Observation of Magnetized Soliton Remnants in the Wake of Intense
Laser Pulse Propagation through Plasmas
L. Romagnani,1,2A. Bigongiari,1,*S. Kar,1S.V. Bulanov,3C.A. Cecchetti,1,7T.Zh. Esirkepov,3M. Galimberti,4
R. Jung,5T.V. Liseykina,6,†A. Macchi,7,8J. Osterholz,5F. Pegoraro,8O. Willi,5and M. Borghesi1
1Centre for Plasma Physics, School of Mathematics and Physics, The Queen’s University of Belfast, Belfast BT7 1NN, United Kingdom
2LULI, E´cole Polytechnique, CNRS, CEA, UPMC, route de Saclay, 91128 Palaiseau, France
3APRC, JAEA, Kizugawa, Kyoto 619-0215, Japan
4Central Laser Facility, Rutherford Appleton Laboratory, Chilton, United Kingdom
5Institut fu ¨r Laser-und Plasmaphysik, Heinrich-Heine-Universita ¨t, Du ¨sseldorf, Germany
6Max-Planck-Institut fu ¨r Kernphysik, Heidelberg, Germany
7Istituto Nazionale di Ottica, CNR, Pisa, Italy
8Dipartimento di Fisica ‘‘E. Fermi,’’ Universita ` di Pisa, Pisa, Italy
(Received 15 January 2010; published 19 October 2010)
Slowly evolving, regularly spaced patterns have been observed in proton projection images of plasma
channels drilled by intense (* 1019Wcm?2) short (? 1 ps) laser pulses propagating in an ionized gas jet.
The nature and geometry of the electromagnetic fields generating such patterns have been inferred by
simulating the laser-plasma interaction and the following plasma evolution with a two-dimensional
particle-in-cell code and the probe proton deflections by particle tracing. The analysis suggests the
formation of rows of magnetized soliton remnants, with a quasistatic magnetic field associated with
vortexlike electron currents resembling those of magnetic vortices.
DOI: 10.1103/PhysRevLett.105.175002 PACS numbers: 52.35.Sb, 52.38.Fz, 52.65.Rr, 52.70.Nc
The generation of coherent and ordered structures is one
of the most prominent features in the dynamics of non-
linear many-body systems . Theoretical and experimen-
tal studies have shown that plasmas interacting with laser
pulses at relativistic intensities provide uniquely favorable
conditions to investigate a broad class of nonlinear phe-
nomena, the most known examples being arguably stimu-
lated Raman and Brillouin scattering, laser filamentation
and self-focusing, or the excitation of large amplitude
wake plasma waves (see  and references therein). A
different but not unrelated class of phenomena which has
more recently attracted a great deal of attention includes
the generation of organized nonlinear entities such as the
so-called electromagnetic (EM) solitons [3–5] and electron
EM solitons have been extensively investigated both
numerically  and in the frame of analytical models ,
and their macroscopic remnants [postsolitons (PSs)]
have been experimentally observed to develop following
the interaction of an intense laser pulse with a rarefied
plasma . Rows of electronvortices  and solitary mag-
netic dipole vortices (MDVs)  have been predicted to
form in the trail of an intense laser pulse propagating in an
underdense plasma. Besides being per se relevant as a
benchmark for nonlinear plasma theoretical models, the
experimental investigation of these phenomena might also
have practical implications, as such nonlinear entities may
contain a sizable fraction of the initial laser pulse energy
[3–5] or be the signature of the development of plasma
In this Letter we present the experimental observation,
employing proton projection imaging (PPI) , of slowly
evolving, localized EM structures, generated following the
interaction at relativistic intensities of a picosecond laser
pulsewith an underdense plasma. These structures initially
appear as a quasiperiodical pattern aligned along the low-
density channel drilled by the laser pulse, and evolve on
a time scale much longer than the pulse duration, re-
maining visible for more than 100 ps after the interaction.
