Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot
The diffraction by a finite and semi-infinite system of plane screens with a slot is considered. The
 problem is solved with the operator approach. The possibility of screens shift in the plane of their displacement
 is taken into consideration Рассмотрена здача дифракции на конечноэ...
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| Cite this: | Electrodynamic Characteristics of Multilayered System
 of Plane Screens with a Slot / M.E. Kaliberda, L.M. Lytvynenko, S.A. Pogarsky // Радиофизика и радиоастрономия. — 2011. — Т. 16, № 2. — С. 177-182. — Бібліогр.: 9 назв. — англ. |
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| author | Kaliberda, M.E. Lytvynenko, L.M. Pogarsky, S.A. |
| author_facet | Kaliberda, M.E. Lytvynenko, L.M. Pogarsky, S.A. |
| citation_txt | Electrodynamic Characteristics of Multilayered System
 of Plane Screens with a Slot / M.E. Kaliberda, L.M. Lytvynenko, S.A. Pogarsky // Радиофизика и радиоастрономия. — 2011. — Т. 16, № 2. — С. 177-182. — Бібліогр.: 9 назв. — англ. |
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| description | The diffraction by a finite and semi-infinite system of plane screens with a slot is considered. The
problem is solved with the operator approach. The possibility of screens shift in the plane of their displacement
is taken into consideration
Рассмотрена здача дифракции на конечноэлементной и полубесконечной системе плоских экранов со щелью. Решение найдено при помощи
операторного метода. Учтена возможность смещения экранов в плоскости их расположения.
Розглянуто задачу дифракції на скінченноелементній та напівнескінченній системі плоских екранів зі щілиною. Розв’язок знайдено за допомогою операторного методу. Враховано можливість зміщення екранів у площині їх розташування.
|
| first_indexed | 2025-12-07T18:48:03Z |
| format | Article |
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Радиофизика и радиоастрономия, 2011, т. 16, №2, с. 177-182
ISSN 1027-9636 © M. E. Kaliberda, L. M. Lytvynenko, and S. A. Pogarsky, 2011
Electrodynamic Characteristics of Multilayered System
of Plane Screens with a Slot
M. E. Kaliberda, L. M. Lytvynenko1, and S. A. Pogarsky
V. Karazin National University of Kharkiv,
4, Svoboda Sq., Kharkiv, 61077, Ukraine
E-mail: Sergey.A.Pogarsky@univer.kharkov.ua
1Institute of Radio Astronomy, National Academy of Sciences of Ukraine
4, Chervonopraporna St., Kharkiv, 61002, Ukraine
Received March 28, 2011
The diffraction by a finite and semi-infinite system of plane screens with a slot is considered. The
problem is solved with the operator approach. The possibility of screens shift in the plane of their displace-
ment is taken into consideration.
Key words: operator equations, transmission and reflection operators, semi-infinite structure
1. Introduction
Multilayered strip structures are widely used in a
number of applications, for example, in the creation of
metamaterials, antennas, selective devices, etc. [1-3].
The practical usage of such structures requires
the knowledge of fundamental solution of electro-
magnetic wave diffraction (in the general case,
with an arbitrary space-time spectrum) by multi-
layered structures. Such problems belong to clas-
sical ones of electrodynamics [4, 5].
Periodic structures with strip conductors have
become an integral part of a number of functional
elements long ago, and in the first place, of antennas
structures. For their synthesis the models, which al-
low to describe the properties of bounded periodic
structures where fields have a continuous space spec-
trum, are essential. Thereupon, the study of charac-
teristics of a semi-infinite and bounded system of
plane screens with a slot is of great interest since the
model where the field is represented as a wave beam,
even two-dimensional, is practically adequate to real
models. In this paper, an approach which describes
the electrodynamic properties of a system of plane
screens with a slot is proposed. This approach is one
of the forms of the so-called semi-inversion method
of the diffraction operator [6-8]. We may demon-
strate the application of such an approach in deter-
mining the scattering operators of a finite-element
and semi-infinite system of screens with a slot.
2. Finite-Element System
Let us place in free space a screen with a slot in
the 0z = plane so that the origin of coordinates be
placed in the middle of the slot. The slot width is 2d.
