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Interaction between parallel Gaussian electromagnetic beams in the ionosphere Agarwal, S. K., & Sharma, A. (2008). Interaction between parallel Gaussian electromagnetic beams in the ionosphere. Journal of Atmospheric and Solar-Terrestrial Physics, 70(7), 980-990. DOI: 10.1016/j.jastp.2008.01.012 Published in: Journal of Atmospheric and Solar-Terrestrial Physics Queen's University Belfast - Research Portal: Link to publication record in Queen's University Belfast Research Portal General rights Copyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made to ensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in the Research Portal that you believe breaches copyright or violates any law, please contact [email protected]. Download date:15. Feb. 2017

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Interaction between parallel Gaussian electromagnetic beams inthe ionosphere

Agarwal, S. K., & Sharma, A. (2008). Interaction between parallel Gaussian electromagnetic beams in theionosphere. Journal of Atmospheric and Solar-Terrestrial Physics, 70(7), 980-990. DOI:10.1016/j.jastp.2008.01.012

Published in:Journal of Atmospheric and Solar-Terrestrial Physics

Queen's University Belfast - Research Portal:Link to publication record in Queen's University Belfast Research Portal

General rightsCopyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or othercopyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associatedwith these rights.

Take down policyThe Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made toensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in theResearch Portal that you believe breaches copyright or violates any law, please contact [email protected].

Download date:15. Feb. 2017

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Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 980–990

Interaction between parallel Gaussian electromagnetic beamsin the ionosphere

Sujeet Kumar Agarwal�, Ashutosh Sharma1

DST Project and Ramanna Fellowship Programme, Department of Education Building, Lucknow University, Lucknow 226 007, India

Received 27 August 2007; received in revised form 24 December 2007; accepted 28 January 2008

Available online 26 February 2008

Abstract

In this paper, the propagation of two initially (z ¼ 0) parallel Gaussian electromagnetic beams, propagating in the

z-direction in ionosphere, has been investigated. The nonlinearity in the dielectric function, responsible for the interaction

between the beams arises from the redistribution of the electron density, caused by the nonuniform distribution of electron

temperature determined by the Ohmic heating of the electrons and the energy loss of electrons on account of collisions. The

wave frequencies have been assumed to be much larger than the electron collision frequency and gyrofrequency. A self-

consistent solution of the electromagnetic wave equation and energy balance equation (considering the solar radiation) has

been obtained in the paraxial approximation. Second-order coupled ordinary differential equations have been obtained for

the distance between the centers of the beams and the beam widths in the x and y directions as a function of the distance of

propagation along the z-axis. Using the available database for the mid-latitude daytime ionosphere, the equations have

been solved numerically for a range of parameters and a discussion of the results has been presented. The physical basis for

the fact that the beams move towards each other, when the resulting irradiance distribution of the two beams has a

maximum in the space between the two beams, has been highlighted.

r 2008 Elsevier Ltd. All rights reserved.

Keywords: Ionosphere; Parallel Gaussian beams; Self-focusing; Nonlinearity

1. Introduction

Recent interest in self-focusing of electromagneticbeams in plasmas has spilled over to the field ofmutual interaction between electromagnetic beamsin a plasma. The cross focusing of coaxial electro-magnetic beams in a plasma, on account of the

dielectric function of either beam depending on theirradiance of both has been investigated for thepredominance of collisional (Gurevich, 1978; Sodhaand Sharma, 2006), ponderomotive (Sodha et al.,1979) and relativistic (Esarey et al., 1988; Gibbon,1990; Kumar et al., 1998; Shvets and Pukhov, 1999;Gupta et al., 2005) nonlinearity. The parametricinstabilities and ponderomotive nonlinearities havebeen authoritatively discussed by Shukla et al.(1975) and Yu et al. (1974).

A relatively recent development in this field is theinvestigation of the interaction between two laserbeams, which are not coaxial (Ren et al., 2000, 2002;

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1364-6826/$ - see front matter r 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jastp.2008.01.012

�Corresponding author. Tel.: +91 9839 705 081.

E-mail address: [email protected] (S.K. Agarwal).1Present address: Institute of Photonics Technologies, Depart-

ment of Electrical Engineering, National Tsing Hua University,

Hsinchu 300 13, Taiwan, ROC.

