Magnetic field of a relativistic nonlinear plasma waveArsen G. Khachatryan Citation: Physics of Plasmas (1994-present) 7, 5252 (2000); doi: 10.1063/1.1316765 View online: http://dx.doi.org/10.1063/1.1316765 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/7/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonlinear plasma wave in magnetized plasmas Phys. Plasmas 20, 083113 (2013); 10.1063/1.4817949 Large amplitude relativistic plasma waves Phys. Plasmas 17, 052303 (2010); 10.1063/1.3418351 CrossField Plasma Acceleration and Potential Formation Induced by Electromagnetic Waves in a RelativisticMagnetized Plasma AIP Conf. Proc. 669, 816 (2003); 10.1063/1.1594055 Slow-time-scale magnetic fields driven by fast-time-scale waves in an underdense relativistic Vlasov plasma Phys. Plasmas 8, 321 (2001); 10.1063/1.1330200 Nonlinear plasma maser driven by electron beam instability Phys. Plasmas 6, 994 (1999); 10.1063/1.873340

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Magnetic field of a relativistic nonlinear plasma waveArsen G. KhachatryanYerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan 375036, Armenia

~Received 17 March 2000; accepted 15 August 2000!

Longitudinal and transverse behavior of the magnetic field of a relativistic nonlinearthree-dimensional plasma wave is investigated. It is shown that the magnetic field of the wave isdifferent from zero and performs higher frequency oscillations compared to the plasma electronfrequency. An increase in the nonlinearity leads to strengthening of the magnetic field. Theoscillations of the magnetic field in the transverse direction arise, which are caused by the phasefront curving of a nonlinear plasma wave. The numerical results conform well to predictions of theanalytical consideration of a weakly-nonlinear case. ©2000 American Institute of Physics.@S1070-664X~00!00212-3#

The progress in the technology of ultrahigh intensity la-sers and high current relativistic charged bunch sources per-mits the use of laser pulses1 or charged bunches2 for theexcitation of strong plasma waves. The excited plasmawaves can be used, for example, for the acceleration ofcharged particles and focusing of bunches.2 The amplitude ofthe longitudinal electric fieldEmax of relativistic plasmawaves excited in cold plasma is limited by the relativisticwave-breaking field3 Erel5@2(g21)#1/2EWB /b, where g5(12b2)21/2 is a relativistic factor,b5vph/c is a dimen-sionless phase velocity of the wave,EWB5mevpevph/e@EWB(V/cm)'0.96np

1/2(cm23)# is the conventional nonrela-tivistic wave-breaking field,vpe5(4pnpe2/me)

1/2 is theelectron plasma frequency,np is the equilibrium density ofplasma electrons,me ande are the mass and absolute valueof the electron charge.

The linear plasma wave theory is valid whenEmax

!EWB . It is well known that a three-dimensional~3D! linearplasma wave in a cold plasma and in the absence of externalfields is potential.4 In this case the magnetic field of plasmawave is zero. The magnetic field is absent also in the one-dimensional nonlinear case due to the symmetry of the prob-lem. However, the magnetic field of nonlinear three-dimensional plasma wave with the relativistic phase velocity(b'1) has not been studied up to now and our aim is toinvestigate this problem.

We shall study nonlinear plasma waves~wake waves!excited in a cold plasma by relativistic electron bunches orintense laser pulses~drivers! and suppose the azimuthal sym-metry of the problem. In this case, for nonzero componentsof plasma electron momentum and an electromagnetic fieldof the wave, we have the following set of equations:5,6

b]Pz

]z2

]ge

]z2b2Ez50, ~1!

b]Pr

]z2

]ge

]r2b2Er50, ~2!

2]Hu

]z1b

]Er

]z1b rNe50, ~3!

¹'Hu1b]Ez

]z1bzNe1ba50, ~4!

b]Hu

]z2

]Er

]z1

]Ez

]r50, ~5!

Ne512a2“'Er2]Ez

]z. ~6!

As usual, Eqs.~1! and ~2! were derived taking into accountthat the curl of the generalized momentum is zero,b2H2rot P50, or in our case,

b2Hu1]Pz

]r2

]Pr

]z50. ~7!

