Nonlinear decay of Langmuir waves into counter-propagating surface ones
This paper presents a study of the nonlinear excitation of surface waves involving a Langmuir wave decay at a plasma-dielectric interface. The Langmuir wave is considered to be incident perpendicularly upon the interface, where it decays into a couple of counter-propagating surface ones. The excit...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Nonlinear decay of Langmuir waves into counter-propagating surface ones / Yu.A. Akimov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 107-109. — Бібліогр.: 3 назв. — англ. |
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Akimov, Yu.A. Azarenkov, N.A. 2015-05-20T16:54:29Z 2015-05-20T16:54:29Z 2006 Nonlinear decay of Langmuir waves into counter-propagating surface ones / Yu.A. Akimov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 107-109. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.35.Mw, 52.40.Db https://nasplib.isofts.kiev.ua/handle/123456789/81794 This paper presents a study of the nonlinear excitation of surface waves involving a Langmuir wave decay at a plasma-dielectric interface. The Langmuir wave is considered to be incident perpendicularly upon the interface, where it decays into a couple of counter-propagating surface ones. The excitation is analyzed for both the undamped and damped Langmuir waves. Исследовано нелинейное возбуждение двух встречных поверхностных волн при распаде ленгмюровской волны, падающей перпендикулярно на границу раздела плазма-диэлектрик. Проанализирована динамика возбуждения поверхностных волн незатухающей и затухающей ленгмюровскими волнами. Дослiджено нелiнiйне збудження двох зустрiчних поверхневих хвиль при розпадi ленгмюрiвської хвилi, що падає перпендикулярно на межу розподiлу плазма-дiелектрик. Проаналiзована динамiка збудження поверхневих хвиль незагасаючою та загасаючою ленгмюрiвськими хвилями. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Nonlinear decay of Langmuir waves into counter-propagating surface ones Нелинейный распад ленгмюровских волн на две встречные поверхностные волны Нелiнiйний розпад ленгмюрiвських хвиль на двi зустрiчнi поверхневi хвилi Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Nonlinear decay of Langmuir waves into counter-propagating surface ones |
| spellingShingle |
Nonlinear decay of Langmuir waves into counter-propagating surface ones Akimov, Yu.A. Azarenkov, N.A. Basic plasma physics |
| title_short |
Nonlinear decay of Langmuir waves into counter-propagating surface ones |
| title_full |
Nonlinear decay of Langmuir waves into counter-propagating surface ones |
| title_fullStr |
Nonlinear decay of Langmuir waves into counter-propagating surface ones |
| title_full_unstemmed |
Nonlinear decay of Langmuir waves into counter-propagating surface ones |
| title_sort |
nonlinear decay of langmuir waves into counter-propagating surface ones |
| author |
Akimov, Yu.A. Azarenkov, N.A. |
| author_facet |
Akimov, Yu.A. Azarenkov, N.A. |
| topic |
Basic plasma physics |
| topic_facet |
Basic plasma physics |
| publishDate |
2006 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Нелинейный распад ленгмюровских волн на две встречные поверхностные волны Нелiнiйний розпад ленгмюрiвських хвиль на двi зустрiчнi поверхневi хвилi |
| description |
This paper presents a study of the nonlinear excitation of surface waves involving a Langmuir wave decay at a
plasma-dielectric interface. The Langmuir wave is considered to be incident perpendicularly upon the interface, where it
decays into a couple of counter-propagating surface ones. The excitation is analyzed for both the undamped and damped
Langmuir waves.
Исследовано нелинейное возбуждение двух встречных поверхностных волн при распаде ленгмюровской волны, падающей перпендикулярно на границу раздела плазма-диэлектрик. Проанализирована динамика возбуждения поверхностных волн незатухающей и затухающей ленгмюровскими волнами.
Дослiджено нелiнiйне збудження двох зустрiчних поверхневих хвиль при розпадi ленгмюрiвської хвилi, що
падає перпендикулярно на межу розподiлу плазма-дiелектрик. Проаналiзована динамiка збудження поверхневих
хвиль незагасаючою та загасаючою ленгмюрiвськими хвилями.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/81794 |
| citation_txt |
Nonlinear decay of Langmuir waves into counter-propagating surface ones / Yu.A. Akimov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 107-109. — Бібліогр.: 3 назв. — англ. |
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2025-11-26T18:38:54Z |
| last_indexed |
2025-11-26T18:38:54Z |
| _version_ |
1850768668078637056 |
| fulltext |
NONLINEAR DECAY OF LANGMUIR WAVES INTO
COUNTER-PROPAGATING SURFACE ONES
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 nonlinear excitation of surface waves involving a Langmuir wave decay at a
plasma-dielectric interface. The Langmuir wave is considered to be incident perpendicularly upon the interface, where it
decays into a couple of counter-propagating surface ones. The excitation is analyzed for both the undamped and damped
Langmuir waves.
