Guiding of a laser beam in a collisionlessmagnetoplasma channel

Arvinder Singh* and Navpreet Singh

Department of Physics, National Institute of Technology Jalandhar, India*Corresponding author: arvinder6@lycos.com

Received March 2, 2011; revised June 10, 2011; accepted June 14, 2011;posted June 17, 2011 (Doc. ID 143504); published July 7, 2011

We present an investigation of laser guiding through an axially nonuniform collisionless magnetoplasma channelformed by the ionizing laser prepulse. With the propagation of a guided laser beam through such a preformedplasma channel, a ponderomotive force on account of the nonuniform intensity distribution of the laser beamcomes into play, which further enhances the plasma channel. Unbalanced diffraction and refraction phenomenathrough such an axially nonuniform collisionless magnetoplasma channel results in alternate convergence anddivergence of the laser beam. Wave equations governing the propagation characteristics of the ionizing prepulseand delayed pulse through an axialy nonuniform collisionless magnetoplasma channel have been solved by themoment theory approach. The effect of the magnetic field and intensity of a guided beam has been seen on thepropagation of the second guided beam in the plasma channel. It is observed that there is an increase in the pro-pagation of a guided beam with the increase in the magnetic field and intensity of the guided beam. This observa-tion is very useful for the fast-ignitor concept, where the guided laser pulse is propagated through the plasmaregion without any absorption up to several Rayleigh lengths. © 2011 Optical Society of America

OCIS codes: 260.5950, 350.5400, 350.5500.

1. INTRODUCTIONThere is currently considerable research interest in the inter-action of intense laser pulses with plasmas, due to the adventof chirped pulse amplification technology, which has resultedin high-power table-top lasers. The major applications of theresearch are x-ray lasers, ultrahigh gradient electron accelera-tors, the fast-ignitor concept and laser-induced lightning[1–17]. Different physical processes are involved in these ap-plications, but the common desire for the success of theseapplications is the long-distance propagation of the laserbeam through the plasma. Diffraction broadening of the laserpulse is one of the main drawbacks for the effective use oflaser energy in these applications, and therefore the guidingof the laser pulse becomes essential for long distance propa-gation in the plasma. Several methods have been proposed toextend the propagation distance of laser pulses beyond thediffraction limit (Rayleigh length, Zr ¼ πr20

λ , where r0 is the spotsize and λ is the laser wavelength) [18].

Self-guiding of intense laser pulses has been studied inmany experiments [19–21]. Self-guiding is expected becauseof self-focusing that occurs when the refractive index ofthe plasma is modified by interaction with an intense laser.For effective guiding, a balance between focusing and defo-cusing should be maintained for long distances. Self-guidingof laser pulses is limited by a number of effects, such as en-ergy depletion by diffraction, scattering, and beam splittinginto filaments at high intensity. These difficulties can beavoided when the laser pulse to be guided does not createits own channel.

There are number of ways for guiding an intense laserpulse, such as capillary discharge and the two-pulse techni-que, as reviewed by Esarey et al. [18]. For optical guidingof laser pulses in plasmas, the plasma channel has to be pre-pared. For perfect guiding, a plasma channel with a density

minimum on the axis and a parabolic increase toward theedge is required. The two-pulse technique is used for guid-ing a laser pulse with high intensity of the order of1014–1020 W=cm2. The physics of guiding the laser pulse isas follows. A plasma channel is created by focusing a laserprepulse on a picosecond time scale via tunnel and impactionization. Since the prepulse has a Gaussian intensity radialprofile, the plasma density that results spontaneously has apeak on the axis and falls off rapidly with radial distance awayfrom the axis before the diffusion sets in. As a result, the re-fractive index is minimum on the axis and increases towardthe edge; hence, the medium behaves as a defocusing medi-um. Therefore, the plasma channel evolved from such a den-sity profile after the laser prepulse is gone should also have anaxial variation of the plasma density.