In Eqs. ~1!–~6! ge5(11Pz21Pr

21a2/2)1/2, bz,r5Pz,r /ge ,andNe5ne /np are, respectively, a relativistic factor, the di-mensionless components of velocity and dimensionless den-sity of plasma electrons,z5(vpe /vph)(Z2vpht), a5nb(z,r )/np , nb is the density of bunch electrons, anda5eE0(z,r )/mecv0 , whereE0 andv0 are the amplitude andfrequency of laser pulse,“'5]/]r 11/r . Also, the follow-ing dimensionless variables have been used: the space vari-ables are normalized onlp/2p5vph/vpe , wherelp is thelinear plasma wavelength, the momenta and velocities—respectively, onmec and the velocity of light and the

PHYSICS OF PLASMAS VOLUME 7, NUMBER 12 DECEMBER 2000

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strengths of electric and magnetic fields—on the nonrelativ-istic wave-breaking fieldEWB . In the general case the ana-lytical consideration of the problem seems impossible. Firstwe consider the weakly-nonlinear case and then present nu-merical results.

In the weakly-nonlinear case we supposeb51 and usethe generalization of the well known expansion7 which wasused to study one-dimensional nonlinear relativistic-plasma-waves:8

u~r ,z!5«u1~r ,C!1«2u2~r ,C!1«3u3~r ,C!1¯,~8!

]C/]z5lp /Lp511«k1~r !1«2k2~r !1¯,

whereu stands for normalized valuesPz,r , Ez,r , or Hu , «5Ez

max!1 is the small parameter,Pz,r , Ez,r , Hu!1, andLp

is the nonlinear plasma wavelength. Substituting expansion~8! into Eqs.~1!–~5! we have

]Pzi

]C2Ezi5S1,i , ~9!

]Pri

]C2Eri 5S2,i , ~10!

2]Hu i

]C1

]Eri

]C1Pri 5S3,i , ~11!

“'Hu i1]Ezi

]C1Pzi5S4,i , ~12!

]Hu i

]C2

]Eri

]C1

]Ezi

]C5S5,i , ~13!

where the subscripti denotes the order of approximation andS125,i are the nonlinear functions ofui 21 ,ui 22 , . . . ,and u1 . In the linear case (i 51) S125,150 and onecan obtain well known solutions~see, e.g., Ref. 9!: Pz1

52R sinC, Pr15(dR/dr)cosC, Ez152R cosC, Er1

52(dR/dr)sinC, andHu150, whereR(r ) depends on the

FIG. 1. The linear plasma wave excited by an electron bunch with a Gaus-sian profile; a050.1, sz52, s r50.5, g510. ~a! The dimensionlessstrength of a longitudinal electric field. 1—Ez(z) at the axis,r 50; 2—r51; 3—r 52. ~b! The strength of the azimuthal magnetic field. 1—r50.2; 2—r 50.75; 3—r 52.

FIG. 2. The nonlinear plasma wave excited by a laser pulse. The pulseparameters area0

253.6, sz52, s r55, g510. ~a! The longitudinal electricfield Ez . r 50,2,4 and 5 in the order of magnitude reduction.~b! The mag-netic field strength. 1—r 52; 2—r 54.

FIG. 3. The radial behavior of the field for the case given in Fig. 2 forz5225. 1—longitudinal electric fieldEz(z5225,r ); 2—magnetic fieldHu(z5225,r ).

5253Phys. Plasmas, Vol. 7, No. 12, December 2000 Magnetic field of a relativistic nonlinear plasma wave

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radial distribution in the exciting source, i.e., ona(r ) ora2(r ). In the second approximation, from the condition ofthe absence of resonance terms~proportional to sinC orcosC), one can find thatk150. Then

S1,25~1/2!]~Pr12 1Pz1

2 !/]C,

S2,25~1/2!]~Pr12 1Pz1

2 !/]r ,

S3,25Pr1~“'Er11]Ez1 /]C!,

S4,25Pz1~“'Er11]Ez1 /]C!,

S5,250,

and for magnetic field strength we obtain the following equa-tion:

@~]/]r !“'21#Hu25]S4,2/]r 2]S3,2/]C. ~14!

