A Three Dimensional Simulation of
Solitary Waves in the Laser Wake
T. Esirkepov1'2, K. Nishihara2, S. Bulanov3, F. Pegoraro4,
F. Kamenets1, N. Knyazev1
1
Moscow Institute of Physics and Technology, Institutskij per. 9, Dolgoprudny 141700, Russia
Institute of Laser Engineering, Osaka University, 2-6 Yamada-oka, Suita, Osaka 565-0871, Japan
3
General Physics Institute ofRAS, Vavilov sir. 38, Moscow 119991, Russia
4
University of Pisa andlNFM, pz.Torricelli 2, Pirn 56100, Italy
Abstract. Three dimensional relativistic electromagnetic sub-cycle solitons were observed in
Particle-in-Cell simulations of an intense short laser pulse propagation in an underdense plasma.
Their structure resembles that of an oscillating electric dipole with a poloidal electric field and a
toroidal magnetic field that oscillate in-phase with the electron density with frequency below the
Langmuir frequency. On the ion time scale the soliton undergoes a Coulomb explosion of its
core, resulting in ion acceleration, and then evolves into a slowly expanding quasi-neutral
cavity.
INTRODUCTION
Particle-in-Cell (PIC) simulation show different types of coherent structures in the
wake of an intense short laser pulse propagating in underdense plasmas [1]. These are
electrostatic wakefields, quasistatic magnetic fields associated with electron fluid
vortices, bubbles in the plasma density associated with electromagnetic solitary waves
and soliton traines. In this paper we discuss the formation and the evolution of
electromagnetic solitary waves ('solitons' for short) in the wake of an intense laser
pulse in the three dimensional (3D) case.
The nonlinear wave evolution in the three dimensional regimes differs drastically
from their evolution in the one- and two-dimensional approximations, as exemplified
by the problem of wave collapse [2] and by the problem of the transverse stability of
solitons [3], Relativistic electromagnetic solitons are now routinely observed in two
dimensional (2D) PIC simulations [4-7] in the wake of an intense short laser pulse
propagating in an underdense plasma. Solitons attract a great attention because they
are of fundamental importance for nonlinear science [8] and are considered to be a
basic component of turbulence in plasmas [9]. Thus the numerical identification of
solitons, among the different kinds of coherent structures formed by an intense laser
pulse in a plasma, stimulated a renewed interest in developing an analytical model
[10,11] and in envisaging ways of detecting solitons experimentally [12]. In principle,
knowing the properties of various types of two-dimensional solitons, one could guess
the topological structure of a three-dimensional soliton. However, the main problem
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that arises here stems from the difficulty of confining the vector electromagnetic fields
inside a finite 3D domain.
As was stressed in Ref. [13], the complexity of the strong electromagnetic wave
interaction with plasmas, due to the high dimensionality of the problem, to the lack of
symmetry and to the importance of nonlinear and kinetic effects, makes analytical
methods unable to provide a detailed description. On the other hand, powerful
methods for investigating the laser-plasma interaction have become available through
the advent of modern supercomputers and the developments of applied mathematics.
In the case of ultra-short relativistically strong laser pulses, simulations with 3D
Particle-in-Cell codes provide a unique opportunity for describing the nonlinear
dynamics of laser plasmas adequately, including the generation of coherent nonlinear
structures such as relativistic solitons.
This has motivated us to perform 3D simulations of the formation and evolution of
laser induced sub-cycle relativistic solitons.
In the following section we briefly summarize the recent developments in the
investigation of relativistic solitons in plasmas. After that, we describe the simulation
code. Then we discuss a 3D relativistic electromagnetic sub-cycle soliton seen in PIC
simulation. In the last section we present a conclusion.
RELATIVISTIC ELECTROMAGNETIC SUB-CYCLE SOLITON
As observed in PIC simulations, the solitons generated in a laser-plasma interaction
are very important for the theory of laser plasma dynamics (more generally - for the
theory of nonlinear waves) and for laser plasma applications because they acquire a
significant part of the laser pulse energy; they exist for many electron plasma wave
periods; their formation and evolution are associated with the change of the laser pulse
frequency; they cause the so called Coulomb explosion of plasma in their locations
producing resulting in the heating of ions.
