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Generation of Solitonlike Structures of Electromagnetic Wave Field During Transillumination of Inhomogeneous Plasmas N.S. Erokhin 1* , Zakharov 1,2 , L.A. Mikhailovskaya 1 1 Space Research Institute of the Russian Academy of Sciences, Moscow, Russia

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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, Russia

1P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia

*e-mail address:

VI International Conference


Achievements, Developments and Perspectives”

June 4-8, 2012, Novosibirsk, Russia



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.


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 /d2 + f() E = 0describing the electromag-netic wave propagation along x-axis. Here  = x / c,is the wave frequency, f() =N2 аndN()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 / p1/2 )exp [ i () ],wherep() = d() / dis the dimensionless wave vector, ()is the wave phase, E0is the typical wave electric field value. In the case of exact solution the following condition must be satisfied f() = [p()]2 – [p()]1/2d2 {[ 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 > 1are 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 ofp(), f(), W() = 1 / [ p() ]1/2where 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 andmin 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 / pminhas increased about two times but max f / min fpractically 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() /  ]2dimensionless 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 andmin f 0.64. Now we have obtained maxp / 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.


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 andd 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()]2and large variations of ef()



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Many thanks you for attention !!