Self-focusing of a Hermite-cosh Gaussian laser beam in a magnetoplasma with rampdensity profileVikas Nanda, Niti Kant, and Manzoor Ahmad Wani

Citation: Physics of Plasmas (1994-present) 20, 113109 (2013); doi: 10.1063/1.4833635 View online: http://dx.doi.org/10.1063/1.4833635 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/20/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Self-focusing of Gaussian laser beam in weakly relativistic and ponderomotive regime using upward ramp ofplasma density Phys. Plasmas 20, 083101 (2013); 10.1063/1.4817019 Self focusing of a quadruple Gaussian laser beam in a plasma Phys. Plasmas 19, 092117 (2012); 10.1063/1.4754696 Non-stationary self-focusing of intense laser beam in plasma using ramp density profile Phys. Plasmas 18, 103107 (2011); 10.1063/1.3642620 Self-focusing of cosh-Gaussian laser beam in a plasma with weakly relativistic and ponderomotive regime Phys. Plasmas 18, 033110 (2011); 10.1063/1.3570663 Self-focusing of intense laser beam in magnetized plasma Phys. Plasmas 13, 103102 (2006); 10.1063/1.2357715

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

37.191.2.38 On: Mon, 12 May 2014 09:50:41

Self-focusing of a Hermite-cosh Gaussian laser beam in a magnetoplasmawith ramp density profile

Vikas Nanda, Niti Kant,a) and Manzoor Ahmad WaniDepartment of Physics, Lovely Professional University, Phagwara 144411, Punjab, India

(Received 21 August 2013; accepted 28 October 2013; published online 26 November 2013)

The early and strong self-focusing of a Hermite-cosh-Gaussian laser beam in magnetoplasma in the

presence of density ramp has been observed. Focusing and de-focusing nature of the Hermite-cosh-

Gaussian laser beam with decentered parameter and magnetic field has been studied, and strong

self-focusing is reported. It is investigated that decentered parameter "b" plays a significant role for

the self-focusing of the laser beam and is very sensitive as in case of extraordinary mode. For mode

indices, m ¼ 0; 1; 2; and b ¼ 4:00; 3:14; and 2:05, strong self-focusing is observed. Similarly

in case of ordinary mode, for m ¼ 0; 1; 2 and b ¼ 4:00; 3:14; 2:049, respectively, strong

self-focusing is reported. Further, it is seen that extraordinary mode is more prominent toward

self-focusing rather than ordinary mode of propagation. For mode indices m ¼ 0; 1; and 2,

diffraction term becomes more dominant over nonlinear term for decentered parameter b ¼ 0. For

selective higher values of decentered parameter in case of mode indices m ¼ 0; 1; and 2,

self-focusing effect becomes strong for extraordinary mode. Also increase in the value of magnetic

field enhances the self-focusing ability of the laser beam, which is very useful in the applications

like the generation of inertial fusion energy driven by lasers, laser driven accelerators, and x-ray

lasers. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4833635]

I. INTRODUCTION

The interaction of light with solids, liquids, gases, and

plasmas occurs very frequently in nature. For over more than

four decades, the nonlinear interaction of laser beams

with matter has been studied intensively by researchers.

Development of short-pulse high intensity lasers of the order

of 1020 W/cm2 makes it possible to investigate the nonlinear

interaction of strong electromagnetic waves with plasmas. A

series of applications of self-focusing of laser beams in

plasmas1,2 like optical harmonic generation,3 laser driven

fusion,4 x-ray lasers and the laser driven accelerators,5 the

production of quasi mono-energetic electron bunches,5 etc.,

create a center of attention of many researchers and scien-

tists. These applications need the laser beams to propagate

over several Rayleigh lengths in the plasmas without loss of

the energy. The propagation of different kinds of laser beams

profile like Gaussian beams,6 Cosh-Gaussian beams,7,8(b)

Hermite-cosh-Gaussian beams (HChG),9 Elliptic Gaussian

beams,10 Bessel beams,11 Leguerre-Gaussiaon beams,12 etc.,

in the plasmas attracts the attention of the researchers.

