J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)593–596
000–000

Fluid Simulation on Subcycle Wave Generation
in Relativistic Laser-Plasma Interactions
HOJO Hitoshi, AKIMOTO Kazuhiro1 and WATANABE Tsuguhiro2
1

Plasma Research Center, University of Tsukuba, Tsukuba 305-8577, Japan
School of Science and Engineering, Teikyo University, Utsunomiya 320-8551, Japan
2
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 12 December 2003 / Accepted: 20 April 2004)

Abstract
The generation of subcycle electromagnetic waves with linear polarization in relativistic laser-plasma interactions
is demonstrated in one-dimensional fuild simulations for a cold plasma. It is shown that the width (FWHM) of thegenerated ultrashort pulse is less than one carrier wavelength, that is, the generated ultrashort pulse is a subcycle
electromagnetic wave with finite propagation speed. The associated wake field following just behind the generated
subcycle electromagnetic wave is demonstrated. The stationary solutions of subcycle waves with circular polarization
are also obtained.
Keywords:
computer simulation, subcycle wave, ultraintense field, laser-plasma interaction, relativistic plasma

1. Introduction

2. Basic equations for simulations

The formation of solitary electromagneic waves inultraintense relativistc laser-plasma interactions has been
recently receiving growing attension, and many theoretical [15] and numerical [6-8] works have been reported. The
simulation studies are based on PIC particle codes. In the onedimensional stationary propagation, Eriskepov et al. [1] havefound an exact solitary-wave solution of relativistic
electromagnetic wave, which is standing called as “postsoliton [8]”. Hojo and Akimoto [2] have also shown
numerically that the same equation describing the above
solution has a soliary wave solution moving with finitevelocity. More recently, solitary-wave solutions of relativistic
electromagnetic wave have been reported for magnetizedplasmas [3], electron-movable ion plasmas [4], and electronpositon plasmas [5].
The aim of this paper is to show the generation of
nonlinear suscycle electromagnetic waves in one-dimensional
fluid simulation on relativistic laser-plasma interactions. The
use of the fluid code is that more acurate computations withless noises would be expected in wave propagation analysis
as compared with particle simulations. In the following
section, we descrive the basic equations used for simulations,and breifly describe stationary subcycle solutions with circularpolarization in section 3, and show the simulation results forthe cubcycle wave generation in section 4.

The starting point is Maxwell equations and fluid
equations of the electron and ion for a cold plasma in one
dimension given by∂
B = −∇ × E ,
∂t(1)

p∂
E = ∇ × B + n − NV ,∂tγ(2)⎛ ∂ pz ∂ ⎞p⎟ p = −E − × B ,⎜ +γ⎝ ∂ t γ ∂z ⎠(3)

∂Ez ,∂z(4)∂∂N + ( NVz ) = 0 ,∂t∂z(5)m∂
V =E,∂tM(6)n = N−where γ = (1 + p2)1/2 and the mass ratio m/M = 1/1836 isassumed. Here, the following normalization was used: ωp0 t→ t, ωp0z /c → z, eE/ωp0 mc → E, eB/ωp0 m → B, P/mc → P,
V/c → V, n/n0 → n, N/n0 → N, where ωp0 = (e2n0 /ε0 m)1/2 and©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: hojo@prc.tsukuba.ac.jp1593

Hojo H. et al., Fluid Simulation on Subcycle Wave Generation in Relativistic Laser-Plasma Interactions

0.1 and ω = 0.97799 (a), 0.79 (b). The solitary-wave
amplitude increases and the pulse-width decreases for thelarger nonlinear frequency shift (i.e., for the smaller ω). We
see that the profiles of φ and n become acute at ξ ≈ 0 for alarge-amplitude wave. In Fig. 2, we show the maximum
amplitudes of a and φ, and the minimum value of n as afunction of ω.

n0 is a constant density. Here, the equation of motion for theion is linearized, although that for the electron is relativisitc.
This approximation for the ion is valid for the present
simulation parameters. In the simulation, the electron density
n is obtained from the Poisson equation, on the other hand,
the ion density N is obtained from the continuity equation.
The numerical scheme is that the spatial derivative is replacedby the central difference, the resultant ordinary differential
equations are solved by the Runge-Kutta method of the thirdorder. We note that a dissipative term – ν ( ∂ 2/ ∂ z 2) p is
introduced to suppress short-wavelength oscillations due tonumerical errors, and that the qualitative feature on thesubcycle wave generation does not depend on the value of ν,
however, the onset time of the generation and the amplitude
of excited wakefield following after the excited subcycle wave
strongly depend on ν.

3. Stationary subcycle wave
In this section, we consider stationary solutions ofsubcycle electromagnetic waves with circular polarization inthe case of immobile ions. If we introduce vector and scalarpotentials, a and φ, defined respectively by∂
E = −∇φ − a ,∂tB=∇× aFig. 1 (a) Subcycle wave for V = 0.1 and ω = 0.97799.

