Electromagnetic wave scattering on a dielectric cylinder in Born approximation
The scattering of the plane electromagnetic wave on a spatially extended, fiber-lake target is considered. The fomula for the scattering cross section is obtained using the approximation analogous to Born one in quantum mechanics.
Збережено в:
| Дата: | 2015 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
|
| Назва видання: | Вопросы атомной науки и техники |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/112356 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Electromagnetic wave scattering on a dielectric cylinder in born approximation / V.V. Syshchenko // Вопросы атомной науки и техники. — 2015. — № 6. — С. 65-67. — Бібліогр.: 5 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-112356 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1123562017-01-21T03:02:50Z Electromagnetic wave scattering on a dielectric cylinder in Born approximation Syshchenko, V.V. Новые и нестандартные ускорительные технологии The scattering of the plane electromagnetic wave on a spatially extended, fiber-lake target is considered. The fomula for the scattering cross section is obtained using the approximation analogous to Born one in quantum mechanics. Розглянуто розсіяння плоскої електромагнітної хвилі на просторово протяжної ниткуватої мішені. Отримано вираз для перерізу розсіяння з використанням наближення, що аналогічно борнівському наближенню у квантовій механіці. Рассмотрено рассеяние плоской электромагнитной волны на пространственно протяженной нитевидной мишени. Получено выражение для сечения рассеяния с использованием приближения, аналогичного борновскому приближению в квантовой механике. 2015 Article Electromagnetic wave scattering on a dielectric cylinder in born approximation / V.V. Syshchenko // Вопросы атомной науки и техники. — 2015. — № 6. — С. 65-67. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 42.25.Bs, 42.25.Fx, 42.81.-i http://dspace.nbuv.gov.ua/handle/123456789/112356 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Новые и нестандартные ускорительные технологии Новые и нестандартные ускорительные технологии |
| spellingShingle |
Новые и нестандартные ускорительные технологии Новые и нестандартные ускорительные технологии Syshchenko, V.V. Electromagnetic wave scattering on a dielectric cylinder in Born approximation Вопросы атомной науки и техники |
| description |
The scattering of the plane electromagnetic wave on a spatially extended, fiber-lake target is considered. The fomula for the scattering cross section is obtained using the approximation analogous to Born one in quantum mechanics. |
| format |
Article |
| author |
Syshchenko, V.V. |
| author_facet |
Syshchenko, V.V. |
| author_sort |
Syshchenko, V.V. |
| title |
Electromagnetic wave scattering on a dielectric cylinder in Born approximation |
| title_short |
Electromagnetic wave scattering on a dielectric cylinder in Born approximation |
| title_full |
Electromagnetic wave scattering on a dielectric cylinder in Born approximation |
| title_fullStr |
Electromagnetic wave scattering on a dielectric cylinder in Born approximation |
| title_full_unstemmed |
Electromagnetic wave scattering on a dielectric cylinder in Born approximation |
| title_sort |
electromagnetic wave scattering on a dielectric cylinder in born approximation |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2015 |
| topic_facet |
Новые и нестандартные ускорительные технологии |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112356 |
| citation_txt |
Electromagnetic wave scattering on a dielectric cylinder in born approximation / V.V. Syshchenko // Вопросы атомной науки и техники. — 2015. — № 6. — С. 65-67. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT syshchenkovv electromagneticwavescatteringonadielectriccylinderinbornapproximation |
| first_indexed |
2025-07-08T03:47:52Z |
| last_indexed |
2025-07-08T03:47:52Z |
| _version_ |
1837049025102086144 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №6(100) 65
ELECTROMAGNETIC WAVE SCATTERING
ON A DIELECTRIC CYLINDER IN BORN APPROXIMATION
V.V. Syshchenko
Belgorod National Research University, Belgorod, Russian Federation
E-mail: syshch@yandex.ru
The scattering of the plane electromagnetic wave on a spatially extended, fiber-lake target is considered. The
fomula for the scattering cross section is obtained using the approximation analogous to Born one in quantum me-
chanics.
PACS: 42.25.Bs, 42.25.Fx, 42.81.-i
INTRODUCTION
The problem of multiple scattering of the electro-
magnetic wave under oblique incidence on the system
of parallel dielectric fibers was considered in [1]. The
scattering on the single fiber had been described there in
the limit of infinitely thin fiber. The wave scattering
cross section in that case possesses the axial symmetry
for small angle of incidence ψ<<1.
The wave scattering cross section by the cylindric
fiber of finite radius is calculated in the present article
using Born approximation. The substantial axial anisot-
ropy of the scattered radiation (especially for large ψ
angles) is demonstrated.
1. BORN APPROXIMATION IN RADIATION
SCATTERING THEORY
The equations for the electric field of a monochro-
matic wave E(r)e-iωt in non-uniform medium created in
the target under passage of the particle satisfies the
equations [2]
(Δ + ε ω2/c2)E = grad div E, (1)
div (εE) = 0. (2)
(where Δ is the Laplasian operator and ε(r) is the dielec-
tric permittivity of the medium for the given frequency
ω) could be easily derived from Maxwell equations
(see, e.g. [2, §68]). The solution of Eqs. (1), (2) for the
case when our nonuniform medium has the structure of
a spatially localized target in vacuum are convenient to
find as the superposition
E(r) = E(0)(r) + E(1)(r), (3)
where E(0) is the electric field of the incident wave that
satisfies the equation
(Δ + ω2/c2)E(0)= 0. (4)
Hence the field E(1) in (3) has to be treated as the
field of the scattered radiation.
