ARTICLE IN PRESS

OpticsOptikOptikOptik 119 (2008) 559–564

0030-4026/$ - se

doi:10.1016/j.ijl

�CorrespondE-mail addr

(A. Singh).

www.elsevier.de/ijleo

Optical guiding of elliptical laser beam in nonuniform plasma

Arvinder Singha,�, Munish Aggarwala, Tarsem Singh Gillb

aDepartment of Applied Physics, NIT Jalandhar, IndiabDepartment of Physics, Guru Nanak Dev University, Amritsar 143005, India

Received 12 October 2006; accepted 15 January 2007

Abstract

Guiding of laser beam in plasma channel formed by short ionizing laser pulse is studied in this paper. When adelayed pulse propagates through this channel diffraction, refraction and self-phase modulation phenomena resultswhich are not balanced resulting in increase/decrease in beam width. These are studied using direct VariationalTechnique. In this paper cross-focussing phenomena is not observed. Parameters like beam width and longitudinal-phase delay of elliptical laser beam are also studied. It has been observed that the propagation of semi-major axis andsemi-minor axis of elliptical beam are independent leading to oscillating guided beam.r 2007 Elsevier GmbH. All rights reserved.

Keywords: Self-focusing; Elliptic; Nonlinear-medium

1. Introduction

Optical guiding of high-intensities laser pulse [26] ledto the development of nonlinear-based short wavelengthlight source, X-ray lasers [1–4], advanced laser fusionschemes [5,6], plasma-based accelerators [7–9]. Plasmachannels [10–13] have been proposed as a mean ofguiding the laser pulses. Intensities of the order of1013–1014 W=cm2 are generally of interest. Guidingof pulse in the plasma is analogue to guiding in opticalfiber, if the refractive index at the beam center can beincreased sufficiently high with respect to the beamedge to balance the effect of diffraction. The two pulsetechnique allows the guiding of laser pulse having highintensity of the order of 1014–1017 W=cm2.

Several experimental methods have been proposed tocreate the plasma channels, e.g. (i) passing a long pulse

e front matter r 2007 Elsevier GmbH. All rights reserved.

eo.2007.01.009

ing author.

esses: arvinder6@lycos.com, singha@nitj.ac.in

through an optic lens to create a line focus in the gas,which ionizes and heat the gas, creating radiallyexpanding hydrodynamics shocks [14], (ii) using a slowcapillary discharge to control the plasma profile [15] and(iii) using the ponderomotive force of an intenserelativistically self-guided laser pulse in a plasma, whichcreates the channel in its own way [16].

The physics of guiding the laser pulse is as follows, aplasma channel is created by focussing a laser prepulseon picosecond-time scale via tunnel and impact ioniza-tion [17]. Since the prepulse has Gaussian intensityradial profile, therefore the plasma density that resultsspontaneously, has a peak on the axis and fall offrapidly with the radial distance away from the axis,before the diffusion of the plasma sets in. In terms of therefractive index, refractive index is minimum on the axisand increases toward edge and hence the mediumbehaves as de-focussing medium.

After the laser prepulse is gone, the plasma created bythe prepulse diffuses radially away from the axis andtherefore the plasma density becomes minimum on the

ARTICLE IN PRESSA. Singh et al. / Optik 119 (2008) 559–564560

axis as compared to the density at the edge of the plasmachannel where the density is maximum. Therefore, therefractive index becomes maximum on the axis anddecreases toward edges and thus the medium behaveslike a focussing medium.

When a delayed second laser pulse is passed throughthe plasma channel, it tends to diverge due to diffractionand converge due to refraction. However, when both theconvergence and divergence parameters are equal, thenthe delayed laser pulse propagates without convergenceand divergence. However, in an axially nonuniformplasma channel this condition cannot be satisfiedthroughout the channel, hence the beam radius changesas it propagates.

Liu and Tripathi studied such phenomena usingparaxial ray approximations (PRA). The paraxialtheory of Akhmanov et al. [18] and Sodha et al. [19]had been the most popular theory. Despite its mathe-matical simplicity it is applicable only to the low-powerlaser beams. The another problem regarding this theorywas contribution only close to the beam axis take partsin the self-focussing mechanism. However, these days,nonparaxial theories are also important. PRA theory forhigher powers fails while the moment and variationalmethod used by Anderson [10–12,20] are more relevantthese days. We have used here the variational technique[20], which not only predict correctly the width of thebeam but also the self-induced longitudinal-phase delayin the plasma channel.

In the present paper, we have examined the self-defocussing of the ionizing short pulse and then the laserguiding in an axially nonuniform plasma channelcreated by the prepulse with the help of the variationalmethod. In Section 2, we develop a variational approachof self-defocussing of prepulse. In Section 3, the problemof laser guiding is dealt with the help of variationalapproach. Discussion of the result is presented inSection 4.

