Soliton-like splashes of the field of electromagnetic wave under its reflectionless propagation

through the inhomogeneous plasma layer

N.S.Erokhin1) and V.E.Zakharov1,2)

(1) Space Research Institute of RAS, Moscow(2) Lebedev Physical Institute of RAS, Moscow

e-mail: nerokhin@iki.rssi.ru

The Fifth International ConferenceSOLITONS, COLLAPSES AND TURBULENCE:

Achievements, Developments and Perspectives

Chernogolovka, August 2009

Abstract. On the basis of exactly solvable mathematical model it is considered the reflectionless propagation of transverse electro-magnetic wave in an inhomogeneous plasma with subwave structures and the rise of soliton-like splashes of wave field with its strong amplification. The spatial profile of splash wave field is depending on the number of free parameters. It is possible the reflectionless passage of electromagnetic wave incident from the vacuum on the plasma through such subwavelength structures which characteristics may be very variable ones.  

The basic equations and their analysis 

Up to the present time it was considered the interaction of electromagnetic waves with inhomogeneous and nonstationary media by the usage of exactly solvable physico-mathematical models [1-4]. Such problems are of the great interest in a number of applications, in particular, for the increasing of efficiency of electro-magnetic radiation absorption during the plasma heating; to explain the possible mechanisms of electromagnetic radiation escape from the sources located in the overdense plasmas in the astrophysics; to find the optimum of inhomogeneous structures characteristics for the purpose to raise the efficiency of absorption coverages and transillumination ones in the radio frequency band and to elaborate the radio transparent coating of antennas. This analysis is important also for the search of optimum of the dielectric permeability distribution in transillumination layer thickness to provide the minimum of reflection of electromagnetic signal and its transmission from antenna covered by the dense plasma [5]. Investigations performed early (see, for example, [2,3]) had shown that it is possible to obtain the

reflectionless passage of electromagnetic waves from the vacuum into the object dielectric coverage in spite of the dielectric permeability jump presence on the intermedia surface. Besides these investigations allow to improve significantly our understanding of electromagnetic fields dynamics in spatio-temporal subwave dielectric structures having the strong spatial dispersion.

Below it is performed the analysis of soliton type splashes origion during the electromagnetic wave reflectionless propagation through the inhomogeneous plasma. In the splashes place the wave amplitude may be strongly amplified by the one order of magnitude and even more. The mathematical model used is based on the exact solution of Helmholtz equation and the problem free parameters number may be arbitrary one. So it allows to change essentially, for example, the number of splashes, their amplitudes and forms, the distances between splashes and so on. Consequently, it is possible to change significantly the spatial profile of transilluminated inhomogeneous plasma structure.

It is necessary to emphasize here that it is studied the subwavelength plasma inhomogeneities with large amplitude variations. So it is impossible to use approximate methods for the analysis of electromagnetic waves interaction with such plasmas.

The exactly solvable models considered may reveal the new features of oscillation dynamics and waves propagation and demonstrate also the interesting opportunities for practical applications under the medium controllable variations. Actually we are considering the resonance tunneling phenomenon of electromagnetic wave over the structured plasma which includes the local opaque region.

Thus in the analogy with papers [6,7] in the cases considered the electromagnetic wave is consistent with the inhomogeneous plasma layer.

Let us consider the interaction of s-polarized electromagnetic wave with the inhomogeneous plasma in the absence of external magnetic field or for the case of wave propagation in the magnetoactive plasma perpendicular to the external magnetic field. For the wave electric field E(x,t) = Re [ F(x) exp ( i t ) ], the function F(x) is the solution of following equation

d2F / dx2 + k02 ef(x) F = 0. (1)

Here k0 = / c is the vacuum wavenumber, is the wave frequency, pe(x) is the

plasma electron langmuir frequency, ef(x) = 1 – [ pe(x) / ]2. In the case of

extraordinary electromagnetic wave propagation across the external magnetic field in magnetoactive plasma the function ef(x) is given by the following formula

ef(x) N2(x) = - ( c2 / ), where N is the refraction index, and c are the

components of dielectric tensor. For the further analysis it is convenient to introduce the dimensionless spatial variable = k0 x and the wavenumber p() = c kx(x) / .

The exact solution of equation (1) is taken like this [1-4]

F() = F0 exp[ i () ] / [1/p()]1/2 , d/d = p(), F0 = const. (2)

So according to (1), (2) we obtain the following connection between the effective dielectric permeability ef(x) and the wave number p()

ef() = [ p() ]2 + (d2p / d2) /2p – 0.75(dp / d)2 /p2. (3)

Let us introduce now the normalized wave amplitude A() [1/p()]1/2. Then the formula (3) may be written like this

d2A / d2 + ef() A - [ 1 / A() ]3 = 0. (4)

Nonlinear equation (4) for the given function ef() determines the spatial profile of

electromagnetic wave amplitude A(). It is necessary to note that even in the case of homogeneous plasma when ef() = const the solution of equation (4) describes (for the

fixed wave frequency) the spatially modulated wave packet with some parameter characterizing the magnitude of amplitude A() variation ! Let us introduce = , where is the parameter determining the typical scale of plasma inhomogeneity. Now we consider the exactly solvable model of localized plasma inhomogeneity for which W() = + / ( 1 + 2 ). So the nondimendional wave number and the effective dielectric permeability are determined by the following expressions

p() = 1 /W2(), ef() = [1/W() ]4 + 2 2 (1 - 3 2) /[ (1 + 2)2 ( + + 2 )]

(5)

with three parameters , , . Let us take as the following = 1.3, = 11, = 0.73. It is necessary to note here that for < c = (0.5 / )0.5 [ ( + ) - 1 / ( + )3 ] 0.5 the

magnitude of ef(0) is less of the unity.

