Generation of Solitonlike Structures of

Electromagnetic Wave Field During

Transillumination of Inhomogeneous Plasmas

N.S. Erokhin1*, Zakharov1,2, L.A. Mikhailovskaya1

1Space Research Institute of the Russian Academy of Sciences, Moscow, Russia1P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow,

Russia*e-mail address: nerokhin@iki.rssi.ru

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]).

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.

Introduction

The analysis performed earlier has shown that it is possible to provide the reflection-less passage of transverse electromagnetic waves from a vacuum through the inhomo-geneous plasma layer with variable enough the plasma permettivity f() profile.

Below by usage of Helmholtz 1D equation the exact solution is investigated to des-cribe the reflectionless propagation of electromagnetic wave through periodically inho-mogeneous wide plasma layer containing subwave structures. Our calculations have shown that the wave field spatial profile in inhomogeneous plasma may be of soliton-like 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 /d2 + f() E = 0 describing the electromag-

netic 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 inhomo-geneous plasma the wave electric field is taken by WKB-expression E() = ( E0 / p

1/2 )

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 }/d2.

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 + Bsin( 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 + Bsin( 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 permet-tivity 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 + Bsin( 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 transillu-

mination 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 func-

tions [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 !!