Long-wavelength transition radiation by relativistic electrons in a pre-wave zone
The spectral-angular density of electromagnetic energy flux through the small detector in the "pre-wave zone" caused by relativistic electron in "forward" direction in the case of normal electron transition through a thin metallic transverse-bounded target is investigated. We sho...
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| author | Dobrovolsky, S.N. Shul’ga, N.F. |
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| citation_txt | Long-wavelength transition radiation by relativistic electrons in a pre-wave zone / S.N. Dobrovolsky, N.F. Shul'ga // Вопросы атомной науки и техники. — 2001. — № 6. — С. 121-124. — Бібліогр.: 12 назв. — англ. |
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| description | The spectral-angular density of electromagnetic energy flux through the small detector in the "pre-wave zone" caused by relativistic electron in "forward" direction in the case of normal electron transition through a thin metallic transverse-bounded target is investigated. We show that detected intensity can be very differ from the spectral-angular density of radiation as well it is measured in the "wave zone" as it is measured in "pre-wave zone" by infinite detector. This distortion is depended from the detector placing in the "pre-wave zone" and from the ratio between transversal size of target and effective transversal diameter of radiation formation region.
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LONG-WAVELENGTH TRANSITION RADIATION
BY RELATIVISTIC ELECTRONS IN A “PRE-WAVE ZONE”
S.N. Dobrovolsky, N.F. Shul'ga
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The spectral-angular density of electromagnetic energy flux through the small detector in the "pre-wave zone"
caused by relativistic electron in "forward" direction in the case of normal electron transition through a thin metallic
transverse-bounded target is investigated. We show that detected intensity can be very differ from the spectral-angu-
lar density of radiation as well it is measured in the "wave zone" as it is measured in "pre-wave zone" by infinite de-
tector. This distortion is depended from the detector placing in the "pre-wave zone" and from the ratio between
transversal size of target and effective transversal diameter of radiation formation region.
PACS: 41.60.-m, 41.90.+e, 41.75.Ht.
1. INTRODUCTION
The transition radiation (TR) from relativistic elec-
trons is widely used in last years in various experimental
investigations. For example, the long-wavelength TR is
explored for diagnostic of electron's beams and as a
source of quasi-monochromatic long-wavelength waves
[1-4]. The thin metallic plates are usually used for pro-
duction the long-wavelength transition radiation from
the ultra-short relativistic electron's bunches. Transition
radiation in the "forward" direction is formed over long
distance for the case of high-energy electrons and mil-
limeter waverange. This distance ("coherence length" or
"pre-wave zone") is determined in the order of value by
the relation λγ 22≈L , where γ is the Lorentz-factor of
electron and λ is the radiated wave length. It can reach
the tens of meters for relativistic electrons and millime-
ter waverange. So, if the detector is placed in the "pre-
wave zone", the strong interference between own elec-
trons fields and radiation fields is observed [5,6]. This
well-known effect was investigated theoretically and ex-
perimentally for case when both detector and metallic
plate have infinite diameter.
Recently the investigation of new effects connected
with finite diameter of targets and detector was present-
ed. So, the effect of targets diameter influence on the TR
intensity was investigated [7-9]. The influence of detec-
tor diameter on the TR intensity was studied in [10] for
"backward" TR in small-angle approximation. It has
been shown in [7-10] that the TR intensity is suppressed
in compare with classical case of infinite diameter of tar-
get and detector, when the target or detector diameter is
less or equal to γ λ . Such effects can be very important
for analysis of modern experiments concerned to the
millimeter and sub-millimeter TR of relativistic ultra-
short electron bunches.
In the present paper we will investigate the long-
wavelength "forward" TR from relativistic electron on a
thin metallic transversally bounded target. We will ana-
lyze the spectral-angular distribution of electromagnetic
energy flux through the small detector within "formation
zone".
