Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers
The scattering of the plane electromagnetic wave on the fiber-lake target is considered. The formula for the scattering cross section is used for building the kinetic equation that describes the propagation of the wave in the beam of parallel fibers. The evolution of intensity of the radiation propa...
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| Опубліковано в: : | Вопросы атомной науки и техники |
|---|---|
| Дата: | 2014 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2014
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| Цитувати: | Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers/ N. F. Shul'ga, V. V. Syshchenko// Вопросы атомной науки и техники. — 2014. — № 5. — С. 115-119. — Бібліогр.: 7 назв. — анг. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859979160046272512 |
|---|---|
| author | Shul'ga, N. F. Syshchenko, V. V. |
| author_facet | Shul'ga, N. F. Syshchenko, V. V. |
| citation_txt | Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers/ N. F. Shul'ga, V. V. Syshchenko// Вопросы атомной науки и техники. — 2014. — № 5. — С. 115-119. — Бібліогр.: 7 назв. — анг. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The scattering of the plane electromagnetic wave on the fiber-lake target is considered. The formula for the scattering cross section is used for building the kinetic equation that describes the propagation of the wave in the beam of parallel fibers. The evolution of intensity of the radiation propagating in the beam of straight and bent fibers is considered.
Рассмотрено рассеяние плоской электромагнитной волны на нитевидной мишени. Полученное выражение для сечения рассеяния использовано при построении кинетического уравнения, описывающего распространение волны в пучке параллельных нитей. Рассмотрена эволюция интенсивности излучения, распространяющегося в пучке прямых и изогнутых нитей.
Розглянуто розсiяння плоскої електромагнитної хвилi на нитевидної мiшенi. Отриманий вираз для перерiзу розсiяння використано для побудови кiнетичного рiвняння, що описує розповсюдження хвилi у пучку паралельних ниток. Розглянуто еволюцiю iнтенсивностi випромiнювання, що розповсюджується у пучку прямих та вигнутих ниток.
|
| first_indexed | 2025-12-07T16:25:09Z |
| format | Article |
| fulltext |
ELECTROMAGNETIC WAVE SCATTERING ON A
DIELECTRIC FIBER AND A SYSTEM OF PARALLEL
FIBERS
N.F.Shul’ga1, V.V.Syshchenko2∗
1A. I.Akhiezer Institute of Theoretical Physics,
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine;
2Belgorod State University, 308015, Belgorod, Russian Federation
(Received April 8, 2014)
The scattering of the plane electromagnetic wave on the fiber-lake target is considered. The formula for the scattering
cross section is used for building the kinetic equation that describes the propagation of the wave in the beam of parallel
fibers. The evolution of intensity of the radiation propagating in the beam of straight and bent fibers is considered.
PACS: 41.20.Jb, 42.25.Bs, 42.25.Fx, 42.81.-i
1. INTRODUCTION
The interaction of electromagnetic (EM) waves with
fiber-like structures is widely used for control of prop-
agation of optical radiation using (see, e.g., recent re-
view [1] and references therein). The present paper
considers the interaction of radiation of arbitrary fre-
quency with fiber-like target. The kinetic equation
describing propagation of the radiation in the system
of parallel fibers (rope) is constructed. It is demon-
strated that not only optical but also hard (up to
X-ray domain) radiation could propagate along the
rope following its bending under some conditions.
2. CROSS SECTION FOR WAVE
SCATTERING ON A FIBER
Scattering of EM wave by charged particle arises from
oscillation of that particles under the action of the
electric field of the incident wave (for incident wave
of low intensity the influence of magnetic field would
be negligibly small) that in its turn leads to emis-
sion of secondary EM waves. This emission is inter-
preted as the scattering of the incident wave; for the
single charged particle the scattering cross section is
described by well-known Thomson formula [2].
Let us consider the incidence of the plane
monochromatic wave on the string of charges under
the angle ψ to the axis of the string (Fig. 1). The
electric field in the incident wave is described by the
formula
E(in)(r, t) = Ae(in) exp[i(k(in)r− ωt)], (1)
where A is the wave amplitude, e(in) is the polariza-
tion vector, k(in) is the wave vector, ω = |k(in)|/c is
the frequency. Scattering of the wave on the string
of charges could be described as the coherent sum
of Thomson scatterings on the charges of the string.