Recently, theoretical investigators focus their attention

on paraxial wave family of laser beams. Hermite-cosh-

Gaussian beam is one of the solutions of paraxial wave

equation and such HChG beam can be obtained in the labo-

ratory by the superposition of two decentered Hermite-

Gaussian beams as Cosh-Gaussian ones. Propagation of

Hermite-cosh-Gaussian beams in plasmas has been studied

theoretically earlier by Belafhal and Ibnchaikh9 and Patil

et al.13,14 The focusing of HChG laser beams in magneto-

plasma by considering ponderomotive nonlinearity has been

theoretically examined by Patil et al.15 and reported the

effect of mode index and decentered parameter on the self-

focusing of the beams. Gill et al.8(a) have recently studied

the relativistic self-focusing and self-phase modulation of

cosh-Gaussian laser beam in magnetoplasma in the absence

of plasma density ramp and reported strong self-focusing of

the laser beams. Gupta et al.,16 in 2007, have investigated

the addition focusing of a high intensity laser beams in a

plasma with a density ramp and a magnetic field and

reported the strong self-focusing of the laser beams. Again

density ramp has also been applied by Kant et al.17,18 to

study the Ponderomotive self-focusing of a short laser pulse

and self-focusing of Hermite-Gaussian laser beams in

plasma. In both the cases, Kant et al. reported strong

self-focusing of the laser beams.

The present work is dedicated to the study of self-

focusing of Hermite-cosh-Gaussian laser beams in colli-

sionless magneto-plasma under plasma density ramp in

applied magnetic field. We derive the equations for beam

width parameter for Hermite-cosh-Gaussian beam profile

propagating in the plasmas in the presence of magnetic field

and plasma density ramp, by applying Wentzel-Kramers-

Brillouin (WKB) approximation and Paraxial approxima-

tion6,19 and hence, solve them numerically by using

Mathematica software. The sensitiveness of decentered

parameter20 is observed, and a small increase in its value

enhances greatly the self-focusing ability of the beam. In

order to simplify the mathematical calculation, only the

transversal components of laser field are evaluated and lon-

gitudinal components are not taken in to consideration in

the present paper. However, while dealing with nonlinear

phenomenon, longitudinal components should be consid-

ered for an exact formulation.21a)E-mail: nitikant@yahoo.com

1070-664X/2013/20(11)/113109/7/$30.00 VC 2013 AIP Publishing LLC20, 113109-1

PHYSICS OF PLASMAS 20, 113109 (2013)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

37.191.2.38 On: Mon, 12 May 2014 09:50:41

II. NON-LINEAR DIELECTRIC CONSTANT

The dielectric constant for the non-linear medium

(collision-less magnetoplasma) with density ramp profile is

obtained by applying the approach as applied by Sodha et al.6

e6 ¼ e60 þ /6ðEE�Þ; (1)

with e60¼1�x2P=xðx7xcÞ;/6ðEE�Þ¼e62A2

0=ð1�xc=xÞ2;x2

P¼4pe2nðnÞ=m; nðnÞ¼n0tanðn=dÞ; x2P0¼4pe2n0=m; xc¼

eB0=mc; e62¼3max2Pð17xc=2xÞ=8Mxðx7xcÞ; and a

¼e2M=6m2x2kBT0. Here, e60 and /6ðEE�Þ represent the lin-

ear and non-linear parts of the dielectric constant, respec-

tively. xP and xc are the plasma frequency and electron

cyclotron frequency, respectively. e;m; and n0 are the mag-

nitude of the electronic charge, mass, and electron density,

“M” is the mass of scatterer in the plasma, “x” is the fre-

quency of laser used, “n¼z=Rd” is the normalized propaga-

tion distance, “Rd” is the diffraction length, “d” is a

dimensionless adjustable parameter, “kB” is the Boltzmann

constant, and “T0” is the equilibrium plasma temperature.

III. SELF-FOCUSING EQUATIONS

Consider the Hermite-cosh-Gaussian laser beam is prop-

agating along the z-direction with the field distribution in the

following form:

Eðr; zÞ ¼ E0

f6ðzÞe

b2

4 Hm

ffiffiffi2p

r

r0fþðzÞ

!e� r

fþðzÞr0þb

2

� �2

þ e� r

fþðzÞr0�b

2

� �2h i

:

(2)

“E0” is the amplitude of Hermite-cosh-Gaussian laser beam

for the central position at r ¼ z ¼ 0; “f6(z)” is the dimen-

sionless beam width parameter, Hm is the Hermite polyno-

mial and “m” is the mode index associated with Hermite

polynomial, and “r0” is the initial spot size of the laser beam.

