J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)275–278
000–000

Stimulated Raman Scattering, Cascade-into-Condensate andAcceleration of Large Relativistic Electromagnetic Solitons inIntense Laser Interaction with an Underdense Plasma
LI Baiwen1,2,3, ISHIGURO Seiji1,2, S̆KORIĆ Milos̆ M.4 and TAKAMARU Hisanori5
1

The Graduate University for Advanced Studies, Toki 509-5292, Japan
2
National Institute for Fusion Science, Toki 509-5292, Japan
3
Institute of Applied Physics and Computational Mathematics, Beijing 100088, P.R.China
4Vinc̆a Institute of Nuclear Sciences, POB 522, Belgrade 11001, Serbia and Montenegro
5Chubu University, 1200 Matsumoto-cho, Aichi 487-8501, Japan(Received: 9 December 2003 / Accepted: 16 March 2004)

Abstract
Stimulated Raman scattering and Cascade-into-Condensate mechanism induced by intense laser in underdense
uniform plasmas are studied by particle simulations. Standing-, backward- and forward-accelerated relativistic
electromagnetic (EM) solitons are observed after multiple interactions. Apart from the inhomogeneity of plasma density
the soliton acceleration depends upon both the laser intensity and the plasma length. The backward and forward
solitons are accelerated towards the plasma-vacuum interface radiating energy in the form of low-frequency EM bursts.
The soliton frequency is about one-half of the unperturbed electron plasma frequency. The transverse electric, magnetic
and electrostatic fields have half-, one- and one-cycle structure in space, respectively.
Keywords:
stimulated raman scattering, raman cascade, electromagnetic solitons, particle simulations

1. Introduction

are generated behind the front of the laser pulse and are madeof nonlinear, spatially localized low-frequency EM fields. A
fairly large part of the laser energy can be transformed intosolitons. The generation and structure of EM solitons wereanalytically investigated and observed by particle simulation
in intense laser interaction with underdense and overdense
plasmas [4-6]. The solitons found in 1D and 2D particle
simulations consist of slowly or non-propagating electron
density cavities with EM field trapped inside and oscillate
coherently with a frequency below the unperturbed plasma
frequency and with the spatial structure corresponding to halfa cycle (subcycle soliton) [5]. In a homogeneous plasma,
solitons were found to exist for a long time, close to theregions where they were generated, and eventually decay bytransforming the soliton energy into fast particles. In
inhomogeneous plasmas, solitons move with the acceleration
proportional to the density gradient toward the low density
side. When a soliton reaches the plasma-vacuum interface, it
radiates away its energy in the form of a short burst of lowfrequency EM radiation [5,6].
In this paper, we present particle simulation results on
SRS, cascade-into-condensate and large relativistic EM
solitons created by linearly polarized intense laser inunderdense homogeneous plasmas. We found that, in addition

Relativistic laser-plasma interactions is a source ofvarious electronic instabilities [1,2]. When a relativistic laser
light propagates in underdense plasma, electron parametricinstabilities, such as, forward and backward stimulated Raman
scattering (F-SRS/B-SRS), and relativistic modulational
instability (RMI), can develop [3]. They do not appear
isolated but are often interconnected. A nonlinear interplay
between B-SRS and F-SRS produces a strong spatial
modulation of the laser pulse and the cascade in its frequency
spectrum. The continuing instability growth through SRS
cascade downshifts the pulse frequency from the fundamental
to the bottom of the light spectra. The spectra of F-SRS
clearly reveal the Raman cascade containing the first antiStokes, first- , second- and higher-order Stokes modes. It getssaturated by the photon condensation mechanism, related tostrong depletion and possible break-up of the laser beam. In
the final stage of the cascade-into-condensate mechanism, thedepleted downshifted laser pulse gradually transforms into a
train of ultra-short relativistic electromagnetic (EM) solitons[3].
Relativistic solitons are EM structure self-trapped by
locally modified plasma refractive index through therelativistic electron mass increase and the electron density
depletion by the ponderomotive force of an intense laser. They

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: lbw@toki.theory.nifs.ac.jp1275