Eq. (1) can be written using (2), (3) and (4) in the
form
2
(1)
2
2
2
(1 ) grad div((1- ) ) .
c
c
ω
ω ε ε
∆ + =
− +
E
E E
(5)
The last equation could be presented in the integral
form,
+′′−′−= ∫ )())(1()()( 2
2
)1( rErrrrE εω
c
G
[ ]} rd ′′′−+ 3)())(1(div grad rErε , (6)
where G(r-r′) is the Green function for Eq. (5),
( ') 3
2 2 3( ')
( / ) 0 (2 )
ie dG
c i
κ
ω κ π
−
− =
− +∫
κ r r
r r .
The asymptotic of that function on large distances r
from the domain where ε(r) is different from unit,
'1( ')
4
f
f
ik r
i
r
eG e
rπ
−
→∞
− → − k rr r
'
4
1 rki
ri
e
r
e
−−→
ω
π
.
where k f = (ω/c)r/r is the wave vector of the scattered
wave, |k f|=|k i |, is needed to find the field of the scat-
tered radiation. Substituting the last formula into (6) and
integrating the second term by parts we obtain
( ) (1)
2
2
( ) ( )
1 ( ) ,
4
f
scattered
r
ik r
f f
e
r c
ω
π
→∞= =
= − − ⋅
E r E r
I k k I
(7)
where
3(1 ( )) ( ) fie d rε −= −∫ k rI r E r . (8)
We see that the integrand in (8) would be nonzero
only in the domain where the dielectric permittivity is
not equal to unit. That illustrates the origin of the scat-
tered radiation from the motion of the electrons in the
medium excited by the incident wave.
To select the polarization of the scattered wave one
have to project (7) to the chosen polarization vector
e f ⊥ k f:
IerEe ⋅=⋅ f
rik
scattered
f cr
e f
2
2
)(
4
1)( ω
π
. (9)
The integral form (6) of the field equation (5) that
leads to (7), (8) is useful for construction of various
approximate solutions. The simplest case is |1 − ε| << 1,
when the difference (1 − ε) in (8) could be treated as a
small perturbation, hence the whole electric field in the
target E(r) in (8) could be replaced by the field E(0) of
the incident wave. This approximation is equivalent to
Born approximation in the quantum theory of scattering
(see, e.g., [3]).
Substituting into (8) the field rkeE ii
ie∝)0( of the
incident plane wave (where ki is the incident wave vec-
tor, ei is its polarization vector, and the wave amplitude
has no matter for computation of the scattering cross
section), calculating the average energy flux of the scat-
tered wave (7) to the solid angle dΩ and dividing it by
ISSN 1562-6016. ВАНТ. 2015. №6(100) 66
the average intensity of the incident wave we obtain the
scattering cross section
2 2( )
2 2(4 )
B
f
d
d c
σ ω
π
= ×
Ω
k I , (10)
where
( )( ) 3(1 ( )) i fiB
i e d rε −= − =∫ k k rI e r
3(1 ( )) i
i e d rε= −∫ qre r , (11)
where q = ki-k f.
The scattering cross section for the radiation with
the polarization e f selected by the detector would be
described by the formula
4 2( )
2 4(4 )
B
f
d
d c
σ ω
π
= ⋅
Ω
e I , (12)
as it could be easy to see from (9).
2. THE SCATTERING ON THE
DIELECTRIC CYLINDER
The formulae (10) and (12) describe the radiation
scattering by the target of arbitrary structure. Consider
now the simplest case of uniform cylinder of the radius
a and the length L → ∞ as the target (Figure), when the
cylinder’s axis makes the angle ψ with the direction of
the wave incidence ki . The integrals in (11) could be
easily calculated after rotation of the coordinate axes to
coincidence of the new axis z′ with the cylinder’s axis.
Then the integration over z′ gives δ-function, and the in-
tegration in the transverse plane gives Bessel function, so
( ) 2 1
||
( )(1 ) (2 ) ( )B
cyl i
J q aa q
q a
ε π δ ⊥
⊥
= −I e , (13)
where ||q and ⊥q are the components of q parallel and
perpendicular to the cylinder’s axis. The presence of the
δ-function expresses the equality of the components of
ki and k f that parallel to the cylinder’s axis, that means
the azimuthal character of the scattering: since the abso-
lute values of the wave vectors ki and k f are also equal,
the scattered radiation will be directed along the surface
of the cone with the axis along the cylinder and the half-
opening angle equal to the incidence angle ψ.
