Slow and fast surface electromagnetic waves in planar structures contained left-handed material
It was studied the properties of electromagnetic surface waves in left-handed material slab bounded by two semi-infinite nonmagnetic media with frequency dependent positive permittivity. It was assumed that all these media are isotropic and non-dissipative. It was shown that the phase velocities of...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
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| Cite this: | Slow and fast surface electromagnetic waves in planar structures contained left-handed material / V.K. Galaydych, N.A. Azarenkov, V.P. Olefir, A.E. Sporov // Вопросы атомной науки и техники. — 2015. — № 4. — С. 306-309. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859518246072352768 |
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| author | Galaydych, V.K. Azarenkov, N.A. Olefir, V.P. Sporov, A.E. |
| author_facet | Galaydych, V.K. Azarenkov, N.A. Olefir, V.P. Sporov, A.E. |
| citation_txt | Slow and fast surface electromagnetic waves in planar structures contained left-handed material / V.K. Galaydych, N.A. Azarenkov, V.P. Olefir, A.E. Sporov // Вопросы атомной науки и техники. — 2015. — № 4. — С. 306-309. — Бібліогр.: 6 назв. — англ. |
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| description | It was studied the properties of electromagnetic surface waves in left-handed material slab bounded by two semi-infinite nonmagnetic media with frequency dependent positive permittivity. It was assumed that all these media are isotropic and non-dissipative. It was shown that the phase velocities of waves are greater than the speed of light, and the group velocities are zero or negative.
Вивчені властивості поверхневих електромагнітних хвиль, що поширюються уздовж шару лівостороннього матеріалу, обмеженого двома напівнескінченими немагнітними середовищами з додатними діелектричними проникностями, що залежать від частоти. Вважалось, що всі середовища є ізотропними та в них відсутні втрати. Показано, що фазові швидкості розглянутих хвиль можуть бути більше швидкості світла, а групові – нульовими або від’ємними.
Изучены свойства поверхностных электромагнитных волн, распространяющихся вдоль слоя левостороннего материала, ограниченного двумя полубесконечными немагнитными средами с положительными диэлектрическими проницаемостями, зависящими от частоты. Предполагалось, что все среды являются изотропны и в них отсутствуют потери. Показано, что фазовые скорости рассматриваемых волн могут быть больше скорости света, а групповые – нулевыми или отрицательными.
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ISSN 1562-6016. ВАНТ. 2015. №4(98) 306
SLOW AND FAST SURFACE ELECTROMAGNETIC WAVES IN
PLANAR STRUCTURES CONTAINED LEFT-HANDED MATERIAL
V.K. Galaydych, N.A. Azarenkov, V.P. Olefir, A.E. Sporov
V.N. Karazin Kharkiv National University, Kharkov, Ukraine
E-mail: viktor.galaydych@gmail.com
It was studied the properties of electromagnetic surface waves in left-handed material slab bounded by two semi-
infinite nonmagnetic media with frequency dependent positive permittivity. It was assumed that all these media are
isotropic and non-dissipative. It was shown that the phase velocities of waves are greater than the speed of light, and
the group velocities are zero or negative.
PACS: 52.35g, 52.50.Dg
INTRODUCTION
A great amount of papers devoted to the studies of
modern artificial materials (metamaterials or left-
handed materials, LHM) were published after first pub-
lications [1, 2]. These papers were mainly devoted to
the optical applications of LHM. But it is necessary to
mention the studies carried out at the Argonne
Wakefield Accelerator Facility; devoted to the investi-
gation of the application of LHM to control the disper-
sion relation in a loaded waveguide [3, 4].
Very often the planar waveguide structures that con-
tain LHM are embedded either by vacuum or by differ-
ent ordinary dielectrics with constant permittivity [5, 6].
The presence of ordinary dielectrics leads to the narrow-
ing of possible wavenumbers range for the eigen surface
electromagnetic waves of such structures as compared
with vacuum bounds.