The general form of wave equation for exponentially

varying field obtained from Maxwell’s equation is given by

r2~E �rðr:~EÞ ¼ �x2

c2e:~E: (3)

In components form, this equation can be written as

@2Ex

@z2þ @

2Ex

@y2� @

@x

@Ey

@yþ @Ez

@z

� �¼ �x2

c2e:~Eð Þ

x; (4a)

@2Ey

@z2þ @

2Ey

@x2� @

@y

@Ex

@xþ @Ez

@z

� �¼ �x2

c2e:~Eð Þ

y; (4b)

@2Ez

@x2þ @

2Ez

@y2� @

@z

@Ex

@xþ @Ey

@y

� �¼ �x2

c2e:~Eð Þ

z: (4c)

In the present case, the variation of magnetic field (~B ¼ B0k̂)

is assumed to be very strong along z-direction of the co-

ordinate system than x-y plane. Thus, the propagating wave

is considered to be transverse in nature in zero order approxi-

mation. This assumption gives rise to the condition

r:~D ¼ 0. The Eqs. (4a)–(4c) are coupled and one cannot

define the propagation vectors which describe the independ-

ent propagation of Ex ; Ey; and Ez. To investigate the inde-

pendent modes of propagation, we multiply y-component

equation by “i” and adding to x-component equation and

also using the condition r:~D ¼ 0, we get equation for extra-

ordinary mode of propagation as

@2A1

@z2þ 1

21þ eþ0

e0zz

� �@2A1

@x2þ @

2A1

@y2

!

þ 1

2

e�0

e0zz� 1

� �@2A2

@x2� @

2A2

@y2

!� i 1� e�0

e0zz

� �@2A2

@x@y

þx2

c2eþ0 þ eþ2

A1A�1

1� xc

x

� �2þ A2A�2

1þ xc

x

� �2

8><>:

9>=>;

2664

3775A1 ¼ 0;

(5)

with A1 ¼ Ex þ iEy; A2 ¼ Ex � iEy; and e0zz ¼ 1�x2P=x

2.

A similar kind of equation can be obtained for ordinary

mode. The functions A1 and A2, having propagation vectors

Kþ and K�, respectively, represent the extraordinary and

ordinary modes of propagation of a magnetoplasma. In order

to study the behavior of one of the modes, other mode can be

considered to be zero. Let us assume A2 � 0, A1 ¼ Aexp½iðxt� kþzÞ� and thereafter introducing additional eiko-

nal, A ¼ A0ðx; y; zÞexp½�ikþðzÞSðx; y; zÞ�, the real and imagi-

nary parts comes out to be;

Real part is

zx2xs sec2 z

dRd

� �k2þc2dRd

� 2

264

375 @S

@z� 1

21þ eþ0

e0zz

� �@S

@x

� �2

þ @S

@y

� �2" #

þ 1

4A20k2þ

1þ eþ0

e0zz

� �

� @2A20

@x2þ @

2A20

@y2

!� 1

2A20

@A20

@x

� �2

þ @A20

@y

!28<:

9=;

264

375� Sx2xs sec2 z

dRd

� �k2þc2dRd

�Szx4x2

s sec4 z

dRd

� �2k4þc4d2R2

d

þzx2xs sec2 z

dRd

� �k2þc2dRd

�z2x4x2

s sec4 z

dRd

� �4k4þc4d2R2

d

þ x2

k2þc2

eþ2

A20

1� xc

x

� �2

0B@

1CA ¼ 0: (6)

113109-2 Nanda, Kant, and Wani Phys. Plasmas 20, 113109 (2013)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

37.191.2.38 On: Mon, 12 May 2014 09:50:41

Imaginary part is

x2xs sec2 z

dRd

� �k3þc2dRd

þzx2xs sec2 z

dRd

� �tan

z

dRd

� �k3þc2dRd

þzx4x2

s sec4 z

dRd

� �4k5þc4d2R2

d

þzx2xs sec2 z

dRd

� �2A2

0k3þc2dRd

� 1

kþA20

0B@

1CA @A2

0

@z

� 1

2kþ1þ eþ0

e0zz

� �@2S

@x2þ @

2S

@y2

!þ 1

2kþA20

1þ eþ0

e0zz

� �@S

@x

� �@A2

0

@x

� �þ @S

@y

� �@A2

0

@y

!" #¼ 0; (7)

with kþ ¼ x=cð Þe1=2þ0 . The solution of Eqs. (6) and (7) for initially Hermite-cosh-Gaussian beam is of the following form:

A20 ¼

E20

f 2þðzÞ

eb2

2 Hm

ffiffiffi2p

r

r0fþðzÞ

!" #2

e�2 r

fþðzÞr0þb

2

� �2

þ e�2 r

fþðzÞr0�b

2

� �2

þ 2e�2r2

f 2þðzÞr

20

þb2

2

� � ; (8)

and

S ¼ x2

2bðzÞ þ y2

2bðzÞ þ /ðzÞ; (9)

where bðzÞ ¼ 2=fþðzÞ 1þ eþ0=e0zzð Þ@fþðzÞ=@z; is the curvature of the wave front and r2 ¼ x2 þ y2� �

: "E0" is the amplitude of

Hermite-cosh-Gaussian laser beam for the central position at r ¼ z ¼ 0; “f6(z)” is the dimensionless beam width parameter;

“r0” is the spot size of the beam. Thus, Eq. (7) becomes,

For m 5 0:

nxs sec2 nd

� �d

� 2 1�xs tannd

� �� �8><>:

9>=>;@2fþ

@n2�

nxs sec2 nd

� �d

1

fþ

@fþ@n

� �2

�nxs sec4 n

d

� �x2

P0eþ0

d2x2e20zz 1þ eþ0

e0zz

� � � nx2s sec4 n

d

� �

d2e0zz 1þ eþ0

e0zz

� �� 2 sec2 nd

� �1�xs tan

nd

� �� �x2

P0eþ0

dx2e20zz 1þ eþ0

e0zz

� � þ2xs sec2 n

d

� �1�xs tan

nd

� �� �

d e0zz 1þ eþ0

e0zz

� �26664

37775@fþ@n

þ1þ eþ0

e0zz

� �2

1�x2P0

x2tan

nd

� �� �f 3þ

2� 2b2ð Þ �8R2

d1eb2

2 1�x2P0

x2tan

nd

� �� �R2

nl1f 3þ

1þ eþ0

e0zz

� �¼ 0; ð10Þ

where R2nl1 ¼ 1� xc=xð Þ2r2

0 eþ0=eþ2E20

� �and Rd1 ¼ kþr2

0;

For m 5 1:

nxs sec2 nd

� �d

� 2 1�xs tannd

� �� �8><>:

9>=>;@2fþ

@n2�

nxs sec2 nd

� �d

1

fþ

@fþ@n

� �2

�nxs sec4 n

d

� �x2

P0eþ0

d2x2e20zz 1þ eþ0

e0zz

� � � nx2s sec4 n

d

� �

d2e0zz 1þ eþ0

e0zz

� �� 2 sec2 nd

� �1�xs tan

nd

� �� �x2

P0eþ0

dx2e20zz 1þ eþ0

e0zz

� � þ2xs sec2 n

d

� �1�xs tan

nd

� �� �

d e0zz 1þ eþ0

e0zz

� �26664

37775@fþ@n

þ1þ eþ0

e0zz

� �2

1�x2P0

x2tan

nd

� �� �f 3þ

2� 2b2ð Þ þ16R2

d1eb2

2 1�x2P0

x2tan

nd

� �� �R2

nl1f 3þ

2� 2b2ð Þ 1þ eþ0

e0zz

� �¼ 0: ð11Þ

113109-3 Nanda, Kant, and Wani Phys. Plasmas 20, 113109 (2013)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

37.191.2.38 On: Mon, 12 May 2014 09:50:41

For m 5 2:

nxssec2 nd

� �d

� 2 1�xs tannd

� �� �8><>:

9>=>;@2fþ

@n2�

nxs sec2 nd

� �d

1

fþ

@fþ@n

� �2

�nxs sec4 n

d

� �x2

P0eþ0

d2x2e20zz 1þ eþ0

e0zz

� � � nx2s sec4 n

d

� �

d2e0zz 1þ eþ0

e0zz

� �� 2 sec2 nd

� �1�xs tan

nd

� �� �x2

P0eþ0

dx2e20zz 1þ eþ0

e0zz

� � þ2xs sec2 n

d

� �1�xs tan

nd

� �� �

d e0zz 1þ eþ0

e0zz

� �26664

37775@fþ@n

�12b2

1þ eþ0

e0zz

� �2

1�x2P0

x2tan

nd

� �� �f 3þ

�32R2

d1eb2

2 1�x2P0

x2tan

nd

� �� �R2

nl1f 3þ

5� 2b2ð Þ 1þ eþ0

e0zz

� �¼ 0: ð12Þ

Equations (10)–(12) are the required equations for beam width

parameter for extraordinary modes of propagation of magneto-

plasma. Similarly, the equations for beam width parameter for

ordinary mode of propagation can be obtained by replacing

xc by � xc and þ sign by � sign in the subscript of f :Similarly, on solving imaginary part, we get the condi-

tion @f6=@z ¼ 0 and f6 ¼ constant for extraordinary and

ordinary modes.

