A novel mechanism for strong magnetic field generation by ultra-intense laser pulses

4 April 1994 PHYSICS LETTERS A

ELSEVIER

Physics Letters A 187 (1994) 67-70

A novel mechanism for strong magnetic field generation by ultra-intense laser pulses Levan N. Tsintsadze 1, Padma K. Shukla lnstitut fiir Theoretische Physik IV, Ruhr- Universit~t Bochum, 44780 Bochum, Germany

Received 15 December 1993; accepted for publication 28 December 1993 Communicated by V.M. Agranovich

Abstract It is shown that ultra-intense short pulse lasers interacting with a preformed plasma can generate strong magnetic fields provided that the spatial gradients of the radiation pressure and the equilibrium electron temperature are nonparallel to each other. This novel mechanism can be a potential candidate for explaining the origin of nonoscillatory intense magnetic fields that have been observed in recent numerical simulations.

In the past, there has been a great deal of interest in the investigation of self-generated magnetic fields in laser-produced plasmas [ 1-8 ]. It is widely thought [1-5 ] that large-scale quasi-stationary toroidal magnetic fields in laser-plasma interactions are generated by the thermoelectric force (or the baroclinic vector Vneo x VTe), which originates in a hot collisionless plasma when the electron temperature gradient V Te is nonparallel to the electron density gradient Vne0. A quasi-stationary magnetic field is also generated in a weakly ionized collisional plasma [6-8 ] due to the noncoUinearity of the gradient of the energy of the high-frequency field and the electron density gradient (or the temperature gradient). Self-generated magnetic fields can have profound influence on transport characteristics, parametric instabilities, as well as absorption of laser light in inertial confinement fusion (ICF) schemes. Stamper [9 ] has presented an extensive review of the magnetic field generation and associated phenomena in laser-plasma interactions.

i Permanent address: Plasma Physics Department, Institute of Physics, Georgian Academy of Sciences, 380077 Tbilisi, Georgia.

The development [ 10 ] of ultra-intense short pulse lasers allows exploration of fundamentally new parameter regimes for nonlinear laser-plasma interaction. In fact, a number of experiments [ 11-13] have been carried out in which plasmas are irradiated by laser beams with intensities up to I22 ~ 1018 W/2m2/cm 2. Here I is the laser intensity and 2~ is the laser wavelength in microns. At such intensities, the electron quiver velocity (Vos = 25.6x/72v cm/s) rapidly approaches the speed of light c, and a host of new phenomena have been predicted [14]. Recent computer simulations by Wilkes et al. [ 15] of the interaction of ultra-intense short pulse lasers with a preformed plasma have exhibited many interesting nonlinear phenomena including hole boring, strong magnetic field generation (,,~ 250 MG), formation of bubbles near the critical surface, as well as a timedependent absorption of the laser light and the generation of very energetic electrons and ions directed into the overdense plasma. Sudan [ 16 ] has argued that the extremely high magnetic fields observed in the numerical simulations of Wilks et al. [ 15 ] cannot be explained on the basis of well-known mechanisms. Rather, he suggests an alter-

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L.N. Tsintsadze, P.K. Shukla / Physics Letters A 187 (1994) 67- 70

native mechanism for the magnetic field generation in terms of dc currents driven by the spatial gradients and temporal variations of the ponderomotive force of the laser light on the plasma electrons. The dc magnetic field is found to be of the same order as the laser oscillating magnetic field. In this Letter, we present a very attractive mechanism for the generation of high magnetic fields by nonuniform intense laser beams in a nonuniform collisionless plasma having equilibrium electron temperature and density gradients. The present mechanism is connected with the relativistic properties of the collisionless plasma; however, its physical scenario is somewhat similar to the previous idea [ 7, 8 ] which focused on a quasi-stationary magnetic field generation in nonrelativistic coUisional plasmas in the presence of small amplitude electromagnetic waves. Clearly, our theory of magnetic field generation holds in the limits when inter-particle collisions are negligible and laser beams are of an arbitrary large amplitude. Let us consider the propagation of powerful circularly polarized laser beams propagating along the z axis in a nonuniform plasma. The vector potential of the laser light is represented as

mentum equation can be written as dtPe = - e E - -ev e x B c

i v (neTe), ne

(2)

