Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)329–332
000–000
J. Plasma Fusion

Kinetic Theory for Spontaneous Magnetic Fieldin Laser Plasma InteractionZHU Shao-ping, CAI Hong Bo and QIAO Bin
Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, P. R. China(Received: 9 December 2003 / Accepted: 29 July 2004)

Abstract
In our previous paper [Phys. Plasmas 10, 4166 (2003)], we pointed out that the experimental results on the axial
spontaneous magnetic field given by Najmudin et al. [Phys. Rev. Lett. 10, 215004 (2001)] can be explained by ourkinetic model [Phys. Plasmas 8, 329 (2001)]. We also found that the azimuthal magnetic field Bθ given by our kinetic
model is much weaker than the axial spontaneous magnetic field, which is not consistent with particle-in-cell simulation results. In the present paper, we reinvestigate the generation mechanism of the spontaneous magnetic fields from
relativistic Vlasov-Maxwell equations. It is found that the azimuthal spontaneous magnetic field Bθ can be same orderas the axial spontaneous magnetic field if the laser beam self-focus effect is considered. It is also obtained that themagnitude of the spontaneous magnetic field is proportional to the plasma density for the case of low plasma density,
but is independent of the plasma density for the case of high plasma density.
Keywords:
fast ignition, collisionless plasma, laser plasma interaction, Vlasov-Maxwell equations,spontaneous magnetic field

For real intense laser plasma interaction, electrons areheated by laser beam. One part of electrons are accelerated
into relativistic velocity region, the other part of electrons arethermal ones. In the other word, there exist two groups ofelectron, thermal one with temperature Tte and relativistic one
with temperature Tre. So, the electrons satisfy the distributionfunction fα 0 = ξ fα M (Tte) + (1 − ξ)fα M(Tre), where fα M(Tte) andfα M(Tre) are nonrelativistic and relativistic distribution function respectively, ξ is the ratio of the number of thermal electrons to the total number of electrons. According to above
ideal and including the effect of finite laser beam, in Ref.
[20] we generalized our kinetic model proposed in Ref. [19]and found that the experimental results on the axial spontaneous magnetic field given by Najmudin et al. [6] can be
explained by our kinetic model. In fact, there is only the axial
spontaneous magnetic field measured in Ref. [6].
We pointed out in Ref. [20] that the azimuthal magnetic
field Bθ given by our kinetic model is much weaker than theaxial spontaneous magnetic field, which is not consistent
with particle-in-cell simulation results. In the present paper,
we reinvestigate the generation mechanism of the spontaneous magnetic fields from relativistic Vlasov-Maxwellequations. We got an equation for the spontaneous magnetic
field by the same method as Ref. [19], but under different
parameter region. It is found that the azimuthal spontaneous

1. Introduction
Attention to the fast ignition scheme for inertial confinement fusion [1] have generated worldwide interest ininteraction of an intense short-pulse laser with plasma [2,3].
Among the various nonlinear effects which may occur inplasma interacting with intense short-pulse laser, the generation of spontaneous magnetic field is one of the most interesting and significant problems because the fields could haveconsiderable influence on nonlinear plasma dynamics, especially on the collimation of relativistic electron beam. The
extremely high spontaneous magnetic field was observed byparticle simulation in the interaction of ultraintense shortlaser pulse with an overdense plasma target [3,4]. Recently,experiment measurement of the spontaneous magnetic field
are reported by Fuchs et al. [5] and Najmudin et al. [6]. So
far, a number of theoretical models for generation of thespontaneous magnetic field are suggested [7-16]. It should be
noted that most of works on the spontaneous magnetic fields
is on the basis of fluid scheme. A kinetic generation mechanism for the spontaneous magnetic field driven by the beatinteraction between two electromagnetic field was proposed
by He [17] in 1983. One of the present authors [18] got an
equation for the spontaneous magnetic field in collisionlessplasma by the same method as Ref. [17], but under different
approximation. In Ref. [19], we discussed the spontaneous
magnetic field from relativistic Vlasov-Maxwell equations.

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: zsp@mail.iapcm.ac.cn1329

Zhu Shao-ping. et al., Kinetic Theory for Spontaneous Magnetic Field in Laser Plasma Interaction

magnetic field Bθ can be same order as the axial spontaneous
magnetic field if the laser beam self-focus effect is considered. It is also obtained that the magnitude of the spontaneous magnetic field is proportional to the plasma density forthe case of low plasma density, but is independent of theplasma density for the case of high plasma density.