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Dong et al., 2002; Sodha et al., 2006a). Themotivation behind these studies, is to use the resultsas a first step towards the understanding of theinteraction between individual filaments of a beamor individual speckles in a random-phase platebeam. Ren et al. (2000) have demonstrated themutual attraction between two skew electromag-netic beams, which may cause the beams to spiralaround each other using the variational method aswell as a three-dimensional particle in cell (PIC)model of laser light in a plasma and considering therelativistic dependence of the electronic mass on thequiver speed of the electrons. The coherent laserbeams, linearly polarized along the same directionare subject to attraction or repulsion depending ontheir being in phase or antiphase. However, thisnonlinearity has been approximated by a quadraticexpression. Ren et al. (2002) took into account therelativistic nonlinearity and the nonlinearity onaccount of density modulations from a plasmawake in an exhaustive study of the interactionbetween two laser beams. It is seen that incoherentbeams and coherent beams, which are linearlypolarized in perpendicular directions or are circu-larly polarized get attracted to each other (whentheir axes are close enough). For coherent beams,linearly polarized but not in mutually perpendiculardirections, the beams attract or repel depending onwhether the beams are in phase or antiphase. In thisanalysis also the relativistic nonlinearity was takenas quadratic in nature. Dong et al. (2002) took intoaccount the relativistic [explained by Hora (1975)and Hauser et al. (1988)] as well as the ponder-omotive nonlinearity for the study of the timeevolution of two interacting beams in a plasma. Thedetailed results of this transient analysis based onthe quadratic nonlinearity and simulation canhardly be compared with the steady-state analysisincorporating saturating nonlinearity (Sodha et al.,2006a, 2007a); however, both the theories lead tosituations for attraction and repulsion of the beams.Shukla et al. (2006) have recently investigatedthe nonlinear interaction between two weakly re-lativistic crossing laser beams in plasmas. A set ofnonlinear mode coupled equations and nonlineardispersion relations have been derived and analyzedfor Raman (backward and forward) scatteringinstabilities as well as Brillouin and modulationself-focusing instabilities. Sodha et al. (2006a) haveinvestigated this phenomenon analytically in theparaxial approximation. The dielectric function wasexpressed as a function of the irradiances of the two

beams for three types of nonlinearities (viz. colli-sional, ponderomotive and relativistic); this expres-sion for the dielectric function was substituted in thewave equation and a solution of the resultingnonlinear equation was obtained in the paraxialapproximation.

The analyses (Ren et al., 2000, 2002; Dong et al.,2002; Shukla et al., 2006) based on numericalsimulation do not bring out the physics of themutual interaction between the beams. Also thecomplexity and the corresponding necessary effortsin the numerical computation account for their notbeing quite appropriate for parametric studies.Following Sodha et al. (2006a) it is interesting toconsider the basic physics of interaction for the caseof incoherent and circularly polarized or cross-polarized coherent beams; the argument can beextended to the general case.

It can be readily seen that the irradiancedistribution resulting from two parallel incoherentbeams at a distance 2x0, having the ratio of axialirradiances k and the widths of the beams as r0 andsr0 has (in between the beams) a maximum when2x0owr0, and a minimum when 2x04wr0; theparameter w as a function of k and s can bedetermined (Sodha et al., 2007a) from the condi-tions dI/dx ¼ 0 and d2I/dx2

¼ 0 [where I(x, wr0) isthe sum of the irradiances of the two beams]. Hencein a nonlinear plasma (in which effective dielectricfunction is an increasing function of the irradiance)the dielectric function has a maximum or minimumin the space between the two beams, depending onwhether 2x0owr0 or 2x04wr0. Since light bendstowards regions of higher refractive index (i.e.higher dielectric function) the beams move towardsthe maximum irradiance region i.e. towards eachother when 2x0owr0. Similarly, when 2x04wr0the beams should repel each other; however, theparaxial theory is not relevant to validating thisresult as it may not be applicable in the regime2x04wr0. Thus, it is not the nature of nonlinearity,but the distance between the axes of the two beams,which determines whether the beams will attract orrepel each other.

On account of the abundance and importance ofnonlinear processes (Gurevich, 1978) in the field ofinteraction of high power electromagnetic beamswith ionospheric plasma, it is natural to investigatethe interaction of close parallel electromagneticbeams in the ionosphere also. Such an investigationis not a straightforward extension of the theoryapplicable to plasmas on account of considerable

ARTICLE IN PRESSS.K. Agarwal, A. Sharma / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 980–990 981

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complexity in the kinetics, collisions and energybalance in the ionospheric plasma. The additionalfactors, which have to be considered, are

(i) the role of solar radiation in the energy balance,(ii) the dependence of the frequency of electro-

n–neutral species, electron–ion and ion–neutralspecies collisions on electron/ion temperatures,

(iii) the dependence of the fraction of energytransferred per collision on electron/ion tem-perature and

(iv) the number of neutral and ionic species.

