Influence of a normal electric field on surfacewave dynamics
This paper presents a study of the nonresonant parametric excitation of counter-propagating surface waves by a uniform in space and varying in time electric pump field, perpendicular to a planar plasma-dielectric interface. The criterion of the wave excitation has been derived and analyzed. Expressi...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Zitieren: | Influence of a normal electric field on surfacewave dynamics / Yu.A. Akimov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 52-54. — Бібліогр.: 2 назв. — англ. |
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| author | Akimov, Yu.A. Azarenkov, N.A. |
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| citation_txt | Influence of a normal electric field on surfacewave dynamics / Yu.A. Akimov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 52-54. — Бібліогр.: 2 назв. — англ. |
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| description | This paper presents a study of the nonresonant parametric excitation of counter-propagating surface waves by a uniform in space and varying in time electric pump field, perpendicular to a planar plasma-dielectric interface. The criterion of the wave excitation has been derived and analyzed. Expressions for the growth rates in the linear stage of the instability are obtained, and the threshold amplitudes of the external electric field above which the parametric instability can occur are found. The spectrum of the excited waves is analyzed as well.
Дослiджено нерезонансне параметричне збудження зустрiчних поверхневих хвиль однорiдним у просторi та змiнним у часi електричним полем накачки, яке є перпендикулярним до планарної межi плазма-дiелектрик. Отримано та проаналiзовано критерiй збудження поверхневих хвиль. Знайдено iнкременти росту поверхневих хвиль на початковiй стадiї нестiйкостi, а також пороговi значення амплiтуди зовнiшнього поля накачки, при перевищеннi яких можливий розвиток параметричної нестiйкостi. Вивчено спектр хвиль, що збуджуються.
Исследовано нерезонансное параметрическое возбуждение встречных поверхностных волн однородным в пространстве и переменным во времени электрическим полем накачки, перпендикулярным плоской границе плазма-диэлектрик. Получен и проанализирован критерий возбуждения поверхностных волн. Найдены инкре- менты роста поверхностных волн на начальной стадии неустойчивости, а также пороговые значения амплиту- ды внешнего электрического поля, выше которых возможно развитие параметрической неустойчивости. Изучен спектр возбуждаемых волн.
|
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INFLUENCE OF A NORMAL ELECTRIC FIELD
ON SURFACEWAVE DYNAMICS
Yu.A. Akimov, N.A. Azarenkov
Karazin Kharkiv National University, 31 Kurchatov av., 61108 Kharkiv, Ukraine,
e-mail: akimov.yury@mail.ru
This paper presents a study of the nonresonant parametric excitation of counter-propagating surface waves by a uni-
form in space and varying in time electric pump field, perpendicular to a planar plasma-dielectric interface. The criterion
of the wave excitation has been derived and analyzed. Expressions for the growth rates in the linear stage of the instability
are obtained, and the threshold amplitudes of the external electric field above which the parametric instability can occur
are found. The spectrum of the excited waves is analyzed as well.
PACS: 52.35.Mw, 52.40.Db
1. INTRODUCTION
Intensive research on surface wave (SW) parametric in-
stability in bounded plasma-like mediums dates from the
1970 – 1980s. These research were connected with the ne-
cessity to solve a problem of the energy input into the work-
ing volumes of plasma installations, including controlled
fusion devices. Therefore, the main attention in these re-
search was focused on the parametric excitation of SWs
owing to the induced scattering of electromagnetic waves
incident from a dielectric or vacuum area on a semibounded
plasma. More recently, these research have received further
development in studies of SWs excited under the irradia-
tion of solid targets by intense, ultrashort laser pulses.
Other research direction on SW parametric instability
is connected with their excitation by an external homoge-
neous electric pump field. Influence of the pump field ly-
ing on the boundary plane is full enough studied, while re-
search on SW excitation in isotropic plasma installations
with a high frequency electric field, oriented perpendicu-
larly to the medium interface, are represented much more
poorly. Our research is devoted to the analysis of a non-
resonant parametric instability of two counter-propagating
SWs in a cold unmagnetized plasma, which is immersed in
a normal (to the medium interface) high frequency electric
field.