In the present work it was considered the planar
waveguide structure that consists of the left-handed
metameterial slab bounded by two non-magnetic plas-
ma-like media which permittivity depends on the wave
frequency and may take the value in the range from 0 to
1.
1. TASK SETTINGS
Let us consider the eigen electromagnetic waves that
propagate along the planar waveguide structure that
contains isotropic LHM slab of thickness ∆ . This LHM
is characterized by effective permittivity and permeabil-
ity which depend on the wave frequency as follow [2]:
2
2
1)(
ω
ω
ωε p−= , (1)
2
0
2
2
1)(
ωω
ωωµ
−
−=
F , (2)
here pω is effective plasma frequency, 0ω is the char-
acteristic frequency of LHM. In the further study we
consider the LHM with / 2 10pω π = GHz,
0 / 2 4ω π = GHz and parameter 0.56F = [5].
This LHM slab is bounded on the both sides by two
different artificial non-dissipative non-magnetic
( 1,2 1µ = ) materials with effective permittivities
2
1
1 2( ) 1 pω
ε ω
ω
= − , (3)
2
2
2 2( ) 1 pω
ε ω
ω
= − , (4)
here 1pω , 2pω is the effective plasma frequencies for
corresponding media. In out study we consider that
1 / 2 2,4pω π = GHz and 2 / 2 3,2pω π = GHz. Such
choice of parameters leads to the existence of the fre-
quency range where ( ) 0ε ω < and ( )1,20 1ε ω< < sim-
ultaneously.
Let restrict our consideration by the surface electro-
magnetic wave that propagates along this structure. Be-
sides, it is assumed that all wave components tend to
zero far away from LHM and possess the dependence
on time t and coordinate z in such form:
3, ( ), ( ) exp[ ( )]E H E x H x i k z tω∝ − , (5)
here x is the coordinate transversal to the wave propa-
gation.
The system of Maxwell equations splits into two
subsystems that describe the waves of H - type and E -
type.
The wave of E -type possesses the dispersion rela-
tion of the following form:
( ) ( ) ( )
( ) ( ) ( ) ( )
2 1 1 2
2 2
1 2 1 2[ ] tanh 0
h h
h h
ε ω κ ε ω ε ω
ε ω ε ω ε ω κ κ
+ +
+ ⋅ ∆ =
, (6)
here, ( )2 2
1,2 3 1,2 1,2h k kε ω µ= − , ( ) ( ) 22
3 kk ωµωεκ −= ,
ck /ω= , c is the speed of light in vacuum.
For E-wave the wave field components normalized
by the )0(yH in the region of LHM can be written as:
( ) xx
y eCeCxH κκ −+= 21 ,
( ) ( ) ( )( )ωεκκ keCeCkxE xx
x /213
−+= , (7)
( ) ( ) ( )( )ωεκ κκ keCeCixE xx
z /21
−−= ,
here 1C and 2C are wave field constants.
In first medium (3) the E-wave field components,
normalized by )0(yH , possess the form:
( ) 1 ,h x
yH x e=
( ) ( )1
3 1/ ,h x
xE x k e k ε= (8)
( ) ( )1
1 1/ .h x
zE x ih e k ε=
In second plasma-like medium (4) the E-wave field
components, normalized by )0(yH , can be written as:
( ) 2h x
yH x Ae−= ,
( ) ( )( )2
3 2/h x
xE x Ak e k ε ω−= , (9)
( ) ( )( )2
2 2/h x
zE x i A h e k ε ω−= − ,
here A is wave field constant. These constants are of
ISSN 1562-6016. ВАНТ. 2015. №4(98) 307
the following form:
( ) ( ) ( )2
1 2 12 ( ) / ( )h
EA h e κε ω ε ω ε ω+ ∆= − Ψ ,
( ) ( )1 1 2 2 1( ) ( ) / ( ) EC h hε ω ε ω ε ω κ ε ω κ= − Ψ , (10)
( ) ( )2
2 1 2 2 1( ) ( ) / ( ) ,EC h h e κε ω ε ω ε ω κ ε ω κ∆= − + Ψ
( )2 2
2 2( 1) ( ) ( 1)E e h eκ κε ω ε ω κ∆ ∆Ψ = + + − .
