IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 8, AUGUST 2013 2251

Sensitiveness of Decentered Parameter forRelativistic Self-Focusing of Hermite-cosh-Gaussian

Laser Beam in PlasmaVikas Nanda, Niti Kant, and Manzoor Ahmad Wani

Abstract— Sensitiveness of the decentered parameter for rel-ativistic self-focusing of Hermite-cosh-Gaussian beam in theplasma is investigated theoretically using Wentzel-Kramers-Brillouin and paraxial ray approximation for mode indices0, 1, and 2. The plot between the beam width parameter andthe normalized propagation distance for different values of thedecentered parameter and intensity has been reported and resultsobtained indicate the dependency of the self-focusing of thelaser beam on the decentered parameter. The selection of thedecentered parameter is more sensitive to self-focusing. For themode indices m = 0 and 1, self-focusing becomes stronger andfor m = 2, self-focusing becomes weaker as the diffraction termbecomes more dominant. Our emphasis is on the selection of thedecentered parameter at which stronger self-focusing of laserbeam is observed which might be very useful in the applications,such as the generation of inertial fusion energy driven by lasers,laser driven accelerators, and so on.

Index Terms— Hermite-cosh-Gaussian beams (HChG), plasmadensity, self-focusing, sensitivity.

I. INTRODUCTION

THE interaction of light with matter is one of the basicphenomena in nature. The advancement in the short

pulse laser technology have enabled experiments using laserpulses focused to extremely high intensity of the order of1020 W/cm2. Self-focusing of laser beams in plasmas [1]–[6]becomes one of the most interesting and fascinating field ofresearch for several decades due to its various applications,such as the generation of inertial fusion energy driven by lasers[7], [9], [12], optical harmonic generation [10], the productionof quasi monoenergetic electron bunches [11], X-ray lasers,and the laser driven accelerators [13]. These applications needthe laser pulse to propagate over several Rayleigh lengths inthe plasmas without loss of energy.

In the plasma three types of self-focusing mechanismsoccur, namely relativistic, ponderomotive, and thermal self-focusing, as the laser pulse propagates through it. The dielec-tric constant of plasma changes greatly with the increase in

Manuscript received March 20, 2013; revised May 18, 2013 and June 3,2013; accepted June 8, 2013. Date of publication July 3, 2013; date of currentversion August 7, 2013.

The authors are with the Department of Physics, Lovely ProfessionalUniversity, Phagwara 144411, India (e-mail: vikasnanda06@yahoo.com;nitikant@yahoo.com; manzoorphysics@gmail.com).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2013.2268164

intensity of the laser beam and it leads to the self-focusing ofthe beam [14], [15]. The variation in the electron density [16]in plasmas is caused by propagating laser pulse of extremelyhigh intensities of the order ranging from 1017–1020 W/cm2

.These high intensity laser pulses provide sufficient energy tothe constituents, such as ions, electrons, and so on, of the plas-mas that causes an electron oscillatory velocity comparable tothe velocity of light. Thus the mass of electron, oscillatingat relativistic velocities in laser field, increases and givesrise to nonlinearity due to which the relativistic self-focusingeffect occurs. The theory of relativistic self-focusing of laserradiation in plasmas was studied by Hora [4]. Self-focusingin the plasma due to ponderomotive forces and relativisticeffects was studied by Siegrist [5]. Relativistic self-focusingand self-channeling of Gaussian laser beam was, recently,reported by Singh et al. by applying moment theory approachto solve the nonlinear differential equation for beam widthparameter and then solved it numerically by Runge-Kuttamethod [8].

Recently, theoretical investigators focus their attentionon paraxial wave family of laser beams. Hermite-cosh-Gaussian (HChG) beam is one of the solutions of paraxialwave equation and such HChG beam can be obtained inthe laboratory by the superposition of the two decenteredHermite-Gaussian beams as cosh-Gaussian ones. Propagationof HChG beams in plasmas was studied theoretically earlierby Belafhal et al. [1] and Patil et al. [17], [18]. The focusingof HChG laser beams in magneto-plasma by consideringponderomotive nonlinearity was theoretically examined byPatil et al. [19] and reported the effect of mode index andthe decentered parameter on the self-focusing of the beams.