Li B. et al., Stimulated Raman Scattering, Cascade-into-Condensate and Acceleration of Large Relativistic Electromagnetic Solitons

to the plasma inhomogeneity, soliton acceleration is controlled
by, both the laser intensity and the plasma length. In the rest,
we just focus on the laser intensity effect. The frequency of
the EM wave trapped inside the soliton is about one half of
the unperturbed plasma frequency. The soliton electric field
has the half-cycle structure in space, while the magnetic fieldand corresponding electrostatic (ES) field have the one-cyclestructure.

t = 0.00 to 227.40t = 0.00 to 227.40
1.0

0.2

0.5

0.1

2. Simulation model

0.5

1.0t = 227.40 to 341.10
0.4
0.2
0.0
0.0

0.5

1.0

0.0
0.0

1.5

1.5|(Ey+cBz)(ω)|2|(Ey-cBz)(ω)|2

0.0
0.0

0.03
0.02
0.01
0.00
0.0

1.5

5
0
0.0

0.5

1.0

1.5

t = 341.10 to 795.90
40
20

0.5

1.0ω/ωο
Fig. 11.010t = 341.10 to 795.90

One-dimension and three-velocity (1D3V) fully
relativistic EM particle-in-cell (PIC) code is used. Simulation
system length is 2700c/ω0 in x direction, in this direction
particles are allowed to move, where c and ω0 are the vacuum
speed of light and the laser frequency, respectively. Plasma
length of 900c/ω0 begins at x = 0 and ends at 900c/ω0; in thefront and rear side of the plasma there are two 900c/ω0 longvacuum regions. Ions are kept immobile as a neutralizing
background. The plasma density and its initial electron
temperature are n = 0.032ncr and Te = 350 eV, where ncr =ω 02 me /4πe2 is the critical density. The number of cells is 10
per 1c/ω0 and 80 particles in each cell. Laser is linearlypolarized with the electric field E0 along the y-direction andthe normalized amplitude β = eE0/me ω0c, where e and me are
the electron mass and charge, respectively. The electrons
which enter vacuum region build a potential barrier thatprevents more electrons of leaving the plasma. For theseelectrons as well as for outgoing EM waves, two additional
damping regions are used. The time, electric field andmagnetic field are normalized to the laser period 2π/ω0,
me ω0c/e and me ω0 /e, respectively; the time is taken zero, t =0, when the laser arrives at the vacuum-plasma boundary.

0.5

t = 227.40 to 341.10

1.500.0

0.5

1.0

1.5

ω/ωο

Spectra (a.u.) of reflected (left) and transmitted (right)
EM waves for β = 0.3.

As shown in Fig. 1, the continuing instability growth throughstimulated Raman cascade downshifts the frequency spectrum
from the fundamental to close to the perturbed electron
plasma frequency. The spectrum of the F-SRS clearly reveals
the linear Raman cascade containing not only the first antiStokes mode, but also the first-, second- and higher-order
Stokes modes. Parametric down-cascade of the laser pulse
into the higher-order B-SRS and F-SRS harmonics saturates
into the Photon Condensate at the bottom of the light
spectrum, related to strong depletion and possible break-up
of the laser beam and relativistic electron heating [4]. The
process is eventually halted bellow the plasma frequency,
which is the cutoff for a laser propagation.

3. Stimulated Raman cascade and thephoton condensation process

4. Acceleration of large amplituderelativistic electromagnetic solitons

When intense laser propagates in underdense plasma, thefirst stage is dominated by B-SRS process; matchingconditions ω0 = ωs + ωek and k0 = ks + kp are well-satisfied,
where ω0, ωs and ωek are the frequency of incident EM wave,
scattered EM wave and electron plasma wave (EPW), while,k0, ks and kp are the corresponding wave number, respectively.
For example, for laser amplitude β = 0.3, the dominant ES
wave is excited EPW with frequency ωpe = 0.18ω0, and thecorresponding backscattered Stokes wave with frequency ωs= 0.82ω0 (see Fig. 1). As time goes on, following the B-SRS,complex nonlinear processes involving F-SRS and RMI
develop, such as, spectral cascade and broadening, which
effectively transfer the laser pulse energy to higher orderscattering modes. A nonlinear interplay between B-SRS andF-SRS produces a strong spatial modulation of the laser pulse
and the down cascade in its frequency spectrum. Along thepropagation beam, there is typically the Raman cascade in
the light spectrum from fundamental laser frequency toward
lower frequencies. The first Stokes line is significant, further
along the propagation, the Stokes mode becomes a new pump,which decays via a secondary Raman scattering, and so on.