IV. RESULTS AND DISCUSSION

The various parameters taken for numerical calculation

are: x ¼ 1:778� 1014rad=s; r0 ¼ 253lm; e ¼ 1:6� 10�19C;m ¼ 9:1� 10�31Kg; and c ¼ 3� 108m=s. The value of in-

tensity in the present case is I0 ¼ 1:84� 1014W=m2� Figure 1

represents the variation of beam width parameter (f6) with

the normalized propagation distance (n) for different values

of decentered parameter, b ¼ 0:00; 3:90; 3:95; and 4:00 for

m¼0. The sensitiveness of decentered parameter is also

observed in the present case and it supports the previous work

of Nanda et al.20 It is clear from the plot that for extraordinary

mode, Fig. 1(a), beam width parameter decreases with the

increase in the values of decentered parameter. For b ¼ 4:00,

beam width parameter falls abruptly at normalized propaga-

tion distance, n ¼ 0:10 and hence self-focusing becomes

strong. In case of ordinary mode, same patterns are observed

but self-focusing is weak as compare to extraordinary

mode of propagation. Gill et al.8(b) have studied self-focusing

and self-phase modulation as well as self-trapping of

cosh-Gaussian beam at various values of decentered parame-

ter (b) and concluded that self-focusing becomes sharper for

b ¼ 2 and occurs at n ¼ 1:45 and the value of beam width

parameter is nearly 0:91 (approximately). In another work,

Gill et al.8(a) reported strong self-focusing effect nearly at

n ¼ 0:65. Patil et al.14 have reported strong self-focusing for

m ¼ 0; b ¼ 2 nearly at n ¼ 0:21. In the present work, in the

presence of density ramp, we report very strong self-focusing

which occurs at n ¼ 0:10 for b ¼ 4:00. In case of ordinary

mode, Fig. 1(b), the self-focusing of beam occurs for

b ¼ 0:00; 3:90; 3:95; and 4:00. For b ¼ 4:00, self-focusing

effect is very strong; however, it is weaker as compared to

extraordinary mode. In the absence of decentered parameter

no self-focusing is observed in both the cases.

FIG. 1. Variation of beam width parameter with normalised propagation dis-

tance for (a) extraordinary and (b) ordinary mode for different values of

decentered parameter. The other various values are taken as m ¼ 0, d ¼ 5,

xr0=c ¼ 150; aE20 ¼ 0:01, xc=x ¼ 0:10, xP0=x ¼ 0:45; and m0=M ¼ 0:01.

113109-4 Nanda, Kant, and Wani Phys. Plasmas 20, 113109 (2013)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

37.191.2.38 On: Mon, 12 May 2014 09:50:41

Figure 2 represents the variation of beam width parameter

(f6) with the normalized propagation distance (n) for different

values of decentered parameter, b ¼ 0:00; 3:04; 3:09; and

3:14 for m¼ 1. It is clear from the plot in Fig. 2(a) that with

the increase in the values of decentered parameter beam width

parameter decreases and hence self-focusing effect is observed.

However, for b ¼ 0 ; diffraction term becomes more dominant

over focusing term and causes the defocusing of beam. Fig.

2(b) describes the same pattern as that in Fig. 2(a); however,

self-focusing is weaker in this case. Patil et al.15 have reported

strong self-focusing for m ¼ 1; b ¼ 2 nearly at n ¼ 1:9(approximately) and for b ¼ 0 and 1, diffraction effect

becomes dominant, both for extraordinary and ordinary modes

of propagation. In the present work, in the presence of density

ramp, we report very strong self-focusing which occurs at n ¼0:10 for b ¼ 3:14 and only for b ¼ 0 defocusing of beam

occurs while for b ¼ 1; 2; and 3, self-focusing of beam occurs

and is very strong for b ¼ 3:14.

Figure 3 represents the variation of beam width parame-

ter with the normalized propagation distance for different

values of decentered parameter, b ¼ 0:00; 1:95; 2:00;and 2:05 for m¼ 2 for extraordinary mode of propagation.