In the z nh= − plane let us place the ( 1)n + -th
screen with a slot, where 1, 2, ... 1,n M= − so that
the y-coordinates of slot centers would differ for
the neighboring screens by the Δ -value. The co-
ordinate system and structure geometry are shown
in Fig. 1. Time dependence of electromagnetic field
expressed as an exponential function of a negative
imaginary power being proportional to circular fre-
quency and time is implied everywhere in the fol-
lowing.
Fig. 1. Finite-element structure
M. E. Kaliberda, L. M. Lytvynenko, and S. A. Pogarsky
Радиофизика и радиоастрономия, 2011, т. 16, №2178
For the E-polarization case, consider the electric
field xE component. Let us represent the incident
field from the half-space 0z > as
( )( , ) ( )exp ( ) d ,i
xE y z q ik y ik z
∞
−∞
= ξ ξ − γ ξ ξ∫
where 2( ) 1 ,γ ξ = −ξ Re 0,γ ≥ Im 0.γ ≥ We sup-
pose that the transmission t and reflection r ope-
rators of a single screen with a slot are known.
Their action on arbitrary function ( )g ζ is described
by expressions
( )( ) ( , ) ( )d ,tg t g
∞
−∞
ξ = ξ ζ ζ ζ∫
( )( ) ( , ) ( )d ,rg r g
∞
−∞
ξ = ξ ζ ζ ζ∫
and
( , ) ( , ) ( ).r tξ ζ = ξ ζ −δ ξ − ζ (1)
Following [7, 8], we denote Fourier amplitudes
of the reflected, transmitted fields and the field
between screens as ( ),a ξ ( ),d ξ ( )nC ξ and ( ),nB ξ
respectively, where 1, ..., 1.n M= − Using expres-
sion (1), these amplitudes in the operator notations
are related as follows
1,a tq q tes B−= − + (2)
1 1 1,C tq tB B= + − (3)
1 ,n n n nC tes C tes B es B+ − −
−= + − 2, ..., 1,n M= −
(4)
1,n n n nB tes C es C tes B+ + −
+= − + 1, ..., 2,n M= −
(5)
1 1 1,M M MB tes C es C+ +
− − −= − (6)
1,Md tes C+ −= (7)
where operator e determines the amplitude variation
of the field that occurs when the coordinate system
is shifted by the distance h along the z-axis toward
field propagation. Operators s± determine the field
amplitude variation that occurs when the coordinate
system is shifted by the Δ -distance either in positive
or negative directions along the y-axis. The system
of equations (2)-(7) may be considered as that of
integral equations.
In the case when the plane waveguide eigen-
waves propagation between layers is possible, pa-
rameter ,kh > π functions ( )nC ξ and ( ),nB ξ where
1, ..., 1,n M= − have singularities in the points
2
sgn( ) 1 ,p
pp
kh
π⎛ ⎞β = −⎜ ⎟⎝ ⎠
where , ..., ,p N N= −
0,p ≠ ,khN ⎡ ⎤= ⎢ ⎥π⎣ ⎦
which correspond to propaga-
tion constants of waveguide eigenwaves. Here no-
tation [ ]⋅ denotes integer part of a number. First, we
assume that the excitation frequency does not coin-
cide with the cutoff frequency of plane waveguide
eigenwaves, i. e. parameter .kh N≠ π Then these
singularities are the poles of first order. Now intro-
duce functions 1 ( )nC ξ and 1 ( )nB ξ as follows
( )
1( )( ) ,
1 exp 2 ( )
n
n
CC
ikh
ξξ =
− γ ξ
( )
1( )( ) .
1 exp 2 ( )
n
n
BB
ikh
ξξ =
− γ ξ
The poles excluding (regularization procedure) yields
1
1 ,a tq q tes GB−= − +
2 1 1 1 2 1 1
1 1 1( ) ( ) ,I e C tq tGB I e B− −− = + − −
2 1 1 1 1
1( ) n n nI e C tes GC tes GB− + −
−− = +
2 1 1( ) ,nes I e B− −− − 2, ..., 1,n M= −
2 1 1 1 2 1 1( ) ( )n n nI e B tes GC es I e C− + + −− = − −
1
1,ntes GB−
++ 1, ..., 2,n M= −
2 1 1 1 2 1 1
1 1 1( ) ( ) ,M M MI e B tes GC es I e C− + + −
− − −− = − −
1
1,Md tes GC+
−=
Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot
179Радиофизика и радиоастрономия, 2011, т. 16, №2
(I is the unity operator).