For the laser amplitude, β = 0.3 case, when SRS throughcascade-into-condensate is saturated, as shown in Fig. 2; aspatially localized, non- propagating electron density cavity
is created. Inside, the EM field is trapped and oscillates
coherently; that is, a large amplitude localized standing
relativistic EM soliton comes into form. In Fig. 3 (top), thefrequency spectra of EM and ES waves trapped inside thesoliton region are plotted. In addition to the laser fundamental
and the excited perturbed EPW, EM component with thefrequency close to 0.13ω0, and corresponding ES component
with the frequency near to 0.87ω0 are observed, one can see
this phenomenon roughly as a 3-wave resonant coupling. The
size of the soliton is about 5λ0 (λ0 is laser wavelength invacuum), close to the electron plasma wavelength λ p [5]. Fig.
3 shows the structure of the soliton, ES field Ex (averagedover λ p) is the one-cycle structure, and the correspondingtransverse electric field Ey (averaged over λ0) is the half-cycle
and the magnetic field Bz is the one-cycle structure. Thespatial EM structure is oscillatory in time, but the ES structure
is not. The explanation comes directly from Maxwell’sequations. The Faraday law gives Bz ~ ∂Ey /∂x; indeed, the x2276

Li B. et al., Stimulated Raman Scattering, Cascade-into-Condensate and Acceleration of Large Relativistic Electromagnetic Solitons

Poisson equation gives the corresponding non-oscillatory ES
field, like Ex in Fig. 3.
However, by increasing the laser amplitude to β = 0.5,
as shown in Fig. 4, the soliton dynamics appears different tothat in the standing soliton β = 0.3 case. The observed, large
localized EM soliton is backward-accelerated towards theplasma-vacuum interface, where it radiates its energy awayin the form of low-frequency EM burst, due to a non-adiabaticinteraction with the plasma boundary. As a result, one
observes a very high transient reflectivity. As we further
increase the amplitude to β = 0.7, as shown in Fig. 4, large
localized EM soliton can be still detected. However, theinteresting feature is that the soliton is now forwardaccelerated. Again, the soliton at the plasma-vacuum (rear)interface irradiates its energy away in the form of lowfrequency EM waves. During the solitary wave radiation very
high transient transmittivity can be detected.
The frequency of backward and forward accelerated
solitons was measured close to 0.5ωpe; the size and thestructure were similar to the standing soliton β = 0.3 case.
Furthermore, the EM energy density profile shows that thefront value is larger than at the rear side of the soliton; still,
for large amplitudes, the difference is greater than that in theβ = 0.3 case. The difference in EM energy appears to bemainly transferred to the soliton, also partly to other processes(e.g. heating). The steep EM energy gradient over the shorttransition layer of the soliton length (c/ωp -classical skindepth) corresponds to the ponderomotive force acting on
plasma electrons. The resulting acceleration will push theelectron cavity with the soliton in the forward direction.
Basically, the longitudinal electron motion is determined by
the balance of the ES field and the ponderomotive force terms[6] . The large flow of relativistic electrons in forward
direction due to SRS, which drives ES fields, getscompensated by a cold (bulk) return current which moves
plasma electrons backwards. Possibly, this is why at moderateintensity solitons are found to move backwards. At larger
amplitudes, the ponderomotive term prevails, the accelerationis reversed and solitons are pushed forwards.

nx/n0

derivative of the Gaussian soliton profile Ey gives Bz in Fig.
3. Similarly, from the Poisson equation, integration over -x of
the Gaussian density cavity (Fig. 2) leads to the ES field Exin Fig. 3. Moreover, PIC data and analytics e.g. eqs. (4)–(7)of [6], show that zero-harmonic term dominates the electron
density perturbation (ponderomotive term). Therefore, the0.04

0.04

0.04

0.03

0.03

0.03

0.02

0.02

0.01

t = 2418.66
0300 600 900

0.01

0.02

t = 2528.59
0300 600 9000.01t = 2608.53
0300 600 900

Ey2+Bz2x(c/ωo)
0.16

0.16

0.16

0.12

0.12

0.12

0.08

0.08

0.04
0.00

0.08

0.04

t = 2418.66
0 225 450 675 900

0.00

0.04

t = 2528.59
0 225 450 675 900

0.00

t = 2608.53
0 225 450 675 900x(c/ω0)
Plasma density n/n0 and EM energy density Ey2 + Bz2
snapshots (averaged over λ0) for laser pulse, β = 0.3.