The soliton seen in PIC simulations is an isolated cavity in the plasma containing an
oscillating electromagnetic field. The size of the cavity is of the order of the plasma
wavelength and the oscillation frequency is below the Langmuir frequency of the
surrounding unperturbed plasma. Exactly one half of a spatial electromagnetic wave is
trapped inside the cavity, therefore we use the term 'sub-cycle'. The dimensionless
amplitude of the electromagnetic field in the soliton is of the order of a=eE/(mecoc)~l
or higher, here E is the magnitude of the electric field inside the soliton, a> is the
soliton frequency, e and me are electric charge and mass of electron, c is the speed of
light. The mass of an electron in such a field ~me(l +a2)m is significantly greater than
its rest mass, therefore we use the term 'relativistic'. The soliton under consideration
moves with a non-relativistic velocity or stands still at the place where it is formed,
and so we sometimes use the term 'slow'.
We use the term 'soliton' instead of 'solitary wave' for brevity, even if the
evolution of these structures can violate their strict definition as solitons in the sense
of the inverse-scattering method in the theory of solitons.
The theory of intense electromagnetic solitons in plasmas has been developed in
many papers, see Refs. [10,11,14], and also references in Ref. [1]. Their best known
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application is the use of electromagnetic 'fast' solitons with group velocity close to the
speed of light for charged particle and photon acceleration. Many analytical models
are based on the so called 'envelope' approximation and predict solitons in the form of
a 'fast' wave packet containing many spatial cycles. Here we primarily discuss the
sub-cycle slow solitons that are observed in Particle-in-Cell simulations. We
emphasize that we have not yet observed 'fast' multi-cycles solitary waves in the PIC
simulations.
One-dimensional (ID) sub-cycle relativistic electro-magnetic solitons in
underdense plasma were observed for the first time in PIC simulation in Ref. [15]. The
solitons are the cavities in the electron density with trapped electro-magnetic field
oscillating with frequency below the unperturbed Langmuir frequency. The
mechanism of soliton formation and the structure of circularly polarized solitons were
investigated in Ref. [16]. An intense laser pulse propagating in an underdense plasma
undergoes Raman scattering. The laser pulse loses its energy rapidly while the number
of photons in the pulse is conserved. Thus the local frequency of the rear part of the
pulse decreases down to the local Langmuir frequency. As a result, the low-frequency
part of the electromagnetic radiation of the laser pulse is trapped inside the cavity in
the electron density, and a sub-cycle soliton is formed. Two "pure" types of onedimensional solitons were found: the linearly polarized L-solitons and circularly
polarized C-solitons.
The exact analytical solution of the electron hydrodynamic - Maxwell equations
representing
one-dimensional
sub-cycle
circularly-polarized
relativistic
electromagnetic soliton was obtained in Ref. [10] in perfect agreement with the results
of ID PIC simulation.
Two-dimensional relativistic electromagnetic solitons were discovered in the PIC
simulations presented in Ref. [4]. Two "pure" types of two-dimensional solitons were
found: S-soliton with transverse electric field and azimuthal magnetic field, and Psoliton with the opposite structure — transverse magnetic field and azimuthal electric
field. In contrast to the electron vortices, which move across a density gradient,
solitons move along the density gradient towards the lowest density. At some critical
density the soliton emits its energy in the form of a low-frequency short
electomagnetic burst [5]. One can imagine "a solitonic gun" when the laser pulse
propagates in the direction perpendicular to the plasma density gradient, and solitons
are emitted in the direction opposite to the density gradient.
The interaction of two 2D S-solitons leads to their merging. The resulting soliton
acquires the total energy of the merged solitons [17]. The interaction of two counterpropagating ID solitons is completely different: they preserve their shape and velocity
but lose their phase, similarly to a well-known "phase shift" in the theory of KdV
solitons.