→ σ1fe' (k1) ·2. Spontaneous magnetic fields

∫

→→ →

eα G (x, t) •∂ ⎞→ →→ ⎟ fα (x, pα , t) = 0∂ pα ⎠→

(1)

j=

Σ∫

eαα(2)∂ fα 0→∂ pα→→⎛
ck 2 · pα ⎞
ω
−
⎜ 2
⎟
⎝
p0α ⎠e' (k 2) ·∂→∂ pα→

Gf (k) =→→
αp ×B
p0α

(3)

cp
→ →→
fα (x, pα , t)d3pα
p0α

(4)

G (x, t) = E(x, t) +→→→→Θsl(q) = 1 +the rest mass of particle, c is the light speed. The other notations in Eqs. (1)-(4) are standard.
We split the particle distribution functions up into theirbackground part nα fα 0, slow-time-scale part fαs (time scalefor spontaneous magnetic field) and fast-time-scale partfα f (time scale for laser oscillation):

→

∫α→

σs

c|q|2+Ω2α→ΣΣ→ →cest(q)· pα e'sσs·→(11)

ω p2α mαΩσs∂ fα 0→∂ pα→

d3 pα→α⎛cq · p ⎞p0α ⎜ Ω −⎟⎝p0α ⎠(12)

where ωpα is the plasma frequwncy. The definition of thenotations in Eqs. (6)-(11) can be found in Ref. [19].
For a laser beam with wave number k0 = 2π/λ0 and2intensity I = I0 exp[−2Σ∫i4π ne 0e3E (q) = −∏ eσsσ1σ2 (q, k1, k2)σsΩΘs (q) σ1, σ2(s)c → →tBs (q) =q × Es (q)Ω→d3 pα→α⎛cq · p ⎞p0α ⎜ Ω −⎟⎝p0α ⎠→∫∂ fα 0→∂ pα→ →→

Θst(q) = 1 −S→(10)ω p2α mα→where fα 0 = ξ fα M(Tte) + (1 − ξ) fα M (Tte) is the background distribution function and nα is the particle number density.
Substituting Eq. (5) into Vlasov equation, splitting the electromagnetic fields up into their slow-time-scale and fasttime-scale parts, and using Fourier-transform technique, we→have the slow-time-scale electric fields E σS (q) (the subscript“s” means slow-time-scale field, the superscript “σS “(σS =l,t) represents longitudinal and transverse fields, respectively)→
and the spontaneous magnetic fields B S(q) in the Fourier representation:

d4k1d4k2(2π)4→Σ Σ Ω |q|c(q · pα )e'sσs ·(5)× Efσ1(k1)Efσ2(k2)δ 4(q − k1 − k 2)

(9)

We have the same definition for the slow-time-scale field. InEq. (6), Θ σS s(q) is the dielectric functions→fα = nα fα 0 + fα s + fα f ,→σf

σfce fσ (k)· pα ⎤ →ck · pα ⎤ → σ⎡e' (k) = ⎢1 −e'(k)+⎥k⎥ fϖ p0α ⎦ϖ p0α ⎦⎣→σf→αrespectively, P0α = P α2 + mα2c2, P α is the momentum, mα isσss⎤ →+ (1 ↔ 2)⎥ d3p (8)⎦Σ E (k)e' (k) ,σ = t, lwhere the index α(α = i,e) represents ions and electrons→→

where e is the charge of proton, k = (k ,ω), q = (q→,Ω) , thesymbol ∫ refers to a slow-time-scale motion caused by the(s)difference beat of two fast-time-scale motions. In Eq. (8), (1↔ 2) represents the preceding term with the indices 1 and 2interchanged. The amplitude of the fast-time-scale electricfield Eσf (k) is defined as follows:→→ →→→cq · pα ⎞⎛p0α ⎜Ω −⎟p0α ⎠⎝→→1 ∂B1 ∂E
4π →∇×E=−, ∇×B=+jc ∂tc ∂tc→→→ σ2fConsider the relativistic Vlasov-Maxwell equationscp∂⎛ ∂+ α • → +⎜p0α∂x⎝ ∂t→ce sσs(q)· pα

1 ⎡⎢
2 ⎣

Πασsσ1σ2 (q, k1, k2) ≡2x +y2[r (z)]

2−2(z − z0)
L2], where r(z) and L

are the transverse and logitudinal dimensions of laser beam,
the laser electric field is usually expressed as→→→→→I02(1 + α 2)→EL(x, t) = E0(x, t)(ex + iε ey)exp(ik 0z − iϖ 0t) + c.c.