The integrands in these equations do not have
singularities and operator G acts on arbitrary func-
tion ( )q ζ as follows
( ) ( )
( )( ) ( )
1 exp 2 ( )
qGq
ikh
ξξ = χ ξ
− γ ξ
( )
0
( )( ) ( )
1 exp 2 ( )
N
p
p N
p
q
ikh =−
≠
⎛
ξ⎜+ χ ξ − δ ξ −β⎜ − γ ξ⎜⎝
∑
a
2
1
ln sgn( ) ,p
p
M
i p
M
⎧ ⎫⎛ ⎞−β⎪ ⎪× + π⎜ ⎟⎨ ⎬⎜ ⎟β −⎪ ⎪⎝ ⎠⎩ ⎭
(8)
where 1 2[ ; ] [ 1;1],M M ⊃ − 2 2
| | ,
2p
p
i p
k h
πσ = −
β
1 2
1 2
1, [ , ],
( )
0 ( , ),
M M
M M
ξ∉⎧
χ ξ = ⎨ ξ∈⎩
1 2
1 2
0, [ , ],
( )
1 ( , ).
M M
M M
ξ∉⎧
χ ξ = ⎨ ξ∈⎩
In the case when the excitation frequency co-
incides with the cutoff frequency of one of the
waveguide waves, i. e. ,kh N= π then in expres-
sion (8) p N≠ ± and integral in the neighborhood
of points N±β should be considered as Cauchy
principal value [9].
3. Semi-Infinite System
Let us use the same notations in this section
as in the case of a finite-element system with
the only difference that in the case of a semi-
infinite structure we should assume .M = ∞ The
structure geometry and coordinate system are
shown in Fig. 2. The Fourier amplitudes of the
reflected field and these between screens are rela-
ted as follows
,a Rq= (9)
1,a tq q tes B−= − + (10)
1 1 1,C tq tes B es B− −= + − (11)
,n nB Res C+= 1, 2, ...,n = (12)
1 ,n n n nC tes C tes B es B+ − −
−= + − 2, 3, ...,n =
(13)
where operator R is the sought for operator of a
semi-infinite structure. Now introduce notations
( , ) ( , ) ( ),R Rξ ζ = ξ ζ + δ ξ−ζ ( ) 12
1 .C C I e
−
= −
The transformations of equations (9)-(12) and re-
gularization procedure, with the use of relation (1),
yield
2 ,Rq tq tes Res GC te GC− += + − (14)
( ) 1
.C I es Res G Rq
−− += + (15)
After substitution C from equation (15) into equa-
tion (14) we obtain operator equation to determine
the operator R,
( ) 1
R t tes Res G I es Res G R
−− + − += + +
( ) 12 .te G I es Res G R
−− +− +
Fig. 2. Semi-infinite structure
2
1 0
( )
d ( ) ( )
M N
p p
p p p
p NpM
p
q
q
=−
≠
⎞
σ β ⎟× ζ + δ ξ −β σ β⎟ζ −β ⎟⎠
∑∫
M. E. Kaliberda, L. M. Lytvynenko, and S. A. Pogarsky
Радиофизика и радиоастрономия, 2011, т. 16, №2180
The amplitudes of field between the n-th and the
( 1)n + -th layers may be obtained from equations
(12) and (13).
For the H-polarization case, instead of relation
(1) use the following relation
( , ) ( , ) ( ),r tξ ζ = ξ ζ + δ ξ −ζ
and introduce the functions
( ){ }
1 ( )( ) ,
( ) 1 exp 2 ( )
n
n
CC
ikh
ξξ =
γ ξ − γ ξ
( ){ }
1 ( )( ) .