The plasma created by the prepulse diffuses radially awayfrom the axis and, therefore, the plasma density becomesminimum on the axis and maximum at the edges of the plasmachannel. Therefore, the refractive index becomes maximumon the axis and decreases toward the edges and, as a result,the medium behaves like a focusing medium.

In this paper, we have considered the propagation of aGaussian laser beam through the collisionless plasma channelformed by the prepulse propagating along the externally ap-plied magnetic field. On a short time scale (t < τE , where τE isthe energy relaxation time) in collisionless plasma, nonlinear-ity and, hence, carrier redistribution in a plane transverse tothe beam propagation arises due to the ponderomotive force.When a delayed intense second laser pulse is passed throughsuch a preformed plasma channel, then it tends to diverge dueto diffraction and converge due to refraction. However, whenboth the convergence and divergence parameters are equal,then the delayed laser pulse propagates without convergenceand divergence, but in an axially nonuniform plasma channel

1844 J. Opt. Soc. Am. B / Vol. 28, No. 8 / August 2011 A. Singh and N. Singh

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this condition cannot be satisfied throughout the channel and,therefore, the beam radius changes as it propagates.

The self-defocusing of the laser prepulse and laser guidingof the second laser pulse in the evolved axially nonuniformplasma channel has been explained by Liu and Tripathi usingparaxial ray approximation [22]. The paraxial theory [23,24]has been the most popular theory for studying the nonlinearpropagation of electromagnetic waves in nonlinear media. De-spite its mathematical simplicity, the problem regarding thistheory is that it takes into account only the regions closeto the beam axis in the self-focusing mechanism. Moment the-ory, which is a global approach, does not suffer from such lim-itations and, hence, gives more accurate results. Momenttheory has been used in this paper to study the optical guidingof laser beams through axially nonuniform collisionless mag-netoplasma channels. The importance of nonparaxiality inself-focusing phenomenon has already been highlighted [25].In Section 2, we develop a moment theory of self-defocusingof ionizing prepulses. In Section 3, we investigate the guidanceof the second pulse in the preformed plasma channel by themoment method approach. Equations of beam width para-meters are solved numerically using the Runge–Kutta method.The results are discussed in Section 4.

2. SELF-DEFOCUSING OF A PREPULSEPropagation of a Gaussian laser beam of frequency ω througha gas along the z axis is considered. The laser ionizes the gasin a time shorter than the laser pulse duration, which is of theorder of picoseconds. The plasma density so formed is mod-eled by Liu and Tripathi [22] as

ω2p ¼ ω2

p0 exp

�−E0

a

jEj�; ð1Þ

where ωp ¼ ½4πnee2

m �1=2 is the electron plasma frequency,ω2p0 is a

constant depending on the neutral particle density (n0), jEj isthe amplitude of the laser field, and E0

a ¼ 23EaðEi

EhÞ1=2, where Ei

is the ionization potential of an atom, Eh is the ionization po-tential of hydrogen, and Ea ¼ m2e5

ℏ4 ≃ 1:7 × 107 esu is the atomicunit of the electric field.

The electric vector E ¼ x̂Aðr; zÞeiðωt−k0zÞ of the laser beamsatisfies the wave equation

∇2E − ∇ð∇ · EÞ þ ω2ϵEc2

¼ 0: ð2Þ

The initial intensity distribution of the prepulse is assumedto be Gaussian and is given by

E · E�jz¼0 ¼ jAj2z¼0 ¼ A200 expð−r2=r21Þ; ð3Þ

where A00 is the maximum amplitude of the electric field atr ¼ 0, i.e., on the axis of the laser beam, r2 ¼ x2 þ y2, andr1 is the initial width of the laser beam at z ¼ 0.

The transverse intensity distribution of the laser beam forz > 0 can be written as [24]

jAj2 ¼ ðA200=f

21Þ expð−r2=r21f 21Þ; ð4Þ

where r1f 1 represents the beam width of the prepulse forz > 0. Therefore, the function f 1 is termed the dimensionless

beam width parameter of the prepulse with f 1 ¼ 1 at z ¼ 0[see Eq. (22)].