= Ot + r e " V , Pe = me(Te)~'eveis t h e relativistic momentum, me(Te) = moG(~) =moK3(~)/K2(~), ~ = moc2/Te, 7e = (1 - v2/¢ 2) -1/2 is the relativistic gamma factor, m0 is the rest mass of the electrons, Te is the electron temperature, and Kn is the Macdonald function of nth order. The electric and magnetic fields are denoted by E and B, respectively. We note that the relativistic momentum strongly depends on the electron energy (or the temperature Te). We separate all the field variables as sums of the slowly and fast varying parts. Thus, Ve = (re) + vef, E = (E) + E f , a n d B = (B) + Bf, where angular brackets denote averaging over the fast time scale z 1/o90 and the subscript f denotes the corresponding quantities associated with the laser light. The slowly varying component of (2) is written as w h e r e dt

e c(Ve) × (n)

Dte(Pe) = - e ( E ) -

+ e - lv(neTe),

(3)

ne

A. = l (j +i~)A±(r,t)exp(ikoz-iogt)+c.c.,

(1) = 0t + (Ve) • V and the relativistic electron ponderomotive force is given by F = -cEV[~me(Te)] + (c2/~)Vme(Te). Here, we have denoted ? = [1 + e 2 A 2 ( r ) / m 2 ( T ) c 4 ] 1/2. We note that the ponderomotive force [ 14 ] orginates from the averaging of the convective and Lorentz forces over the fast oscillations. The averaged electric and magnetic fields are related by Faraday's law,

w h e r e Dte

where J , # are the orthogonal unit vectors transverse to the z axis, k0 is the laser wave number, o90 is the laser frequency, and A± is the slowly varying amplitude of the laser vector potential. The plasma motion in the presence of electromagnetic fields is described by the hydrodynamic equations that have been derived in Ref. [ 17 ]. We suppose that the parameters of the plasma are such that the magnetic Raynold number Rm = 4rtaa2/c2to > > 1, where a = nee2/mevei is the electrical conductivity of the plasma, a is the characteristic scale length of the inhomogeneity or the magnetic field extent, to is a typical rise (or expansion) time of the self-induced magnetic fields, ne is the electron number density, e is the electron charge, me is the electron mass, vei is the electron-ion momentum collision frequency, and c is the speed of light. Accordingly, the frictional force between the electrons and ions can be neglected. The electron viscous force can also be neglected by assuming that the collisional mean free path l is much smaller than a. In such a situation, the electron mo-

V x (E) = - l o t ( B ) . ¢

(4)

Eqs. (3) and (4) are closed with the help of Amprre's law, V )< (B) -----

41ten ¢

((/)i) - (/)e)),

(5)

where n -- ne = ni is the quasi-neutral plasma number density and the mean velocity v i of the nonrelativistic ion fluid is determined by miDti(vi) = e(E) + e(/.si) x ( n ) c

l v ( n ~ i ), tti

(6)

L.N. Tsintsadze, P.K. Shukla / PhysicsLetters A 187(1994) 67-70 where Dti = (gt + (vi) • ~7, m i (Ti) is the i o n mass

(temperature), and the ponderomotive force acting on the ion fluid is ignored because is smaller than the electron ponderomotive force F. We now consider the generation of nonstationary as well as stationary magnetic fields in plasmas. For the nonstationary case, we take the curl of (3), assume that the inertial forces are small in comparison with the ponderomotive force, and use (4) and (5) to obtain

69

Taking some typical parameters, viz. n ,~ 1 0 2 2 - 1 0 23 cm -a, a ,,~ 10 -2 cm, T¢ ~ 1 keY, eA.L/moc 2 ,,~ ½ and to ' ~ 1 0 - 9 S-1 , we find that the magnetic field strength (IBI) ~ 5T~eA2to/2a 2mgc3 could be of the order of several megagauss. On the other hand, at ultra-relativistic temperatures (viz. Te > > m0c2), we have me(Te) = 4Te/moc 2 > > m0. Accordingly, for small amplitude laser beams, we obtain from (9)


x

C3

C VTe x Vn¢ + " ~ V m e ( T e ) x ~ y , +en¢

(7)

where the ions are assumed to be immobile. Let us analyze Eq. (7) in detail. Considering the electron energy equation (e.g., Eq. (1.5) in Ref. [ 17 ] ), we find that the energy exchange between the electrons and ions is neglected in the limit l/a > > (mo/mi) 1/2, whereas the electron heat flow is disregarded for to < < moaZvei/T¢. Then, the electron energy equation leads to the adiabatic law

neL(~) = const,

(8)

where L(~) = [~/K2(~)]exp[-~G(~)]. At nonrelativistic temperatures (viz. Te < < m0c2), we have G(~) = 1 + 5Te/2moc 2 and (8) yiel0s n~T~"3/2 = const, while at ultra-relativistic temperatures (viz. Te > > m0c2) we have G(~) = 4T¢/moc 2 and the adiabatic law is similar to the photon gas neTe 3 = const. It turns out that for the adiabatic compression, (7) reduces to C3