Fig.1. Incidence of the EM wave on the string of
charges in the z direction causes their oscillations
in the transverse plane
The electric field of the wave scattered by n-th
charge of the string could be presented on large dis-
tance from it, R0 ≫ |rn|, as [3]
En =
e2A
mω2R0
e−iω(t−R0/c)
[[
e(in),k(out)
]
,k(out)
]
× exp
[
i(k(in) − k(out))rn
]
, (2)
where e and m are charge and mass of the oscillating
particle (electron), k(in) is the wave vector of the scat-
tered wave, rn is the radius vector of the n-th charge.
After summation of the contributions of large num-
ber of oscillating electrons forming the string, N ≫ 1,
and calculation the intensity of the scattered radia-
tion in ratio to the incident energy flux density we
obtain the cross section of the wave scattering by the
∗Corresponding author E-mail address: syshch@bsu.edu.ru
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2014, N5 (93).
Series: Nuclear Physics Investigations (63), p.115-119.
115
string of charges:
dσ
dΩ
= 2π
(
e2
m
)2
N
a
[
k(out), e(in)
]2
ω2c2
×
∞∑
j=−∞
δ
(
k
(in)
∥ − k
(out)
∥ − 2π
a
j
)
, (3)
where a is the period of the string, k
(in)
∥ and k
(out)
∥
are the components of the wave vectors of the inci-
dent and scattered waves k(in) and k(out) parallel to
the string axis.
In the limiting case of small a, a < π/k(in) = λ/2,
where λ is the wavelength (that means the transition
from the discrete string of charges to the uniform
fiber) the only nonzero term in the sum over j in (3)
will be the j = 0 term. The cross section averaged
over polarizations will be equal to
dσ
dΩ
= πr20Ln
2
e
1 +
(
ck
(out)
z
ω
)2
δ
(
k
(in)
∥ − k
(out)
∥
)
,
(4)
where ne = 1/a is the number of electrons per unit
length of the fiber, r0 = e2/mc2 is the electron clas-
sical radius, L = Na is the total length of the fiber.
Delta-function in (4) reflects the fact that the
components of the wave vectors of the incident and
scattered waves parallel to the fiber axis are equal to
each other:
k
(in)
∥ = k
(out)
∥ =
ω
c
cosψ. (5)
Since the absolute values of the wave vectors k(in) and
k(out) are also equal, the angular distribution of the
scattered radiation has the shape of empty cone with
the half-opening angle equal to the incidence angle ψ
of the wave to the fiber. Such a behavior permits de-
scriptive interpretation: the incident wave produces
a perturbation in the fiber that moves along the fiber
with the velocity c/ cosψ. This superluminal motion
generates the radiation analogous to Cherenkov one.
Similar effect is described, particularly, in [4], where
the radiation arising during the incidence a wave of
dielectric permittivity of a substance (exited by the
laser pulse) on the uniform charged fiber is consid-
ered.
The second term in the factor 1+c2(k
(out)
z )2/ω2 in
(4) leads to the azimuthal asymmetry of the scatter-
ing cross section. This asymmetry is negligibly small
for small angles of incidence ψ ≪ 1, and the scatter-
ing becomes isotropic over azimuth angle in respect
to the fiber axis. Hereafter we shall neglect the az-
imuthal asymmetry and put
1 + c2
(
k(out)z
)2
/ω2 ≈ 2. (6)
The account of the finite thickness of the fiber leads
to the increase of the azimuthal asymmetry [5].
3. KINETICS OF THE RADIATION
PROPAGATION THROUGH THE ROPE
OF PARALLEL FIBERS
Consider now the scattering of the radiation inci-
dent on the system of parallel fibers (rope) randomly
arranged in the plane transverse to the fiber axis
(Fig. 2). Evolution of the radiation intensity angular
distribution during the propagation along such fiber
should be described by a kinetic equation analogous
to one describing the multiple scattering of particles
in amorphous medium [6]. However, our problem is
more complicated because of dependence of the scat-
tering cross section on the fiber not only on the scat-
tering angle (in respect to the direction of incidence),
but also on the direction of incidence itself (in respect
to the fiber axis). So, the dependence of the radiation
intensity I on the z coordinate would be described by
the kinetic equation
∂
∂z
I(ϑ, z) = (7)
=
ns
L
∫
[I(ϑ− χ, z)dσ(ϑ− χ,χ)− I(ϑ, z)dσ(ϑ,χ)],
where ϑ is the 2-dimensional angle that describes the
radiation propagation direction in respect to the z
axis, dσ(ϑ,χ) cross section of the wave scattering to
the 2-dimensional angle χ from the initial direction
ϑ, ns is the number of fibers per unit cross section
of the rope. The first term in the right-hand side of
(7) describes the income of the radiation intensity in
the direction ϑ because of scattering off the direction
ϑ−χ on the angle χ. The second term describes the
outcome of the radiation intensity in the direction ϑ
because of the scattering off that direction.