This paper is dedicated to study the sensitiveness of thedecentered parameter for relativistic self-focusing of HChGbeams in plasmas. We derive the equations for beam widthparameter for HChG beam and solve them numericallyby applying Wentzel-Kramers-Brillouin approximation andParaxial approximation [20], [21] for mode indices 0, 1, and2 and observed the enhancement in the self-focusing of thelaser beams as the beam width parameter decreases withthe normalized distance for the optimum sensitive values ofthe decentered parameter. For the sake of simplicity, onlythe transversal components of laser field are evaluated andlongitudinal components are not taken in to considerationin this paper. However, longitudinal components should be

0093-3813 © 2013 IEEE

2252 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 8, AUGUST 2013

Fig. 1. Variation of beam width parameter (f) with the normalized propagation distance (ξ) for different values of decentered parameters and for: (a) αE20 = 3,

(b) αE20 = 5, (c) αE2

0 = 7. The other parameters are taken as m = 0, ωr0/c = 3, m0/M = 0.02 and ωpo/ω = 0.4.

taken for an exact formulation, while dealing with nonlinearphenomena [22].

II. FIELD DISTRIBUTION OF HCHG LASER BEAMS

The field distribution of HChG laser beams propagating inthe plasma along z-axis is of the following form:

E(r, z) = E0

f (z)

[Hm

((√

2r)

r0 f (z)

)]

eb24

(e−

(r

r0 f (z) + b2

)2

+ e−

(r

r0 f (z) − b2

)2)

(1)

where E0 is the amplitude of HChG laser beam for the centralposition at r = z = 0, f(z) is the dimensionless beam widthparameters, Hm is the Hermite polynomial of mth order, r0 isthe spot size of the beam, and b is the decentered parameterof the beam.

III. NONLINEAR DIELECTRIC CONSTANT

The dielectric constant for the nonlinear medium (collision-less plasma) is obtained by applying the approach given bySodha et al. [21]

ε = ε0 + φ(E E∗)with

ε0 = 1 − ω2p/w

2

ω2p = 4πn0e2/m

m = m0γ,w2p = w2

po/γ,w2po = 4πn0e2/m0

and

γ = 1/

√1 − v2/c2, (2)

where ω0 and ∅ are the linear and nonlinear parts of thedielectric constant, respectively, is the plasma frequency, eis the electronic charge, m0 is the rest mass of the electron,ω is the frequency of the incidents laser beam, and n0 is theequilibrium electron density.

NANDA et al.: SENSITIVENESS OF DECENTERED PARAMETER FOR RELATIVISTIC SELF-FOCUSING 2253

Fig. 2. Variation of beam width parameter (f) with normalized propagation distance (ξ) for different values of the decentered parameter and for: (a) αE20 = 3,

(b) αE20 = 5, (c) αE2

0 = 7. The other parameters are taken as m = 1, ωr0/c = 3, m0/M = 0.02 and ωpo/ω = 0.4.

In case of collision-less plasma, ponderomotive force causesthe nonlinearity in the dielectric constant and hence, nonlinearpart of the dielectric constant can be written as [21] follows:

φ(E E∗) = ω2po/γω2

[1 − ex p

(−3m0γα1 E E∗

4M

)](3)

with α1 = e2 M/6m20γ

2ω2kbT0 where M is the mass of thescatterer in the plasma, Kb is the Boltzmann constant, and T0isthe equilibrium plasma temperature.

IV. EVOLUTION OF BEAM WIDTH PARAMETER

For isotropic, nonconducting and nonabsorbing medium (forset of values J = 0 and ρ = 0 with μ = 1, Maxwell’s equationsgive the following wave equation:

∇2 �E − ε

c2

∂2 �E∂ t2 + �∇

(�E �∇(ε)

ε

)= 0 (4)

Consider a plane polarized wave with electric field vectorgiven by the following:

�E = A(r, z)e(ωt−kz) (5)

where A(r, z) is the complex amplitude of the electric field.Thus

∂2 �E∂ t2 = −ω2 �E . (6)

Hence (4) becomes

∇2 �E + k2 �E + �∇(

�E �∇(ε)

ε

)= 0 (7)

where k2 = εω2/c2, and for (1/k2)∇2(ε) << 1, (7) can bewritten as follows:

2254 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 8, AUGUST 2013

Fig. 3. Variation of beam width parameter (f) with normalized propagation distance (ξ) for different values of the decentered parameter and for: (a) αE20 = 3,

(b) αE20 = 5, (c) αE2

0 = 7. The other parameters are taken as m = 2, ωr0/c = 3, m0/M = 0.02 and ωpo/ω = 0.4.