The wave of H-type possesses the dispersion relation
of the following form:
( ) ( )
( ) ( )
2 1 1 2
2 2
1 2 1 2[ ] tanh 0.
h h
h h
µ ω κ µ µ
µ ω µ µ κ κ
+ +
+ ∆ =
(11)
In the region of LHM slab the H-wave field compo-
nents, normalized by the )0(yE , can be written as:
( ) 1 2
x x
yE x D e D eκ κ−= + ,
( ) ( ) ( )( )3 1 2 /x x
xH x k D e D e kκ κ µ ω−= − + , (12)
( ) ( ) ( )( )1 2 /x x
zH x i D e D e kκ κκ µ ω−= − − ,
here 1D and 2D are wave field constants.
In the first medium the H-wave field components,
normalized by the )0(yE , can be expressed as:
( ) xh
y exE 1= ,
( ) ( )13 /1 µkekxH xh
x −= , (13)
( ) ( )11 /1 µkeihxH xh
z −= .
In the second plasma-like half-space the H-wave field
components, normalized by )0(yE , can be written as:
( ) xh
y BexE 2−= ,
( ) ( )23 /2 µkeBkxH xh
x
−−= , (14)
( ) ( )22 /2 µkehBixH xh
z
−= ,
here B is wave field constant. These constants are of
the following form:
( ) ( )2
1 2 12 ( ) /h
HB h e κµ µ ω µ+ ∆= − Ψ ,
( )1 1 2 2 1( ) ( ( ) ) / HD h hµ ω µ ω µ κ µ κ= − Ψ , (15)
( )2
2 1 2 2 1( ) ( ( ) ) / HD h h e κµ ω µ ω µ κ µ κ∆= − + Ψ ,
2 2
2 2( 1) ( ) ( 1)H e h eκ κµ ω µ κ∆ ∆Ψ = + + − .
2. MAIN RESULTS
The numerical solutions of dispersion relations (6)
and (11) for E - and H -waves are presented at the
Fig. 1. Numbers of curves 1, 2 correspond to the waves
of H -type and curves marked by the numbers 3, 4 cor-
respond to the waves of E -type.
It is important to notice that the central metamaterial
slab demonstrates left-handed properties ( 0)( <ωε and
0)( <ωµ simultaneously) for the normalized frequency
01 / 1,5ω ω< Ω = < . The letters on the Fig. 1a,b,c cor-
respond to the curves ( )2ξ ε ω= Ω , ( )1ξ ε ω= Ω ,
( ) ( )ξ ε ω µ ω= Ω , respectively. The waves of the sur-
face type can exist in the area right to the lines marked
by the letters a, b and higher of line marked by the letter
c. The line L corresponds to the light in vacuum, i.e.
ξ = Ω .
Fig. 1. The dependence of the normalized frequency
0/ωω=Ω on the dimensionless wave number
3 0/k cξ ω= at the thickness 0 / 2cω ∆ = for H1,2-modes
(lines 1, 2) and E1,2 -modes (lines 3, 4)
Further we shall study the dependence of phase
3/phV kω= and group 3grV d dkω= velocities on the
problem’s parameters. Now, the Fig. 1 shows that three
modes can have phase velocity higher than light speed
in vacuum.
Phase velocities of both 1,2E -modes (see curves 3, 4
of the Fig. 1) monotonously decreases with the increase
of wave frequency. They do not change qualitatively its
behavior with the increase of thickness of LHM layer
(see Fig. 2). Phase velocity of 2H -mode (curve 2 of the
Fig. 1) monotonously increases with the increase of
wave frequency and also does not change qualitatively
its behavior with increasing of thickness of LHM layer.