Fig.2. The rope of straight fibers; the initial wave
propagates along the z axis
However, it is convenient to re-write the kinetic equa-
tion (7) in slightly different form in order to use the
cross section (4) specifics, when only the azimuthal
scattering around the fiber axis takes the place. Let
us re-write the cross section (4) with (6) in the form
σ(θ(in),θ(out)) =
1
L
dσ
d2θ(out)
= (8)
= 2πr20n
2
e
c
ω
δ
(
cos θ(in) − cos θ(out)
)
,
where θ is the 2-dimensional angle that describes the
radiation propagation direction in respect to the fiber
116
axis. Introducing the spherical coordinates d2θ′ =
sin θ′dθ′dϕ and integrating over the polar angle θ′ us-
ing the delta-function in (8), we obtain the kinetic
equation that takes into account the azimuthal char-
acter of the wave scattering on the fiber:
∂
∂ξ
I(θ, φ, ξ) = (9)
= ns
∫ π
−π
[I(θ, φ− ϕ, ξ)σϕ(θ)− I(θ, φ, ξ)σϕ(θ)] dϕ,
where the azimuthal angle ϕ is referenced from the
incident wave azimuth, and the azimuthal scattering
cross section
σϕ(θ) ≈ 2πr20n
2
ec/ω (10)
in our approximation depends neither on the angle of
incidence θ nor on the azimuthal scattering angle ϕ.
However, the equation (9) remains valid in general
case.
Azimuthal isotropy of the cross section (10) per-
mits the further simplification of the kinetic equation
(9):
∂
∂ξ
I(θ, φ, ξ) = (11)
= nsσϕ
∫ π
−π
[I(θ, φ− ϕ, ξ)− I(θ, φ, ξ)] dϕ.
The integration of the second term in the right-hand
side of the equation gives the factor 2π, and the inte-
gration of the first term leads to azimuthal averaging
of the radiation intensity for the given θ. So,
∂
∂ξ
I(θ, φ, ξ) = −2πnsσϕ
{
I(θ, φ, ξ)− ⟨I(θ, φ, ξ)⟩φ
}
,
(12)
where
⟨I(θ, φ, ξ)⟩φ =
1
2π
∫ π
−π
I(θ, φ, ξ) dφ.
Let us rewrite the kinetic equation (12) in the
finite-difference form:
I(θ, φ, ξ +∆ξ) = I(θ, φ, ξ)− (13)
−∆ξ · 2πnsσϕ
{
I(θ, φ, ξ)− ⟨I(θ, φ, ξ)⟩φ
}
.
Equation(13) demonstrates that the evolution of the
intensity distribution happens as follows: the initial
intensity is weakened by the factor (1−∆ξ · 2πnsσϕ)
at every step ∆ξ along the fiber, but at the same time
for every value of the angle θ the azimuthal symmet-
rical “ring” (proportional to the average intensity for
the given θ) is added to the existing intensity. So, the
initial distribution with the shape of a sharp peak
directed under the angle ψ to our fibers gradually
transforms into the ring of the radius ψ in the space
of angles. This behavior is illustrated by Fig. 3 where
the evolution of the initial intensity distribution of
Gaussian form
I(θx, θy) = I0 exp[−((θx − ψ)2 + θ2y)/2σ
2], (14)
(where θx = θ cosφ, θy = θ sinφ, σ is the disper-
sion) is presented. The evolution had been simulated
numerically using the finite differences equation (13).