2. LINEAR SWS
We consider a semibounded homogeneous dissipative
plasma bounded by a dielectric. Let the z-axis be di-
rected along the wave propagation direction, while the x-
axis is perpendicular to the plasma-dielectric interface. The
plasma occupies the half-space x > 0, whereas the dielec-
tric occupies the x < 0 region. In the considered structure,
the wavenumber kz and frequency ω of SWs propagating
along the plasma-dielectric interface are known to be con-
nected by the following relation [1]
k2
z = k2 εpεd
εp + εd
. (1)
In this expression, k = ω/c is the vacuum wavenumber
with c being the speed of light in vacuum. The dielectric
permittivities of the mediums are denoted by εd for the di-
electric and by εp = 1 − ω2
pe/ω2 for the plasma with the
electron plasma frequency ωpe, respectively.
According to linear dispersion relation (1), two
counter-propagating SWs may exist in the considered
structure with the same frequency. Their wavenumbers are
±kz . The fields of these waves can be represented in the
following form
W± =
1
2
[
W± exp(−iωt) + W
∗
± exp(iωt)
]
exp(−γt),
(2)
where W = (Ex, Ez,Hy). The factor exp(−γt) in ex-
pression (2) describes a linear attenuation of the SWs with a
damping rate γ. The rate of collisional and resonant damp-
ing of the considered waves is [1]
γ =
ν
2
εd(1 − εp)
ε2
p + εd
−
√
− ε2
p
εp + εd
πηkωε2
pε
2
d
ε2
d − ε3
p − εpεd(1 − εp)
,
(3)
with ν � ω being the electron collision frequency. Here,
the parameter η = (dεp/dx)−1
x=x0
characterizes the plasma
density inhomogeneity in the narrow non-uniform transi-
tion layer at the resonant point x0, where εp(x0) = 0.
According to the linear theory [1], spatial distribution
of the SW fields is given by
in the plasma (x > 0)
E±x = ±i
kz
κp
E± exp(−κpx ± ikzz),
E±z = E± exp(−κpx ± ikzz),
H±y = i
kεp
κp
E± exp(−κpx ± ikzz),
in the dielectric (x < 0)
E±x = ∓i
kz
κd
E± exp(κdx ± ikzz),
E±z = E± exp(κdx ± ikzz),
H±y = −i
kεd
κd
E± exp(κdx ± ikzz),
(4)
where E+ and E− are the Ez-field amplitudes both the
waves, propagating in the positive and negative directions
of the z-axis. Here, κ2
p,d = k2
z − k2εp,d characterize pene-
tration depths of the wave fields into the plasma and dielec-
tric, accordingly.
3. NONLINEAR DISPERSION RELATION
We study the parametric excitation of electromagnetic
SWs by a uniform in space and variable in time electric
field, oriented perpendicularly to the plasma-dielectric in-
terface:
E0 =
1
2
[
E0 exp(−iω0t) + E
∗
0 exp(iω0t)
]
,
E0 = (E0, 0, 0). (5)
To consider this field to be uniform in the plasma region,
the values
qx =
∣∣E−1
0 ∂E0/∂x
∣∣ , qz =
∣∣E−1
0 ∂E0/∂z
∣∣ , (6)
which characterize the inhomogeneity of amplitude E0,
must be much less than the respective SW values, qx � κp,
qz � kz , over the SW skin depth, 1/κp.
The efficiency of SW excitation by field (5) is provided
by the reciprocity of the waves under study and by the ful-
filment of the spatial synchronism condition with the pump
field for them: 0 = kz + (−kz). Note that, generally,
the temporary synchronism of the three interacting waves,
ω0 = ω+ω, is not obligatory [2]. In what follows, we con-
sider nonresonant excitation of SWs, when its frequencies
are equal to ω = ω0/2 + Δω with Δω being a frequency
detuning.