In this case, for b ¼ 1:95; 2:00; and 2:05, beam exhibits self-

focusing effect for both extraordinary and ordinary modes,

however, relatively self-focusing is weaker in case of ordi-

nary mode as compared to that in case of extraordinary mode

of propagation. While for b ¼ 0:00 beam gets defocused

both for extraordinary and ordinary mode of propagation.

Patil et al.15 have reported strong self-focusing for m ¼ 2;b ¼ 0 and 1 nearly at n ¼ 2:3 (approximately) and for b ¼ 2,

diffraction effect becomes dominant, both for extraordinary

and ordinary modes of propagation. In the present work, in

the presence of density ramp, we report very strong self-

focusing which occurs at n ¼ 0:10 for b ¼ 2:05, in case of

extraordinary mode, Fig.3(a) and for b ¼ 2:049, in case of

ordinary mode of propagation Fig. 3(b).

FIG. 3. Variation of beam width parameter with normalised propagation dis-

tance for (a) extraordinary and (b) ordinary mode for different values of

decentered parameter. The other various values are taken as m ¼ 2, d ¼ 5,

xr0=c ¼ 150; aE20 ¼ 0:01, xc=x ¼ 0:10, xP0=x ¼ 0:45; and m0=M ¼ 0:01.

FIG. 2. Variation of beam width parameter with normalised propagation dis-

tance for (a) extraordinary and (b) ordinary mode for different values of

decentered parameter. The other various values are taken as m ¼ 1, d ¼ 5,

xr0=c ¼ 150; aE20 ¼ 0:01, xc=x ¼ 0:10, xP0=x ¼ 0:45; and m0=M ¼ 0:01.

113109-5 Nanda, Kant, and Wani Phys. Plasmas 20, 113109 (2013)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

37.191.2.38 On: Mon, 12 May 2014 09:50:41

Figure 4 represents the variation of beam width parame-

ter with the normalized propagation distance for mode indi-

ces m ¼ 0; 1; 2 and decentered parameter b ¼ 3:29, both for

extraordinary, Fig. 4(a), and ordinary mode of propagation

of beam, Fig. 4(b). It is reported that in case of extraordinary

mode, for m ¼ 2, self-focusing occurs earlier at n ¼ 0:07.

While for mode indices m ¼ 0 and 1, self-focusing occurs at

n ¼ 0:16 and 0:11, respectively. In case of ordinary mode,

self-focusing is strong for m ¼ 2; b ¼ 3:29 and occurs at

n ¼ 0:08 and thereafter beams start defocusing for higher

values of decentered parameter.

Figure 5(a) represents the variation of beam width

parameter with normalised propagation distance for extraor-

dinary mode for different values of magnetic field parameter

given as xc=x ¼ 0:05; 0:10 and 0:15. It is investigated that

for m ¼ 0 and b ¼ 3:95, with the increase in the values of

magnetic field parameter, beam width parameter decreases

and hence self-focusing ability of beam enhances to a greater

extent. For xc=x ¼ 0:15, strong self-focusing is observed

than for xc=x ¼ 0:05; and 0:10. Gupta et al.16 have investi-

gated the addition focusing of a high intensity laser beam in

a plasma with density ramp and a magnetic field and

reported strong focusing which occurs at n ¼ 0:8 for mag-

netic field 45 MG. However, in our case self-focusing occurs

at n ¼ 0:1 for xc=x ¼ 0:15. Similar patterns are observed in

case of mode indices m ¼ 1 and 2, for optimized values of

other parameter as taken in case of m ¼ 0. Whereas, in case of

ordinary mode of propagation, Fig. 5(b), beam width parame-

ter increases with the increase in the value of magnetic field

parameter and hence decreases self-focusing of laser beams.

V. CONCLUSION

From the above results, we conclude that the presence

of density ramp enhances the self-focusing effect to a greater

extent and also it occurs earlier with normalized propagation

FIG. 5. Variation of beam width parameter with normalised propagation dis-

tance for (a) extraordinary and (b) ordinary mode of propagation for differ-

ent values of xc=x. The other various values are taken as m ¼ 0, b ¼ 3:95,

d ¼ 5, xr0=c ¼ 150; aE20 ¼ 0:01, xP0=x ¼ 0:45; and m0=M ¼ 0:01.