The solution of this equation is

Hu25 f 1~r !1 f 2~r !cos~2C!, ~15!

where f 1,2(r ) satisfies two equations:

~D'2r 2221! f 15~1/2!~d/dr !@R~D'21!R#,

~D'2r 2221! f 25~1/2!@~dR/dr !D'R2R~d/dr !D'R#;

hereD'5¹'(d/dr) is the transverse part of the Laplacian.In the ith approximationHu i;cos(iC). The dependencek2(r ) can be obtained in the third approximation. So, theweakly-nonlinear theory predicts that the magnetic field of a3D nonlinear plasma wave is not zero, in opposition to the3D linear case and to the 1D nonlinear one, and oscillates atthe harmonics of the linear plasma frequency; the nonlinearwavelength changes in the radial direction.

We have solved Eqs.~1!–~6! numerically choosing theGaussian profile of the driver both in longitudinal and trans-verse directions:

A~z,r !5A0exp@2~z2z0!2/sz2#exp~2r 2/s r

2!, ~16!

whereA(z,r ) stands fora or a2. Figure 1 shows a three-dimensional linear plasma wave excited by a relativisticelectron bunch. The linear numerical solution obtainedagreed well with the predictions of the linear theory.2 As isseen in Fig. 1~b!, in the linear case the magnetic field excitedin plasma is localized in the range occupied by the bunch andin the wakeEz

max@Humax'0, which corresponds to the linear

theory.In the nonlinear regime the behavior of plasma waves is

qualitatively changed. In Fig. 2 one can see a nonlinearplasma wave that is excited by an intense laser pulse. Themain difference here from the linear case is the change ofshape and length of the wave with the radial coordinater@see Fig. 2~a!#. The magnetic field strength in the nonlinearplasma wave as shown in Fig. 2~b! is different from zero andhas a magnitude comparable to that of other components ofthe field. The magnitude of higher frequency oscillations~ascompared to the plasma frequency! performed by the mag-

netic field alongz grows in proportion to the nonlinearity.Such a behavior of the magnetic field is a purely nonlineareffect. Indeed, in the linear caseHu50, i.e., the contributionof momentum components at the plasma frequency in theexpression~7! is compensated. The nonlinearity of the waveimplies a rise of higher harmonics inPz andPr . Accordingto ~7!, the rise of the magnetic field is due to these harmonicsand this accounts for frequent oscillations seen in Fig. 2~b!,which conforms with the weakly-nonlinear analytical consid-eration presented above. On the other hand, the nonzeromagnetic field means, according to~7!, that the motion ofplasma electrons in the nonlinear wave is turbulent (rotPÞ0). The degree of turbulence~the measure of which isHu)grows in proportion to the nonlinearity.

It is easy to see that due to the dependence of the wave-length onr, the field in the radial direction grows more cha-otic as the distance from the driver increases. In fact, theoscillations of the plasma for differentr are ‘‘started’’ be-hind the driver with nearly equal phases but different wave-lengths@see Fig. 2~a!#. As uzu increases, the change of phasein the transverse direction~for fixed z) becomes more andmore marked. This leads to a curving of the phase front andto ’’oscillations’’ in the transverse direction.6,10 The radialbehavior of the longitudinal electric field and magnetic fieldof the nonlinear plasma wave is presented in Fig. 3. Quali-tatively the radial dependence of the field differs from that ofthe linear case by the change of sign and ‘‘steepening’’ offields alongr and that is connected with the curvature of thewave phase front.

So, we have found that the magnetic field of nonlinearplasma wave is different from zero and performs higher fre-quency oscillations as compared to the plasma frequency.The latter qualitatively differs from the linear case and theone-dimensional nonlinear one. The obtained numerical re-sults agree well with the predictions of weakly-nonlineartheory.

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5254 Phys. Plasmas, Vol. 7, No. 12, December 2000 Arsen G. Khachatryan

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