2. THE TRANSITION RADIATION AND
OWN ELECTRON ELECTROMAGNE-
TIC FIELD
We will consider the case of normally electron pas-
sage through the center of thin metallic disc with radius
a . We assume that disk is fairly thin, i. e. the thickness
of target is less than radiated wavelength λ , but much
more than "penetration length" of electron field into a
metal. Let's the electron is moved along the OZ axis. We
will interest by the energy flux through the small plate
(detector), which placed at the distance r from the tar-
get center. Firstly, we should calculate the electromag-
netic fields, which are formed after electron passage
through the target. The summary electric field in for-
ward (concerning electrons passage) direction is consist-
ed of the own electron field ),()( te rE and TR field
),( trE′ :
),(),(),( )( ttt e rErErE ′+=+ . (1)
The Fourier component of the ),()( te rE field with re-
spect to time is determined by the expression
( ) ( )
−
=
v
iz
v
K
v
i
v
K
v
c
v
ee ω
γ
ω ρ
γγ
ω ρ
ργ
ω
ω exp12
01
vρrE , (2)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, 121-124. 121
where e - is an electron's charge, v=v , ρ is transver-
sal coordinate ( zeρr ⋅+= z , ze is unit vector of OZ
axis), 0K and 1K are McDonald's functions of zero and
first kind.
In the relativistic case (when 1> >γ ) the electron's
own field can be considered as transversal. It can be rep-
resented with accuracy of 1−γ by expression
( ) ( )
≈
v
iz
v
K
v
ee ω
γ
ω ρ
γ
ω
ρω exp2 12
ρrE . (3)
We assume that target is an ideal thin metallic
screen, therefore the part of own electron field which
fall into target is fully reflected and the part which
missed the target is diffracted onto space 0>z . So, we
can write for summary electric field and TR field the
next conditions
)()()()( )()( rErErE ee a ωωω ρ −Θ=′+ , 0=z ,
(4)
)()()( )( rErE ea ωω ρ−Θ−=′ , 0=z ,
where )(xΘ - is a Heaviside step function: 1)( =Θ x , if x
≥0, and 0)( =Θ x , if x < 0. The "forward" radiation field
),( trE′ is propagating in positive direction of OZ axis as
a wave packet of free electromagnetic waves
Now, by using the equations (3) and (5) we obtain
the expression for the "forward" and "backward" radia-
tion fields in the point zeρr ⋅+= z
( )
−±×
×
+
⋅
−=′ ∫
∞
22
0
2
2
1
2
exp
),()(
2)(
χω
γ
ωχ
γ
ωχχ ρ
χχ
ρω
ciz
v
v
aaFJ
d
v
e ρrE
, (5)
where
∫
+=
a
vKJd
vvF
0
11
2
2 )()( γ
ω ρχ ρρρ
γ χ
ω
γ
ωχ . (6)
The 1J is Bessel function of first kind. The formula (5)
is describing the "forward" TR field which appeared af-
ter relativistic electron passage through a thin metallic
disk. This expression is correct under all distances
greater than radiated wavelength from the target.
3. SPECTRAL-ANGULAR DENSITY
OF ELECTROMAGNETIC ENERGY FLUX
Now let's consider the electromagnetic energy flux
that crossed over the entire observation time through the
small detecting plate that is perpendicular to the vector
r and placed at the distance r from the target. We
should calculate the value of Poynting vector. The flux
of the Poynting vector is given by
∫ ×= dtdcS σ
π
2)(
4
nHE , (7)
where r/rn = and σ2d is an elementary area with nor-
mal vector n . Here the integration is over the surface of
detecting plate and the time of observation. We express
the field H through E and expand these fields into
Fourier integrals by time. After that, we can write sum-
mary energy flux in the "forward" direction on a unit of
frequency and on the unit of solid angle by expression
2)(
2
2
),(),(
4
zzcr
dd
dS e ρEρE ωωπω
′+=
Ω
, 0>ω . (8)
Here ϕϑϑ ddd sin=Ω ( ϑ and ϕ is the polar and the
azimuthal angles). We can calculate the transition radia-
tion intensity and energy flux through the detecting plate
by using the equations (5). So, the spectral-angular den-
sity of electromagnetic flux is followed
2)()(2
24
2
cos rade SSy
c
e
dd
dS −⋅=
Ω
ϑ
πβω , (9)
where
( ) [ ]11
1
1)( expsin −−−= βϑ γγ iqyKS e ,
∫
−
+
⋅
=
−
1
0
2
22
12)( 1exp
),()sin( xiq
x
uxFxyJdxxS rad
γ
γϑ
β ,
∫
+=
u
wxJwKdww
x
xuxF
0
11
2
)()(1)(),( γ
γ
γγ ,
1−⋅⋅= cry ω , ϑcosyq = , 1)( −⋅⋅⋅= γω cau , 1−⋅= cvβ .