In this communication the following simplifyingassumptions have been made:

(i) the interaction lasts for periods less than theelectron density relaxation time tN and hencethe electron density is constant; the variation oftN with ionospheric height has been given inTable 13 of Gurevich’s1 book (tN lies between52 s and 3.7� 104 s for heights 100–500 km);

(ii) the ion–neutral species collision frequencynembd̄einei, where nei is the electron–ion colli-sion frequency and d̄ei is the mean fraction ofexcess energy transferred by the electron,during a collision with an ion (this leads tothe ion temperature and temperature of theneutral species to be the same); this is specifi-cally true for an ionospheric height of 150 kmfor which computations have been made and

(iii) the heat capacity (p number density) of theneutral species is much larger than that of ionsand electrons (this ensures a constant tempera-ture neutral species heat sink); this is true forthe ionosphere, at least up to 1000 km [Tables 1and 2 Gurevich (1978)].

A paraxial theory of interaction between twoinitially parallel electromagnetic Gaussian beams inthe ionosphere has been presented here. Theredistribution of the electron density, caused bythe nonuniform distribution of electron tempera-ture, determined by the Ohmic heating of theelectrons and the energy loss of electrons on accountof collisions results in the nonlinear dielectricfunction, which is responsible for the interactionbetween the beams. The electron energy loss bythermal conduction has been neglected on accountof the electron cyclotron frequency being muchgreater than the electron collision frequency ne. TheOhmic heating of the electrons by the nonuniformfield of the beams and the loss of electron energy bythe collisions gives rise to a nonuniform distributionof the electron temperature, which creates pressuregradients of the electron and ion gases. Thebalancing of these pressure gradients by the spacecharge field in the steady-state results in a nonuni-form field-dependent distribution of electron densityand hence the dielectric function. The presentanalysis to determine the space-dependent distribu-tion of the electron temperature and thereby thedielectric function is similar to a recent study ofSodha et al. (2006b), which investigates self-focusingof a single beam in a collisional plasma withsignificant thermal conduction.

The introduction may be concluded by a passingreference to the important fact that the stochasticpulsations are suppressed in laser plasma interac-tion by laser beam smoothing, using a random-phase plate (Hora, 2006), in which the thermalcollisions wash out the density ripples (Hora andAydin, 1992; Glowacz et al., 2006). The randomiza-tion of phase of neighboring beams also leads tosmoothing of the beam as shown experimentally forlaser plasma interaction by Labaune et al. (1992)and Hora (2006). This topic is also important for

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Table 1

Dependence of a0, a1, a2 and a3 on height

Height (km) a0 a1 a2 a3

80 5.80� 103 �1156.90 218.90 �7.60

90 8.30� 102 �226.40 47.80 �2.10

100 2.15� 102 �60.60 13.70 �0.60

110 1.10� 102 �0.10 7.70� 10�5 �1.10� 10�8

120 3.08� 101 �10.10 2.63 �0.10

130 9.90 �3.20 1.00 0.00

150 �2.90 4.50 1.20 �0.10

200 0.05 0.20 �0.05 0.00

Table 2

Dependence of b0, b1 and b2 on height

Height (km) b0 b1 b2

80 141,666.66 977,954.54 �13,257.57

90 25,983.33 144,990.15 �2200.75

100 9116.66 41,827.42 �611.36

110 3465.00 9762.04 �144.31

120 1949.00 4782.40 �72.50

130 1070.83 1951.70 �29.35

150 761.16 606.23 �6.40

200 516.73 �16.68 6.31

S.K. Agarwal, A. Sharma / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 980–990982

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suppression of self-focusing in generation of plasmablocks for laser-driven fusion (Badziak et al., 2004,2005; Hora et al., 2007).

In general the magnetic field plays an importantpart in the magnetized plasmas (Shukla and Stenflo,1984). However, this communication considers thecase of the wave frequency much higher than theelectron cyclotron frequency and in this approxima-tion the beams can to a satisfactory degree beassumed to be linearly polarized. Since the electroncyclotron frequency is much higher than theelectron collision frequency, thermal conductionby electrons can be neglected in the energy balanceequation. Gurevich has shown that in the iono-sphere the ponderomotive nonlinearity is 10�4–10�3

times the collisional nonlinearity (Gurevich, 1978,Table 3).