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 0
θ
x
θ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 5
θ
x
θ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 10
θ
x
θ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 20
θ
x
θ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 30
θ
x
θ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 50
θ
x
θ
y
I
, a
rb
itr
ar
y
un
its
Fig.3. Evolution of the initial intensity distribution (14) with the dispersion σ = 0.001 and the angle
of incidence ψ = 0.005 during propagation along the rope of straight fibers for the depths ∆ξ · t, where
t = 0, 5, 10, 20, 30, 50. Coordinate step is ∆ξ = 0.1/2πnsσϕ, the direction of the fibers corresponds to the
angles θx = θy = 0
117
Fig.4. The rope of bent fibers; the initial wave
propagates along the fibers
The finite differences equation (13) permits to
investigate the evolution of the radiation inten-
sity under propagation in the smoothly bent rope.
Our approach is similar to the so-called split op-
erator method used for numerical solution of the
time-dependent Schrödinger equation (see, e.g., [7]).
Namely, the azimuthal scattering around the current
direction of the fibers on the small part of the rope
according to the finite-difference formula (13) is al-
ternated with the changing the direction of the fibers
for the given value.
Let us consider the rope bent in the plane (x, z)
by the constant angle α per each step ∆ξ along the
rope (Fig. 4). The initial distribution I(ϑx, ϑy) is
described by the Gaussian function of the form (14)
with the angles ϑx and ϑy referenced from the di-
rection of the fixed axis z as arguments, in contrast
to the case of the straight rope considered above.
Let the maximum of the initial distribution coincides
with the local direction of the fibers.
The results of simulation of the evolution of that
initial distribution are presented on Fig. 5; the rope
bent per every step ∆ξ is α = 0.0002. We see that
along with the significant spreading of the angular
distribution the drift (shift of the maximum of the
distribution) following the rope bent. So the con-
ditions under which the rope “leads” the radiation
after its bent are possible.
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 0
ϑ
x
ϑ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 20
ϑ
x
ϑ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 30
ϑ
x
ϑ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 40
ϑ
x
ϑ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 50
ϑ
x
ϑ
y
I
, a
rb
itr
ar
y
un
its
−0.01
0
0.01
−0.01
0
0.01
0
1
2
t = 60
ϑ
x
ϑ
y
I
, a
rb
itr
ar
y
un
its
Fig.5. Intensity distribution for the radiation propagating in the bent rope (Fig. 4) for the distances ∆ξ · t
along the rope, where t = 0, 20, 30, 40, 50, 60; the coordinate step ∆ξ is the same as on Fig.3. The position
of the maximum for t = 0 coincides with the local direction of the fibers axes
4. CONCLUSIONS
The scattering of the electromagnetic wave under its
oblique incidence on the linear string of charges is
considered. The condition of validity of the approxi-
mation of the uniform fiber for that string is found.
The scattering in the last case would happen only in
the directions making the same angle with the fiber
axis as the incidence angle. In the specific case of
the small angle of incidence and the infinitely thin
fiber the scattering cross section would possess the
azimuthal symmetry around the fiber axis.
The cross section obtained is used for constructing
the kinetic equation that describes the multiple scat-
tering of the radiation in the system of parallel fibers
(rope). The numerical procedure that simulates the
wave propagation in the smoothly bent rope based
on the finite difference form of the kinetic equation
is developed. The existence of the conditions under
which the maximum of the radiation angular distri-
bution drifts following the rope bent is demonstrated.
118
ACKNOWLEDGEMENTS
The work was supported by the Ministry of Educa-
tion and Science of the Russian Federation (the State
assignment N 3.500.2014/K).
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3. N.F. Shul’ga, V.V. Syshchenko, N.V. Soboleva
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ÐÀÑÑÅßÍÈÅ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÉ ÂÎËÍÛ ÍÀ ÄÈÝËÅÊÒÐÈ×ÅÑÊÎÉ ÍÈÒÈ È
ÑÈÑÒÅÌÅ ÏÀÐÀËËÅËÜÍÛÕ ÍÈÒÅÉ
Í.Ô.Øóëüãà, Â.Â.Ñûùåíêî
Ðàññìîòðåíî ðàññåÿíèå ïëîñêîé ýëåêòðîìàãíèòíîé âîëíû íà íèòåâèäíîé ìèøåíè. Ïîëó÷åííîå âûðà-
æåíèå äëÿ ñå÷åíèÿ ðàññåÿíèÿ èñïîëüçîâàíî ïðè ïîñòðîåíèè êèíåòè÷åñêîãî óðàâíåíèÿ, îïèñûâàþùåãî
ðàñïðîñòðàíåíèå âîëíû â ïó÷êå ïàðàëëåëüíûõ íèòåé. Ðàññìîòðåíà ýâîëþöèÿ èíòåíñèâíîñòè èçëó÷å-
íèÿ, ðàñïðîñòðàíÿþùåãîñÿ â ïó÷êå ïðÿìûõ è èçîãíóòûõ íèòåé.
