Transcript БЕЗОТРАЖАТЕЛЬНОЕ ПРОХОЖДЕНИЕ ВОЛНЫ ЧЕР

Generation of Solitonlike Structures of
Electromagnetic Wave Field During
Transillumination of Inhomogeneous Plasmas
N.S. Erokhin1*, Zakharov1,2, L.A. Mikhailovskaya1
1Space
1P.N.
Research Institute of the Russian Academy of Sciences, Moscow, Russia
Lebedev Physical Institute of the Russian Academy of Sciences, Moscow,
Russia
*e-mail
address: [email protected]
VI International Conference
“SOLITONS, COLLAPSES AND TURBULENCE:
Achievements, Developments and Perspectives”
June 4-8, 2012, Novosibirsk, Russia
Abstract.
By usage of exact solution for Helmholtz equation it is investigated the reflectionless
propagation of electromagnetic wave through the thick inhomogeneous plasma layer (so
called the wave barrier transillumination). On the basis of numerical calculations it has
been shown that in the inhomogeneous plasma layer under the reflectionless
propagation the wave amplitude spatial profile may has the solitonlike structure.
Moreover for the case of relatively small variations of local effective plasma
permettivity the large modulations both wave amplitude and wave vector may be
observed in this system. It is important to note here that the transilluminated plasma
layer may contains wide enough wave opacity zones and plasma inhomogeneity may
includes the large number of plasma density subwave structures.
It is revealed that by the change of physico-mathematical model incoming parameters it
is possible to vary significantly the plasma inhomogeneity characteristics including
plasma layer thickness, the number of small-scale structures and nevertheless the full
transillumination of gradient barriers by electromagnetic wave will take place. It is very
important also that due to the plasma dielectric permeability gradients the specific wave
cuttoff frequency determined by inhomogeneity profile may appears.
It is analized the possible spatial profiles of electromagnetic wave amplitude, the
plasma effective dielectric permeability, the wave vector and the plasma density spatial
distribution in the inhomogeneous layer under the incoming parameters variations.
Sometimes the wave dynamics is very sensitive to small changes of incoming
parameters.
The exactly solvable physico-mathematical models for electromagnetic waves
interaction with the inhomogeneous plasma are of the greate interest to numerical
applications, for example, to study the features of electromagnetic radiation interactions
with inhomogeneous dielectrics including plasmas. In particular, it is important to
realize the electromagnetic radiation tunneling through gradient wave barriers in the
problem of dense plasma heating up to very high temperatures, to transilluminate
opaque plasma layers in the communication tasks and for the development of new
methods of dense plasma diagnostics.
It is interesting also for the understanding of physical mechanism realizing the radiation
escape from the sources placed deeply in the overdense plasma in astrophysics. This
task is important for the elaborations of absorbing coverings and transillumination ones
in the radiophysics, to elaborate the thin radiotransparent fairings for antennas and so (
see, for example, papers [1-7]).
Introduction
The effective transillumination of inhomogeneous plasma structures for incident
electromagnetic waves is very important for such applications as dense plasma heating
by the powerful electromagnetic radiation, the understanding of mechanisms for
escape of radiation from sources placed in a high density astrophysical plasma, to
prepare the efficient of antireflecting and absorbing coatings in radiophysics and so
on. The exactly solvable models of gradient wave barriers transillumination are
interesting for investigations the new features of wave amplification in
inhomogeneous plasma, the plasma instabilities dynamics including waves generation
and their nonlinear interactions in the plasma flows presence. New features may
appears in the interaction of electromagnetic waves with charged particles under the
small scale plasma inhomogeneities presence and for the very short wave impulse
evolution in inhomogeneous plasmas. Additional features may appears for
electromagnetic waves interaction with the inhomogeneous chiral plasmas.
In the reflectionless wave passage problem, it is of large interest to seek an optimum
spatial profile of the dielectric function that allows a minimum coefficient of reflection
and/or an efficient transmission of electromagnetic signals from antennas with a high
density plasma layer on their surface. It should be noted that the exactly solvable
models considered must demonstrate fundamentally new features of the wave
dynamics and can also demonstrate various interesting practical applications when the
medium parameters are varying significantly on small scales.