∂2 �E∂z2 + ∂2 �E

∂r2 + 1∂ �Er∂r

+ εω2

c2�E = 0. (8)

We know that the solution of (8) is of the form

�E = A(r, z)ei

(ωt− z

c

√ω2− ω2

poγ

). (9)

Substituting this value in (8), we obtain the following:

2ik∂ A

∂z= ∂2 A

∂r2 + 1∂ A

r∂r+ k2 A

ε0φ(AA∗). (10)

The solution of (10) is of the form

A(r, z) = Aop(r, z)e−iks(r,z) (11)

where A0p and S are the real function of r and z, respectively.Using (11) in (10) and separating real and imaginary parts, weobtain the following.

Real part is

2∂S

∂z+

(∂S

∂z

)2

= 1

k2 Aop

(∂2 Aop

∂r2 + 1

r

∂ Aop

∂r

)+ φA2

op

ε0. (12)

Imaginary part is

2∂ Aop

∂z+ 1

rAop

∂S

∂r+ Aop

∂2S

∂r2 + 2∂S

∂r

∂ Aop

∂r= 0. (13)

The solution of (12) and (13) is

A2op = E2

0

f 2(z)

[Hm

(√2r

r0 f

)]2

eb22

⎛⎜⎜⎝e

−2

(r

r0 f (z) + b2

)2

+e−2

(r

r0 f (z) − b2

)2

+ 2e−(

2r2

r20 f 2(z)

+ b22

)⎞⎟⎟⎠ (14)

andS = r2

2β(z) + ϕ(z) (15)

NANDA et al.: SENSITIVENESS OF DECENTERED PARAMETER FOR RELATIVISTIC SELF-FOCUSING 2255

where β(z) = (1/ f (z))∂ f/∂z is an arbitrary function of z.Using these values in (12), we obtain the equation governingthe evolution of beam with parameter

For m=0

∂2 f

∂ξ2 =[

4 − 4b2 − 6αE20ω2r2

0 m0ω2po

ω2γ c2 Me

b22

]1

f 3 (16)

For m=1

∂2 f

∂ξ2 =[

4 − 4b2 − 12αE20ω2r2

0 m0ω2po

ω2γ c2Me

b22 (b2 − 2)

]1

f 3(17)

For m=2

∂2 f

∂ξ2 =[−8b2 − 24αE2

0ω2r20 m0ω

2po

ω2γ c2Me

b22 (5 − 2b2)

]1

f 3(18)

where ξ = z/Rd, similarly (13) gives the condition, ξ = 0,f =constant.

V. RESULTS AND DISCUSSION

We solved (18) and (19) numerically and analyze thesensitiveness of the decentered parameter for relativistic self-focusing of HChG beams in the plasmas for first three modeindices. The variation of beam width parameter f with thenormalized propagation distance ξ for mode indices m =0, 1, and 2, for different intensity parameters αE2

0 = 3, 5,and 7 (corresponding intensities are 2.2 × 1017 W/cm2,3.7 × 1017 W/cm2, and 5.2 × 1017 W/cm2, respectively)is analyzed for different parameters given as ω r0/c = 3,r0 = 3 × 10−6 m, ω = 3 × 1014 rad/s, m0/M = 0.02 andωpo/ω = 0.4. From Fig. 1(a), it is clear that with the littleincrease in the value of the decentered parameter b, the beamwidth parameter f decreases greatly for αE2

0 = 3. Hence,self-focusing of laser beam becomes more and more strongerand it is obvious from the figure that small decimal changein the value of the decentered parameter greatly affects thebeam width parameter. In addition, the decentered parameteris very sensitive and its proper selection decides the focusingor the defocusing effect at different intensities. In Fig. 1(b) and(c), the variation of beam width parameter f with normalizedpropagation distance ξ for intensity parameters αE2

0 = 5 and7 is observed for various parameters taken to be similar as thatin Fig. 1(a). The plot obtained clarifies that for mode index,m = 0, the beam gets more and more focused due to thedominance of the nonlinear self-focusing term. For the valuesof the intensity parameters αE2

0 = 3, 5, and 7, self-focusingbecomes stronger for the decentered parameter b = 0.92, 0.74,and 0.52, respectively. Hence, for the laser of high intensity,self-focusing occurs at low values of the decentered parameter.

The variation of the beam width parameter f with thenormalized propagation distance ξ , for mode index m = 1, atdifferent values of various parameter taken similar to previousvalues as taken in case of mode index m = 0 is shown inFig. 2. From Fig. 2(a), it is concluded that for certain valuesof the decentered parameter b, focusing of laser beam occurs.