Fig. 2. The dependence of the phase velocity /phV c
of E-wave (curve 3 on Fig. 1) on the frequency
0/ωω=Ω for different LHM slab thickness
0 / 0,5;1.5;2cω ∆ =
But 1H -mode (see curve 1 at Fig. 1) shows qualita-
tively different behavior (Fig. 3). If LHM slab thickness
is rather small (see line marked by the number 0.5 at the
Fig. 3) the H-wave decreases its phase velocity with the
increase of wave frequency. When the LHM slab thick-
ness increases its value (see lines marked by the num-
bers 1.5 and 2.0 at the Fig. 3) there is a qualitative tran-
sition of the behavior of phase velocity versus frequen-
cy: phase velocity of the 1H -mode increases with the
increase of the wave frequency.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 308
Fig. 3. The dependence of the phase velocity /phV c of
1H -wave (line 1 on Fig.1) on the frequency 0/ωω=Ω
for different LHM slab thickness 0 / 0,5;1.5;2cω ∆ =
The carried out study shows that the considered sur-
face waves can be either slow or fast depending on the
choice of operating frequency.
It was shown that it is possible to change the phase
velocity of the considered waves at fixed frequency by
the variation of LHM slab thickness ∆ . Such depend-
ence is called the geometric dispersion and it is present-
ed on Fig. 4.
Fig. 4. The dependence of the phase velocity /phV c on
LHM slab thickness ∆ for: 1 – 1H -wave (see curve 2
on Fig. 1, 1,64Ω = ); 2 – 2H -wave (see curve 3 on
Fig. 1, 1,23Ω = ); 3 – 1E -wave (see curve 4 on Fig. 1,
1,154Ω = )
Fig. 5. The spatial distribution of electric field wave
amplitudes versus transversal coordinate x for differ-
ent modes 1 – 1H -mode ( 3 4,0k = , 1,19258Ω = ) and
2 – 2H -mode ( 3 4,0k = ; 1,193586Ω = )
The normalized phase velocity for 2E -wave linearly
changes from 0.5 to 3.2 at the same parameters.
The dependence of electric wave field amplitude on
the transversal coordinate x for the considered modes
has symmetric or anti-symmetric character (for example
Fig. 5).
When the wavelength of the surface wave increases
the penetration depth of the considered waves increases
sharply, and the wave becomes essentially a bulk wave
(Fig. 6).
Fig. 6. The spatial distribution of electric field wave
amplitudes versus transversal coordinate for 2H -mode
for different wavenumber k3 = 4.0 and k3 = 1.2
Fig. 7. The dependence of the normalized group velocity
/grV c on of the frequency 0/ωω=Ω for LHM slab
thickness 0 / 0.5cω ∆ = (numbers of lines corresponds
to the Fig. 1)
It is necessary to mention, that the group velocity
value of the considered waves can be either zero, or
small, or high (Fig. 7).
The Fig. 7 shows that the considered modes are for-
ward and backward, slow and fast. It is shown that in
the considered waveguide structure the H-modes with
zero group velocity can exist. We supposed that such
waves can be used in accelerating technology.
CONCLUSIONS
It was found the possibility of existence of the sur-
face electromagnetic waves that propagate along the
LHM planar slab that bounded by two semi-infinite
plasma-like media.
It was found four modes (two E-modes and two H-
modes) with different wave field structure. It was shown
that three modes of this four can have phase velocity
higher than light speed in vacuum.
It was also shown that waves considered possess
both positive and negative frequency dispersion of the
phase velocity.
It was obtained that one of the H-waves can be
backward (phase and group velocity directed opposite-
ly). The absolute value of the group velocity of H-waves
can be quite low, down to zero.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 309
It was shown that is possible to change the phase ve-
locity of waves considered at fixed frequency by varia-
tion of LHN slab thickness.