Starting from nonlinear Maxwell’s equations and the
equation of plasma electron motion in the fields of weakly
nonlinear SWs we can write the following set of equations
for the SW fields in the plasma region
∇× E± − ikH± = 0, (7)
∇× H± + ikεpE± = (4π/c)J±, (8)
where the right-hand side of Eq. (8) is governed by a non-
linear current, J±, to second order in the amplitudes of the
excited waves
J± = i
e3n0 exp(2iΔωt)
2m2ω2ω0
[
∇(E0E
∗
∓) − ω2
ω2
pe
E0(∇E
∗
∓)
]
.
(9)
The first item in this expression is a current density of the
volume charges of the plasma, whereas the second one
characterizes a current of the surface charges, induced at
the dielectric border. It can be easily shown that, since, in
the system under consideration, the electric pump field is
perpendicular to the medium interface, the surface current
to second order is identically equal to zero.
If we substitute the linear values of SW and pump fields
(4), (5) into expression (9) for the nonlinear current, we ob-
tain its components
J±x =
e3n0kz
2m2ω2ω0
×
× E0E
∗
∓ exp(−κpx ± ikzz + 2iΔωt), (10)
J±z = −i
e3n0k
2
z
2m2ω2ω0κp
×
× E0E
∗
∓ exp(−κpx ± ikzz + 2iΔωt). (11)
Solving (7) and (8) together with (9), it is possible to
get the following expressions for the fields of the nonlinear
SWs in the plasma region
H±y = i
kεp
κp
C± exp(−κpx ± ikzz), (12)
E±x = ±i
kz
κp
C± exp(−κpx ± ikzz)−
− i
kεp
4π
c
J±x, (13)
E±z = C± exp(−κpx ± ikzz)−
− i
kεp
4π
c
J±z, (14)
where C± are constants determined by boundary condi-
tions.
Integrating equation (7) and (8) over x between−δ and
δ, and then letting δ tend to zero, one can obtain the bound-
ary conditions for the SW fields
H±y(x = −0) = H±y(x = +0),
E±z(x = −0) = E±z(x = +0).
}
(15)
The continuity ofH±y is a result of the absence of a surface
charge current at the medium interface.
Applying boundary conditions (15) to fields in the
plasma (12), (14), and dielectric (4), we can derive the con-
stants C± characterizing amplitude values of weakly non-
linear SW fields (12)–(14)
C± = E± +
ek2
zω2
pe
2cmκpkεpω2ω0
E0E
∗
∓ exp(2iΔωt), (16)
as well as the nonlinear dispersion relation for the consid-
ered SWs
k
(
εp
κp
+
εd
κd
)
E± = − ek2
zω2
pe
2cmκ2
pω
2ω0
E0E
∗
∓ exp(2iΔωt).
(17)
The left-hand side of Eq. (17) is a dispersion relation
of the linear SWs, while its right-hand side is a response
of the tangential component of the nonlinear current Jz ,
caused by the SW interaction with the pump field.
4. RESULTS AND DISCUSSION
Now we consider the SW parametric excitation by
the pump field in the weak interaction framework, when
|∂ ln E±/∂t| � |ω|. In this approach the dynamical equa-
tions [2], corresponding to nonlinear dispersion equation
(17), for the excited wave amplitudes can be written as fol-
lows
∂E±
∂t
= iαE0E
∗
∓ exp(2iΔωt), (18)
α =
eω
2cmω0
εp(1 − εp)ε2
d
(εp − εd)(ε2
p + εd)
√−(εp + εd)
,
where the coefficient α characterizes the efficiency of the
SW interaction with the pump field.
The solution of equation (18) is threshold in nature:
E± = [C1± ch(βt) + C2± sh(βt)] exp(iΔωt), (19)
β =
√
α2|E0|2 − Δω2.