(6)E0(x, t) =(7)⎡ x2 + y2(z − z 0)2 ⎤exp ⎢−−⎥2L2 ⎦⎣ [r(z)]
2330

(13)

Zhu Shao-ping. et al., Kinetic Theory for Spontaneous Magnetic Field in Laser Plasma Interaction

where ε = 0, ±1 for linearly polarized (LP) and circularly
polarized (CP) laser respectively. It should be noted that→
this E L(x→,t) is not correct because it does not satisfy the→Poisson equation ∇·E L = 0. Here we write the field in theform→→→→→→

EL(x, t) = [E0(x, t)(ex + iε ey) + (Ezr + iEzi)ez]
exp(ik 0z − iϖ 0t) + c.c.

(14)

By Poisson equation, we have approximately

Ezr = −

1 ⎡ ∂E0
1 ∂ 2E0 ⎤+⎢ε⎥k0 ∂ x∂ z ⎦k0 ⎣ ∂ y(15)1 ⎡ ∂E0ε ∂ 2E0 ⎤−⎥⎢
k0 ∂ y∂ z ⎦k0 ⎣ ∂ x

Ezi =(16)Fig. 1 The dependence of Bz(x = 0, y = 0, z = 0) on the ratioof the number of thermal electron to the total electrons ξ, where CP laser with λ = 1.05 µm, I0 = 6.7 × 1018
W/cm2 is used and the other parameters is given asr(z) = 5λ, L = ∞, plasma density ne = 0.028nc and Tte = 5
keV.

From Eqs. (13) and (14), considering the expression for thelaser electric fields, after a lengthy calculation, we find thespontaneous magnetic field for the approximate region→→of vti|q | << Ω << vte |q | (vtα is the thermal velocity of particle.),→ →B(x, t) = ne[ξ + 2(1 − ξ )β3]→→→

(ε E0Ezrex + E0Eziey − ε E 20 ez),(17)

and the spontaneous magnetic field under the condition of→→vte|q | << Ω << c|q |,→ →⎤⎡1(ξ ne − ∇2)B(x, t) = 2ne ⎢ ξ + (1 − ξ )β 3⎥⎦⎣2→zr x.

(18)

2→0 z→zi y∇ × [∇ × (ε E0E e + E0E e − ε E e )]In Eqs. (17) and (18), the subscript “s” has been omitted forsimplicity, andβ 3 = me2 c2

1 ⎛1|pe|2 ⎞∫ p ⎜⎝ 2 − 3p ⎟⎠ f (T )d p ,
20e

20e

3eM

ree(19)and the normalized transform is introduced as→→→→Fig. 2 The dependence of Bz(x = 0, y = 0, z = 0) on the plasma density ne, where CP laser with λ = 1.05 µm, I0 =6.7 × 1018 W/cm2 is used and the other parameters isgiven as r(z) = 5λ, L = ∞, ξ = 0.6 and Tte = 5 keV.

t → t / ϖ0 , x → cx / ϖ0 ,E→E→ m cϖmecϖ 0 →,B→B e 0 .
ee(20)
Ref. [6] and found that the theoretical expectation has a good
agreement with the experiment results given in Ref. [6]. We
here do not pay more attention to the solution of Eq. (17) andfocus our attention to the solution of Eq. (18).
Figure 1 shows the dependence of the magnitude of the
spontaneous magnetic field on the ratio of the number ofthermal electrons to the total number of electrons ξ, where
CP laser with λ = 1.05 µm, I0 = 6.7 × 1018 W/cm2 is used andthe other parameters is given as r(z) = 5λ, L = ∞, plasma density ne = 0.028nc (nc is the critical density) and Tte = 5 keV.
The temperature of relativistic electrons Tre is determined by
Wilks’s scaling law Tre = 511(γ − 1) keV. It can be seen thatthe absolute magnitude of the spontaneous magnetic field

Equation (17) had been obtained in our previous paper. From
Eq. (17), we can find that the magnitude of the spontaneous
magnetic field is proportional to the plasma density. But Eq.
(18) shows that the magnitude of the spontaneous magnetic
field is proportional to the plasma density in the case of low
plasma density and is independent of he plasma density forthe case of relative dense plasma density. It should be notedthat Eqs. (17) and (18) are suitable for both LP and CP lasers.

3. Solutions of spontaneous magnetic field
equations
In Ref. [20], we calculated the spontaneous magnetic
field given by Eq. (17) using the experimental parameters in3331

Zhu Shao-ping. et al., Kinetic Theory for Spontaneous Magnetic Field in Laser Plasma Interaction

Fig. 3 The spatial profile of Bθ(x = 0, y = 0, z = 1.96), where
LP laser with λ = 1.05 µm, I0 = 4.7 × 1018 W/cm2 is used
and the other parameters is given as r(z) = r0(1 + arsin(kr z))/(1 + ar), r0 = 4λ, ar = 0.5, kr = 12/λ, L = 5λ, plasma density ne = 0.2nc, ξ = 0.6 and Tte = 5 keV. The
magnitude of magnetic field is in the unit of MG.