PACS: 52.35.Mw, 52.40.Db
1. INTRODUCTION
At present, properties of surface waves (SWs) in
bounded plasma-like structures are a subject of intensive
theoretical and experimental research. Directions of the
nonlinear effect studies determining properties of SWs in
plasma waveguides are wide enough. Analysis of the
SW dynamics in three-wave interactions is main among of
them.
High-frequency volume waves, such as Langmuir
waves, are well known to be essential for unmagnetized
plasma. Their propagation in bounded plasma plays an im-
portant role in SW dynamics. They are responsible for the
resonant damping of the SWs in inhomogeneous transient
layers, as well as for their nonlinear damping, caused by
the generation of a SW second harmonic at frequencies,
close to the plasma one. Moreover, the Langmuir waves
incident upon a dielectric surface can effectively decay into
two SWs [1].
It should be marked that the results presented in paper
[1] are restricted by the consideration of a dielectric with
the permittivity εd = 3, when the Langmuir waves decay
into a couple of the potential SWs. Thus, the decay of the
Langmuir waves into the electromagnetic SWs that takes
place at εd < 3 has been left without appropriate attention.
Our paper is devoted to this aspect of the parametric in-
teraction of the Langmuir and surface waves. We consider
a resonant parametric instability of the counter-propagating
electromagnetic SWs at their interaction with the Langmuir
wave incident normally upon a dielectric surface.
2. LINEAR SWS
Let us consider a semibounded homogeneous dissipa-
tive plasma bounded by a dielectric. Let the plasma occupy
the half-space x > 0, whereas the dielectric occupies the
x < 0 region. The wavenumber kz and frequency ω of the
SWs propagating along the plasma-dielectric border (the
z-axis) are well known to be connected by the following
relation [2]
k2
z = k2 εpεd
εp + εd
. (1)
In this expression, k = ω/c is the vacuum wavenumber,
c is the speed of light in vacuum, εp = 1 − ω2
pe/ω2 is
the dielectric permittivity of the plasma with ωpe being the
electron plasma frequency, and εd is the permittivity of the
dielectric.
According to linear dispersion relation (1), the SWs
are reciprocal in the considered structure. It means that
two counter-propagating SWs may exist with the same fre-
quency ω and opposite wavenumbers±kz . Their fields can
be represented in the following form
W± =
1
2
[
W± exp(−iωt) + W
∗
± exp(iωt)
]
, (2)
whereW = (Ex, Ez,Hy).
Spatial distribution of the SW fields in the plasma and
dielectric is given by [2]
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),
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),
(3)
whereE+ andE− are theEz(x = 0)-field amplitudes both
the waves, propagating in the positive and negative direc-
tions of the z-axis. Here, κ2
p,d = k2
z − k2εp,d characterize
penetration depths of the wave-fields into the plasma and
dielectric.
3. NONLINEAR DISPERSION EQUATION
We study the parametric excitation of the electromag-
netic SWs by the Langmuir wave propagating perpendic-
ularly to the plasma-dielectric interface. In the hydrody-
namical approach of a cold plasma, such a Langmuir wave
is described by
E0 =
1
2
[
E0 exp(−iω0t) + E
∗
0 exp(iω0t)
]
, ω0 = ωpe
E0 = {E0[exp(−ik0x) − exp(ik0x)], 0, 0} , (4)
with ω0 = ωpe. Efficiency of this interaction is provided
by the reciprocity of the SWs under study and by the spatial
synchronism of the three waves, 0 = kz + (−kz). Another
condition characterizing the efficiency of such an interac-
tion is the temporary synchronism of the considered waves,
ω0 = ω + ω. It is provided by the fact that the maximum
frequency, ωmax = ωpe/
√
1 + εd, above which the SW
existence is impossible, exceeds the half-frequency of the
Langmuir wave, ωmax ≥ ω0/2, in the range εd ≤ 3. It en-
ables the effective interaction of the surface and Langmuir
waves.