2. Analysis

2.1. Radial distribution of electron density

On account of the nonuniform radial distributionof the beam irradiance, there is a correspondingdistribution of electron and ion temperatures, whichcreates pressure gradients in the electron and iongas; in the steady state these gradients are balancedby the space charge field. This phenomenon, alongwith the assumption of charge neutrality andefficient energy exchange between the ions andneutral species leads to an expression for theelectron density in terms of the electron temperatureTe; thus, following earlier workers (Sodha et al.,2007b) one obtains

Ne

Ne00¼

T e00 þ T

T e þ T, (1)

where the suffix e refers to electrons and the suffixdouble zero refers to the regions, which are notaffected by the beams. T is the temperature of theneutral species and N refers to the electron density.Eq. (1) is based on assumptions (ii) and (iii), asgiven in Section 1; these assumptions are indeedvalid at the ionospheric height of 150 km (for whichcomputations have been made), as indicated inSection 1.

2.2. Propagation of two initially parallel Gaussian

electromagnetic beams in the ionosphere

The dielectric function, corresponding to alinearly polarized electromagnetic wave of fre-

quency much higher than the collision and cyclo-tron frequencies of electrons and propagating alongthe direction of geomagnetic field in the ionosphereis given by

�j ¼ �rj � i�ij , (2a)

where suffixes r and i indicate real and imaginaryparts and j refers to the jth beam

�rj ¼ 1�o2

p0

o2j

!Ne

Ne00

� �, (2b)

�ij �o2

p0

o2j

!ne0oj

� �, (2c)

where e is electronic charge, m is the electronic mass,op0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4pe2Ne00=mÞ

pand ne0 are respectively the

plasma frequency and the electron collision fre-quency in the absence of the field of the waves.

When eij51, its dependence on the temperature ofthe electrons and the irradiance can be neglected asa good approximation.

The irradiance of the two Gaussian beamspropagating along the z-axis can at z ¼ 0, berepresented by

E1E�1 ¼ E210 exp �

ðxþ x0Þ2

r210�

y2

r210

� �(3a)

and

E2E�2 ¼ E220 exp �

ðx� x0Þ2

r220�

y2

r220

� �, (3b)

where E1 and E2 are the amplitudes of the electricvectors of the two beams, 2x0 is the distancebetween the axes of the beams at z ¼ 0 and r10and r20 are the widths of the two beams at z ¼ 0.

The electron temperature Te is a function ofPajEjE

�j [aj being a parameter (defined later) and

Ej being the amplitude of the electric vector of thejth beam of frequency oj]. Since EjE

�j is a function

of x and y, one can expand Te and Ne and hence erj[referring to Eqs. (1) and (2b)] in powers of x and y

and in the paraxial approximation retain terms onlyup to those having x2 and y2. Thus in the paraxialapproximation Te, Ne and erj can be expressed as

T e ¼ T e0 �x

r10T e1x �

x2

r210T e2x �

y2

r210Te2y, (4)

Ne ¼ Ne0 � xNe1x � x2Ne2x � y2Ne2y

ARTICLE IN PRESSS.K. Agarwal, A. Sharma / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 980–990 983

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and

�rj ¼ �0j �x

r10�1xj �

x2

r210�2xj �

y2

r210�2yj , (5)

where Te0, Te1x, Te2x, Te1y, Ne0, Ne1x, Ne2x, Ne2y, e0j,e1xj, e2xj and e2yj are functions of z, E1E

�1 and E2E

�2.

In the steady state, the amplitude of the electricvector Ej satisfies the scalar wave equation

r2Ej þo2

j

c2�jEj ¼ 0. (6)

Following earlier workers (Sodha et al., 2006a),the solution of the wave equation in the paraxialapproximation, consistent with the dependence ofthe dielectric function on transverse coordinates,expressed by Eq. (5) is

EjE�j ¼

E2j00

f jxðxÞf jyðxÞ

ffiffiffiffiffiffiffiffiffiffiffi�0jð0Þ

�0jðxÞ

s

� exp �fx� ð�1Þjx0F jðxÞg2

r2j0f2jxðxÞ

�y2

r2j0f2jyðxÞ

" #,

(7a)

EjE�j ¼

E2j0

f jxðxÞf jyðxÞ

exp �fx� ð�1Þjx0FjðxÞg2

r2j0f 2jxðxÞ

�y2

r2j0f 2jyðxÞ

" #,

(7b)

EjE�j ¼ A2

j0ðxÞ

exp �fx� ð�1Þjx0FjðxÞg2

r2j0f 2jxðxÞ

�y2

r2j0f 2jyðxÞ

" #.