The refractive index (ϵ) of the plasma may be written as

ϵ ¼ ϵ0 þ ϕðE · E�Þ; ð5Þ

where

ϵ0 ¼ ϵjr¼0 ¼ 1 −ω2po

ω2 exp

�−E0af 1A00

�; ð6Þ

and ϕðE · E�Þ is the nonlinear part of the dielectric constant,given as

ϕðE · E�Þ ¼ ω2po

ω2

�exp ·

�−E0af 1A00

�− exp

�−E0af 1A00

er2

2r21f 21

��: ð7Þ

Now using E ¼ x̂Aðr; zÞeiðωt−k0zÞ in the wave Eq. (2) and ne-glecting ∂2A

∂z2by using the Wentzel–Kramers–Brillouin (WKB)

approximation, we get

2ik0∂A∂z

− ∇2⊥A −

ω2

c2ϕðE · E�ÞA ¼ 0: ð8Þ

Neglecting ∂2A∂z2

implies that the solution of Eq. (8) can beobtained for a slowly converging/diverging beam.

Now Eq. (8) can be rewritten as

∂A∂z

þ i2k0

∇2⊥Aþ iPðA · A�ÞA ¼ 0; ð9Þ

where

PðA · A�Þ ¼ ω2

2k0c2ϕðE · E�Þ; PðA · A�Þ ¼ k0

2ϵ0ϕðE · E�Þ:

ð10Þ

According to the definition of a second-order moment, themean-square radius of the laser beam is given by

ha2i ¼RR ðx2 þ y2ÞA · A�dxdy

I0; ð11Þ

Differentiating twice Eq. (11) with respect to z and substitut-ing the values of ∂A

∂z and ∂A�∂z from Eq. (9), we get

d2ha2idz2

¼ 2k20I0

�1k0

ZZj∇⊥Aj2dxdy

þZZ

ðx2 þ y2Þ1=2jAj2 ∂pðjAj2Þ

∂rdxdy

�: ð12Þ

Equation (12) is a second-order differential equation thatgoverns the evolution of the mean-square beam radius ofthe laser prepulse with the distance of propagation. However,it is more convenient to express it in terms of the invariants I0and I2. For this purpose, we integrate the second integral inEq. (12) and, by taking the values of I2, FðA · A�Þ, and QðA ·A�Þ as per the Eqs. (15)–(17), we arrive at the followingequation:

A. Singh and N. Singh Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. B 1845

d2ha2idz2

¼ 4I2I0

−4I0

ZZQðA · A�Þdxdy; ð13Þ

where I0 and I2 are the invariants of Eq. (9) [26]:

I0 ¼ZZ

jAj2dxdy; ð14Þ

I2 ¼ZZ

12k20

ðj∇⊥Aj2 − FðA · A�ÞÞdxdy; ð15Þ

where

FðA · A�Þ ¼ 1k0

ZA·A�

0PðA · A�ÞdðA · A�Þ; ð16Þ

QðA · A�Þ ¼�A · A�PðA · A�Þ

k0− 2FðA · A�Þ

�: ð17Þ

From Eqs. (4), (11), and (14), it can be shown that

I0 ¼ πr21A200; ð18Þ

ha2i ¼ r21f21: ð19Þ

From Eq. (19), we derive the following equation:

d2ha2idz2

¼ 2r21f 1

�d2f 1dz2

þ 1f 1

�df 1dz

�2�: ð20Þ

Substituting the value of I0 from Eq. (18) into Eq. (13) andusing Eq. (20), we get the following equation:

d2f 1dξ2 þ 1

f 1

�df 1dξ

�2¼ 2k20

πA200f 1

�I2 −

ZZQðjAj2Þdxdy

�; ð21Þ

where ξ ¼ ðz=k0r21Þ is the dimensionless distance of propa-gation.