( ) 1 exp 2 ( )
n
n
BB
ikh
ξξ =
γ ξ − γ ξ
Relation (8) should be rewritten in the form
( ) ( ){ }
( )( ) ( )
( ) 1 exp 2 ( )
qGq
ikh
ξξ = χ ξ
γ ξ − γ ξ
( ){ }
( )( )
( ) 1 exp 2 ( )
q
ikh
⎛ ξ+ χ ξ⎜⎜ γ ξ − γ ξ⎝
2
10
( )
( ) d
MN
p p
p
p N pM
p
q
=−
≠
σ β
− δ ξ −β ζ
ζ −β∑ ∫
2 2
1 1
1 1
1 1( 1) (1)( 1) d ( 1) d
1 1
M M
M M
q q−
⎞σ − σ ⎟−δ ξ + ζ − δ ξ − ζ
⎟ζ + ζ − ⎠
∫ ∫
0
( ) ( )
N
p p p
p N
p
q
=−
≠
+ δ ξ −β σ β∑
2
1
ln sgn( )p
p
M
i p
M
⎧ ⎫⎛ ⎞−β⎪ ⎪× + π⎜ ⎟⎨ ⎬⎜ ⎟β −⎪ ⎪⎝ ⎠⎩ ⎭
1 2
1
1
1( 1) ( 1) ln
1
Mq i
M−
⎧ ⎫⎛ ⎞+⎪ ⎪+δ ξ + σ − − π⎨ ⎬⎜ ⎟− −⎪ ⎪⎝ ⎠⎩ ⎭
1 2
1
1
1( 1) (1) ln ,
1
Mq i
M
⎧ ⎫⎛ ⎞−⎪ ⎪+δ ξ − σ + π⎨ ⎬⎜ ⎟−⎪ ⎪⎝ ⎠⎩ ⎭
where ,
2p
p
i
kh
σ = −
β
1
1 .
4
i
kh±σ = ∓
4. Numerical results
Fig. 3 shows the dependences of transmission τ
and reflection ρ coefficients of a double screen with
a slot as functions of screen shift Δ along the y-axis
for the cases of normal incidence, 0 90 ,ϕ = ° and of
an angle of incidence 0 30ϕ = ° (see insert of Fig. 3).
The distance between screens is 5 .h k= With such
value of parameter kh, the 01TE -wave can propa-
gate between screens. The dependences of excita-
tion factors of right t+ and left t− waveguides are
shown in Fig. 4 as functions of parameter .Δ The
transmission, reflection and waveguide excitation fac-
tors are calculated from the formulas
1
2 2
0
0 1
( ) ( cos ) 1 d ,
sin
a
kd −
πρ = ξ + δ ξ − ϕ −ξ ξ
ϕ ∫
1
2 2
0 1
( ) 1 d ,
sin
d
kd −
πτ = ξ − ξ ξ
ϕ ∫
2
0sin
t
kdkh
± π=
ϕ
( ){ }
22
1
1
( ) 1 exp 2 ( ) .
N
p
p p
p p
C ikh
±
=±
−β
× β − γ β
β∑
Fig. 3. Dependences of transmission (solid line) and
reflection (dashed line) coefficients for the case of
an angle of incidence 0 30 ,ϕ = ° and transmission (dot-
ted line) and reflection (dashed-dotted line) coefficients
for the case of normal incidence, 0 90 ,ϕ = ° vs. Δ of
a double screen with a slot. The structure parameters
are kh 5,= d h 1=
Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot
181Радиофизика и радиоастрономия, 2011, т. 16, №2
In the case of normal incidence, the curves of
transmission and reflection coefficients are symmet-
rical with respect to line 0,hΔ = and the curve of
right-waveguide excitation factor equals to the curve
of left-waveguide excitation factor and is reflected
symmetrically over the same line. In the case of angle
of incidence 0 30 ,ϕ = ° when 0,hΔ < the transmit-
ted field is practically absent. In this case, virtually all
energy of scattered field is consumed by plane
waveguide excitation and the values of right-waveguide
excitation factor are significantly greater than those
of left-waveguide excitation factor. When the second
screen center approaches the 0y = plane, the inci-
dent field passes through the slots into half-space
,z h< − and the transmission coefficient increases.