The analysis performed earlier has shown that it is possible to provide the reflectionless passage of transverse electromagnetic waves from a vacuum through the inhomogeneous plasma layer with variable enough the plasma permettivity f() profile.
Below by usage of Helmholtz 1D equation the exact solution is investigated to describe the reflectionless propagation of electromagnetic wave through periodically inhomogeneous wide plasma layer containing subwave structures. Our calculations have
shown that the wave field spatial profile in inhomogeneous plasma may be of solitonlike one. Moreover in the case of relatively small variations of effective plasma
dielectric permeability it may be observed the large modulations both of wave vector
and wave field amplitude. It will be considered the dependence of spatial profiles of
these characteristics on choice of problem incoming parameters. It is important to note
also that due to subwave plasma inhomogeneity the cuttoff frequency may appears.
Basic equations and their investigation
Let us consider Helmholtz equation d2E /d2 + f() E = 0 describing the electromagnetic wave propagation along x-axis. Here  =  x / c,  is the wave frequency, f() =
N2 аnd N() is the index of plasma refraction determined by components of dielectric
permeability. For the reflectionless propagation of electromagnetic wave in inhomogeneous plasma the wave electric field is taken by WKB-expression E() = ( E0 / p1/2 )
exp [ i () ], where p() = d() / d is the dimensionless wave vector, () is the
wave phase, E0 is the typical wave electric field value. In the case of exact solution the
following condition must be satisfied f() = [p()]2 – [p()]1/2 d2 {[ 1 / p() ]1/2 }/d2.
It is necessary to note here that connection between p(), f() is the nonlinear one. Let
us investigate the following exactly solvable model of reflectionless transillumination
of inhomogeneous plasma by taking for dimensionless wave vector p() such function
p() =  / [ A + Bsin( 2 ) ], where , , B = ( A2 – 1 )1/2, A > 1 are the problem
incoming parameters. Substituting this p() into Helmholtz equation we obtain the
plasma dielectric permeability f() = 2 + ( 2 - 2 ) / [ A + Bsin( 2) ]2. The spatial
profiles of p(), f(), W() = 1 / [ p() ]1/2 where W() is the normalized wave
amplitude are given in the Fig.1 for the following choice of incoming parameters  =
0.695,  = 0.7, А = 1.5. So we have pmax  1.814, pmin  0.266, max f  0.489 and min
f  0.443, pmax / pmin  6.82, max f / min f  1.104.
Fig. 1a. Profiles of wave vector and field amplitude.
Fig. 1b. Profile of plasma dielectric permeability.
So in this case of plasma inhomogeneity we have obtain small variation of f() but
the wave vector modulation is large enough. For the choice of incoming parameters
 = 0.69,  = 0.7, А = 1.9 calculations result to pmin  0.197, max f  0.489 and
pmax / pmin  12.274, max f / min f  1.106. Thus now the magnitude of pmax / pmin
has increased about two times but max f / min f practically is unchanged.
Fig. 2a. Profiles of linear plasma dielectric permeability L() and nonlinear one
f() = L() +  W()2 .
The case of periodical plasma inhomogeneity described by the following model for
field amplitude W( ) =  +  [ 1 + cos (   ) ]4 / 16 with parameters  = 1,  = 6.5,
 =  / b, b = 10 is shown on the Fig. 2a by plotts of linear plasma dielectric permettivity L() and nonlinear one f() = L() +  W()2 when the cubic nonlinearity is
taken into account with  = 0.04. It is seen that the linear dielectric function the L()
has far deeper wells L() = - 2.08.
According to the Fig. 2a the nonlinear dielectric function f() has more weaker
opaque regions even when the nonlinearity parameter  is small. In certain plasma
sublayers the profiles of L() and f() are rather close to one another. Hence due to
the nonlinearity and the resonance tunneling an electromagnetic wave may propagates
through inhomogeneous plasma without reflection and strong electromagnetic field
splashes are generated in some plasma sublayers.