By using Eqs. (5), (7), (10), (15)–(17), and (21), we get thefollowing equation for the beam width parameter f 1:

d2f 1dξ2 ¼ 1

f 31þ�r1ωp0

c

�2 E0

a

A00

1r41f

41

I1 −1f 1

�df 1dξ

�2; ð22Þ

where

I1 ¼Z

∞

0re

− r2

2r21f 21 exp

�−E0af 1A00

er2

2r21f 21

�r2dr:

The initial conditions for numerically solving Eq. (22) aref 1 ¼ 1, and df 1

dξ ¼ 0 at ξ ¼ 0.

3. GUIDED PROPAGATION OF SECONDDELAYED PULSEThe plasma moves radially away from the axis after the pas-sage of the prepulse. The second pulse is guided through thechannel, the density profile of which has been given by Liuand Tripathi [22]:

ω2p=ω2 ¼ α1ðzÞ þ α2ðzÞr2; ð23Þ

where α1ðzÞ and α2ðzÞ are monotonically decreasing functionsof z:

α1 ≤ ω2p0=ω2e−ðE0

a=A00Þf 1 ; α2 ≤ α1=r20f 21:

The electric field vector of the second laser pulse propagat-ing through the magnetoplasma along the static magnetic fieldB0 coinciding with the z axis can be expressed as

E ¼ x̂E0ðr; zÞeiðωt−k0zÞ: ð24Þ

The initial intensity distribution of the main beam isassumed to be Gaussian:

E0 · E�0jz¼0 ¼ E2

00 expð−r2=r22Þ; ð25Þ

where r2 ¼ x2 þ y2, and r2 is the initial width of the guidedlaser beam at z ¼ 0.

For z > 0,

E0 · E�0 ¼ ðE2

00=f22Þ expð−r2=r22f 22Þ; ð26Þ

where r2f 2 represents the beam width of the guided laserpulse for z > 0. Therefore, the function f 2 is termed the di-mensionless beam width parameter of the guided laser pulsewith f 2 ¼ 1 at z ¼ 0 [see Eq. (33)].

The nonuniform Gaussian intensity redistributes the car-riers from the high field region to the low field region underthe effect of ponderomotive force. The modified electron den-sity may be written as [24]

N0e ¼ Ne exp

�−

α0�1 − ωc

2ω

��1 − ωc

ω

�2 · E0 · E�

0

�; ð27Þ

where α0 ¼ e2

16kBTmω2, e is the electronic charge, kB is theBoltzmann constant, T is the equilibrium temperature of col-lisionless plasma, m is the mass of electrons, ω is the angularfrequency of the guided laser beam, and ωc ¼ eB0

mc is the elec-tron cyclotron frequency.

Therefore, the refractive index of the collisionless magne-toplasma under the effect of ponderomotive force is given as

ϵ ¼ 1 −ω2p

ω2

�1 − ωc

ω

� exp

�−

α0�1 − ωc

2ω

��1 − ωc

ω

�2 · E0 · E�

0

�

¼ 1 −ω2p

ω2

�1 − ωc

ω

� exp½−β0E0 · E�0�; ð28Þ

1846 J. Opt. Soc. Am. B / Vol. 28, No. 8 / August 2011 A. Singh and N. Singh

where

β0 ¼α0�1 − ωc

2ω

��1 − ωc

ω

�2 :

The refractive index of the collisionless magnetoplasma canalso be expressed as

ϵ ¼ ϵ0 þ ϕðE0 · E�0Þ; ð29Þ

where

ϵ0 ¼ 1 −α1�

1 − ωcω

� exp

�−β0E2

00

f 22

�; ð30Þ

ϕðE0 · E�0Þ ¼

α1�1 − ωc

ω

� exp

�−β0E2

00

f 22

�

−ðα1 þ α2r2Þ�

1 − ωcω

� exp

�−β0E2

00

f 22e− r2

r22f 22

�: ð31Þ

Now, by using Eqs. (24) and (29) in wave Eq. (2) andneglecting ∂2E0

∂z2by using the WKB approximation, we get

2ik0∂E0

∂z− ∇2

⊥E0 þ

ω2

c2ϕðE0 · E�

0ÞE0 ¼ 0: ð32Þ

By following the same procedure as in Section 2 for theprepulse, we get the following equation for the beam widthparameter f 2 of the second delayed pulse:

d2f 2dξ2 ¼ 1

f 32−

2k20

ϵ0f 2�1 − ωc

ω

��α1 · β0E2

00

r22f62

I2 þα2 · β0E2

00

r22f62

I3 þα2f 22

I4

�

−1f 2

�df 2dξ

�2; ð33Þ

where

I2 ¼Z

∞

0exp

�−β0E2

00

f 22e

−r2

r22f 22

�e−2r2

r22f 22r3dr;

I3 ¼Z

∞

0exp

�−β0E2

00

f 22e

−r2

r22f 22

�e−2r2

r22f 22r5dr;

I4 ¼Z

∞

0exp

�−β0E2

00

f 22e

−r2

r22f 22

�e

−r2

r22f 22r3dr:

We have numerically solved Eq. (33) by taking initial con-ditions f 2 ¼ 1, df 2

dξ ¼ 0 at ξ ¼ 0, and, for a model α2 [22],

α2 ¼2

R2dð1þ ξ2Þ1=2 :

4. DISCUSSIONThe differential equations in Eqs. (22) and (33) for the beamwidth parameter f 1 of the laser prepulse and f 2 of the guidedlaser pulse, respectively, have been solved numerically for thefollowing set of parameters:

ωp0

c¼ 0:33 × 106 m−1; ω ¼ 1:778 × 1015 s−1;

λ ¼ 1:06 μm; r1 ¼ r2 ¼ 18 μm;

and for different values of magnetic field, prepulse, andguided beam intensities.

The results are depicted in the form of graphs in Figs. 1–5.Equation (22) is a second-order nonlinear ordinary differentialequation that governs the variation of beam width parameterf 1 of the laser prepulse with a dimensionless distance of pro-pagation (ξ). There are three terms on the right-hand side ofEq. (22); the first term is due to diffraction divergence and thesecond term is due to nonlinear refraction, which is respon-sible for the self-defocusing of the prepulse. The initial contri-bution due to the third term at ξ ¼ 0 is zero, which evolveswith the distance of propagation and, hence, counteractsthe diffraction and self-defocusing of the prepulse.

Figure 1 presents the variation of the beam width para-meter f 1 of the prepulse with a dimensionless distance of pro-pagation (ξ). The dotted curve corresponds to the case whenonly the diffraction divergence term is present and the othertwo terms are ignored. The dashed–dotted curve correspondsto the case when only the first two terms, diffraction diver-gence and nonlinear refraction, are considered. The solidcurve corresponds to the case when all the three terms aretaken into account. From Fig. 1, it is observed that, in allthe cases, the beam width parameter f 1 increases monotoni-cally and leads to defocusing of the ionizing pulse.

Fig. 1. Variation of beam width parameter f 1 of a prepulse ofintensity 4:1 × 1017 W=cm2 with normalized distance of propagation(ξ) for the set of parameters ωp0

c ¼ 0:33 × 106 m−1, ω ¼ 1:778 × 1015 s−1,and r1 ¼ 18 μm.

A. Singh and N. Singh Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. B 1847

It is clear from the dashed–dotted curve that there is asharp defocusing of the ionizing pulse when the first twoterms are taken. This is due to the fact that the first diffractingterm is supplemented by the second refractive-induced defo-cusing term.

When the prepulse propagates, there is finite contributionfrom the last term 1

f 1ðdf 1dξ Þ2, which prevents the prepulse from

sharp defocusing, as evident from the solid curve of Fig. 1 and,hence, makes significant contribution in the formation of aplasma channel. Because of defocusing of the laser prepulse,the intensity of the beam decreases and, as a result, ionization/density of the plasma decreases and we get an axially nonuni-form plasma channel.