FIG. 4. Variation of beam width parameter with normalised propagation

distance for (a) extraordinary and (b) ordinary mode for mode indices

m ¼ 0; 1; and 2. The other various values are taken as b ¼ 3:29, d ¼ 5,

xr0=c ¼ 150; aE20 ¼ 0:01, xc=x ¼ 0:10, xP0=x ¼ 0:45; and m0=M ¼ 0:01.

113109-6 Nanda, Kant, and Wani Phys. Plasmas 20, 113109 (2013)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

37.191.2.38 On: Mon, 12 May 2014 09:50:41

distance. For mode indices m ¼ 0; 1 and 2, self-focusing is

more commonly observed for higher values of decentered

parameter b viz. 4.00, 3.14, and 2.05, respectively, both for

ordinary and extraordinary modes of propagation of laser

beams. However, for ordinary mode, self-focusing is a little

bit weaker than extraordinary mode. Decentered parameter

decides the focusing/defocusing nature of Hermite-cosh-

Gaussian beam as for b¼ 0, defocusing of beam occurs and

for m ¼ 1, weaker self-focusing effect is observed. The de-

pendence of beam width parameter on decentered parameter

is previously also investigated by Patil et al.15 For b ¼ 3:29,

self-focusing of Hermite-cosh-Gaussian laser beam is found

to occur earlier at n ¼ 0:07 for mode index m ¼ 2 for both

modes of propagation. Further, sensitiveness of decentered

parameter is observed which strongly supports our previous

work.20 Also with the increase in the value of magnetic field

parameter, self-focusing ability of the laser beam increases

abruptly. All this happens due to the presence of plasma den-

sity ramp and magnetic field. Thus, plasma density ramp

plays a very vital role to the self-focusing of the Hermite-

cosh-Gaussian laser beam, and it enhances the self-focusing

effect. Present study may be useful for the scientist working

on laser-induced fusion.

ACKNOWLEDGMENTS

This work was supported by a financial grant from CSIR,

New Delhi, India, under Project No. 03(1277)/13/EMR-II.

1H. Hora, Z. Phys. 226, 156 (1969).2M. R. Siegrist, Opt. Commun. 16, 402 (1976).3S. Banerjee, A. R. Valenzuela, R. C. Shah, A. Maksimchuk, and D.

Umstadter, Phys. Plasmas 9, 2393 (2002).4K. A. Brueckner and S. Jorna, Rev. Mod. Phys. 46, 325 (1974).5J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E.

Lefebvre, J. P. Rousseau, F. Burgy, and V. Malka, Nature 431, 541

(2004).6M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self-Focusing of LaserBeams (Tata-McGraw-Hill, New Delhi, 1974).

7S. D. Patil, M. V. Takale, S. T. Navare, V. J. Fulari, and M. B. Dongare,

Opt. Laser Technol. 44, 314 (2012).8T. S. Gill, R. Kaur, and R. Mahajan, Laser Part. Beams 29, 183 (2011); T.

S. Gill, R. Mahajan, and R. Kaur, Phys. Plasmas 18, 033110 (2011).9A. Belafhal and M. Ibnchaikh, Opt. Commun. 186, 269 (2000).

10T. S. Gill, N. S. Saini, S. S. Kaul, and A. Singh, Optik 115, 493 (2004).11P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, Opt. Commun.

222, 107 (2003).12A. Thakur and J. Berakdar, Opt. Express 18, 27691 (2010).13S. D. Patil, M. V. Takale, and M. B. Dongare, Opt. Commun. 281, 4776

(2008).14S. Patil, M. Takale, V. Fulari, and M. Dongare, J. Mod. Opt. 55, 3529

(2008).15S. D. Patil, M. V. Takale, S. T. Navare, and M. B. Dongare, Laser Part.

Beams 28, 343 (2010).16D. N. Gupta, M. S. Hur, and H. Suk, Appl. Phys. Lett. 91, 081505

(2007).17N. Kant, S. Saralch, and H. Singh, Nukleonika 56, 149 (2011).18N. Kant, M. A. Wani, and A. Kumar, Opt. Commun. 285, 4483

(2012).19S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. Usp.

10, 609 (1968).20V. Nanda, N. Kant, and M. A. Wani, IEEE Trans. Plasma Sci. 41, 2251

(2013).21L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).

113109-7 Nanda, Kant, and Wani Phys. Plasmas 20, 113109 (2013)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

37.191.2.38 On: Mon, 12 May 2014 09:50:41