(7c)

Further the beam width parameters fjx, fjy

corresponding to x and y directions, respectivelyand parameter Fj representing the axial separationof jth beam from z-axis are given by

d2f jx

dx2¼ �

r2

�0j f jx

�2xjf2jx �

pjs2j

r2f 2jx

!�

1

2�0j

d�0j

dx

df jx

dx,

(8a)

d2f jy

dx2¼ �

r2

�0j f jy

�2yjf2jy �

pjs2j

r2f 2jy

!�

1

2�0j

d�0j

dx

df jy

dx

(8b)

and

d2F j

dx2¼ �

r2

�0j

�2xjF j þ ð�1Þj �1xj

2ffiffiffimp

� ��

1

2�0j

d�0j

dxdFj

dx,

(8c)

where x ¼ ðc=o1r210Þz is the dimensionless distanceand r ¼ (r10o1/c) is the initial dimensionless widthof the first beam,

pj ¼ o21=o

2j ; sj ¼ r210=r2j0; m ¼ x2

0=r210

and at z ¼ 0, Fj ¼ fjx ¼ fjy ¼ 1.Eq. (7) indicates that the curve x ¼ (�1)jx0Fj(x),

y ¼ 0 represents the axis of the jth beam.In view of Eq. (8c) the parameter SFj, repre-

senting the axial separation between the beams isgiven by

d2

dx2X

Fj ¼ � r2X 1

�0j

�2xjF j þ ð�1Þj �1xj

2ffiffiffimp

� �

�X 1

2�0j

d�0j

dxdF j

dx. (8d)

It is evident from Eq. (8d) that for the beams toattract each other

X 1

�0j

�2xjF j þ ð�1Þj �1xj

2ffiffiffimp

� �40. (9)

2.3. Evaluation of e0j, e1xj, e2xj and e2yj

From Eq. (7) the irradiance of jth beam at a point(x, y), for any value of z can in the paraxialapproximation be expressed as

EjE�j �

E2j0

f jxf jy

exp �x20F

2j

r2j0f2jx

!1þ xð�1Þj

2x0Fj

r2j0f2jx

!(

�x2

r2j0f2jx

1�2x2

0F2j

r2j0f2jx

!�

y2

r2j0f2jy

). (7d)

Further, from Eqs. (1), (2), (4) and (5), oneobtains

�0j ¼ 1� Opj

ðT e00 þ TÞ

ðT e0 þ TÞ, (10a)

�1xj ¼ Opj

Te1xðT e00 þ TÞ

ðT e0 þ TÞ2, (10b)

�2xj ¼ Opj Te2x þT2

e1x

ðTe0 þ TÞ

� �ðT e00 þ TÞ

ðTe0 þ TÞ2(10c)

ARTICLE IN PRESSS.K. Agarwal, A. Sharma / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 980–990984

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and

�2yj ¼ Opj

Te2yðTe00 þ TÞ

ðTe0 þ TÞ2, (10d)

where O ¼ o2p0=o

21.

Te0, Te1x, Te2x and Te2y depend upon the axialirradiances and widths of the two beams. Thus, e0j,e1xj, e2xj and e2yj can be obtained as functionsof z, fjx, fjy and Fj for given axial irradiances E2

j00

(at z ¼ 0) at a given height in the mid-latitudedaytime ionosphere, from a knowledge of thedependence of Te0, Te1x, Te2x and Te2y on theirradiance

PajEjE

�j and the beam widths.

2.4. Evaluation of electron temperature

The energy balance for the electrons in theionosphere (Gurevich, 1978) can be expressed as

Qþ J � E ¼ Qþe2Nene2m

PEjE

�j

o2j

¼ Ne

Xnemdem þ neid̄ei

n o�

3

2kBðTe � TÞ, (11a)

where J � E is the time-averaged Ohmic loss per unitvolume, Q is the net power per unit volume gainedby the electrons [above the thermal energy (3/2)NekBTe00] from solar radiation, on account of excessenergy of electrons produced by photo ionization,nem is the frequency of electron collision with themth neutral species, dem is the fraction of excessenergy of an electron (above that of neutral species)lost in a collision with the mth neutral species, nei isthe electron collision frequency with ions and d̄ei isthe mean fraction of excess energy of an electron(above that of an ion) lost in a collision with an ion.

The temperature of ions and neutral species istaken to be the same and independent of the field ofthe beams in view of the assumptions (ii) and (iii) ofSection 1. Further the electron cyclotron frequencyoc5oj, but it is much greater than the electroncollision frequency ne and hence the electron energyloss by thermal conduction has been neglected. Inthe absence of the waves (i.e. in undisturbedionosphere) J � E ¼ 0 and Eq. (11a) reduces to

Q ¼3

2kBðTe00 � TÞNe00

Xnem0dem0 þ nei0d̄ei0

n o,

(11b)

where suffix zero indicates the values in the absenceof the beams, which have been tabulated by

Gurevich (1978). From Eqs. (11a) and (11b) oneobtains

X ajEjE�j

2000¼

1

ne

Xnemdem þ neid̄ei

n o T e

T� 1

� ��

�Ne00

Ne

Xnem0dem0 þ nei0d̄ei0

n o

�Te00

T� 1

� ��, (12)

where aj ¼ ð2000e2=3mo2j kBTÞ.