Equation (33) is also a second-order nonlinear ordinary dif-ferential equation that governs variation of the beam widthparameter f 2 of the guided pulse with a dimensionless dis-tance (ξ) of propagation. The first term on the right-hand sideof Eq. (33) is due to diffraction divergence and the secondterm is due to nonlinear refraction, which counteracts the dif-fraction and is also responsible for the guidance of the laserpulse. There is a third term, 1

f 2ðdf 2dξ Þ2; the initial contribution of

this term is zero as df 2dξ ¼ 0 at ξ ¼ 0. However, as the beam pro-

pagates, this term supports the second nonlinear refractiveterm and, hence, contributes significantly to the guidanceof the laser pulse.

Figure 2 presents the variation of the beam width para-meter f 2 of the second guided pulse with the dimensionlessdistance of propagation ξ for three values of the guided pulseintensities, 1:8 × 1017, 3:25 × 1017, and 5:0 × 1017 W=cm2 at afixed value of prepulse intensity, 4:1 × 1017 W=cm2, and mag-netic field ωc

ω ¼ 0:4. Now, as per the definition of the Rayleighlength, the propagation distance at which the value of f 2 be-comes 1.414 is taken as the guiding length of the laser beam. Itis observed, that for a second guided pulse intensity1:8 × 1017 W=cm2, f 2 varies in the range 0:1824 < f 2 < 1:414as ξ goes from 0 to 31, as shown by the dotted curve. Also,f 2 varies in the range 0:1817 < f 2 < 1:414 as ξ goes from 0to 45.25 for guided pulse intensity 3:25 × 1017 W=cm2, asshown by the dashed–dotted curve. It is further observed thatf 2 varies in the range 0:133 < f 2 < 1:414 as ξ goes from 0 to51.70 for a guided pulse intensity 5:0 × 1017 W=cm2, as shown

by the solid curve. So, as we increase the intensity of theguided pulse from 1:8 × 1017 to 5:0 × 1017 W=cm2, laser guidingincreases from 31 Rayleigh lengths to 51 Rayleigh lengths.This is due to the dominance of the nonlinear self-focusingterm over the diffraction term with the increase in intensity.Because of the increase in the intensity of the guided pulse,the ponderomotive force becomes more prominent and,hence, increases the self-focusing of the guided pulse, whichfurther leads to an increase in the laser guiding.

Figure 3 presents the variation of the beam width para-meter f 2 of the second guided pulse with the dimensionlessdistance of propagation (ξ) for three values of the first ionizinglaser prepulse intensity, 0:9 × 1017, 1:8 × 1017, and 4:1 ×1017 W=cm2 at a fixed value of second delayed laser pulse in-tensity, 5:0 × 1017 W=cm2. It is observed from Fig. 3 that, for aprepulse intensity of 0:9 × 1017 W=cm2, f 2 varies in the range0:159 < f 2 < 1:414 as ξ goes from 0 to 28.10, as shown by thedotted curve, and that, at a prepulse intensity of1:8 × 1017 W=cm2, f 2 varies in the range 0:122 < f 2 < 1:414as ξ goes from 0 to 36, as shown by the dashed–dotted curve.It is further observed that, for a prepulse intensity of4:1 × 1017 W=cm2, f 2 varies in the range 0:133 < f 2 < 1:414as ξ goes from 0 to 51.70, as shown by the solid curve. Thisis because the second nonlinear refractive term in Eq. (33) isvery sensitive to the prepulse intensity. So, as we increase theprepulse intensity, the nonlinear refractive term dominatesover the diffractive term and results in an increase in the laserguidance. Therefore, from the analysis it is predicted that, aswe increase the intensity of the prepulse from 0:9 × 1017 to4:1 × 1017 W=cm2, laser guidance increases from 28.10Rayleigh lengths to 51.70 Rayleigh lengths.