ÐÎÇÑIßÍÍß ÅËÅÊÒÐÎÌÀÃÍÈÒÍÎ�I ÕÂÈËI ÍÀ ÄIÅËÅÊÒÐÈ×ÍIÉ ÍÈÒÖI ÒÀ
ÑÈÑÒÅÌI ÏÀÐÀËÅËÜÍÈÕ ÍÈÒÎÊ
Ì.Ô.Øóëüãà, Â.Â.Ñèùåíêî
Ðîçãëÿíóòî ðîçñiÿííÿ ïëîñêî¨ åëåêòðîìàãíèòíî¨ õâèëi íà íèòåâèäíî¨ ìiøåíi. Îòðèìàíèé âèðàç äëÿ ïå-
ðåðiçó ðîçñiÿííÿ âèêîðèñòàíî äëÿ ïîáóäîâè êiíåòè÷íîãî ðiâíÿííÿ, ùî îïèñó¹ ðîçïîâñþäæåííÿ õâèëi ó
ïó÷êó ïàðàëåëüíèõ íèòîê. Ðîçãëÿíóòî åâîëþöiþ iíòåíñèâíîñòi âèïðîìiíþâàííÿ, ùî ðîçïîâñþäæó¹òüñÿ
ó ïó÷êó ïðÿìèõ òà âèãíóòèõ íèòîê.
119
|
| id | nasplib_isofts_kiev_ua-123456789-80494 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:25:09Z |
| publishDate | 2014 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Shul'ga, N. F. Syshchenko, V. V. 2015-04-18T14:53:40Z 2015-04-18T14:53:40Z 2014 Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers/ N. F. Shul'ga, V. V. Syshchenko// Вопросы атомной науки и техники. — 2014. — № 5. — С. 115-119. — Бібліогр.: 7 назв. — анг. 1562-6016 PACS: 41.20.Jb, 42.25.Bs, 42.25.Fx, 42.81.-i https://nasplib.isofts.kiev.ua/handle/123456789/80494 The scattering of the plane electromagnetic wave on the fiber-lake target is considered. The formula for the scattering cross section is used for building the kinetic equation that describes the propagation of the wave in the beam of parallel fibers. The evolution of intensity of the radiation propagating in the beam of straight and bent fibers is considered. Рассмотрено рассеяние плоской электромагнитной волны на нитевидной мишени. Полученное выражение для сечения рассеяния использовано при построении кинетического уравнения, описывающего распространение волны в пучке параллельных нитей. Рассмотрена эволюция интенсивности излучения, распространяющегося в пучке прямых и изогнутых нитей. Розглянуто розсiяння плоскої електромагнитної хвилi на нитевидної мiшенi. Отриманий вираз для перерiзу розсiяння використано для побудови кiнетичного рiвняння, що описує розповсюдження хвилi у пучку паралельних ниток. Розглянуто еволюцiю iнтенсивностi випромiнювання, що розповсюджується у пучку прямих та вигнутих ниток. The work was supported by the Ministry of Education and Science of the Russian Federation (the State assignment N 3.500.2014/K). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Электродинамика Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers Рассеяние электромагнитной волны на диэлектрической нити и системе параллельных нитей Розсiяння електромагнитноi хвилi на дiелектричнiй нитцi та системi паралельних ниток Article published earlier |
| spellingShingle | Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers Shul'ga, N. F. Syshchenko, V. V. Электродинамика |
| title | Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers |
| title_alt | Рассеяние электромагнитной волны на диэлектрической нити и системе параллельных нитей Розсiяння електромагнитноi хвилi на дiелектричнiй нитцi та системi паралельних ниток |
| title_full | Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers |
| title_fullStr | Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers |
| title_full_unstemmed | Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers |
| title_short | Electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers |
| title_sort | electromagnetic wave scattering on a dielectric fiber and a system of parallel fibers |
| topic | Электродинамика |
| topic_facet | Электродинамика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80494 |
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