2. Self-defocusing of ionizing laser pulse

Consider the propagation of a Gaussian laser beam offrequency o0 through a gas along z-axis. The laserionizes the gas via tunnel ionization in a time shorterthan the pulse duration. Leemans et al. [14] have givenan expression for the rate of increase of plasma density.It has been found that the plasma density builds uprapidly in about 20 ps once the laser intensity crosses6:0� 1013 W=cm2. In the experiment performed byDurfee and Milchberg [21], the laser power density isin the range 1013–1014 W=cm2. For a Gaussian beam ofradius a0. We expect a high-density plasma within a fewpicoseconds in the axial region r5r0. For r greater thansome value, there would be no ionization. Durfee and

Milchberg [21] in there code, have indeed obtainedsharp density fallout beyond r�r0=2. Such a sharpplasma density profile is modelled by Liu and Tripathi[17] as

o2p ¼ o2

p0exp

�E0ajEj

� �, (1)

where

E 0a ¼2

3Ea

ffiffiffiffiffiffiffiffiffiffiffiffiffiEi

Eh

� �s(2)

and o2p0

is a constant depending on the neutral particledensity. Ei is the ionization potential of the atom, Eh isthe ionization potential of hydrogen and

Ea ¼ 4p2m2e5=h4. (3)

The dielectric function of the plasma may be written as[17]

� ¼ �0 þ �2r2, (4)

� ¼ 1�o2

p0

o2o

!exp½�E 0a=jEj�, (5)

where

�0 ¼ �jr¼0 ¼ 1�o2

p0

o20

!exp½�E0a=jE00j�, (6)

�2 ¼q�qjEj

qjEjqr2

����r¼0

¼o2

p0

2o20

E0aE00a2ðzÞ

exp�E0a

A

� �. (7)

In the slowly varying approximation, we obtain thefollowing nonlinear Schrodinger equation (NLSE):

2ikqE

qzþr2

?E þo2

0

c2�2r

2jEj ¼ 0. (8)

Eq. (8) has been solved by Liu and Tripathi [17]using PRA. As mentioned earlier, when high-powerlaser beams are involved in the laser plasma experiment,one must follow methods based on invariants ofnonlinear Schrodinger equation (8) and the moment ofelectromagnetic field. The latter is used to determine thecondition of self-focussing/defocussing as well as beamwidth of laser beam in nonlinear medium. Hence inspiteof its nonlinear elegance, the description of phase bymoment theory is still an open question.

Another approach involves the invariant of nonlinearSchrodinger equation is the variational approach usedby the Anderson et al. to describe the self-focussing inthe plasma [12] and optical fiber [20]. Variationalmethod which though approximately analytical butfairly general in nature, has been used in number ofinvestigations. We have used variational approach tosolve Eq. (8). Despite its limitation like lack of finer

ARTICLE IN PRESSA. Singh et al. / Optik 119 (2008) 559–564 561

detail of phase description, inability to explain singu-larity and collapse. The method is fairly used in theinvestigation of wave propagation problems even whenhigh-power laser beams are involved. We reformulatethis equation into variational problem corresponding toLagrangian L so that vanishing of functional derivativegives Eq. (9) viz..

L ¼ rqE

qr

��������2

þ ikr EqE%

qz� E%

qE

qz

� ��

o2

c2�2r

2jEj2. (9)

Thus the solution to the variational problem

dZZZ

Ldxdydz ¼ 0 (10)

also solves the nonlinear Schrodinger equation. Using.

E ¼ A½z� expr2

2a2ðzÞþ iqðzÞr2

� �(11)

as a trial function in Eq. (9) and carrying out theintegration, we obtain the reduced problem with hLi as.

hLi ¼ jAj21

2þ 2q2a4

� �þ ik A

qA%

qz� A%

qA

qz

� �a2

2

þ kjAj2qq

qza4 �

o20

c2�2jAj

2 a4

2. ð12Þ

Taking variation with respect to A, A%, a, q, etc. andfollowing the procedure of Anderson [16], and alsousing the normalization.

z ¼zc

o0a20

(13)

we arrive at the following:

dðA2a2nÞ

dz¼ 0, (14)

q ¼1

an

dan

dz, (15)

d2an

dz2¼

1

2a3n

1þopa0

c

� 2a2n expð�E0a=jEjÞ � 2a2n

dan

dz

� �2" #

,

ð16Þ

dfdz¼

1

2a2n

1þ 4a2n

dan

dz

� �2

þ 2a20a

2n

dq

dz

"

�1

2

opa0

c

� 2expð�E0a=jEjÞ

#. ð17Þ

The normalized beam width ðan ¼ a=a0Þ of prepulse isstudied by using second-order nonlinear differentialequation (16). The first two terms on the right-hand sideof equation (16) represents diffraction and defocussingphenomena and play an important role in determiningthe beam width. These terms are responsible for theaxial nonuniformity in the index of refraction of theplasma channel.