The module of normalized directional patterns of trans-
mitted field is shown in Fig. 5. The directional pattern
of transmitted field is calculated from the formula
( )1 4( ) 2 cos( ) sin( ) , [0, ].iD kd e− πϕ = π − ϕ ϕ ϕ∈ π
The value of parameter Δ is chosen so that the
reflection coefficient in the case of normal inci-
dence coincide with the reflection coefficient for
the case of an angle of incidence 0 30 .ϕ = ° As the
figure shows, when we change the incidence angle
by 60° the main lobe rotates approximately by 10 .°
Fig. 6 shows the dependences of transmission τ
and reflection ρ coefficients of a structure of six
layers and reflection coefficient ∞ρ of a semi-infi-
nite structure as functions of parameter .Δ As in the
case of a double screen, when parameter Δ is
decreased, the reflection coefficient is also decreased
while the transmission coefficient increased. But in
the six-element structure case, the quasi-passband
becomes narrower than in the two-element struc-
ture case. The reflection coefficient of a finite-ele-
ment structure approaches the reflection coefficient
Fig. 4. Dependences of right (solid line), left (dashed
line) waveguide excitation factors for the case of an
angle of incidence 0 30 ,ϕ = ° and right (dotted line),
left (dashed-dotted line) waveguide excitation factors
for the case of normal incidence, 0 90 ,ϕ = ° vs. Δ of
a double screen with a slot. The structure parameters
are kh 5,= d h 1=
Fig. 5. Module of normalized directional patterns of trans-
mitted field of a double screen with a slot for the cases
0 30ϕ = ° (dashed curve) and 0 90ϕ = ° (solid curve)
Fig. 6. Dependences of reflection coefficient of a semi-
infinite structure (solid line), and transmission
(dashed-dotted line) and reflection (dashed line) coef-
ficients of a six-element structure vs. ,Δ 0 90ϕ = °.
The structure parameters are kh 5,= d h 1=
M. E. Kaliberda, L. M. Lytvynenko, and S. A. Pogarsky
Радиофизика и радиоастрономия, 2011, т. 16, №2182
of a semi-infinite structure when the number of
screens is increased.
Fig. 7 shows the module of normalized directional
patterns of transmitted field for a structure of six
layers. Normalization is performed by the maximum of
module of the directional pattern when 0.Δ = Varia-
tion of parameter Δ leads to rotation of the antenna
pattern main lobe. In our case, one can observe the
rotation by the angle of 27° with respect to normal,
and simultaneous decreasing of main lobe level by no
more than 7 %. The further increasing of parameter
Δ leads to significant decreasing of lobe level. This
follows from the dependence of transmission coeffi-
cient of this structure as a function of parameter .Δ
5. Conclusions
The diffraction by the finite-element and semi-
infinite systems of plane screens with a slot is solved.
The presented approach may be applied to solve
the synthesis problems of antenna devices with
controlled antenna patterns, excitation of open
periodic structures, and creation of metamaterials.
References
1. L. M. Lytvynenko and S. L. Prosvirnin, “Wave reflection
by a periodic layered metamaterial”, The European Phys-
ical Journal Applied Physics, vol. 46, 32608-(p1-p6), 2009.
2. X. H. Wu, A. A. Kishk, and A. W. Glisson, “A Trans-
mission Line Method to Compute the Far-Field Radia-
tion of Arbitrary Hertzian Dipoles in a Multilayer Struc-
ture Embedded With PEC Strip Interfaces”, IEEE Trans.
Antennas Propag., vol. 55, no. 11, pp. 3191-3198, 2007.
3. A. K. Bhattacharyya, “Analysis of Multilayer Infinite
Periodic Array Structures with Different Periodicities
and Axes Orientations”, IEEE Trans. Antennas Propag.,
vol. 48, no. 3, pp. 357-369, 2000.
4. B. Noble, Methods Based on the Wiener-Hopf Tech-
nique for the Solution of Partial Differential Equa-
tions, Oxford: Pergamon, 1958, 263 p.
5. T. B. A. Senior, “Diffraction by a semi-infinite metallic
sheet”, Proc. R. Soc. A., vol. 213, No. 1115. pp. 436-458,
1952.
6. K. Schwarzschild, “Die Beugung und Polarisation
des Lichts durch einen Spalt. I”, Math. Ann., vol. 5,
pp. 177-247, 1902.
7. L. M. Lytvynenko and S. L. Prosvirnin. Spectral scatte-
ring operators in diffraction by plane screens, Kyiv:
Naukova Dumka, 1984, 239 p. (in Russian).