Based on nonlinear Maxwell’s equations and the equa-
tion of plasma electron motion in the fields of weakly non-
linear SWs, we can write the following set of equations for
the SW fields in the plasma region
∇× E± − ikH± = 0,
∇× H± + ikεpE± = (4π/c)J±,
}
(5)
where the right-hand sides of Eqs. (5) is governed by a
nonlinear current
J± =
ieω2
pe
8πmω2ω0
[
(E0∇)E
∗
∓ + (E
∗
∓∇)E0−
− E0 × (∇× E
∗
∓) +
ωω0
ω2
pe
E
∗
∓(∇E0)
]
. (6)
It can be easily shown that, since, in the system under con-
sideration, the Langmuir wave possesses the perpendicular
to the medium interface electric field-component only, a
surface current to second order is equal to zero.
Substituting linear fields of the SWs (3) and pump wave
(4) into nonlinear current (6) and solving then equation set
(5) together with the boundary conditions, consisting of the
continuity of H±y and E±z fields at the interface x = 0,
one can derive the nonlinear dispersion equation of the con-
sidered SWs
k
(
εp
κp
+
εd
κd
)
E± =
2ek0(κ2
p + k2
z)
cmωκp(k2
0 + 4κ2
p)
E0E
∗
∓. (7)
The left-hand side of Eq. (7) is a dispersion relation for the
linear SWs, while its right-hand side is a response of the
nonlinear current J±, caused by the SW interaction with
the Langmuir wave.
4. SW AMPLITUDE DYNAMICS
We consider the Langmuir wave amplitude to be small,
when the SW amplitudes undergo a little change over their
period, |∂ ln E±/∂t| � |ω|. In this approach, the dynam-
ical equations [3] for the excited wave amplitudes can be
written, using (7), as follows
∂E±
∂t
= αE0E
∗
∓ exp(−γ0t) − γE±, (8)
α = − 2e
cm
εdε
2
pkk0
(εd + ε2
p)[4ε2
pk
2 − (εp + εd)k2
0]
,
where the coefficient α characterizes efficiency of the SW
interaction with the pump wave. The factor exp(−γ0t), in
(8), describes the Langmuir wave attenuation. In the cold
plasma considered here, it is mainly caused by electron col-
lisions:
γ0 = ν/2, (9)
where ν is the the electron collision frequency. The item,
−γE±, in (8), describes the linear attenuation of the SWs,
E± ∝ exp(−γt), with a damping rate γ. The rate of colli-
sional and resonant attenuation of the SWs [2] is
γ =
ν
2
εd(1 − εp)
ε2
p + εd
−
√
− ε2
p
εp + εd
πηkωε2
pε
2
d
ε2
d − ε3
p − εpεd(1 − εp)
,
(10)
The parameter η = (dεp/dx)−1
x=x0
characterizes the plasma
density inhomogeneity in the narrow transient layer at the
resonant point, x0, where εp(x0) = 0.
4.1 LANGMUIR WAVE OF A CONSTANT
AMPLITUDE
When there is a steady source maintaining the Lang-
muir wave amplitude to be constant, γ0 = 0. In this case,
the solution of Eq. (8) is threshold in nature,
E± = E±(0) exp(−γt)×
×
[
ch(βt) +
αE0
β
E∗
∓(0)
E±(0)
sh(βt)
]
, β = |αE0|. (11)
So, a threshold value of the pump wave amplitude, above
which the SW excitation is possible, can be written as
|E0|th = γ/|α|. (12)
When the pump wave amplitude exceeds threshold value
(12), a simultaneous growth of both the SW amplitudes ap-
pears with the rate
γNL = γ (|E0|/|E0|th − 1) . (13)
Thus, an increase in the pump wave amplitude, as well as
a decrease of the linear SW damping rate, leads to an in-
crease of the nonlinear growth rate, γNL.
Mark that influence of k0 on the nonlinear growth rate,
γNL, and threshold, |E0|th, is governed by the dependence
α(k0). It can be easily shown that the parameter α(k0) is
negative. Its absolute value reaches the maximum at the
following wavenumber of the Langmuir wave
k0(max) = k
√
− 4ε2
p
εp + εd
, (14)
At this wavenumber, threshold (12) has the minimum,
while nonlinear growth rate (13) attains the maximum. The
numerical analysis (fig.1) shows that, with an increase in
the permittivity of the dielectric, εd, the minimum of |E0|th
decreases and shifts towards larger wavenumbers of the
pump wave. At that, the SW excitation is the most effective
at wavenumbers of the Langmuir wave close to 5 ωpe/c.
Mark, the obtained results are fair at the initial stage,
t < 1/γ0, of a weakly damped pump wave decay also,
when γ0 � γNL. At this stage, the amplitude |E0| can be
treated as a constant.