The expression (9) describes the energy flux in "for-
ward" direction through the small detecting plate that is
seemed on the solid angle Ωd and placed into point
zeρr ⋅±= z . In other words, it is a spatial distribution of
electromagnetic energy flux for waves with the frequen-
cy ω . It is obviously that spectral-angular density of
electromagnetic energy flux is composed from the term
of energy flux of own electron field ("electron's term"),
from the term of energy flux of TR field ("radiation
term") and the interference term. The terms of equation
(9) depends on distance from the target and on target's
diameter. Further we will analyze the angular distribu-
tion of spectral-angular density of "forward" electro-
magnetic energy flux for various distances from the tar-
get.
4. DISCUSSION
Let's consider the angular distribution of electromag-
netic energy flux (9).
While the transversal size of target is essentially
greatly than transversal diameter of formation zone (
1> >u ), the function F is near to unit. This case corre-
sponds to the problem of transition radiation by rela-
122
tivistic electron on the infinite plate. So, for the dis-
tances γ λ>z we can to simplify the formula (9)
2)(
0
)(
0
2
222
2
cos rade SSy
c
e
dd
dS +=
Ω
ϑ
γπβω
, (10)
where
( )1
1
1)(
0 sin −−= ϑ γβ yKS e ,
∫
∞ −
−
+
=
0
2
22
1
12)(
0
2
cos
exp
1
)sin(
uyi
u
uyJ
duuS rad
γ
ϑϑγ
.
The term of radiation field energy flux in the equa-
tion (10) depend on the value 22
cos
γ
ϑy
. It is the ratio be-
tween the longitudinal detector-target distance and the
longitudinal "coherence length" ("pre-wave zone")
ωγ cl 22= . Such dependence for )(
0
redS from parameter
lz / is caused by the small diameter of detector. Let's
explain this fact in detail. Toward this end we consider
the spectral-angular density of electromagnetic energy
"forward" flux in case when the detector is a transversal-
ly-infinite plate. It represented by the following expres-
sion [7]
2)()(
22
2
2
2
sin
sin rade SS
c
e
dd
dS
∞∞−
∞ −
+
=
Ω γϑ
ϑ
πω
, (11)
where
( )vizS e ωexp)( =∞ , ( )cizS rad ωϑcosexp)( =∞ .
The module of )(radS∞ doesn't depend on target-detector
distance, whereas the module of )(
0
radyS sufficiently de-
pends on this distance. Let’s note that the spectral-angu-
lar density of TR does not depend from the distance to
the target in the classical case when the detector is an in-
finite plate. Following reason caused this situation. The
fields of TR are formed by the radiation from the target
surface with λ γ effective diameter. So, the interference
between radiation from various elementary radiators on
the target's surface is appeared. Thanks to this interfer-
ence the term )(
0
radS depends on the target-detector dis-
tance. Within the limits of "pre-wave zone" (i.e., when
λγ 22≤z ) only some part of whole radiated field is fall
into detector. The radiation source is not dot-like in this
case. In the "wave zone" (i.e., when λγ 22> >z ) the
module of )(
0
radyS doesn’t depends from the target-de-
tector distance. The radiation source is seemed as a dot-
like source in this case. The above-considered effect oc-
curred when the detector is small plate and it is absents
when the detector is an infinite plate.
The whole spectral-angular distribution (10) has also
the target-detector distance dependence that is caused by
the interference term. It is the interference between own
electron field and TR fields, which is sufficient within
the "coherence length" ωγ cl 22= .