8. M. E. Kaliberda, L. M. Lytvynenko, and S. A. Pogarskii,
“Operator Method in the Analysis of Electromagnetic
Wave Diffraction by Planar Screens”, J. Comm. Tech.
Electron., vol. 54, no. 9, pp. 975-981, 2009.
9. L. M. Lytvynenko and S. L. Prosvirnin, “Transverse
slot in a plane waveguide”, Radiotekhnika i Electro-
nika, vol. 22, no. 7, pp. 1321-1326, 1977, (in Russian).
Электродинамические характеристики
многослойной системы плоских экранов
со щелью
М. Е. Калиберда, Л. Н. Литвиненко,
С. А. Погарский
Рассмотрена здача дифракции на конечноэле-
ментной и полубесконечной системе плоских эк-
ранов со щелью. Решение найдено при помощи
операторного метода. Учтена возможность сме-
щения экранов в плоскости их расположения.
Електродинамічні характеристики
багатошарової системи плоских екранів
зі щілиною
М. Є. Каліберда, Л. М. Литвиненко,
С. О. Погарський
Розглянуто задачу дифракції на скінченноеле-
ментній та напівнескінченній системі плоских ек-
ранів зі щілиною. Розв’язок знайдено за допомо-
гою операторного методу. Враховано можливість
зміщення екранів у площині їх розташування.
Fig. 7. Module of normalized directional patterns of trans-
mitted field of a six-element structure for h 0.28Δ = −
(solid line), h 0Δ = (dotted line) and h 0.28Δ =
(dashed line), 0 90ϕ = °. The structure parameters are
kh 5,= d h 1=
|
| id | nasplib_isofts_kiev_ua-123456789-98217 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-9636 |
| language | English |
| last_indexed | 2025-12-07T18:48:03Z |
| publishDate | 2011 |
| publisher | Радіоастрономічний інститут НАН України |
| record_format | dspace |
| spelling | Kaliberda, M.E. Lytvynenko, L.M. Pogarsky, S.A. 2016-04-10T16:49:49Z 2016-04-10T16:49:49Z 2011 Electrodynamic Characteristics of Multilayered System
 of Plane Screens with a Slot / M.E. Kaliberda, L.M. Lytvynenko, S.A. Pogarsky // Радиофизика и радиоастрономия. — 2011. — Т. 16, № 2. — С. 177-182. — Бібліогр.: 9 назв. — англ. 1027-9636 https://nasplib.isofts.kiev.ua/handle/123456789/98217 The diffraction by a finite and semi-infinite system of plane screens with a slot is considered. The
 problem is solved with the operator approach. The possibility of screens shift in the plane of their displacement
 is taken into consideration Рассмотрена здача дифракции на конечноэлементной и полубесконечной системе плоских экранов со щелью. Решение найдено при помощи
 операторного метода. Учтена возможность смещения экранов в плоскости их расположения. Розглянуто задачу дифракції на скінченноелементній та напівнескінченній системі плоских екранів зі щілиною. Розв’язок знайдено за допомогою операторного методу. Враховано можливість зміщення екранів у площині їх розташування. en Радіоастрономічний інститут НАН України Радиофизика и радиоастрономия Распространение, дифракция и рассеяние электромагнитных волн Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot Электродинамические характеристики многослойной системы плоских экранов со щелью Електродинамічні характеристики багатошарової системи плоских екранів зі щілиною Article published earlier |
| spellingShingle | Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot Kaliberda, M.E. Lytvynenko, L.M. Pogarsky, S.A. Распространение, дифракция и рассеяние электромагнитных волн |
| title | Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot |
| title_alt | Электродинамические характеристики многослойной системы плоских экранов со щелью Електродинамічні характеристики багатошарової системи плоских екранів зі щілиною |
| title_full | Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot |
| title_fullStr | Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot |
| title_full_unstemmed | Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot |
| title_short | Electrodynamic Characteristics of Multilayered System of Plane Screens with a Slot |
| title_sort | electrodynamic characteristics of multilayered system of plane screens with a slot |
| topic | Распространение, дифракция и рассеяние электромагнитных волн |
| topic_facet | Распространение, дифракция и рассеяние электромагнитных волн |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/98217 |
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