the
current induced on the surface of a perfectly conducting
cylinder and propagating along the cylinder at the speed
of light in vacuum. The strength of this field has
integrable singularity (turns to infinity) strictly along
the surface of the cylinder. The peculiarity is due to the
guiding properties of a perfectly conducting cylinder.
The spatial structure (directional pattern) of this elec-
tromagnetic radiation does not depend on the energy of
the bunch and is determined by the geometry of the cy-
lindrical antenna under consideration. The amplitude of
the radiation field decreases monotonically with increas-
ing polar angle . The maximum of the radiation field
in the direction characteristic of transition radiation
01/ is never formed. This is due to the strong dis-
tortion of the radiation field in this direction by radia-
tion propagating along the surface of the cylindrical
conductor. The shape of the emitted electromagnetic
pulse has been determined.
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Article received 12.10.2021
ПЕРЕХОДНОЕ ИЗЛУЧЕНИЕ РЕЛЯТИВИСТСКОГО ЭЛЕКТРОННОГО СГУСТКА
НА ПОЛУБЕСКОНЕЧНОМ МЕТАЛЛИЧЕСКОМ ЦИЛИНДРЕ
В.А. Балакирев, И.Н. Онищенко
Рассмотрено переходное излучение релятивистского электронного сгустка, возникающее при его столкновении с
торцом полубесконечного идеально проводящего цилиндра. Электронный сгусток движется вдоль оси полубесконечно-
го цилиндра. Получены выражения для напряженности поля электромагнитного излучения в волновой зоне. Исследова-
но влияние направляющих свойств идеально проводящего цилиндра на диаграмму направленности переходного излуче-
ния.
ПЕРЕХІДНЕ ВИПРОМІНЮВАННЯ РЕЛЯТИВІСТСЬКОГО ЕЛЕКТРОННОГО ЗГУСТКА
НА НАПIВНЕСКIНЧЕННОМУ МЕТАЛЕВОМУ ЦИЛІНДРІ
В.А. Балакiрев, I.М. Онiщенко
Розглянуто перехідне випромінювання релятивістського електронного згустка, що виникає при його зіткненні з тор-
цем напівнескінченного ідеально провідного циліндра. Електронний згусток рухається вздовж осі циліндра. Отримано
вирази для напруженості поля електромагнітного випромінювання в хвильовій зоні. Досліджено вплив направляючих
властивостей ідеально провідного циліндра на діаграму направленості перехідного випромінювання.
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