Two SurfaceWave Decay Of LangmuirWaves
Yu.A. Akimov¤ and N.A. Azarenkov¤
¤Karazin Kharkiv National University, 31 Kurchatov av., 61108 Kharkiv, Ukraine
Abstract. This report presents a study of the resonant excitation of two counter-propagating surface waves under a Langmuir
wave decay. The Langmuir wave is considered to be incident perpendicularly to a plasma-dielectric interface. The surface
wave excitation is studied at the presence of a steady source of the Langmuir wave. Expressions for the growth rates and
threshold amplitudes of the Langmuir wave above which the instability can occur are found. It is analyzed also the excitation
dynamics of counter-propagating surface waves by a damped pump wave.
Keywords: Surface waves, Langmuir waves, decay instability
PACS: 52.35.Mw, 52.40.Db
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. The analysis of SW dynamics in three-wave interactions is one of them.
High-frequency volume waves, such as Langmuir waves, are well known to be essential for isotropic plasma.
Their propagation in bounded plasma plays an important role in SW dynamics. They are responsible for the resonant
damping of 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, Langmuir waves incident upon a dielectric
surface can effectively decay into two SWs .
It should be marked that the results presented in paper  are restricted by the consideration of a dielectric with the
permittivity ed = 3, when the Langmuir waves decay into a pair of the potential SWs. Thus, a decay of the Langmuir
waves into the electromagnetic SWs that takes place at ed < 3 has been left without appropriate attention. Our paper
is devoted to this aspect of parametric interaction of volume and surface waves. We consider resonant parametric
instability of counter-propagating electromagnetic SWs at their interaction with the Langmuir wave incident normally
upon a dielectric surface.
Let us consider a semibounded homogeneous dissipative isotropic plasma bounded by a dielectric. Let the plasma
occupies the half-space x > 0, whereas the dielectric occupies the x < 0 region. The wavenumber kz and frequency w
of SWs propagating along the plasma-dielectric border (the z-axis) are well known to be connected by the following
In this expression, k = w=c is the vacuum wavenumber, c is the speed of light in vacuum, ep = 1 ¡w2
the dielectric permittivity of the plasma with wpe being the electron plasma frequency, ed is the permittivity of the
According to the linear dispersion relation (1), the SWs in the considered structure are reciprocal. It means that
two counter-propagating SWs may exist with the same frequency
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 interaction of the counter-propagating SWs with the damped pump
wave does not result in an unlimited growth of the SW amplitudes, as it takes place at a decay of the Langmuir wave
with a constant amplitude. It is explained by the fact that the energy transferred from the Langmuir wave to the SWs
exponentially decreases in time. Thus, after the complete damping of the Langmuir wave, the SW amplitudes will
damp with the linear rate, jE§(t !¥)j!h§(¥)jE§(0)jexp(¡gt): The parameters h§(t) = exp(gt)jE§(t)j=jE§(0)j
are the ratios of the nonlinear SW amplitudes, jE§(t)j, to the amplitude values of linear waves, jE§(0)jexp(¡gt), and
characterize efficiency of the three-wave interaction. At t !¥,
where F = F0 ¡F+ ¡F¡, with F0;§ = arg E0;§ being the interacting wave phases. By virtue of the considered SW
equivalence and their reciprocity, it is naturally to set in (19) jE+(0)j = jE¡(0)j and h§(t) ´h(t).
Analysis of the parameter h(¥) shows that dynamics of the SW amplitudes significantly depends on initial phases
of the interacting waves, F(0). The temporal dependences of h(t) and F(t) are presented in fig. 1b for different initial
values F(0). One can see that a SW amplitude saturation level, h(¥), decreases with an increase of the initial phase,
F(0), deviation from zero. Moreover, at F(0) !p, the interacting waves have opposite phases and h(¥) ! 0. It
means that, at F(0) !p, 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.
As it was marked at the analysis of the Langmuir wave decay of a constant amplitude, the influence of the
wavenumber k0 on the decay instability dynamics, at k0rde ¿ 1, is governed by the dependence a(k0). At that, the
damping rate of the Langmuir wave, g0, can be considered as independent of its wavenumber. Thus, one can conclude
that, at the damped Langmuir wave decay, the greatest efficiency of the SW excitation is reached at k0 = k0(max) also,
when the coupling coefficient of the SWs with pump wave, a(k0), is maximum.
1. A. G. Sitenko, V. N. Pavlenko and S. M. Revenchuk, Preprint No. 79-76P, Inst. of Theoretical Physics, Ukranian Academy of
2. A. N. Kondratenko, Plasma Waveguides, Atomizdat, Moscow, 1976.
3. J. Weiland and H. Wilhelmsson, Coherent Nonlinear Interaction of Waves in Plasmas, Pergamon, Oxford, 1976.
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