The obtained results can be useful for the plasma
electronics modeling and improvement of acceleration
technology.
REFERENCES
1. V.G. Veselago. The electrodynamics of substances
with simultaneously negative values of ε and μ //
Soviet Physics Uspekhi. 1968, v. 10, p. 509-514.
2. J.B. Pendry, A.J. Holden, W.J. Stewart, I. Youngs.
Extremely low frequency plasmons in metallic mi-
crostructures // Phys. Rev. Lett. 1996, v.76, p. 4773.
3. S. Antipov, W. Liu, W. Gai, J. Power, L. Spentzou-
ris. Double-negative metamaterial research for ac-
celerator applications // Nucl. Instr. Meth. 2007,
v. A579, iss. 3, p. 915-923.
4. S. Antipov, L. Spentzouris, W. Gai, M. Conde,
F. Franchini, et al. Observation of wakefield genera-
tion in left-handed band of metamaterial-loaded
waveguide // J. Appl. Phys. 2008, v. 104, p. 014901.
5. I.V. Shadrivov, A.A. Sukhorukov, Yu.S Kivshar.
Guided modes in negative-refractive-index wave-
guides // Phys. Rev. 2003, E 67, p. 057602-4.
6. V.K. Galaydych, N.A. Azarenkov, V.P. Olefir,
A.E. Sporov. Surface electromagnetic waves in left-
handed material slab embedded in plasma-like me-
dia// Problems of Atomic Science and Technology.
Series “Plasma Physics» (20). 2014, № 6, p. 112-
115.
Article received 23.06.2015
МЕДЛЕННЫЕ И БЫСТРЫЕ ПОВЕРХНОСТНЫЕ ЭЛЕКТРОМАГНИТНЫЕ ВОЛНЫ В ПЛОСКИХ
СТРУКТУРАХ, СОДЕРЖАЩИХ ЛЕВОСТОРОННИЙ МАТЕРИАЛ
В.К. Галайдыч, Н.А. Азаренков, В.П. Олефир, А.Е. Споров
Изучены свойства поверхностных электромагнитных волн, распространяющихся вдоль слоя левосторон-
него материала, ограниченного двумя полубесконечными немагнитными средами с положительными ди-
электрическими проницаемостями, зависящими от частоты. Предполагалось, что все среды являются изо-
тропны и в них отсутствуют потери. Показано, что фазовые скорости рассматриваемых волн могут быть
больше скорости света, а групповые – нулевыми или отрицательными.
ПОВІЛЬНІ ТА ШВИДКІ ПОВЕРХНЕВІ ЕЛЕКТРОМАГНІТНІ ХВИЛІ В ПЛАСКИХ СТРУКТУРАХ,
ЩО МІСТЯТЬ ЛІВОСТОРОННІЙ МАТЕРІАЛ
В.К. Галайдич, М.О. Азарєнков, В.П. Олефір, О.Є. Споров
Вивчені властивості поверхневих електромагнітних хвиль, що поширюються уздовж шару лівосторон-
нього матеріалу, обмеженого двома напівнескінченими немагнітними середовищами з додатними діелект-
ричними проникностями, що залежать від частоти. Вважалось, що всі середовища є ізотропними та в них
відсутні втрати. Показано, що фазові швидкості розглянутих хвиль можуть бути більше швидкості світла, а
групові – нульовими або від’ємними.
