Focusing of Hermite-cosh-Gaussian laser beams incollisionless magnetoplasma

S.D. PATIL, M.V. TAKALE, S.T. NAVARE, AND M.B. DONGAREDivision of Nonlinear Optics and Holography Laboratory, Department of Physics, Shivaji University, Kolhapur, India

(RECEIVED 26 February 2010; ACCEPTED 8 April 2010)

Abstract

This paper presents an investigation of the focusing of Hermite-cosh-Gaussian laser beams in magneto-plasma byconsidering ponderomotive nonlinearity. The dynamics of the combined effects of nonlinearity and spatial diffractionis presented. To highlight the nature of focusing, plot of beam-width parameter vs. dimensionless distance ofpropagation has been obtained. The effect of mode index and decentered parameter on the self-focusing of the beamshas been discussed.

Keywords: Hermite-cosh-Gaussian beams; Nonlinear plasma physics; Paraxial; Self-focusing

1. INTRODUCTION

Interaction of intense laser with plasmas is a major area ofexperimental and theoretical study due to its relevance tooptical harmonic generation (Sprangle & Esarey, 1991;Milchberg et al., 1995; Zhou et al., 1996), X-ray generation(Eder et al., 1994), inertial confinement fusion (Tabak et al.,1994; Deutsch et al., 1996; Hora, 2004), and laser drivenaccelerators and particle beams (Sprangle et al., 1998;Umstadter et al., 1996; Mora & Antonsen, 1996; Andreevet al., 1997, 1998; Amiranoff et al., 1998; Sari et al.,2005; Neff et al., 2006; Zhou et al., 2007; Chen et al.,2008; Niu et al., 2008). So far the main thrust of the theoreti-cal and experimental investigations on self-focusing of alaser beam has been directed toward the study of the propa-gation characteristics of a Gaussian beam (Akhmanovet al., 1968; Hora, 1969; Sodha et al., 1974, 1976; Esareyet al., 1997; Umstadter, 2001; Sharma et al., 2003, 2004).Nevertheless, a few papers have been published on the self-focusing of cylindrical off modes of beams (Jones et al.,1988), super Gaussian beams (Nayyar, 1986; Grow et al.,2006; Fibich, 2006), degenerate modes of beams(Karlsson, 1992), elliptic Gaussian beams (Anderson et al.,1980; Cornolti et al., 1990; Gill et al., 2000, 2004; Saini &Gill 2006), Bessel beams (Johannisson et al., 2003),

cosh-Gaussian beams (Patil et al., 2009), and Hermite-Gaussian beams (Takale et al., 2009).

Apart from these, great interest has recently been evincedin paraxial beams family, usually known as Hermite-cosh-Gaussian (HChG) beams (Belafhal & Ibnchaikh, 2000).In relatively recent studies, the propagation of various HChGsin complex optical systems and turbulent atmosphere, hasbeen investigated in detail (Ibnchaikh et al., 2001; Wang &Lu, 2001; Ji & Lu, 2002; Lu & Qing, 2001; Luo & Lu,2002; Yu et al., 2002; Gu et al., 2004, 2005; Mei et al.,2004, 2005; Qui et al., 2004; Zhao et al., 2004, 2005;Zeng et al., 2005, 2006; Du & Zhao, 2006, 2007; Tanget al., 2006; Bai et al., 2009, 2010; Yang et al., 2009).Recently, we have investigated its propagation in semicon-ductors (Patil et al., 2008a, 2008b). A review of the literaturehighlights the fact that the propagation characteristics ofHChGs in plasma have not been studied to a significantextent; as an exception, the beam propagation in isotropicplasma has been studied (Patil et al., 2007).