In order to compare the results with those of Liu andTripathi of (Eq. (15) Ref. [17]), we find here anadditional last term on the right-hand side of equation(16). This term though initially zero evolves with thepropagation there by counteracting the diffraction andself-defocusing. In spite of that, overall result isdefocusing of an with almost constant slope. Thiscontrast well with those of Liu and Tripathi where veryfast defocusing of ionizing prepulse is displayed.Additional feature of present investigation is thatlongitudinal-phase modulation of the prepulse is pre-dicted by Eq. (17).

3. Guided propagation of laser beam

Once the prepulse is over, plasma formed startsmoving radially away from the axis. Now we will studythe dynamics of the delayed pulse in this phase.According to Durfee and Milchberg [21] the densityprofile of the plasma on the arrival of the delayed secondpulse is given by

o2p

o2¼ a1ðzÞ þ a2ðzÞr2. (18)

Here a1ðzÞ and a2ðzÞ are monotonically decreasingfunction of z.

If we express the electric field of second pulse as

E ¼ AðZÞ exp�x2

2a2þ

y2

2b2

� �� �expðiq1x

2 þ iq2y2Þ,

(19)

where a, b are width parameters of the beam,respectively, in the x, y direction. The amplitude A aswell beam widths a, b are real functions of z. This beamis elliptical Gaussian in nature.

In slowly varying envelope approximations we havenonlinear Schrodinger wave equation as

�2ikqE

qzþ r2

?E þo2

c2a2ðzÞðx2 þ y2ÞE ¼ 0, (20)

where

k ¼ocð1� a1ðzÞÞ

1=2; k0 ¼ kðz ¼ 0Þ. (21)

Here r2?E is a spatial dispersion that spreads

the beam in the transverse direction, whereas the lastterm is the nonlinear term which compresses the beam.

ARTICLE IN PRESSA. Singh et al. / Optik 119 (2008) 559–564562

The dynamics of self-focussing is determined by therelative competition between spatial dispersion andattracting nonlinearity.

Eq. (20) can be reformulated into the variationalproblem corresponding to Lagrangian L so as to makeqhLi=qZ ¼ 0 equivalent to Eq. (20), viz.

L2 ¼qE

qx

��������2

þqE

qy

��������2

� ik EqE%

qz� E%

qE

qz

� �

þo2

c2a2ðx2 þ y2ÞjE2j. ð22Þ

Using the averaging procedure similar to the proceedingsection to get the reduced variational problem. Weobtain the equations for the normalized beam width andlongitudinal-phase delay hLi as follows:

hLi ¼jA2jpab

4ðq2

1a2 þ q2

2b2Þ þjA2jp16

a

bþ

b

a

� �

�ikpab

4AqA%

qz� A%

qA

qz

� �

�jA2jKpab

8a2 qq1

qzþ b2 qq2

qz

� �

þo2

c2a2jA2j

pab

16ða2 þ b2

Þ, ð23Þ

d2a

dz2¼

a40

a3ð1� a1Þ�

o2rpE0a

ð1� a1ÞA00exp

�E0aA00

� �a, (24)

d2b

dz2¼

b40

b3ð1� a1Þ

�o2

rpE0a

ð1� a1ÞA00exp

�E0aA00

� �b, (25)

dfdz¼

a20

4

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� a1Þ

p 1

a2þ

1

b2

� ��

1

16

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� a1Þ

pa20

�o2rp

E0aA00

� �exp

�E 0aA00

� �ða2 þ b2

Þ ð26Þ

and.

d2r2

dz2¼

2

ð1� a1Þa40

a2þ

b40

b2

� �� 2

o2rp

ð1� a1ÞE0aA00

� �

� exp�E0aA00

� �ða2 þ b2

Þ þ 2da

dz

� �2

þ 2db

dz

� �2

,

ð27Þ

where r2 ¼ a2 þ b2 and z ¼ Zc=o0a20 is the normalized

distance of the propagation.Eqs. (24) and (25) represents the behavior of the semi-

major axis ‘a’ and semi-minor axis ‘b’, respectively.Eqs. (24) and (25) are solved numerically and results aredisplayed in Figs. 2 and 3, respectively. The long-itudinal-phase delay is also studied from Eq. (26) whichis also a function of semi-major and semi-minor axis ofelliptical Gaussian laser beam. Eq. (27) describes thecombined behavior of ‘a’ and ‘b’ of the beam which

further depends on the da=dz and db=dz which althoughinitially do not contribute but later evolves with distanceof propagation z.

We have chosen the following set of parameters forcomputation: .

E 0aA00¼ 0:5; orp ¼

opa0

C¼ 5:0,

a1 ¼ 0:4; a0 ¼ 0:002 and b0 ¼ 0:0012.