In an electron-ion plasma ID and 2D solitons evolve into post-solitons in an ion
time-scale due to the ion acceleration by the time-averaged electrostatic field inside
the soliton. This effect leads to the formation of slowly expanding bubbles in plasma
density [7].
Presumably sub-cycle soliton cavities were detected recently in the brilliant
experiment of Dr. M. Borghesi et al., [12]. In this experiment a new powerful
diagnostic tool - proton imaging - was applied. The intense laser pulse is divided into
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two beams, one interacts with the target under investigation and produces a plasma,
the other irradiates a thin metal foil producing fast protons with a quasi-thermal energy
spectrum corresponding to the effective temperature 3-8 MeV. This proton beam
passes through the target, and is then captured by a detector, which consists of many
thin films. Each film of the detector captures the protons with the corresponding
energy because the proton slowing down in the matter depends on its energy. So, we
obtain a "proton image" in each film. The initial transverse size of the proton beam is
tens of microns, the size of the image formed is up to several millimeters. Inside the
soliton the ponderomotive pressure of the trapped electromagnetic radiation acts on
the electrons and results in the formation of the quasistatic electric field. The electric
fields in the target under study deflect the protons and therefore the density of protons
in different parts of the image is changed accordingly. This allows us to reconstruct
the structure of the strong quasistatic electric fields in the target.
3D PARTICLE-IN-CELL CODE «REMP»
We use three-dimensional massively parallel and fully vectorized Relativistic
Electro-Magnetic Particle-mesh code (REMP), written by one of the authors (T.E.).
The code is based on Particle-in-Cell method [18], and exploits 'Density
Decomposition' — a new scheme of the electric current density assignment, described
in Ref. [19]. The scheme guarantees the exact conservation of the local electric
charge, and is valid for an arbitrary form-factor (shape) of a quasi-particle. Currently
we use the second-order polynomial form-factor (parabolic spline). This technique
allows us to significantly reduce unphysical numerical effects of the PIC method such
us numerical heating, sampling noise, numerical collisions, etc. This advantage is very
important for identifying hydrodynamic-like long-living coherent structures such as
solitons and electron vortices.
The code is written in Fortran-90. It is parallelized via Message Passing Interface
(MPI) using domain decomposition method in the framework of the symmetric SIMD
(single image - multiple data) model. The code is fully vectorized. The electric current
density (or electric charge density) assignment is vectorized using the modified presorting method and advantages of Density Decomposition scheme. Interprocessor
communications are optimized using vector algorithm of binary sortkey sorting.
We developed the system of automatic visualization of data. This system is
optimized for processing large three-dimensional data. The system uses various
techniques to visualize three-dimensional data such as vector and scalar fields,
distribution functions, phase spaces, etc. Currently it is based on Interactive Data
Language package (IDL, RSI inc.), which works on a PC or graphics workstation. We
routinely use animations, which are very desirable to reveal the features and
mechanisms of a quickly varying physical process.
The code is tested and used on CRAY-T3E 1200 (CINECA, Bologna, Italy), SGI
Origin-2000 (SNS, Pisa, Italy), Hitachi SR-8000, NEC SX-5 (ILE, Osaka University,
Japan), and Intel-based PC or clusters under Windows or Linux.
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3D SOLITON SEEN IN PIC SIMULATION
In this section we present the results of three-dimensional Particle-in-Cell
simulation of laser induced 3D sub-cycle relativistic electromagnetic solitons.