4.2 DAMPED LANGMUIR WAVE
Account for the Langmuir wave damping complicates
solution of coupled equation set (8),
E± =
{
C1±I1/2(ξ) + C2±K1/2(ξ)
}×
× exp[−(γ0/2 + γ)t], ξ =
|αE0|
γ0
exp(−γ0t).
(15)
Fig.1. Influence of k0 and εd on threshold |E0|th (12) for
the plasma with (dω2
pe/dx)−1
x=x0
ω3
pe/c = 0.001 and
ν/ωpe = 0.001
Fig.2. Evolution of the normalized amplitudes, η, and
phases, Φ, under |αE0|/γ0 = 3.0
Here, I1/2 and K1/2 are the modified cylindrical Bessel
and McDonald functions of the order 1/2. The constants
C1± and C2± are determined by initial values of the SW
amplitudes and their derivatives.
Analysis of the last expression demonstrates that inter-
action of the counter-propagating SWs with the damped
pump wave does not result in an unlimited growth of the
SW amplitudes. Moreover, after the complete damping of
the Langmuir wave, the SW amplitudes damp with the lin-
ear rate, |E±(t → ∞)| → η±(∞)|E±(0)| exp(−γt). The
parameters η±(t) = exp(γt)|E±(t)|/|E±(0)| are the ra-
tios of the nonlinear SW amplitudes, |E±(t)|, to the ampli-
tude values of linear waves, |E±(0)| exp(−γt), and char-
acterize efficiency of the three-wave interaction. By virtue
of the considered SW equivalence and their reciprocity, it
is naturally to set |E+(0)| = |E−(0)| and η±(t) ≡ η(t).
Analysis of the parameter η(t) shows that dynamics of
the SW amplitudes significantly depends on initial phases
of the interacting waves. The temporal dependences of η(t)
andΦ(t) = Φ0(t)−Φ+(t)−Φ−(t), withΦ0,± = arg E0,±,
are presented in fig.2 for different initial values Φ(0). One
can see that a SW amplitude saturation level, η(∞), de-
creases with an increase of the initial phase, Φ(0), devia-
tion from zero. Moreover, at Φ(0) → π, the interacting
waves have opposite phases and η(∞) → 0. It means that,
at Φ(0) → π, nonlinear interaction of the SWs and pump
wave does not result in a growth of the SW amplitudes. On
the contrary, it leads to their damping.
In the considered case of a damped Langmuir wave,
the influence of the wavenumber k0 on the decay instability
dynamics is also governed by the dependence α(k0). Thus,
one can conclude that, at the damped Langmuir wave de-
cay, the greatest efficiency of the SW excitation is reached
at k0 = k0(max), when the coupling coefficient of the SWs
with the pump wave, α(k0), has the maximum.
REFERENCES
1. N.B. Aleksić, A.G. Zagorodny, V.I. Zasenko. Reso-
nant interaction of electron waves in semibounded plasma:
Preprint. Kiev: Institute of Theoretical Physics, ITP-90-
28P, 1990.
2. A.N. Kondratenko. Plasma Waveguides. Moscow: “At-
omizdat”, 1976.
3. J. Weiland, H. Wilhelmsson. Coherent Nonlinear Inter-
action of Waves in Plasmas. Oxford: “Pergamon”, 1976.
НЕЛИНЕЙНЫЙ РАСПАД ЛЕНГМЮРОВСКИХ ВОЛН НА ДВЕ ВСТРЕЧНЫЕ ПОВЕРХНОСТНЫЕ
ВОЛНЫ
Ю.А. Акимов, Н.А. Азаренков
Исследовано нелинейное возбуждение двух встречных поверхностных волн при распаде ленгмюровской вол-
ны, падающей перпендикулярно на границу раздела плазма-диэлектрик. Проанализирована динамика возбужде-
ния поверхностных волн незатухающей и затухающей ленгмюровскими волнами.
НЕЛIНIЙНИЙ РОЗПАД ЛЕНГМЮРIВСЬКИХ ХВИЛЬ НА ДВI ЗУСТРIЧНI ПОВЕРХНЕВI ХВИЛI
Ю.О. Акiмов, М.О. Азарєнков
Дослiджено нелiнiйне збудження двох зустрiчних поверхневих хвиль при розпадi ленгмюрiвської хвилi, що
падає перпендикулярно на межу розподiлу плазма-дiелектрик. Проаналiзована динамiка збудження поверхневих
хвиль незагасаючою та загасаючою ленгмюрiвськими хвилями.
|