And using initial conditions .

d2r2

dz2

����z¼0¼ 0.

4. Discussion

In earlier investigations Gill [22], Liu and Tripathi[17], the guided laser beam propagated only a fewRayleigh lengths. Particularly in the case of Liu andTripathi [17], guided propagation was less than 1.3Rayleigh length as z grows from 0 to 3. However, thepropagation of Gaussian beam extended to 12 Rayleighlength in case of Gill [22]. This was due to additionalterm existing on the right-hand side of normalized beamwidth Eq. (18) of Gill [22]. The discrepancy arises due tofact that Liu and Tripathi [17] used PRA which hasinherent limitation. However, Gill [22] used variationalapproach and one additional term like nonlinearrefraction term became available to counteract thediffraction affects and thereby extending the range ofguided propagation. With the availability of pulse shapetechnology it is possible to tailor the plasma densityprofile by prepulse and extend the propagation of thedelayed pulse over hundred of Rayleigh length.

The phenomena of cross-focussing of semi-major axisand semi-minor axis are oftenly observed [23,24], Inthese process, self-focussing of major axis leads todefocussing of semi-minor axis and vice versa. In ourpresent investigation, self-focussing of both axis viz.semi-major axis and semi-minor axis are observed(Fig. 4), however, the phenomena of cross-focussing isnot observed.

Eqs. (16), (24), (25) and (27) are solved numericallyand the results are displayed in the form of the graphs. Itis obvious from the Fig. 1 that the beam width of theprepulse increases monotonically leading to de-focuss-ing of the ionizing pulse. This is due to the fact that thefirst diffractive term on the right-hand side of Eq. (16) issupplemented by the refractive-induced defocussing

ARTICLE IN PRESS

0.5 1 1.5 2 2.5 3ζ

2

4

6

8

ζ

an

Fig. 1. Variation of prepulse pulse an with normalized distance of

propagation z for the following set of parameters: a0 ¼ 0:002 cm,

b0 ¼ 0:0012 cm, a1 ¼ 0:4, orp ¼ 5, E0a=A00 ¼ 0:5.

0.5 1 1.5 2 2.5 3ζ

0.0008

0.0012

0.0014

0.0016

0.0018

0.002

ζ

a

Fig. 2. Variation of beam widths a with normalized distance of

propagation z for the same set of parameters as in Fig. 1.

0.5 1 1.5 2 2.5 3ζ

0.0006

0.0008

0.0012

ζ

b

Fig. 3. Variation of beam widths b with normalized distance of

propagation z for the same set of parameters as in Fig. 1.

0.5 1 1.5 2 2.5 3ζ

0.0005

0.00075

0.00125

0.0015

0.00175

0.002

ζ

a,b

Fig. 4. Variation of beam widths a (solid line) and b (dotted

line) with normalized distance of propagation z for the same

set of parameters as in Fig. 1.

0.5 1 1.5 2 2.5 3ζ

1×10-6

2×10-6

3×10-6

4×10-6

5×10-6

ζ

ρ

Fig. 5. Variation of effective beam radius r with z for the same

set of parameters as in Fig. 1.

0.5 1 1.5 2 2.5 3ζ

-1

-0.5

0.5

1

φ re

g

ζ

Fig. 6. Variation of freg of the prepulse pulse (solid line) and

delayed pulse (dotted line) with normalized distance of

propagation z for the same set of parameters as in Fig. 1.

A. Singh et al. / Optik 119 (2008) 559–564 563

second term. However, with the propagation of thebeam, there is finite contribution from the last term onthe right-hand side of Eq. (16), which prevents the beamfrom the steep defocussing.

Figs. 2 and 3 represent the propagation of the semi-major axis ‘a’ and semi-minor axis ‘b’, respectively, ofthe guided laser beam. It is evident from the Figs. thatboth the axis shows oscillatory character during thepropagation of the guided laser beam in the plasmachannel. It is also observed from Fig. 4 that there is nocross-focussing during propagation. This is due to the

fact that differential equations (24) and (25) for ‘a’ and‘b’ are not nonlinearly coupled. Therefore their propa-gation are independent of each other. Due to thenonoccurrence of cross-focussing phenomena, theguided beam propagates over hundred of Rayleighlength (Fig. 5) and thus supports the experimentallyobserved results [25]. Such theoretical prediction forlong propagation of the guided laser beam has not beenreported so far.

Further Eqs. (17) and (26) are solved numerically tocalculate the regularized phase ½freg ¼ fðzÞ �fðzÞjnonlinearity¼0� in the absence of nonlinearity. It is

ARTICLE IN PRESSA. Singh et al. / Optik 119 (2008) 559–564564

obvious from Fig. 6 that the freg is always negative. Thissupports the earlier work of Karlsson et al. [12].

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