Here, the parameter β characterizes a growth of the SW
amplitudes at their interaction with the pump field. The
constants in solution (19) have the following form
C1± = E±(0) exp (−iφ±(0)) , (20)
C2± = C1∓
α|E0|
|β| sign sin[φ+(0) + φ−(0) − φ0],
where φ±(0) = arg E±(0) and φ0 = arg E0.
Taking into account the linear SW attenuation (3), the
necessary condition of their excitation can be written as
|E0| > |E0|th =
√
(γ2 + Δω2)/α2. (21)
This condition imposes a restriction on the minimum value
of the pump field amplitude, above which the SW excita-
tion is possible, |E0|th. Under smaller amplitudes of the
pump field, the SW damping dominates over the growth
of their amplitudes due to the parametric instability and re-
sults in a decreasing of both the SW amplitudes in time.
When the pump field exceeds threshold value (21), a simul-
taneous growth of both the SW amplitudes appears. This
growth is characterized by the nonlinear rate
γNL =
√
α2|E0|2 − Δω2 − γ. (22)
Thus, an increase in the pump field amplitude, |E0|, as well
as a decrease of the linear damping rate, γ, leads to an in-
crease of the nonlinear growth rate, γNL. The maximum
growth rate is reached in the case of resonant excitation
(Δω = 0) [2], when both SWs are excited with the fre-
quencies ω = ω0/2.
The numerical analysis (fig.1) shows that threshold
value (21) decreases, as the frequency ω0 increases or the
frequency detuning, Δω, decreases. Thus, the considered
pump field can excite a spectrum of the SWs (fig.1). A
width of this spectrum, Δω+(E0) − Δω−(E0), is deter-
mined by the values Δω±(E0), at which |E0|cr = |E0|
and the nonlinear growth rate γNL vanishes.
Fig.1. Influence of the pump field frequency, ω0, and
frequency detuning, Δω, on the threshold amplitude,
|E0|th, for the plasma with ν/ωpe = 0.001 and
(dω2
pe/dx)−1
x=x0
ω3
pe/c = 0.001, bounded by fused silica
with εd = 3.78
According to the results shown in fig.1, an increase in
the frequency and amplitude of the pump field leads to an
increase of the excited SW spectrum width, until the half-
frequency of the pump field, ω0/2, comes close to the max-
imum value of the SW frequency, ωmax = ωpe/
√
1 + εd,
above which the SWs do not exist [1]. It explains behav-
ior of the curve in fig.1 corresponding to ω0/ωpe = 1.0.
At that value of the parameter ω0/ωpe, the half-frequency
of the pump field, ω0/2, exceeds the maximum SW fre-
quency, ωmax, (in the presented calculations, ωmax ≈
0.457 ωpe). As a result, the excitation of the SWs with
frequencies above the half-frequency of the pump field,
ω > ω0/2, becomes impossible. Further increase of ω0
results in a decreasing of the SW spectrum, until the fre-
quency ω0/2 + Δω−(E0) reaches the value ωmax, after
which the excitation of any SWs becomes impossible.
REFERENCES
1. A.N. Kondratenko. Plasma Waveguides. Moscow: “At-
omizdat”, 1976.
2. J. Weiland, H. Wilhelmsson. Coherent Nonlinear Inter-
action of Waves in Plasmas. Oxford: “Pergamon”, 1976.
ВЛИЯНИЕ НОРМАЛЬНОГО ЭЛЕКТРИЧЕСКОГО ПОЛЯ НА ДИНАМИКУ ПОВЕРХНОСТНЫХ ВОЛН
Ю.А. Акимов, Н.А. Азаренков
Исследовано нерезонансное параметрическое возбуждение встречных поверхностных волн однородным в
пространстве и переменным во времени электрическим полем накачки, перпендикулярным плоской границе
плазма-диэлектрик. Получен и проанализирован критерий возбуждения поверхностных волн. Найдены инкре-
менты роста поверхностных волн на начальной стадии неустойчивости, а также пороговые значения амплиту-
ды внешнего электрического поля, выше которых возможно развитие параметрической неустойчивости. Изучен
спектр возбуждаемых волн.