10-5

t = 1138.09 to 2047.69

t = 1138.09 to 2047.69|Ey(ω)|2|Ex(ω)|2Fig. 2

10-6
10-7

10-8

0.0

0.3

0.6

0.9

1.2

10-2
10-3-410-510-610
0.0

0.3

0300 600 900

0.6

0.9

1.2

ω/ω00.4
0.2
0.0
-0.2
-0.4

t = 2538.58
Bz

t = 2598.54
Ey

0.10
0.05
0.00
-0.05
-0.10

0300 600 900

0.10
0.05
0.00
-0.05
-0.10

t = 2588.54

0300 600 900

t = 2558.57
0.4

t = 2638.51
0.10

0.05
0.00
-0.05
-0.10

0.2
0.0
-0.2
-0.4

0.05
0.00
-0.05
-0.10

0300 600 900x(c/ω0)

Ey2+Bz2

Fig. 3Bz

t = 2628.52
0.10
EyExExω/ω00300 600 900x(c/ω0)0300 600 900x(c/ω0)

Spectra (a.u.) of EM and ES waves inside the soliton,and profiles of ES field, Ex, transverse electric Ey andmagnetic field Bz, for β = 0.3.

5. Summary and conclusions
Our other results do not comply with some of the earlier
explanations apart from the standing soliton case [5,6].
Solitons move with an acceleration proportional to the density
gradient towards the low density side due to theinhomogeneity of plasma density [5]. However, we found thatin underdense homogeneous plasmas, apart from standing
solitons, by varying the laser intensity, we could also detect
the backward and forward accelerated solitons. Different
plasma length and laser amplitude can change the complex
details of the relativistic laser-plasma interaction. For
example, the SRS cascade, photon condensation, electron
acceleration and heating, and so on; can largely affect thesoliton formation, its structure, acceleration and dynamics. To
our knowledge, our results on acceleration of large relativistic
EM solitons in the underdense homogeneous plasma layer

0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1 t = 939.68
0.1 t = 679.85
0.1 t = 879.72
0.00.00.0-450 0 450 9001350 -450 0 450 9001350 -450 0 450 9001350

Ey2+Bz2x(c/ω0)
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2 t = 709.83
0.2 t = 919.69
0.2 t = 979.65
0.0
0.00.0-450 0 450 9001350 -450 0 450 9001350 -450 0 450 9001350

x(c/ω0)Fig. 4The EM energy density Ey2 + Bz2 snapshots for β = 0.5(top) and for β = 0.7 (bottom).

3277

Li B. et al., Stimulated Raman Scattering, Cascade-into-Condensate and Acceleration of Large Relativistic Electromagnetic Solitons[2] K. Estabrook et al., Phys. Fluids 26, 1892 (1985); ibid.,
B 1, 1282 (1989).
[3] K. Mima et al., Phys. Plasmas 8, 2349 (2001); C. D.
Decker et al., Phys. Rev. E 50, R3338 (1994).
[4] R.N. Sudan et al., Phys. Plasmas 4 (5), 1489 (1997); S.
Poornakala et al., ibid., 9, 1820 (2002); T. Zh. Esirkepovet al., JETP Lett. 68, 36 (1998).
[5] S.V. Bulanov et al., Phys. Rev. Lett. 82, 3440 (1999); T.
Esirkepov et al., ibid., 89, 275002 (2003); Y. Sentoku etal., ibid., 83, 3434 (1999).
[6] Lj. Hadz̆ievski, M.S. Jovanović, M.M. S̆korić and K.
Mima, Phys. Plasmas 9, 2569 (2002).

have not been presented in the past theoretical and simulation
studies [5]. Furthermore, despite a large number of earlier
works, a clear interpretation of the nature of the accelerationof soliton in uniform plasmas appears to be lacking.
One of us (M.M.S̆) acknowledges a partial support of
the Ministry of Science and Technologies of the Republic ofSerbia, Project No. 1964. We thank Lj. Hadz̆ievski forstimulating discussions.

References[1] D.W. Forslund et al., Phys. Fluids 18, 1002 (1975); ibid.,
18, 1017 (1975).

4278