Recently, focusing of dark hollow Gaussian electromag-netic beams in plasma has been reported (Sodha et al.,2009a, 2009b; Misra & Mishra, 2009a). Also, ring formationin electromagnetic beams in a magneto-plasma has beengiven by Misra and Mishra (2009b). Focusing of a ringripple on Gaussian electromagnetic beams in a magneto-plasma has been exploited (Misra & Mishra, 2009c). Sodhaet al. (2009c) studied the growth of a ring ripple on aGaussian electromagnetic beam in plasma with relativistic-ponderomotive nonlinearity. Thanks to such recent

343

Address correspondence and reprint requests to: S.D. Patil, Division ofNonlinear Optics and Holography Laboratory, Department of Physics,Shivaji University, Kolhapur 416 004, India. E-mail: sdpatil_phy@rediff-mail.com

Laser and Particle Beams (2010), 28, 343–349.#Cambridge University Press, 2010 0263-0346/10 $20.00doi:10.1017/S0263034610000297

development in the paraxial approach. But for the sakeof simplicity, we have exploited the self-focusing ofcylindrical HChGs in collisionless magneto-plasma underparaxial approach given by Akhmanov et al. (1968) anddeveloped by Sodha et al. (1974, 1976). To illustrate thenature of focusing/defocusing, the dependence of focusingparameter on the distance of propagation has been given.

2. BASIC FORMULATION

We consider the propagation of HChG laser beams through amagneto-plasma along the z direction, which is also thedirection of the static magnetic field. The wave equation isconsidered to comprise two field configurations, the extra-ordinary mode and the ordinary mode represented as

A1 = Ex + iEy and A2 = Ex − iEy

where E is the electric vector of the wave, which satisfies thewave equation,

∇2E −∇(∇ · E) + v2

c2(1 · E) = 0. (1)

In component form, Eq. (1) becomes

∂2Ex

∂z2+ ∂2Ex

∂y2− ∂

∂x

∂Ey

∂y+ ∂Ez

∂z

( )= −v2

c2(1 · E)x, (2a)

∂2Ey

∂z2+ ∂2Ey

∂x2− ∂

∂y

∂Ex

∂x+ ∂Ez

∂z

( )= −v2

c2(1 · E)y, (2b)

and

∂2Ez

∂x2+ ∂2Ez

∂y2− ∂

∂z

∂Ex

∂x+ ∂Ey

∂y

( )= −v2

c2(1 · E)z. (2c)

We assume the variation of field in the z-direction to be morerapid than in the x-y plane so that the wave field can be con-sidered as transverse in the zero order approximation andhence no space-charge is generated. Then, ∇.E ¼ 0 that gives

∂Ez

∂z= − 1

1zz1xx

∂Ex

∂x+ 1xy

∂Ey

∂x+ 1yx

∂Ex

∂y+ 1yy

∂Ey

∂y

( ). (3)

On multiplying Eq. (3) with +i and adding Eq. (2a) weobtain

∂2A1

∂z2+ 1

21 + 10+

10zz

( )∂2

∂x2+ ∂2

∂y2

( )A1

+ 12

10+10zz

− 1

( )∂

∂x+ i

∂

∂y

( )2

A2 +v2

c2

× [10+ +F+(A1A∗1, A2A∗

2)]A1 = 0

(4)

and

∂2A2

∂z2+ 1

21 + 10−

10zz

( )∂2

∂x2+ ∂2

∂y2

( )A2

+ 12

10−10zz

− 1

( )∂

∂x− i

∂

∂y

( )2

A2 +v2

c2

× [10− +F−(A1A∗1, A2A∗

2)]A2 = 0,

(5)

where

1+ = 10+ +F+. (6)

Here

10+ = 1 −v2

p

v(v+ vc),

F+ = 12A20

[1 − vcv

]2

and

10zz = 1 −v2

p

v2;

with

12 =v2

p

v(v− vc)3m0

8Ma 1 − vc

2v

( ),

vp = 4pN0e2

m0

( )1/2

,

vc =eB0

m0c,

and

a = e2M

6m20v

2kBT0.

Here 10 and F are the linear and nonlinear parts of dielectricconstant, respectively, vp and vc are the plasma frequencyand electron cyclotron frequency, respectively; e, mo, andNo being the magnitude of the electronic charge, rest mass,and electron density, M is the mass of scatterer in theplasma, v is the frequency of laser used, kB is theBoltzmann’s constant, and T0 is the equilibrium plasmatemperature.

Although Eqs. (4) and (5) are coupled with each other, thecoupling is not strong. To a good approximation we canassume that one of the two modes is zero and so the behaviorof the other mode can be studied. On assuming A2 ≈ 0,

Patil et al.344

Eq. (4) for A1 gives

∂2A1

∂z2+ 1

21 + 10+

10zz

( )∂2

∂y2

( )A1

+ v2

c2× [10+ +F+(A1A∗

1)]A1 = 0.