The factor 2000 (EMH/m, MH being mass of ahydrogen atom) has been introduced for numericalconvenience. Since all the terms on the right-handside of Eq. (12) are known functions (Gurevich,1978) of electron temperature Te (at a given height),this equation expresses a useful relationship be-tween the electron temperature Te and the irradi-ance ajEjE

�j .

At 150 km, the significant ionic species in theionosphere are NO+, O2

+ and O+, with concentra-tions, tabulated by Gurevich (1978) (Table 2);further d̄ei is given by

d̄ei ¼2m

MHnOþ=16

þ nOþ2=32

� �þ nNOþ=30 n o

,

(13)

where 16, 30 and 32 being the molecular/atomicmasses of the ionic species O+, NO+ and O2

+,respectively in atomic mass units, nOþ , nOþ

2and nNOþ

being the fractional concentrations of ions, tabu-lated by Gurevich (1978) as functions of height.

The significant neutral species at 150 km heightare N2, O2 and O. Gurevich (1978) has described thetemperature dependence of nem and dem correspond-ing to these neutral species. The electron–ioncollision frequency nei is however given by

nei ¼ nei0Ne

Ne00

Te

T e00

� ��3=2(14)

and electron collision frequency ne is given by

ne ¼ nei þX

nem. (15)

The mid-latitude daytime model of the iono-sphere and the data base on the collision processes[collated by Gurevich (1978)] can be used to obtaina relationship between ne,

Pnemdem þ neid̄ei

and

the electron temperature Te. The relationships(best fit) can be represented empirically for a given

ARTICLE IN PRESSS.K. Agarwal, A. Sharma / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 980–990 985

Author's personal copy

height as

Xnemdem þ neid̄ei ¼ a0 þ a1

Te

T e00

� �þ a2

T e

T e00

� �2

þ a3T e

Te00

� �3

, (16a)

Xnemdem þ neid̄ei ¼ A1 �

x

r10A2 �

x2

r210A3 �

y2

r210A4

(16b)

and

ne ¼ b0 þ b1T e

Te00

� �þ b2

Te

T e00

� �2

, (17a)

ne ¼ B1 �x

r10B2 �

x2

r210B3 �

y2

r210B4, (17b)

where

A1 ¼ a0 þ a1Te0

Te00

� �þ a2

T e0

T e00

� �2

þ a3Te0

T e00

� �3

,

(18a)

A2 ¼T e1x

T e00a1 þ a2

2Te0

Te00þ a3

3T2e0

T2e00

!, (18b)

A3 ¼ a1Te2x

Te00þ a2

2T e0T e2x � T2e1x

T2e00

þ a33T e0ðT e0Te2x � T2

e1xÞ

T3e00

, (18c)

A4 ¼T e2y

T e00a1 þ a2

2T e0

T e00þ a3

3T2e0

T2e00

!, (18d)

B1 ¼ b0 þ b1Te0

Te00

� �þ b2

T e0

T e00

� �2

, (19a)

B2 ¼Te1

Te00b1 þ b2

2Te0

T e00

� �, (19b)

B3 ¼ b1Te2x

Te00þ b2

2Te0Te2x � T2e1x

T2e00

(19c)

and

B4 ¼Te2y

Te00b1 þ b2

2Te0

T e00

� �. (19d)

Tables 1 and 2 represent the values of theparameters a0, a1, a2, a3, b0, b1 and b2, which have

been evaluated for 80–200 km height using databaseof Gurevich (1978).

Substituting for (Ne/Ne00), EjE�j , ne,

Pnemdemþ

neid̄eiÞ and Te from Eqs. (1), (7d), (17b), (16b) and(4), respectively in Eq. (12), expanding the terms inpowers of x and y, equating the coefficients of x, x2,y2 and the term independent of x and y on bothsides of the equation one obtains

Z1 � B1

XKj ¼ 0, (20a)

Z2B1 � Z1B2 þ B21

Xð�1ÞjKj

2sjF jffiffiffimp

f 2jx

¼ 0, (20b)

