J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)333–336
000–000

Spontaneous Magnetic Field and High Harmonics inLaser-Dense Plasma Interaction
ZHENG Chunyang, LIU Zhanjun, ZHU Shao-ping and HE Xiantu
Institute of Applied Physics and Computational Mathematics, Beijing
P.O. Box 8009, Beijing 100088, PRC(Received: 9 December 2003 / Accepted: 7 April 2004)

Abstract
The high harmonics in the reflection spectra from short intense laser pulse interaction with overdense plasmas
were observed by the particle-in-cell (PIC) simulations. The simulation reveal that the quasistatic magnetic field andtransverse density modulation can influence the electron dynamics, which is in responsible of high harmonic generation.
Keywords:
high harmonics, particle-in-cell, quasistatic magnetic field, electron dynamics

1. Introduction

2. PIC simulation results of the highharmonic emission

The generation of high harmonics in the process of solid
target irradiated by short intense laser has been successfully
observed in experimentally and in simulations [1-2]. A
mechanism for high harmonics emission was proposed by
Bulanov et al. [3], based on phase modulation by the laser
light upon reflection from an oscillating plasma-vacuum
interface, viz. oscillating mirror model, agreed with particlein-cell (PIC) results [4].
Recent experiments also report the effect of initial scale
length of plasma to the high harmonic emission [5-6]. The
experiments of harmonics have also been used to estimate themagnitude of self-generated magnetic field.
The generation and features of high harmonics involves
a number of laser and plasma parameters. In the existence ofpre-plasma on the front of solid target, transverse density
modulation occurs and can be enhanced substantially due tolaser filamentation instability [7]. This type of instability was
found to be stronger in the direction perpendicular to the laser
polarization direction. So for p-polarization laser in oursimulations, the transverse density is not evident comparing
to s-polarization laser. The generation of dipole magnetic fieldnear the critical density can be attributed to the joint
interaction of filamentation instability and Weibel instability.
So, the forward relativistic electron streams can acquire
perpendicular velocity component due to the action of theLorentz force. In this paper, we present our PIC simulation
results, which demonstrate that the density modulation andmagnetic field play important role to high harmonic emission.

We first introduce the basic physics model and picture.
Properties of the plasma response are strongly dependent on→ →the polarization of the laser in relativistic regime. Let A (r , t)→→and ϕ be the vector and scalar potential, E = –∂ A /∂ t – ∇ϕ,→with Coulomb gauge ∇·A = 0, and from Ampere’s and→ →Poisson’s equation, the equations for A (r , t) and ϕ are writtenas→→→∇ 2A – 12 ∂ A2 = – 4cπ J ⊥ ,c ∂t2∇ ϕ = – 4πρ ,2→(1)

where J ⊥ is the electric current perpendicular to the z
direction (light propagates along the z direction), ρ the charge
density and c the light velocity.
→ →→→→→
Define A (r , t) = R[e p A0 exp i(kz – ω t)], e p = e y ± α ie x. The
laser pulse is linearly polarized when α = 0. The momentum
equation is as follow
→

→→

→

d (P – e A ) = e[∇ϕ – (∇A ) · v→ ] ,dt(2)

where v is electron velocity, and –e the charge of an electron.
In the perpendicular direction→→P⊥ = eA .

(3)

In the parallel directiond P = e ∂ϕ – e ∂ A 2 .
(4)∂ z 2γ ∂ z
dt z
Relativistic factor for electron is γ, γ = (1 + Pz2 + A2)1/2 =2
1 – v 2z .
1+ASource term for the electromagnetic field is J ⊥ ∝ δn ·©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: zhengcy@iapcm.ac.cn
1333

Zheng C. et al., Spontaneous Magnetic Field and High Harmonics in Laser-Dense Plasma Interaction

1 – v 2z

2→

1 + A P ⊥ , and the high order terms areresponsible for high harmonic generation. If the laser pulse islinearly polarized, A2 contains a term oscillating at twice thelaser frequency. It is well known that when the laser is normal
incidence, only “odd” laser harmonics are generated, and ppolarized “even” laser harmonics can be generated when thelaser is incident under an angle to the surface. This picture
can be explained by “oscillating mirror model” [4]. For
circular polarized laser, A2 is not dependent on fast time
variance, and the ponderomotive force is determined by theshape of pulse. High harmonics cannot be generated throughoscillating mirror mechanism.
Two dimensional (2D) configuration corresponding toeqs. (1)–(4) is considered. A 2D fully relativistic andelectromagnetic PIC code is used to study the nonlinearbehavior. That λ0 = 1 µm is the laser wavelength in vacuum,Fig. 1

Spatial-tme profile of ne at x = 0. Time is in laser cycle
unit and length is in laser wavelength unit.

(a)

(d)

(b)

(e)

(c)

(f)

Fig. 2 Contours of plasma density (a-c) and laser field Ey (d-f) at t = 4, 10, 14 laser cycle.