Or(B) = b-~Vme(Te) × Vy,

~eVTex

v e 2 A 2 (r) m2c4

Furthermore, for the general case, the magnetic field strength is found to be c 2 [m¢(T¢)-molc2 1 (IBI) ~ - T e a to -. e Te y

(12)

Here, super-strong magnetic fields may be produced by ultra-intense laser beams in nonuniform plasmas. Next, we focus our attention on the generation of stationary magnetic fields. Here, we include the ion dynamics and neglect the electron and ion inertial forces in comparison with the electron ponderomotive force. Adding (3) and (6) and assuming that the ponderomotive force dominates over the thermal force, we obtain from the radial component (Vez) -- (Viz)

= -co~ -

[(G(~)2:] -

G(~)

)

,

(13)

where O9cc = e(Bo)/moc is the electron cyclotron frequency and Be is the azimuthal component of the induced magnetic field. F r o m Amp6re's law, we have

(9)

where we have neglected the nonlinear Lorentz force term by assuming that 47tnea/toc(IBI) > > 1. Eq. (9) governs the evolution of self-generated magnetic fields in the presence of arbitrarily large amplitude nonuniform laser beams in plasmas with equilibrium electron temperature gradient. For nonrelativistic temperatures and finite but small amplitude laser beams, (9) takes the form Or(B) ---- 5 C

(11)

(B) x

.

(10)

(V¢z)- (Viz)-~

0 (rCOce), r Or

~2

(14)

where 2¢ = c/cop¢ is the collisionless electron skin depth and Ogpc = (4rtne2/mo) 1/2 is the electron plasma frequency. Combining ( 13 ) and (14) we obtain

O (rtOce) = _o92r(ff_~[G(~)(1 + p2)1/2] Ogce~--~

_

1

o G(~))

(1 + p2)1/2 a r

(15)

70

L.N. Tsintsadze, P.K. Shukla / Physics Letters A 187 (1994.) 67-70

where p = e A ± ( r ) / m o G ( ~ ) c 2. Eq. (15) can be numericaUy investigated for a given laser b e a m profile. F o r example, for a Ganssian beam, viz. A 2 (r) = A E o e x p ( - r E / r 2 ) , where r0 is the effective b e a m radius, we obtain [ 18 ] for a cold plasma cote _ x ~ [ _ 2 q ( ~ ) COpe r

-I-2q(0)-~2q(?)-f(~)]U2, (16)

where ~ = r/ro, q(~) = [1 + p 2 ( ~ ) ] l / 2 , f ( ~ ) = l n l [ q ( ~ ) - 1] [ q ( 0 ) + 1 ] / [ q ( ~ ) + 1 ] [ q ( 0 ) - 1]l, and p(O) = e A x o / m o C 2 = vos/c. The electron gyrofrequency CO¢¢for p ( 0 ) = 3 a r o u n d r = r0 is o f the order o f COpe. This result indicates that super-strong magnetic fields, as observed in computer simulations [ 15], might be generated by ultra-strong short laser pulses, which have inhomogeneous radial profiles. To summarize, we have presented a novel mechanism for spontaneous generation o f magnetic fields by nonuniform intense laser beams in inhomogeneous collisionless plasmas. It is shown that super-strong magnetic fields can be produced on account o f nonparallel gradients o f the electron temperature and the laser b e a m profiles in hot plasmas. Spontaneously generated strong magnetic fields could significantly affect the propagation o f intense short laser pulses through the plasma, and they can also alter the transport properties as well as could be used as a basis for the foundation o f new diagnostic methods for laser produced plasmas. This work was supported by the C o m m i s s i o n o f the European Economic C o m m u n i t y (Brussels) through the H u m a n Capital and Mobility program under con-

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