The graphs on Fig. 1 show the angular distribution of
electromagnetic energy flux (10) under various distances
from the target. The formula (10) showed that the detec-
tor is placed closely to the electron's trajectory (i.e.,
when λ γρ ≤ ) the own electron's field gives the main
contribution to the summary electromagnetic energy
flux. This situation corresponds to the small-angle re-
gion on Fig. 1. If the detector is placed far from the elec-
tron's trajectory (i.e., when λ γρ ≥ ) the main contribu-
tion in the whole electromagnetic energy flux is caused
by the TR fields. This case corresponds to the great-an-
gle region on Fig. 1.
Fig. 1. Angular distribution of spectral-angular
density of electromagnetic energy flux in "forward" di-
rection for electron with 30=γ . Solid curve corre-
sponds case when 50=y , dotted curve - 300=y and
dashed curve - 3000=y . W(Θ)=cπ2e-2(dS/dωdΩ) and Θ
is marked in a degree scale
The angular distribution on Fig. 1 have showed that
if the detector is placed in the "pre-wave zone" ( 50=y ,
300=y ), the summary electromagnetic flux is strongly
oscillate in the large-angles region. These oscillations
are caused by the interference both between own elec-
tron's field and TR fields, and radiation fields from the
various elementary radiators on the target's surface. Both
this interference effects are decreased under increasing
of target-detector distance. So, within the "wave zone" (
3000=y ) these effects are diminished.
Now consider the case when the target's diameter is
near or less than λ γ . In this case the function F has
strong oscillations under 1≈u and 1< <F under 1< <u .
The suppression of "radiation term" in the equation (9)
is occurred. The additional distortion of summary elec-
tromagnetic energy flux through the small detector will
be observed.
Fig. 2 shows the angular dependence of electromag-
netic energy flux (9) in the "pre-wave zone" for various
diameters of target. The detected summary spectral-an-
gular electromagnetic energy "forward" flux is strongly
distorted at the large angles when target diameter is near
to the value λ γ .
5. CONCLUSIONS
We have considered the long-wavelength transition
radiation from relativistic electron on a thin metallic
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
123
disk in the "pre-wave zone". The general formula for
spectral-angular density of electromagnetic energy flux
through the small detector in the "forward" direction is
obtained. We have analyzed the spectral-angular distri-
bution of "forward" electromagnetic energy flux through
the small detector.
Fig. 2. Angular distribution of spectral-angular
density of electromagnetic energy flux in "forward" di-
rection for electron with 30=γ on the finite-diameter
target. Dotted curve corresponds to the case when
5.0=u , solid curve - 2=u . The distance from the tar-
get 300=y , W(Θ)=cπ2e-2(dS/dωdΩ) and Θ is marked in
a degree scale
It is shown that within the limits of "pre-wave zone"
the interference between radiation fields from various el-
ementary radiators and interference between own elec-
tron field and whole radiation field are occurred. Both
the spectral-angular density of whole electromagnetic
energy flux and spectral-angular density of TR is depend
on distance from the target.
We have analyzed the angular distribution of electro-
magnetic energy flux in the "pre-wave zone". The strong
oscillation of intensity of electromagnetic energy flux is
observed in the intermediate range of observation an-
gles. For small angles (when detector is placed nearly
from the electron trajectory) and for big angles (when
detector is placed at the length much more than λ γ from
the electron trajectory) the own electron's field energy
flux and the TR energy flux is dominated correspond-
ingly.
For the case when the target diameter is equal or less
than transversal size of "pre-wave zone" λ γ the suffi-
cient suppression of TR field contribution to the whole
electromagnetic energy "forward" flux is observed.
A number of experiments concerning long-wave-
length TR of relativistic electron bunches were per-
formed in the last years [1-4,6,11,12]. For this experi-
ments conditions (electrons with 10010 ÷≈γ , long-
wavelength TR with .101 cm÷≈λ ) the longitudinal (
λγ 22≈l ) and transversal ( λ γ≈⊥l ) size of "pre-wave
zone" are the macroscopic values and can be compared
with distance to the detecting setup and transversal tar-
get's size. So, the considered interference effects can
play important role in such experiments.