2334

Zheng C. et al., Spontaneous Magnetic Field and High Harmonics in Laser-Dense Plasma Interaction

ω 0 = 2π c/λ0 and nc is the critical density. The simulation size
is 12λ0 long in the z-direction with 2.4λ0 of vacuum beenpreceded, and the plasma density is ramped from 0.45nc to
9nc over a distance of 1.6λ0, then a plasma plateau with 7λ0
length and 9nc density exists. The plasma density in the xdirection is uniform with 6λ0 wide. One cell has each 30
electrons and ions, and ions are immobile as a neutralized
background. The grid size and time step are chosen as dx =dz = 0.01λ0, dt = 0.005τ (τ = 2π /ω0) in order to ensure thatthe first several order harmonics can be detected.
The laser pulse with intensity I = 1019 W/cm2 linearlypolarized in the y-direction is incident normally withtransversely uniform and longitudinally semi-infinite, whichrises up in about three laser cycles with a Gaussian profile.
From Fig. 1, when the pre-plasma exists, the dominate
motion of density oscillation by the linearly polarized laser
pulse is in longitudinally direction with the frequency 2ω0,density modulation in transverse occurs later with the orderof one laser wave length due to the laser filamentation as can
be clearly seen in Fig. 2.
The profile of current density vector and self-generated
magnetic field is given in Fig. 3 at two time intervals, 7 and11 laser cycle. The Weibel instability of plasma with hot andcold electron streams develope in the same time as the laser
filamentation instability. The Weibel instability cause thebreak up of relativistic electron beams into filaments guided

by quasistatic magnetic fields with transverse scale length on
the order of the local plasma skin depth ~λ0 near the criticaldensity [8-9]. The peak magnitude of the ring-like magnetic
field <By > is about 20 MG. Electron trajectories can beaffected through the “vz cross <By >” force to induce atransverse oscillation. For relativistic electron, the gyro-radiusis about micron order. In Fig. 3, the electron current density
vector is given. The relativistic electrons move in different
direction in transverse at different position because sign ofthe magnetic field <By > is by turns in negative or positive, sochange the sign of “vz cross <By >” force. The electrons in thelongitudinal direction mainly oscillate in “2ω0” frequency, so
the static magnetic field generates harmonic current at “2ω0”along x-direction. For s-polarized light, the “ω0” term ofdensity variation vanishes. High harmonics in “even mode”
can be generated through this transverse oscillating
mechanism. The “vy cross Bx” term (here, vy is the electron
quiver velocity, and Bx is the laser magnetic field) force also
induces a transverse “2ω0” oscillation when the s-polarizedlaser oblique incident on the vacuum-plasma interface
explained by “oscillating mirror model” [4]. Normal incidence
laser is under an angle to the transverse modulation surface(see Fig. 2(b-c)), even harmonic can also be generated throughthis mechanism. It is difficult to distinguish mechanisms ofeven harmonic generation due to density modulation ormagnetic field mechanisms.

(a)

(b)

(c)

(d)

Fig. 3 Contours of magnetic field (a-b) and current density vector (c-d) at t = 7, 11 laser cycle.

3335

Zheng C. et al., Spontaneous Magnetic Field and High Harmonics in Laser-Dense Plasma Interaction(a)(b)(c)(d)Fig. 4 Reflected light spectra (a-b) and electron current spectra (at z = 3λ, x = 0.25λ) (c)(d).

current can strongly influence the transverse electron
dynamics, which, in addition to the transverse density
modulation, is in responsible of high harmonic generation.

The reflected light was Fourier analyzed to obtain theharmonic components, which can be seen in Fig. 4. The fields
Ex, Ey and the electron current density are recorded in (z =3λ0, x = 0.25λ0). The even modes of reflected light in theperpendicular direction of incident laser were given. Even
harmonics were generated by transverse 2ω0 oscillating
current.
For the p-polarized laser pulse, the incident fields were(Ex, By). Although the same laser and plasma parameters wereused, the transverse modulation of density and field is notevident, so weak high harmonics are detected.

References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]

3. Conclusion
The high harmonics in the reflection spectra from shortintense laser pulse interaction with overdense plasmas areobserved by the PIC simulations. The simulation reveal thequasistatic magnetic field generated by the longitudinal

4336

P. Gbbon, Phys. Rev. Lett. 76, 50 (1996).
J.M. Rax et al., Phys. Plasmas, 7, 1026 (2000).
S. Bulanov et al., Phys. Plasmas 1, 745 (1994).
R. Lichters et al., Phys. Plasmas 3, 3425 (1996).
M. Zepf et al., Phys. Rev. E58, R5253 (1998).
A. Ishizawa, IEEE J. Quantum Electron. 37, 384 (2001).
K. Nishihara et al., Proc. SPIE 3886, 90 (1999).
M. Tatarakis et al., Phys. Plasmas 9, 2244 (2002).
A.M. Pukhov and J. Meyer-ter-Vehn, Phys. Rev. Lett. 79,2686 (1997); Y. Sentoku et al., Phys. Rev. E65, 046408(2002).