These selected values of the decentered parameter depend onthe initial intensity of laser beam. From Figs. 1 and 2, it isclear that in case of mode index m = 0, for intensity parameterαE2

0 = 3, self-focusing is stronger at b = 0.92, however, incase of mode index m = 1, for intensity parameter αE2

0 =3, self-focusing of beam becomes stronger at b = 1.287.In comparison with the results obtained for m = 0 and 1,it is clear that for m = 1, self-focusing occurs at highervalues of the decentered parameter. From Figs. 1 and 2, wecan clearly see the sensitiveness of the decentered parameteron self-focusing. Thus, self-focusing of HChG laser beamcan be controlled by the mode indices and the decenteredparameter, as the decentered parameter is more sensitive toself-focusing. These results support the results obtained byPatil et al. [18], [19]. For m = 1, with the increase invalue of the decentered parameter b, focusing term becomesdominant for all taken values of intensity. It would be quiteinteresting to see the effect of the decentered parameter on thefocusing/defocusing nature of HChG beam for higher valuesof mode indices.

The variation of beam width parameter f with the normal-ized propagation distance ξ for m = 2 and other parametersare same as taken previously is shown in Fig. 3. From theplot it is concluded that for mode index m = 2, self-focusingeffect of laser beam reduces greatly with the increase in thevalue of the decentered parameter and the defocusing termin (18) becomes dominant over the nonlinear self-focusingterm. Again the selection of values of the decentered parameterplays a very crucial role by virtue of which at least weak self-focusing effect is observed and thereafter beam get defocused.We observed that for m = 2, self-focusing is weak for intensityparameters αE2

0 = 3, 5, and 7 as compared with that in caseof mode indices m = 0 and 1 for the same parameters.

VI. CONCLUSION

In this investigation, we studied the sensitivity of the decen-tered parameter for relativistic self-focusing of HChG laserbeam in the plasma. We derived the equation of beam widthparameter by using paraxial ray approach and investigatedthe sensitiveness of the decentered parameter on the self-focusing. The focusing/defocusing phenomena of HChG laserbeam in the plasma for mode indices m = 0, 1, and 2,can be controlled and made stronger with the decenteredparameter b, as the value of b was more sensitive to self-focusing. In addition, self-focusing became stronger with theincrease in selected values of the decentered parameter at aparticular intensity for mode indices m = 0 and 1. For m= 2, we observed that opposite results indicating that self-focusing effect becomes weaker. Thus, for m = 2, HChG beamexhibited diffraction effect and self-focusing effect was weakerfor all taken values of laser intensity. The results obtained inthis paper were added to the sensitiveness of the decenteredparameter to self-focusing. We reported the selection of thedecentered parameter as the decentered parameter was moresensitive to self-focusing. This paper helped the investigatorsto choose the intensity parameter as per their requirementby the proper selection of the decentered parameter led to

2256 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 8, AUGUST 2013

substantial improvement in the focusing quality which maybe useful in inertial fusion energy driven by lasers.

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Vikas Nanda received the master’s degree in philos-ophy in polymer-ion beam interaction from Anna-malai University, Tamil Nadu, India, in 2008. Heis currently pursuing the Ph.D. degree with LovelyProfessional University, Phagwara, India.

His current research interests include self-focusingof a short pulse laser in plasmas and clusters.

Niti Kant received the Ph.D. degree in laser-plasmainteraction from IIT Delhi, Delhi, India, in 2005.

He has been an Associate Professor with thePhysics Department, Lovely Professional Univer-sity, Phagwara, India, since 2007. He was a Post-Doctoral Fellow with POSTECH, Gyeongsangbuk-do, Korea, from 2005 to 2007. He has supervisedten M.Phil. students and is now supervising threePh.D. students. His current research interests includeultrashort intense lasers interaction with plasmas,laser plasma based accelerators, and THz radiation.

Manzoor Ahmad Wani received the M.Sc. degreein physics from Kashmir University, Srinagar, India,in 2006, and the master’s degree in philosophy inphysics from Lovely Professional University, Phag-wara, India, in 2011. His dissertation was entitled,“Self-Focusing of Hermite-Gaussian Laser Beam inPlasma Under Density Transition.”

He is currently a Research Student. His currentresearch interests include self-focusing of Hermite–Gaussian laser beams in plasma under plasmadensity ramp.