B21Z3 þ Z2B2B1 � Z1ðB3B1 þ B2

� B31

X sjKj

f 2jx

1�2msjF

2j

f 2jx

!¼ 0 (20c)

and

Z4B1 � B4Z1 � B21

X sjKj

f 2jy

¼ 0, (20d)

where

A ¼ ða0 þ a1 þ a2 þ a3Þðt� 1Þ

ðtþ 1Þ,

Z1 ¼ A1ðt1 � 1Þ � Aðt1 þ 1Þ,

Z2 ¼ ðA1 � AÞt2 þ A2ðt1 � 1Þ,

Z3 ¼ ðA1 � AÞt3 þ A3ðt1 � 1Þ � A2t2,

Z4 ¼ ðA1 � AÞt4 þ A4ðt1 � 1Þ,

Kj ¼ajE

2j0=2000

f jxf jy

exp �msjF

2j

f 2jx

!; j ¼ 1; 2, (21)

t1 ¼ ðTe0=TÞ; t2 ¼ ðT e1x=TÞ; t3 ¼ ðT e2x=TÞ,

t4 ¼ ðTe2y=TÞ and t ¼ ðT e00=TÞ.

Thus, for a given set of parameters, the depen-dence of the parameters Te0, Te1x, Te2x and Te2y onajE

2j00 (at z ¼ 0) may be evaluated from the above-

coupled equations.

3. Computations for the interaction between identical

and incoherent beams

The mid-latitude daytime model of the iono-sphere detailed by Gurevich (1978) and the databaseon collision processes (Gurevich, 1978) has beenused for numerical computations. The values of theparameters a0, a1, a2, a3, b0, b1 and b2 consistentwith the empirical relationships represented byEqs. (16a) and (17a) have been evaluated and

ARTICLE IN PRESSS.K. Agarwal, A. Sharma / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 980–990986

Author's personal copy

tabulated (Tables 1 and 2) for different heights(80–200 km) using database of Gurevich (1978).

Using C++ programming the computations havebeen made to investigate the simultaneous propaga-tion of two initially parallel identical and incoherentGaussian electromagnetic beams with r10 ¼ r20 ¼ r0,a1E2

100 ¼ a2E2200 ¼ aE2

00 and wave frequencyo1 ¼ o2 ¼ o ¼ 5� 107 rad/s. Eqs. (20) have beensolved numerically to determine the dependence ofthe parameters Te0, Te1x, Te2x and Te2y [i.e. theelectron temperature Te as defined in Eq. (4)] onaxial irradiance aE2

0 for a given set of parameters;Eqs. (10) have been used to evaluate the dependenceof the parameters e0j, e1xj, e2xj and e2yj [or thedielectric function defined by Eq. (5)] on axialirradiance aE2

0. From the knowledge of the depen-dence of the dielectric function on axial irradiancethe variation of the beam width parametersf1x ¼ f2x ¼ fx and f1y ¼ f2y ¼ fy, and the parameterF1 ¼ F2 ¼ F (representing the separation of eitherbeam from the z-axis) with dimensionless distanceof propagation x ¼ ðc=or20Þz has been determinedfor the set of parameters described above.

4. Numerical results and discussion

It has been discussed in the introduction that thebeams will move towards each other, when theresulting irradiance distribution of the two beamshas a maximum in the space between the two beams.

The numerical computations have been made toinvestigate the interaction of identical Gaussianbeams. In case of identical beams Eqs. (20b) and(20c) give

T e1x ¼ 0 (22a)

and

T e2x

T e00B1 ðA1 � AÞ

Te00

T

��

þ a1 þ a22Te0

T e00þ a3

3T2e0

T2e00

!ðt1 � 1Þ

)

�Z1 b1 þ b22T e0

T e00

� ��

¼ B31

2K

f 2x

1�2mF2

f 2x

!, (22b)

where

K ¼aE2

0=2000

f xf y

exp �mF 2

f 2x

!.

In view of Eqs. (22a) and (10b) the condition (9)of the attraction between the two beams reduces to

2OTe2xðT e00 þ TÞF

�0ðTe0 þ TÞ240, (23a)

where

�0 ¼ 1� OðTe00 þ TÞ

ðT e0 þ TÞ.

It is evident from Eq. (22b) and condition (23)that the beams bend towards each other when

mof 2

x

2F2(23b)

at z ¼ 0, mo0.5.The dependence of (2e2x/e0) on m ¼ x2

0=r20 isillustrated in Fig. 1.

Evidently, under the condition (23) the resultingirradiance distribution of the two beams also has amaximum in the space between the two beams.

Fig. 2 illustrates the dependence of F, (x0F beingthe distance of the axis of the beam from x ¼ 0 axisin the xz-plane) on r2, the parameter whichcharacterizes the initial width of the beam. It isseen that the bending of the beams towards eachother increases with increasing r2. This can bereadily explained by the fact that increasing r2 leadsto higher electron temperature and associatednonlinearity.