ВПЛИВ НОРМАЛЬНОГО ЕЛЕКТРИЧНОГО ПОЛЯ НА ДИНАМIКУ ПОВЕРХНЕВИХ ХВИЛЬ
Ю.О. Акiмов, М.О. Азарєнков
Дослiджено нерезонансне параметричне збудження зустрiчних поверхневих хвиль однорiдним у просторi
та змiнним у часi електричним полем накачки, яке є перпендикулярним до планарної межi плазма-дiелектрик.
Отримано та проаналiзовано критерiй збудження поверхневих хвиль. Знайдено iнкременти росту поверхневих
хвиль на початковiй стадiї нестiйкостi, а також пороговi значення амплiтуди зовнiшнього поля накачки, при
перевищеннi яких можливий розвиток параметричної нестiйкостi. Вивчено спектр хвиль, що збуджуються.
|
| id | nasplib_isofts_kiev_ua-123456789-110385 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:21:31Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Akimov, Yu.A. Azarenkov, N.A. 2017-01-04T08:25:08Z 2017-01-04T08:25:08Z 2007 Influence of a normal electric field on surfacewave dynamics / Yu.A. Akimov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 52-54. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 52.35.Mw, 52.40.Db https://nasplib.isofts.kiev.ua/handle/123456789/110385 This paper presents a study of the nonresonant parametric excitation of counter-propagating surface waves by a uniform in space and varying in time electric pump field, perpendicular to a planar plasma-dielectric interface. The criterion of the wave excitation has been derived and analyzed. Expressions for the growth rates in the linear stage of the instability are obtained, and the threshold amplitudes of the external electric field above which the parametric instability can occur are found. The spectrum of the excited waves is analyzed as well. Дослiджено нерезонансне параметричне збудження зустрiчних поверхневих хвиль однорiдним у просторi та змiнним у часi електричним полем накачки, яке є перпендикулярним до планарної межi плазма-дiелектрик. Отримано та проаналiзовано критерiй збудження поверхневих хвиль. Знайдено iнкременти росту поверхневих хвиль на початковiй стадiї нестiйкостi, а також пороговi значення амплiтуди зовнiшнього поля накачки, при перевищеннi яких можливий розвиток параметричної нестiйкостi. Вивчено спектр хвиль, що збуджуються. Исследовано нерезонансное параметрическое возбуждение встречных поверхностных волн однородным в пространстве и переменным во времени электрическим полем накачки, перпендикулярным плоской границе плазма-диэлектрик. Получен и проанализирован критерий возбуждения поверхностных волн. Найдены инкре- менты роста поверхностных волн на начальной стадии неустойчивости, а также пороговые значения амплиту- ды внешнего электрического поля, выше которых возможно развитие параметрической неустойчивости. Изучен спектр возбуждаемых волн. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Influence of a normal electric field on surfacewave dynamics Вплив нормального електричного поля на динамiку поверхневих хвиль Влияние нормального электрического поля на динамику поверхностных волн Article published earlier |
| spellingShingle | Influence of a normal electric field on surfacewave dynamics Akimov, Yu.A. Azarenkov, N.A. Basic plasma physics |
| title | Influence of a normal electric field on surfacewave dynamics |
| title_alt | Вплив нормального електричного поля на динамiку поверхневих хвиль Влияние нормального электрического поля на динамику поверхностных волн |
| title_full | Influence of a normal electric field on surfacewave dynamics |
| title_fullStr | Influence of a normal electric field on surfacewave dynamics |
| title_full_unstemmed | Influence of a normal electric field on surfacewave dynamics |
| title_short | Influence of a normal electric field on surfacewave dynamics |
| title_sort | influence of a normal electric field on surfacewave dynamics |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110385 |
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