Fig. 4 The spatial profile of Bz(x = 0, y = 0, z = 0.98), where
LP laser with λ = 1.05 µm, I0 = 4.7 × 1018 W/cm2 is used
and the other parameters is given as r(z) = r0(1 + arsin(kr z))/(1 + ar), r0 = 4λ, ar = 0.5, kr = 12/λ, L = 5λ, plasma density ne = 0.2nc, ξ = 0.6 and Tte = 5 keV. Themagnitude of magnetic field is in the unit of MG.

reduces as the number of relativistic electrons increases.
Figure 2 shows the dependence of the magnitude of the spontaneous magnetic field on plasma density ne, where CP laser
with λ = 1.05 µm, I0 = 6.7 × 1018 W/cm2 is used and the otherparameters is given as r(z) = 5λ, L = ∞, ξ = 0.6 and Tte = 5keV. It is clear that the magnitude of the spontaneous magnetic field increases as the plasma density grows up and isindependent of the plasma density as ne is greater than about
0.3nc.
The spatial profile of the spontaneous magnetic field isshown in Fig. 3 and Fig. 4, where LP laser with λ = 0.5 µm,
I0 = 4.7 × 1018 W/cm2 (The parameters for laser beam is sameas that given in Fuchs et al.’s experiment [5]) is used and theother parameters is given as r(z) = r0(1 + ar sin(krz))/(1 + ar),r0 = 4λ, ar = 0.5, kr = 12/λ, L = 5λ, plasma density ne = 0.2nc,
ξ = 0.6 and Tte = 5 keV. The peak magnitude of azimuthal
magnetic field Bθ is about 70 MG, which is consistent withexperimental result given by Fuchs et al. [5]. As shown Fig.
4, the peak magnitude of Bz is about 0.7 MG and much smaller than Bθ. For the case of CP lasers where ε ≠ 0, Bz should
be enhanced and has the same order of magnitude as Bθ.

of very low plasma density, but is independent of the plasma
density for the case of high plasma density.
This work was supported by National Hi-Tech InertialConfinement Fusion Committee of China, National Natural
Science Foundation of China No.10135010, Special Fundsfor Major State Basic Research Projects in China, andScience Foundation of CAEP.

References[1] M. Tabak et al., Phys. Plasmas 1, 1626 (1994).
[2] A.B. Borisov et al., Phys. Rev. Lett. 68, 2309 (1992).
[3] S.C. Wilks et al., Phys. Rev. Lett. 69, 1383 (1992).
[4] A. Pukhov et al., Phys. Rew. Lett. 76, 3975 (1996).
[5] J. Fuchs et al., Phys. Rew. Lett. 80, 1658 (1998).
[6] Z. Najmudin et. al., Phys. Rev. Lett. 87, 215004 (2001).
[7] R.N. Sudan, Phys. Rev. Lett. 70, 3075 (1993).
[8] V.K. Tripathi et al., Phys. Plasmas 1, 990 (1994).
[9] R.J. Mason et al., Phys. Rev. Lett. 80, 524 (1998).
[10] A.D. Steiger et al., Phys. Rev. A 5, 1467 (1972).
[11] T. Lehner, Phy. Scripta 49, 704 (1994).
[12] V.Yu. Bychenkov et al., Laser Part. Beams 14, 55(1996).
[13] A. Pukhov et al., Phys. Rew. Lett. 76, 3975 (1996).
[14] Z.M. Sheng et al., Phys. Rev. E 54, 1833 (1996).
[15] V.I. Berezhiani et al., Phys. Rev. E 55, 995 (1997).
[16] M.G. Haines, Phys. Rev. Lett. 87, 135005 (2001).
[17] X.T. He, Acta Math. Sin. 32, 325 (1983).
[18] S.P. Zhu, Phys. Fluids B 5, 1024 (1993).
[19] S.P. Zhu et al., Phys. Plasmas 8, 321 (2001).
[20] S.P. Zhu et al., Phys. Plasmas 10, 4166 (2003).

4. Summary
The generation mechanism of the spontaneous magnetic
fields is reinvestigated from relativistic Vlasov-Maxwell
equations. It is found that the azimuthal spontaneous magnetic field Bθ can be same order as the axial spontaneous magnetic field if the laser beam self-focus effect is considered. It
is also obtained that the magnitude of the spontaneous magnetic field is proportional to the plasma density for the case4332