http://ufn.ru/en/articles/1968/4/d/
http://ufn.ru/en/articles/1968/4/d/
http://www.rsphysse.anu.edu.au/nonlinear/papers/IlyaShadrivov.shtml#PRE_2003_67_57602
http://www.rsphysse.anu.edu.au/nonlinear/papers/AndreySukhorukov.shtml#PRE_2003_67_57602
http://www.rsphysse.anu.edu.au/nonlinear/papers/YuriKivshar.shtml#PRE_2003_67_57602
INTRODUCTION
1. TASK SETTINGS
2. MAIN RESULTS
REFERENCES
МЕДЛЕННЫЕ И БЫСТРЫЕ ПОВЕРХНОСТНЫЕ ЭЛЕКТРОМАГНИТНЫЕ ВОЛНЫ В ПЛОСКИХ СТРУКТУРАХ, СОДЕРЖАЩИХ ЛЕВОСТОРОННИЙ МАТЕРИАЛ
ПОВІЛЬНІ ТА ШВИДКІ ПОВЕРХНЕВІ ЕЛЕКТРОМАГНІТНІ ХВИЛІ В ПЛАСКИХ СТРУКТУРАХ, ЩО МІСТЯТЬ ЛІВОСТОРОННІЙ МАТЕРІАЛ
|
| id | nasplib_isofts_kiev_ua-123456789-112209 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-25T20:43:26Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Galaydych, V.K. Azarenkov, N.A. Olefir, V.P. Sporov, A.E. 2017-01-18T19:30:44Z 2017-01-18T19:30:44Z 2015 Slow and fast surface electromagnetic waves in planar structures contained left-handed material / V.K. Galaydych, N.A. Azarenkov, V.P. Olefir, A.E. Sporov // Вопросы атомной науки и техники. — 2015. — № 4. — С. 306-309. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.35g, 52.50.Dg https://nasplib.isofts.kiev.ua/handle/123456789/112209 It was studied the properties of electromagnetic surface waves in left-handed material slab bounded by two semi-infinite nonmagnetic media with frequency dependent positive permittivity. It was assumed that all these media are isotropic and non-dissipative. It was shown that the phase velocities of waves are greater than the speed of light, and the group velocities are zero or negative. Вивчені властивості поверхневих електромагнітних хвиль, що поширюються уздовж шару лівостороннього матеріалу, обмеженого двома напівнескінченими немагнітними середовищами з додатними діелектричними проникностями, що залежать від частоти. Вважалось, що всі середовища є ізотропними та в них відсутні втрати. Показано, що фазові швидкості розглянутих хвиль можуть бути більше швидкості світла, а групові – нульовими або від’ємними. Изучены свойства поверхностных электромагнитных волн, распространяющихся вдоль слоя левостороннего материала, ограниченного двумя полубесконечными немагнитными средами с положительными диэлектрическими проницаемостями, зависящими от частоты. Предполагалось, что все среды являются изотропны и в них отсутствуют потери. Показано, что фазовые скорости рассматриваемых волн могут быть больше скорости света, а групповые – нулевыми или отрицательными. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Приложения и технологии Slow and fast surface electromagnetic waves in planar structures contained left-handed material Повільні та швидкі поверхневі електромагнітні хвилі в пласких структурах, що містять лівосторонній матеріал Медленные и быстрые поверхностные электромагнитные волны в плоских структурах, содержащих левосторонний материал Article published earlier |
| spellingShingle | Slow and fast surface electromagnetic waves in planar structures contained left-handed material Galaydych, V.K. Azarenkov, N.A. Olefir, V.P. Sporov, A.E. Приложения и технологии |
| title | Slow and fast surface electromagnetic waves in planar structures contained left-handed material |
| title_alt | Повільні та швидкі поверхневі електромагнітні хвилі в пласких структурах, що містять лівосторонній матеріал Медленные и быстрые поверхностные электромагнитные волны в плоских структурах, содержащих левосторонний материал |
| title_full | Slow and fast surface electromagnetic waves in planar structures contained left-handed material |
| title_fullStr | Slow and fast surface electromagnetic waves in planar structures contained left-handed material |
| title_full_unstemmed | Slow and fast surface electromagnetic waves in planar structures contained left-handed material |
| title_short | Slow and fast surface electromagnetic waves in planar structures contained left-handed material |
| title_sort | slow and fast surface electromagnetic waves in planar structures contained left-handed material |
| topic | Приложения и технологии |
| topic_facet | Приложения и технологии |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112209 |
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