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6. A.R. Mkrtchyan, L.A. Gevorgian, B.V. Khachatryan,
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7. N.F. Shul'ga, S.N. Dobrovolsky. Theory of rela-
tivistic electron transition radiation in a thin metal
target // JETP. 2000, v. 90, №4, p. 579-583.
8. N.F. Shul'ga, S.N. Dobrovolsky. About transition ra-
diation by relativistic electrons in a thin target in a
millimeter range of waves // Phys. Lett A. 1999,
v. 259, p. 291-294.
9. N.F. Shul'ga, S.N. Dobrovolsky, V.V. Syshchenko.
About influence of target's shape on the transition radia-
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Physica. 2001, №3, p. 121-125.
10. V.A. Verzilov. Transition radiation in the pre-wave
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Phys. Rev. D. 1996, v. 53, p. 1684-1689.
12. P. Gorham, D. Saltzberg, P. Schoessow et al. Radio-
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124
S.N. Dobrovolsky, N.F. Shul'ga
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
PACS: 41.60.-m, 41.90.+e, 41.75.Ht.
1. INTRODUCTION
2. THE TRANSITION RADIATION AND OWN ELECTRON ELECTROMAGNE-TIC FIELD
3. SPECTRAL-ANGULAR DENSITY
OF ELECTROMAGNETIC ENERGY FLUX
4. DISCUSSION
Now consider the case when the target's diameter is near or less than . In this case the function has strong oscillations under and under . The suppression of "radiation term" in the equation (9) is occurred. The additional distortion of summary electromagnetic energy flux through the small detector will be observed.
Fig. 2 shows the angular dependence of electromagnetic energy flux (9) in the "pre-wave zone" for various diameters of target. The detected summary spectral-angular electromagnetic energy "forward" flux is strongly distorted at the large angles when target diameter is near to the value .
5. CONCLUSIONS
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79439 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T13:00:20Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Dobrovolsky, S.N. Shul’ga, N.F. 2015-04-01T19:56:17Z 2015-04-01T19:56:17Z 2001 Long-wavelength transition radiation by relativistic electrons in a pre-wave zone / S.N. Dobrovolsky, N.F. Shul'ga // Вопросы атомной науки и техники. — 2001. — № 6. — С. 121-124. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 41.60.-m, 41.90.+e, 41.75.Ht. https://nasplib.isofts.kiev.ua/handle/123456789/79439 The spectral-angular density of electromagnetic energy flux through the small detector in the "pre-wave zone" caused by relativistic electron in "forward" direction in the case of normal electron transition through a thin metallic transverse-bounded target is investigated. We show that detected intensity can be very differ from the spectral-angular density of radiation as well it is measured in the "wave zone" as it is measured in "pre-wave zone" by infinite detector. This distortion is depended from the detector placing in the "pre-wave zone" and from the ratio between transversal size of target and effective transversal diameter of radiation formation region. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics of high energies in matter and strong fields Long-wavelength transition radiation by relativistic electrons in a pre-wave zone Длинноволновое переходное излучение релятивистских электронов в “пре-волновой зоне” Article published earlier |
| spellingShingle | Long-wavelength transition radiation by relativistic electrons in a pre-wave zone Dobrovolsky, S.N. Shul’ga, N.F. Electrodynamics of high energies in matter and strong fields |
| title | Long-wavelength transition radiation by relativistic electrons in a pre-wave zone |
| title_alt | Длинноволновое переходное излучение релятивистских электронов в “пре-волновой зоне” |
| title_full | Long-wavelength transition radiation by relativistic electrons in a pre-wave zone |
| title_fullStr | Long-wavelength transition radiation by relativistic electrons in a pre-wave zone |
| title_full_unstemmed | Long-wavelength transition radiation by relativistic electrons in a pre-wave zone |
| title_short | Long-wavelength transition radiation by relativistic electrons in a pre-wave zone |
| title_sort | long-wavelength transition radiation by relativistic electrons in a pre-wave zone |
| topic | Electrodynamics of high energies in matter and strong fields |
| topic_facet | Electrodynamics of high energies in matter and strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79439 |
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