The azimuthal character of the scattering permits
clear interpretation as a manifestation of Cherenkov
mechanism. Indeed, the incident wave produces a per-
turbation in the medium that moves along the fiber with
the phase velocity ν = ω/(ki)|| = c/cosψ > c. This super-
luminal motion generates the radiation analogous to
Cherenkov one. The half-opening angle of the Cheren-
kov cone cosθ = c/v is just equal to ψ.
Substitution of (13) into (10) gives (using the rule of
δ-function squaring, [δ(q ||)]2 = δ(q ||)·L/2π) the cross sec-
tion
2 2
2
2
4 2 1
||
2
( )
(1 ) ( ) .
f i
d
d c
J q aLa q
q a
σ π ω
ε δ ⊥
⊥
= × ×
Ω
× −
k e
(14)
The cross section corresponding to the polarization
e f is described by the formula
4 2
4
2
4 2 1
||
2
( )
(1 ) ( ) .
f i
d
d c
J q aLa q
q a
σ π ω
ε δ ⊥
⊥
= ⋅ ×
Ω
× −
e e
(15)
In the limiting case of infinitely thin fiber, a → 0,
Eq. (15) agrees with the corresponding result of [1] as
well as the results of [4].
Direction diagrams for the radiation scattered by uniform cylinder; the incident radiation is directed along
the z axis. Dashed curve represents the relative intensity of the scattered radiation according to (16), the solid curve
gives the same in the limit of infinitely thin fiber when ( ) 211 →⊥′⊥′ aqaqJ , the cone represents the azimuthally
symmetric scattering. The angle of incidence is ψ = 0.2 radian (upper row) and ψ = 0.4 radian (lower row); other
parameters are aω/c = 1 (left column) and aω/c = 3 (right column)
ISSN 1562-6016. ВАНТ. 2015. №6(100) 67
The scattering cross section for unpolarized radia-
tion results from (14) after averaging over polarizations
of the incident wave:
2 2
2
2 2
2
4 2 1
||
( )
4
( )
(1 ) ( ) .
f z
d k
d c c
J q aLa q
q a
σ π ω ω
ε δ ⊥
⊥
= + × Ω
× −
(16)
The directional diagram of the scattered radiation is
presented on Figure the relations
| |2 sin sin ,
2
q
c
ω φψ⊥ =
| |( ) cos 2asin sin sin ,
2f zk
c
ω φψ =
(where the angle φ is measured from the direction
( )⊥ik ) had been used for plotting.
The only source of azimuthal asymmetry in (16) in
the limit a → 0 is the factor 2 2 2( ) /f zk cω+ that tends
to 2ω2/c2 under ψ → 0. The account of the finite cylin-
der radius leads to the increase of the azimuthal asym-
metry via the factor ( ) aqaqJ ⊥⊥1 . The analogous be-
havior had been demonstrated in [5] for the transition
radiation by fast charged particles on the fiber-like tar-
gets.
CONCLUSIONS
The scattering of the electromagnetic wave under its
oblique incidence on the linear extended target is con-
sidered. The formulae for the scattering cross section
are obtained using Born approximation. The azimuthal
character of the scattering is demonstrated as well as the
axial asymmetry around the target axis for the finite
angle of incidence and the finite target thickness. The
results could be used for improving the kinetic theory of
propagation of the radiation through the system of paral-
lel fibers [1].
ACKNOWLEDGEMENTS
This work was supported by the Russian Science
Foundation Grant (project N 15-12-10019).
REFERENCES
1. N.F. Shul’ga, V.V. Syshchenko // Problems of
Atomic Science and Technology. Series “Nuclear
Physics Investigations”. 2014, № 5, p. 115.
2. L.D. Landau and E.M. Lifshitz. Electrodynamics of
Continuous Media. Pergamon Press, 1960.
3. L.D. Landau and E.M. Lifshitz. Quantum Mechan-
ics: Non-Relativistic Theory. Pergamon Press, 1977.
4. M. Kerker, D.D. Cooke, J.M. Carlin // Journal of the
Optical Society of America. 1970, v. 60, p. 1236.
5. N.F. Shul’ga, V.V. Syshchenko // Phys. Lett. 2003,
A 313, p. 307.
Article received 16.06.2015
РАССЕЯНИЕ ЭЛЕКТРОМАГНИТНОЙ ВОЛНЫ НА ДИЭЛЕКТРИЧЕСКОМ ЦИЛИНДРЕ
В БОРНОВСКОМ ПРИБЛИЖЕНИИ
В.В. Сыщенко
Рассмотрено рассеяние плоской электромагнитной волны на пространственно протяженной нитевидной
мишени. Получено выражение для сечения рассеяния с использованием приближения, аналогичного бор-
новскому приближению в квантовой механике.
РОЗСІЯННЯ ЕЛЕКТРОМАГНІТНОЇ ХВИЛІ НА ДІЕЛЕКТРИЧНОМУ ЦИЛIНДРI
У БОРНIВСЬКОМУ НАБЛИЖЕННІ
В.В. Сищенко
Розглянуто розсіяння плоскої електромагнітної хвилі на просторово протяжної ниткуватої мішені. Отри-
мано вираз для перерізу розсіяння з використанням наближення, що аналогічно борнівському наближенню у
квантовій механіці.
|