Figure 4, presents the variation of the beam width para-meter f 2 of the second guided pulse with the dimensionlessdistance of propagation (ξ) and for fixed values of prepulseand guided pulse intensities 4:1 × 1017 and 5:0 × 1017 W=cm2,respectively, and ωc

ω ¼ 0:0, 0.4. It is observed from Fig. 4 that,for ωc

ω ¼ 0:0, i.e., without any magnetic field, f 2 varies in therange 0:0891 < f 2 < 1:414 as ξ goes from 0 to 25.4, as shownby the dotted curve. For ωc

ω ¼ 0:4, i.e., in the presence of a mag-netic field, f 2 varies in the range 0:133 < f 2 < 1:414 as ξ goesfrom 0 to 51.70, as shown by the solid curve. Therefore, fromthe analysis it is predicted that, as we apply the magnetic field,

Fig. 2. Variation of beam width parameter f 2 of a guided pulse withnormalized distance of propagation (ξ) for three different values ofguided pulse intensities, 1:8 × 1017 W=cm2 for the dotted curve, 3:25 ×1017 W=cm2 for dashed–dotted curve, and 5:0 × 1017 W=cm2 for the so-lid curve at a fixed value of prepulse intensity 4:1 × 1017 W=cm2 andfor the set of parameters ωp0

c ¼ 0:33 × 106 m−1, ω ¼ 1:778 × 1015 s−1,and r1 ¼ r2 ¼ 18 μm.

Fig. 3. Variation of beam width parameter f 2 of a guided pulse withnormalized distance of propagation (ξ) for three different values ofprepulse intensities, 0:9 × 1017 W=cm2 for the dotted curve, 1:8 ×1017 W=cm2 for the dashed–dotted curve, and 4:1 × 1017 W=cm2 forthe solid curve at a fixed value of guided pulse intensity 5:0 ×1017 W=cm2 and for the set of parameters mentioned in Fig. 2.

1848 J. Opt. Soc. Am. B / Vol. 28, No. 8 / August 2011 A. Singh and N. Singh

there is an increase in the laser guidance from 25.4 Rayleighlengths to 51.70 Rayleigh lengths. Similarly, from Fig. 5, it isevident that, as we increase the magnetic field from ωc

ω ¼ 0:2 toωcω ¼ 0:6, there is an increase in the laser guidance from 35.0 to59.75 Rayleigh lengths.

In our present investigations, the guided laser pulse propa-gates through a plasma region created by the prepulse, with-out any absorption up to several Rayleigh lengths, as isevident from Figs. 2–5.

A new route to the achievement of ignition in laser-inducedfusion has been provided by the fast-ignitor concept, in whichan intense laser pulse is guided through a channel to the edgeof the core, where it is absorbed, producing a large flux ofmegaelectron-volt electrons [3]. These electrons then deposittheir energy within the core, thus heating it to the ignition tem-perature before hydrodynamic disassembly can take place. Inthis process, the intense channeling pulse must propagatethrough plasma regions without being absorbed and withoutbeam break-up due to filamentation. So, it is very importantfor the fast-ignition scheme to understand if such propagation

is possible. In the present paper, such propagation has beenpredicted.

ACKNOWLEDGMENTSThe authors are thankful to the Ministry of Human Resourcesand Development of India for providing financial assistancefor carrying out this work.

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Fig. 4. Variation of beam width parameter f 2 of a guided pulse withnormalized distance of propagation (ξ) for fixed values of prepulseand guided pulse intensities 4:1 × 1017 and 5:0 × 1017 W=cm2, respec-tively, at two different values of magnetic field of ωc

ω ¼ 0:0 for thedotted curve and ωc

ω ¼ 0:4 for the solid curve, and for the set of para-meters mentioned in Fig. 2.

Fig. 5. Variation of beam width parameter f 2 of a guided pulse withnormalized distance of propagation (ξ) for fixed values of prepulseand guided pulse intensities of 4:1 × 1017 and 5:0 × 1017 W=cm2, re-spectively, at three different values of magnetic field, ωc

ω ¼ 0:2 forthe dotted curve, ωc

ω ¼ 0:4 for the dashed–dotted curve, and ωcω ¼ 0:6

for the solid curve, and for the set of parameters mentioned in Fig. 2.

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