Simulations were performed with the code REMP, described above. In the simulation
the laser pulse is gaussian and linearly polarized along the z-axis, its dimensionless
amplitude is a=eE/(meOJc)=l
corresponding to the peak intensity
I=L38xl018W/cm2xm2/A2, its FWHM size is 8Ax5Ax5A. The laser pulse propagates
along the j-axis. The focal plane of the laser pulse is placed in front of the plasma slab
at the distance of 3A. The density of plasma is n=036ncr, where ncr=meG?/(4ne2) —
critical density. Ions and electrons have the same absolute charge, the mass ratio is
mi/me=1836. The simulation grid is 660x400x400, the mesh size is 0.05A. The plasma
slab length is 13A. The total number of quasiparticles is 425-106. The boundary
conditions along the y- and z-axes are periodic and absorbing for electromagnetic
radiation and for the particles along the x-axis. The simulation were performed on 16
processors of the NEC SX-5 vector supercomputer at CMC, Osaka University. In the
figures the time and space units are the period 2x/a) and the wavelength /I of the
incident radiation, respectively. The structure of 3D soliton is clearly seen in the
animations produced from the data (420 stills with the period 0.05).
FIGURE 1. Iso-surface of the electromagnetic energy density corresponding to the dimensionless
value of (E2+B2)/8^0.Q9/8xzl t = 33.75-2^/0).
0,5
0.0
-0,5
-1,0
30
35
40
ut/2-n
45
50
55
FIGURE 2. Electric field component Ez minimal and maximal values in the soliton versus time.
Ill
FIGURE 3. The soliton structure. Upper raw: arrows represent planar electric field, background is
transverse magnetic field. Bottom: electron density. Columns (a),(c) are cross-sections at y=0;
(b),(d) - at x=14.5. (a),(b) correspond to t=39.3; (c),(d) - to t=40.2.
The laser pulse undergoes a weak self-focusing in the plasma slab. Behind the laser
pulse we see a soliton train and one isolated soliton in Fig. 1. These structures have
oscillating electric field polarized in the same direction as the laser pulse, Fig. 2. The
frequency of the oscillations is below the Langmuir frequency of the surrounding
unperturbed plasma cos- 0.93Cfye- Therefore the soliton oscillation is not-resonant with
the plasma waves and radiation loss due to the wave transformation is not significant.
Figures 3 and 4 show the structure of the isolated soliton. The soliton resembles an
oscillating electric dipole. Although we see a strong electrostatic component, the
soliton is electromagnetic as one can see directly from the oscillations of the electric
and magnetic fields in the soliton shown in Fig. 4. The electric field in the soliton is
poloidal and the magnetic field is toroidal. This structure of the electromagnetic field
can be considered as that of the lowest eigenmode of a cavity resonator with a
deformable wall, and thus we call this structure a Transverse Magnetic (TM) soliton.
We note that since the cavity crosses the equatorial plane twice within one oscillation
period, the field at the equatorial plane oscillates with frequency 2co$. The 3D soliton
can be described by the 4-component electromagnetic potential (q>,A) and by the 4component electric current density (pj)9 which are both confined inside a finite
domain. In the equatorial plane the structure of the three dimensional soliton is similar
to that of a two dimensional s-soliton, while that in the perpendicular planes is similar
to a two dimensional p-soliton. Considering the soliton as a wave packet, it has only
half of one cycle in space, so we use the term "sub-cycle soliton". We also note that
the soliton is azimuthally symmetric but in the x-z plane the symmetry is perturbed, as
seen in Fig. 3. This may be due to the fact that the laser pulse has propagated through
the plasma in the j-direction. Namely the soliton dynamics is slightly affected by the
quasistatic magnetic field and by the remnants of the density filament [20] induced by
the laser pulse.
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t=40.2
t=39.3
t=40.2
t=39.3
B
FIGURE 4. Three dimensional structure of the electric and magnetic fields in the soliton during the
soliton period. Electric field is poloidal, magnetic field is toroidal.
FIGURE 5. Ion acceleration inside the soliton during 10 soliton periods, ion momentum is
expressed in mec.
On the ion time-scale the soliton evolves into the post-soliton. The magnitude of the
soliton decreases, as shown in Fig. 2, and ions acquire momentum, see Fig. 5. As the
last stage we see a slowly expanding (v~3-10~3c) cavity in plasma density. In the
soliton train almost isolated structures considered as solitons tend to merge and form a
foam of bubbles with relatively high-density (2-5ncr) walls.