The plotts of functions p(), W(), ef() are given in the Fig.1a-Fig.1c. According to

the Fig.1a the minimum of p() 6.6 10-3 is in the point = 0, where the wave field amplitude W() amplification is 9.46 (see the Fig.1b).

Fig.1b.

Fig.1c.

But far enough from the splash we have W() . The profile of effective dielectric permeability has the nonmonotonous behaviour: in the field splash center we have ef(0) 0.953 (that is the maximum of ef() ) and there are two plasma regions where

ef() < 0. So ef() = 0 in the points 0.58 and 3.87. The minimum value

of ef() equal - 0.46 is in the points 1.2. Thus for this case of parameters

choice there is the localized (2 ~ 1 ) plasma inhomogeneity with very large wave field amplification at its center where ef 0.953. It is necessary emphasize here the

nonlocal connection of wave field splash place with points in which ef() = 0.

The model considered may be generalized to the case of n arbitrary disposed wave field splashes with amplitudes Am = ( + m) if we take the function W() like this

W() = + m m / [ 1 + m2 ( - m )2 ], where 1 m n, m is the center of splash

with number m. These plasma structures are subwavelength ones for the choice m >

1.The other model of exact solution for electromagnetic wave field splash in the inhomogeneous plasma having the exponential splash field decreasing outside splash center is realized under the following choice of function W() : W() = + / ch , = . So the spatial profile of effective plasma permeability is described by the following expression

ef() = 1 / W4() + 2 ( 2 - ch2 ) / [ ch2 { + ch } ].

Let us take now > 0 , > 0. The condition ef(0) < 1 is fulfilled for the choice

< c = ( 1 + / )1/2 [ 1 – 1 / ( + )4 ]1/2.

In the case = 1.3 , = 11 , = 0.73 the plotts of functions W(), p(), ef() are

shown in the Fig.2. As it seen from the Fig.2 in the splash center we have min p = p(0) 0.0066, the wave field amplification in the splash is 9.46. The ef() profile has

nonmonotonous behaviour and max ef() = ef(0) 0.477 is in the point = 0. In the

splash region there are two subregions 0.9 < < 4 where ef() < 0 with minimum

magnitude of ef() - 0.314.

So according to the exact solution of Helmholtz equation given above the strong enough spatial dispersion of ef() changes significantly the classical conceptions

between profiles of wave number p() and effective dielectric permeability ef(). It is

necessary to note also that it is possible more common case of plasma inhomogeneity with n arbitrary placed wave field splashes having their amplitude ( + m) under the

following choice of function W() : W() = + m m / [ ch { m ( - m ) } ] where

1 m n, m is the center of splash with number m.

Let us take now p() = 1 - / ( 1 + 2 ), = , = t , = 0.99 , = 0.04. We may consider the spatial profiles of wave electric field

F(x, t) = E(x,t) / E0 = [ 1 / p() ]1 / 2 cos [ (, ) ] , where

(, ) = ( 1 / )( - arctg ) - ,

for the different moments of nondimensional time . In the Fig.3 such profiles are presented for the choice 1 = 0, 2 = 3 / 2. The large splashes of wave field are

observed at the n = n, but for times n = ( n + 0.5 ) these splashes aren’t so strong.

Fig.2a.

  

Fig.2b.

Fig.2c.

Fig.3.

Let us present now the analysis results of wave field splashes raise in the case of periodical inhomogeneity in the magnetoactive plasma. For example, we may take the following exactly solvable model

W() = + 0.25 ( 1 + cos )2.

The plotts of functions W(), p(), ef() are given in the Fig.4 for the interval

< 15 and the incoming parameters choice = 1.2 , = 8, = 0.8 , = . In this case we have :

min p() = p(0) = 0.012 , max p() = 0.694 , max W = 9.2 , min W = 1.2 ,

max ef() = 4.734 , ef(1.5) =0.014 , ef(0) = 0.557 , min ef() = 0.01.

Since in this example we have max ef() > 1 such situation corresponds to the

extraordinary electromagnetic wave propagation across the external magnetic field in the magnetoactive plasma.

Fig.4a.

Fig.4b.

Fig.4c.

Conclusions  

On the basis of exactly solvable model of Helmholtz equation it is considered the reflectionless propagation of electromagnetic wave in the inhomogeneous plasma and the raise of soliton type splashes of wave field in some plasma sublayers conditioned by the strong decreasing of wave number p() magnitude.

Such splashes are possible both for the s-polarized electromagnetic wave of given frequency in the plasma without the external magnetic field and also for the extraordinary electromagnetic wave propagating across the external magnetic field in the magnetoactive plasma.

The connection of wave field amplitude with the effective dielectric permeability is describing by the nonlinear equation and the effect of strong spatial dispersion of ef()

is very important for subwavelength structures of the plasma inhomogeneity. The spatial profiles of wave fields are characterizing by a number of free parameters.

It is possible the nonreflection passage of electromagnetic wave incident on the plasma layer from vacuum and structures characteristics may be very variable ones. In particular, the nonreflection passage of electromagnetic wave through the inhomogeneous plasma may be described with special presentation of p() including some finite but arbitrary function which may contains the stochastic spatial variations. It is possible to take into account also the effects of cubic nonlinearity in ef().

References 

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