For the case considered above the graph of v() = [ pe() /  ]2 dimensionless plasma
density is given below in the Fig.2b where pe() is the electron langmuir frequence
of inhomogeneous plasma. According to the Fig. 2b large amplitude modulations of
the plasma density take place in plasma layer.
It is necessary to note also that the electromagnetic wave may propagates through
inhomogeneous plasma without reflection both in the presence and absence of an
external magnetic field and independently on the plasma layer thickness. In this model
the plasma layer thickness may be increased to n times where n = 2, 3 … is a whole
number but the reflectionless passage of electromagnetic wave will take place. The
plasma layer may has fairly thick opaque regions where f() < 0.
Fig. 2b. The graph of nondimensional plasma density v().
The case of transillumination of inhomogeneous magnetoactive plasma is shown in the Fig.3
for p() =  / [A + Bsin( 2 )] and incoming parameters choice  = 0.8,  = 0.78, А =
2. So we have max p  4.835, min p  0.143, max f  2.3 and min f  0.64. Now we
have obtained max p / min p  33.8, max f / min f  3.588. Therefore in this variant of
exactly solvable model there is very large variation of wave vector p() but effec-tive
dielectric permeability f() has the more moderate modulation. According to Fig.3 the spatial
profiles of functions f() and p() are the solitonlike structures.
Fig. 3a. The plot of effective dielectric premeability of inhomogeneous
magnetoactive plasma.
Here it is necessary to note that for plasma inhomogeneities with sufficiently smooth
spatial profiles of wave vector p() in the presence of large amplitude subwave
structures the spatial profile of effective dielectric premeability ef() may has strong
qualitative differences from plot of p() .
Fig. 3b. The plot of wave vector in the inhomogeneous magnetoactive plasma.
CONCLUSION
The considered above exactly solvable models of electromagnetic waves (EW)
propagation in the inhomogeneous plasma with large amplitude subwave structures
have demonstrated various possibilities of reflectionless EW passage (the transillumination effect) through plasma layers of any thickness. The typical features of such
transillumination of gradient barriers may be conditioned by the following.
Firstly, in the dependence on incoming parameters choice the large variations of both
the wave vector p() and the wave field amplitude W() may be obtained but the
plasma dielectric permeability ef() may has small enough changes on the EW
trajectory. The opposite case of large variability of ef() for small enough
modulations in p() and W() may take place also.
Secondly, calculations have revealed that in the external magnetic field absence the
wave vector p() may be larger the unity ( p > 1 ) in some plasma sublayers. It is
meaning that local Cherenkov resonance interaction of transverse electromagnetic
wave with of fast charged particle fluxes becomes possible. So the instability like
beam one may occurs in the inhomogeneous plasma resulting to EW generation.
Thirdly, the analysis performed has shown the possibility of inhomogeneous plasma
transillumination in the presence of opaque sublayers with ef() < 0 and according to
classical conceptions such regions must cause the strong reflection of EW incident on
the plasma. It is interesting to note also that in the reference frame of exactle solvable
models the wave vector p() may includes a some arbitrary function f() and p()
expression may results to the automatic satisfaction of nonreflection conditions
performing for the wave fields at the plasma-vacuum boundaries namely p() = 1 and
d p()/d  = 0.
The spatial structures of p(), W() and ef() may be solitonlike one.
Finally it is necessary to note the following. In the common case plotts of funtions p()
and ef() may have quite different behaviour. For example, let us consider the case of
wave vector p() as the sum of two step-like functions given below with parameters
 = 0.67, 1 = - 0.4, 2 = 0.25, 1 = 0.46, 2 = 0.65, b1 = 4, b2 = 12. The plot of functions [p()]2 , ef() are given in the Fig. 4 and we see their differences.
Fig. 4. Smoth profile of [p()]2 and large variations of ef()
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Many thanks you for attention !!