(7)

In Wentzel-Kramers-Brillouin approximation, the planewave solution of Eq. (7) can be written as

A1 = Aexp[i(vt − k+z)], (8)

where k+ ¼ (v/c)10+1/2 and A is the complex amplitude.

We substitute Eq. (8) in Eq. (7) and neglect ∂2A/∂z2 to get

− 2ik+∂A

∂z+ 1

21 + 10+

10zz

( )∂2

∂x2+ ∂2

∂y2

( )A

+ v2

c2F+(AA∗)A = 0.

(9)

Taking A ¼ A0(r, z) exp(2ik+S(r, z)), one can express thesolution for A0

2 and S as

A20 = E2

0

f 2Hm

��2

√r

fv0

( )[ ]2

× expb2

2

( )exp −2

r

fv0+ b

2

( )2[ ]{

+ exp −2r

fv0− b

2

( )2[ ]

+ 2exp − 2r2

f 2v20

+ b2

2

( )[ ]}(10)

and

S = r 2

2b(z) + w(z), (11)

where b(z) ¼ (1/f )(df/dz) and w(z) are an arbitrary functionof z.

Following approach given by Akhmanov et al. (1968) andits extension by Sodha et al. (1974, 1976), the differentialequations for the beam-width parameter of HChG beams ina collisionless magneto-plasma are formulated as below:

1f0+

d2f0+dz2

= 1 + 10

10zz

( )2

R−2d

[

− 1 + 10

10zz

( )R−2

n expb2

2

( )]1

f 40+

,

(12)

1f1+

d2f1+dz2

= 1 + 10

10zz

( )2

R−2d − 2(b2 − 2)

[

× 1 + 10

10zz

( )R−2

n expb2

2

( )]1

f 41+

,

(13)

and

1f2+

d2f2+dz2

= 1 + 10

10zz

( )2

R−2d − 4(5 − 2b2)

[

× 1 + 10

10zz

( )R−2

n expb2

2

( )]1

f 42+

,

(14)

where Rd ¼ kv02 is the diffraction length, Rn ¼ v0|1 2 (vc/

v)|(10/12)1/2 is the self-focusing length and subscripts 0,1, and 2 on f represent mode index m with + representingextraordinary mode.

The first terms on the right-hand side of Eqs. (12)–(14)represents spatial dispersion and are responsible for diffrac-tion divergence. On the other hand, second terms, respect-ively, are nonlinear in nature leading to self-focusing of thebeam. Obviously, the self-focusing/defocusing of the beamis possible, when second terms on right-hand side of eachexpression are negative/positive according to the desirabilityof decentered parameter values. Usually, diffraction of laserbeam is more dominant than it’s self-focusing due to theabsorption and scattering losses in the nonlinear medium.For the desirability of appreciable self-focusing of thebeam, corresponding analytical solutions of Eqs. (12)–(14)obtained under the condition Rn , Rd are as below:

f 20+ = 1 + 1 − Rd

Rn

( )2 exp(b2/2)[1 + (10/10zz)]

{ }

× [1 + (10/10zz)]2h2,

(15)

f 21+ = 1 + 1 − 2(b2 − 2)

Rd

Rn

( )2 exp(b2/2)[1 + (10/10zz)]

{ }

× [1 + (10/10zz)]2h2

(16)

and

f 22+ = 1 + 1 − 4(5 − 2b2)

Rd

Rn

( )2 exp(b2/2)[1 + (10/10zz)]

{ }

× [1 + (10/10zz)]2h2,

(17)

where h ¼ z/Rd is the dimensionless distance of propa-gation. Similar equations can be obtained for the ordinarymode by replacing vc with 2vc and changing the subscriptsfrom + to 2 on f in respective expressions.

3. RESULTS AND DISCUSSION

In this section, we will illustrate the self-focusing and defo-cusing of HChG beams in collisionless magneto-plasma forthe first three mode indices. The behavior of beam-widthparameters f+ and f2 for both the modes of collisionlessmagneto-plasma with the dimensionless distance of propa-gation h for various decentered parameters (b-values) isexamined by numerical estimates. The following typical

Focusing of Hermite-cosh-Gaussian laser beams 345

parameters are chosen for the purpose of numerical calcu-lation:

aE20 = 8 × 10−3[1 − (vc/v)]2, l = 1.55mm,

v0 = 500mm, vp/v = 0.77, vc/v = 0.1.