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CONCLUSION
In conclusion we have provided the demonstration of the existence of 3D sub-cycle
relativistic electromagnetic solitons in a collisionless cold plasma. The solitons are
generated as a consequence of the frequency downshift of an intense laser pulse
propagating in an underdense plasma. A substantial part of the pulse energy is
transformed into solitons, approximately 25-30% of the incident laser pulse. The core
of the soliton is positively charged on average in time and the soliton undergoes a
Coulomb explosion in an ion time-scale. This process results in heating of the plasma
ions. Then the soliton evolves into a post-soliton, which is a slowly expanding quasineutral cavity in the plasma. Merging solitons can form a slowly expanding foam.
ACKNOWLEDGMENTS
We appreciate the help of ILE computer group and Cyber Media Center of Osaka
University (Japan). This work was supported by Japan Society for the Promotion of
Sciense, Istituto Nazionale di Fisica della Materia (Italy), Russian Ministry of science
and technology, Russian Fund of Basic Research.
REFERENCES
1. Bulanov, S. V., et al., in Reviews of Plasma Physics, New York: Kluwer Academic / Plenum
Publishers, 2001, edited by V. D. Shafranov, Vol. 22, p. 227.
2. Zakharov, V. E., Sov. Phys. JETP 35, 908 (1972).
3. Kadomtsev, B. B., and Petviashvili, V. I., Doklady ANSSSR 192, 753 (1970); Zakharov, V. E., and
Rubenchik, A. M., Sov. Phys. JETP 38, 494 (1974).
Bulanov, S. V., et al., Phys. Rev. Lett 82, 3440 (1999).
Sentoku, Y., et al., Phys. Rev. Lett. 83, 3434 (1999).
Naumova, N. M., et al., Phys. Plasmas 8,4149 (2001).
Naumova, N. M., et al., Phys. Rev. Lett. 87, 185004 (2001).
Novikov, S. et al., Theory of Solitons: the Inverse Scattering Method, New York: Consultants
Bureau, 1984; Taniuti, T., and Nishihara, K., Nonlinear Waves, Boston: Pitman Advanced
Publishing Program, 1983.
9. Mima, K., et al., Phys. Plasmas 8, 2349 (2001).
10. Esirkepov, T. Zh., et al., JETP Lett. 68, 36 (1998).
11. Farina, D., et al., Phys. Rev. E 62,4146 (2000); Farina, D., and Bulanov, S. V., Phys. Rev. Lett. 86,
5289 (2001); Plasma Phys. Rep. 27, 680 (2001); Poornakala, S., et al., Phys. Plasmas 9,1820
(2002).
12. Borghesi, M., et al., Phys. Rev. Lett. 88, 135002 (2002).
4.
5.
6.
7.
8.
13. Dawson, J. M., and Lin, A. T., in: Basic Plasma Physics, edited by M. N. Rosenbluth and R. Z.
Sagdeev, North-Holland, Amsterdam, 1984, Vol. 2, p. 555; Dawson, J. M., Phys. Plasmas 6,4436
(1999).
14. Gersten, J. I., and Tzoar, N., Phys. Rev. Lett. 35, 934 (1975); Kozlov, V. A., et al., Sov. Phys. JETP
49, 75 (1979); Kaw, P. K, et al., Phys. Rev. Lett. 68, 3172 (1992).
15. Bulanov, S.V., et al., Phys. Fluids B 4, 1935 (1992).
16. Bulanov, S.V., et al., Plasma Phys. Rep. 21, 600 (1995).
17. Bulanov, S.V., et al., Physica D 152-153, 682-693 (2001).
18.Hockney, R. W., andEastwood, J. W., Computer Simulation Using Particles, McGraw-Hill Inc.,
1981.
19. Esirkepov, T.Zh., Comput. Phys. Comm. 135, 144 (2001).
20. Kuznetsov, A.V., et. al., Plasma Phys. Rep. 27, 3 (2001).
114