(7)with ∇ · a = 0, and considering a circularly polarized modewith ax + iay = a(ξ ) exp{iω [(1 – V 2) t – Vξ]}, we can obtain[1]d2dξd22dξ2

γ =a=

φ=1a21−V γ − bbγ −b22−ω a,(8)(9),2 221+ a +V b =1+φ +V b,(10)Fig. 1 (b) Subcycle wave for V = 0.1 and ω = 0.79.

where ξ = z – Vt and V is the propagationg speed normalized
by c. We note that ω being normalized by ωpe comes up to 1
when a → 0, and the nonlinear frequency shift is proportional
to 1 – ω.
For V = 0, Eqs. (8)-(10) are reduced to2⎡⎤
1 ⎛ da ⎞ ⎥2 2⎢a=aγ−ωγ+⎟22 ⎜⎢γ ⎝ dξ ⎠ ⎥dξ⎣⎦d2(11)with γ = (1 + a2)1/2 [2]. Equation (11) does not reduce to thestationary form of the nonlinear Schrödinger equation withcubic nonlinearity even to the third order in a due to theexistence of (da/dξ )2. However, it can be shown that its exact
solitary-wave solution [1] is reduced to the waveform of sech(ξ ) in the lowest order of a.
We now solve Eqs. (8)-(10) numerically with theboundary condition in which a and φ vanish to zero at ⏐ξ⏐→ ∞. In Fig.1, we show the solitary-wave solutions for V =Fig. 22594

The wave amplitudes amax , φmax and nmin vs. ω for V =0.1.

Hojo H. et al., Fluid Simulation on Subcycle Wave Generation in Relativistic Laser-Plasma Interactions

4. Simulation results
In this section, we show the results from fluid simulaions
based on the equations discussed in section 2. The initial laser
pulse is excited at the left-hand side (z = 0) in the simulation
area and the boundary condition at the right-hand side is theout-going wave condition. The initial density profile at t = 0
is given byn(z) = N(z) = {1 + tanh[(z – 300)/10]}/2 .

(12)That is, a laser pulse is generated in the vacuum region andis launched into a plasma region. The initial laser pulse withlinear polarization is assumed to have the waveform ofGaussian pulse given by

E x (t ) = E0 exp ⎡− (t − t0 ) 2 / D 2 ⎤ sin(ω t ) .
⎣⎦

(13)

We show the simulation result on subcycle wave
generation in Fig. 3, where E0 = 0.8, ω = 2, D = 10, t0 = 20,
ν = 2.5 × 10-4, ∆z = 0.05, and ∆t = 0.0025 (in the normalizedvalues). We see that, for the launched laser pulse into aplasma, the steepening of the wave form takes place, and thenthat the laser pulse is separeted into two parts of a newlygenerated ultrashort pulse with large amplitude and the smallamplitude longer pulse in a plasma. The FWHM of the newlygenerated ultrashort pulse in Ex is about 0.8λ at t = 1400, λbeing the carrier wavenumber of a laser pulse. Therefore, thegenerated pulse is a subcycle electromagnetic wave. On theother hand, the small-amplitude longer pulse is seemed to beexpanding with time due to the dispersion effect in a plasma.
The speed of the generated subcycle wave is V/c = 0.57 at t =1200, and is gradually decreasing with time, but tends to besaturated. The gradual slowdown of the speed is supposed tobe due to the slow decorrelartion with the small-amplitude
longer pulse. The feature of this generated subcycle wave isdifferent from that obtained in the particle simulations [1,68], where the generated subcyle wave is almost standing.
We show the wake field Ez associated with Ex at t = 1200
in Fig. 4. We see that the wake field follows just behind thenewly-generated subcycle wave. The amplitude of the wakefield is about Ez ≈ 0.1. The wake field is a electrostatic plasma
wave and it is often used for plasma-particle acceleration suchas the laser wake field acceleration. The close-up figure of Exand Ez at t = 1200 is shown in Fig. 5, where the value of Ez
is maltiplied by ten times. We suppose that the complex
behavior of the wakefield depends on the phase relation
between the wakefield (Ez) and the large-amplitude part of anincident laser pulse (Ex), which is developing into a subcycle
pulse. However, this problem is considered to be a future
research subject.
Finally, as the intense subcycle electromagnetic wave isshown to accelerate charged particles very efficiently [9], webelieve in that the present study on the generation of subcycle
electromagnetic waves is useful in charged particle
acceleration researches.

Fig. 3 The temporal behavior of lase pulse Ex .

Fig. 4 The profile of wake field Ez at t = 1200.

Fig. 5 The close-up of Ex and Ez at t = 1200.

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Hojo H. et al., Fluid Simulation on Subcycle Wave Generation in Relativistic Laser-Plasma Interactions

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