The results are plotted in the form of graphs.Figure 1 displays the variation of beam-width parameters

f+ and f2 with the dimensionless distance of propagation h

for m ¼ 0 with various decentered parameters b ¼ 0, 1,2. This leads to the self-focusing effect for all b-values and

sharp focusing occurs at b ≥ 3, which is not shown in thefigure for the sake of clarity. From this figure it is observedthat higher the decentered parameter is, the sharper the self-focusing effect is. This shows that the additional self-focusing is observed for higher decentered parameters butordinary mode of collisionless magneto-plasma is less sus-ceptible for self-focusing than extraordinary mode.

Figure 2 plots the beam-width parameters ( f+ and f2) withthe dimensionless propagation distance h of HG beams (i.e.,for b ¼ 0) in collisionless magneto-plasma. It depicts that for

Fig. 1. (Color online) Beam width parameters ( f+ and f2) versus the dimen-sionless propagation distance (h) of ChG beam (m ¼ 0) in a collisionlessmagnetoplasma for extraordinary (solid curves) and ordinary (dashedcurves) modes. The red, blue and cyan curves are for the decentred parametervalues of 0, 1, and 2, respectively.

Fig. 2. (Color online) Beam width parameters ( f+ and f2) versus the dimen-sionless propagation distance (h) of HG beams (b ¼ 0) in a collisionlessmagnetoplasma for extraordinary (solid curves) and ordinary (dashedcurves) modes. The red, blue and cyan curves are for the mode indexvalues of 0, 1, and 2, respectively.

Fig. 3. (Color online) Beam width parameters ( f+ and f2) versus the dimen-sionless propagation distance (h) of H1ChG beam (m ¼ 1) in a collisionlessmagnetoplasma for extraordinary (solid curves) and ordinary (dashedcurves) modes. The red, blue and cyan curves are for the decentred parametervalues of 0, 1, and 2, respectively.

Fig. 4. (Color online) Beam width parameters ( f+ and f2) versus the dimen-sionless propagation distance (h) of H2ChG beam (m ¼ 2) in a collisionlessmagnetoplasma for extraordinary (solid curves) and ordinary (dashedcurves) modes. The red, blue and cyan curves are for the decentred parametervalues of 0, 1, and 2, respectively.

Patil et al.346

extraordinary as well as ordinary modes of magneto-plasma,Hermite Gaussian beams defocuses for m ¼ 1 mode.However, they exhibit self-focusing effect for m ¼ 0 and2. In this case also, ordinary mode is less prone forself-focusing.

In Figure 3, we have plotted beam width parameters ( f+and f2) as a function of dimensionless distance of propa-gation of H1ChG beam (i.e., for m ¼ 1) in a collisionlessmagneto-plasma. We see that H1ChG beam gives defocusingfor b ¼ 0 and 1, while for b ¼ 2, we obtain self-focusing forboth extraordinary and ordinary modes.

The variation of beam width parameters ( f+ and f2) withthe dimensionless propagation distance h of H2ChG beam(i.e., for m ¼ 2) is displayed in Figure 4. One sees that forboth modes of collisionless magneto-plasma, the beamfocuses for b ¼ 0 and 1, while it defocuses for b ¼ 2. Notethat in all figures, solid curves defines extraordinary modeand dashed curves are for ordinary mode. If we compareall figures, it is found that the self-focusing/defocusing ofHChG beams is dependent on the mode index m and decen-tered parameter b.

4. CONCLUSION

In conclusion, the differential equations for beam-width par-ameters are established under paraxial approximation byinvestigating the propagation of Hermite-cosh-Gaussianbeams, taking into account the ponderomotive mechanismin a collisionless magneto-plasma. The analytical solutionsare obtained for the same. In addition to adopting a generalsource beam, our formulation also incorporates the additionalself-focusing effect in collisionless magneto-plasma byexploiting beam’s decentered parameter. Furthermore, thepropagation analysis of HChG beams under higher-ordernonlinearity and nonparaxiality is quite interesting. Thepresent results should be useful in understanding collision-less heating of the plasma useful for laser fusion.

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