Fig. 3 represents the dependence of F on theinitial separation of the beams; it can be seen thatwith decrease of the parameter m ¼ x2

0=r20 i.e. withbeams getting initially closer, the interaction of the

ARTICLE IN PRESS

0.15

0.10

0.05

0.000.0 0.1 0.2 0.3 0.4 0.5 0.6

μ

(2ε 2

x)/ε

0

αE∞2 = 100

αE∞2 = 50

αE∞2 = 10

Fig. 1. Critical condition for the bending of the identical beams

towards each other at z ¼ 0: variation of (2e2x/e0) with m ¼ x20=r20

at z ¼ 0. The dependence of (2e2x/e0) on m corresponding to

various beam irradiances aE200 ¼ 10, 50, 100 are, respectively

shown by dotted, dash-dotted and solid curves. For the attraction

between the two beams (2e2x/e0)40 or mo0.5.

S.K. Agarwal, A. Sharma / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 980–990 987

Author's personal copy

beams gets stronger. This is qualitatively an obviousconclusion.

Fig. 4 illustrates the dependence of the parameterF on the axial irradiance aE2

00 of the two beams(taken as equal). It is seen that the bending of thebeams towards each other increases with increasingirradiance of the beams. The greater irradiance ofthe beams leads to greater nonlinearity and henceincreased bending of the beams.

Fig. 5 shows the axes of the two beams; for thespecific case of equal irradiances aE2

00 ¼ 100, width

r2 ¼ 10 and frequency o ¼ 5� 107 rad/s of the twoincoherent beams. It is seen that the axes aresymmetrical about z-axis in the xz-plane.

Fig. 6 represents the variation of fx and fy withdistance of propagation x. It is seen that as xincreases, fx and fy become significantly different.This indicates that a symmetrical Gaussian beamturns into an elliptical Gaussian beam on account ofthe interaction with the other beam.

The computations for other heights can be readilymade by using Tables 1 and 2 for the relevantcoefficients.

ARTICLE IN PRESS

1.5

1

0.5

0

-0.5

F

-1

0 5 10 15 20 25

c

b

a

Fig. 2. Dependence of the parameter F representing the axial

separation of either beam from z-axis with distance of propaga-

tion x ¼ ðc=or20Þz for different values of the parameter

r2 ¼ ðr20o2=c2Þ, corresponding to aE2

00 ¼ 100, o ¼ 5� 107 rad/s

and m ¼ x20=r20 ¼ 0:3. For the daytime mid-latitude ionospheric

height of 150 km, the variation of F for r2 ¼ 5, r2 ¼ 10 and

r2 ¼ 15 are, respectively shown by the curves indicated as a, b,

and c.

1.5

1

0.5

0

-0.5

-1

0 5 10 15 20 25

F

I II III

Fig. 3. Dependence of the axial separation F of either beam from

z-axis on distance of propagation x ¼ ðc=or20Þz for various m ¼x20=r20

of the beams, corresponding to aE2

00 ¼ 100,

o ¼ 5� 107 rad/s and r2 ¼ 10. The variation of F for m ¼ 0.05

(I), m ¼ 0.2 (II) and m ¼ 0.3 (III) is shown for the daytime mid-

latitude ionospheric height of 150 km.

1.5

1

0.5

0

-0.5

-1

0 5 10 15 20 25

F

a

b

c

Fig. 4. Variation of the parameter F representing the axial

separation of either beam from z-axis with distance of propaga-

tion x ¼ ðc=or20Þz for different values of the axial irradiance aE200

of the beams, corresponding to r2 ¼ 10, o ¼ 5� 107 rad/s and

m ¼ x20=r20 ¼ 0:3. For the daytime mid-latitude ionospheric height

of 150 km curves correspond to (a) aE200 ¼ 100, (b) aE2

00 ¼ 50 and

(c) aE200 ¼ 20; respectively.

1.5

1

0.5

0

-0.5

-1

0 5 10 15 20 25

F &

-F

Fig. 5. Variation of parameters F and �F representing the axial

separation from z-axis (or axes) of the two identical beams with

distance of propagation x ¼ ðc=or20Þz, corresponding to

aE200 ¼ 100, r2 ¼ 10, o ¼ 5� 107 rad/s and m ¼ x2

0=r20 ¼ 0:3 for

the daytime mid-latitude ionospheric height of 150 km.

S.K. Agarwal, A. Sharma / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 980–990988

Author's personal copy

Acknowledgments

The authors are grateful to Department ofScience and Technology, Government of India forfinancial support. They are also highly grateful toProf. M.S. Sodha for suggesting the problem andvaluable discussion.

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