Simulations of the laser-driven plasma evolution carried
out with a two-dimensional (2D) particle-in-cell (PIC)
code show the development of EM solitons and their
evolution into PSs inside laser-generated plasma channels
. Most noticeably the simulations indicate that, besides
exhibiting properties typical of solitons (e.g., trapping of
EM radiation in their inside and spatial localization
mechanism), these structures are also accompanied by
vortexlike electron currents and quasistatic magnetic field
patterns similar to those of MDVs. Extrapolating the re-
sults from 2D PIC simulations to the three-dimensional
Particle tracing (PT) simulations  of the probe proton
deflections show that such a 3D field distribution produces
synthetic proton images consistent with the experimental
The experiment was carried out at the Rutherford
Appleton Laboratory (RAL), employing the VULCAN
Nd-glass laser system operating in the chirped pulse
PRL 105, 175002 (2010)
22 OCTOBER 2010
? 2010 The American Physical Society
amplification (CPA) mode. A first laser pulse (1:054 ?m
wavelength, 1.2 ps duration, and delivering ?30 J energy
on target in linear polarization, hereafter named CPA1)
was focused at intensities * 1019Wcm?2onto a super-
sonic helium (He) jet (2 mm aperture diameter nozzle
driven at 50 bar pressure). The main pulse was always
preceded by a pedestal (? 300 ps duration and ?106con-
trast ratio) capable of preionizing the interaction region.
The electron density profile (linearly ramping along the
laser axis up to its peak value ?1:5 ? 1020cm?3over a
distance of ?400 ?m and remaining constant after that,
see also ) was diagnosed by in situ frequency-doubled
optical interferometry with picosecond time resolution.
Comparison of the electron density profile with a separate
characterization of the neutral gas density in the gas jet
 indicates full preionization of the gas.
The EM fields generated in the He plasma in the trail of
the CPA1laser pulse were diagnosed employing a laser-
driven transverse proton probe, arranged in a point pro-
jection geometry with single-shot temporal multiframe
ing a second laser pulse (CPA2) (with similar parameters
foil, proton target) and it was detected employing a multi-
layer stack of radiochromic films (RCFs). The distance
of the proton target from the CPA1propagation axis was
l ? 3 mm, and from the RCF pack it was L ? 3 cm, giving
a projection magnification M ’ L=l ? 10. Spatial and tem-
poral resolutions of a few ?m and of a few ps, respectively,
Typical PPI experimental data are shown in Fig. 1. In the
images, regions of a darker (lighter) color compared to the
background proton signal correspond to regions of accu-
mulation (depletion) of the probe protons, and for our
experimental conditions the proton density variations re-
flect the EM field gradients in the probed plasma. The laser
pulse, linearly polarized in the z direction (i.e., along the
normal to the page, z being the symmetry axis of the proton
beam), is impingingfromthe left and bythe earlier probing
time t ? 3:5 ps [1(a)] it has already exited the field of view
to the right-hand side. In the proton images the channel
drilled by the CPA1laser pulse into the ionized He jet is
displayed as a lighter color region along x delimited by two
dark lines [1(a)]. An additional dark line is visible near the
channel axis on the left-hand side of the images.
Longitudinal (i.e., reflecting a dependence of the deflecting
fields on x) modulations first appear in the channel at
probing times t * 5 ps [1(b)], and later they evolve into
a row of localized bubblelike structures aligned along the
plasma channel and particularly evident on the right-hand
side of the image 1(c). The localized structures slowly
expand, remaining visible until the latest observation
times (t * 140 ps) [Fig. 2(a)], when they have evolved
into a cloud of irregularly distributed bubbles. Whenever
the plasma channel has split into secondary filaments,
bubble structures are also observed inside some of the
In order to infer on the nature of the observed patterns,
the interaction of the laser pulse with the He plasma and
the following plasma evolution were modeled with 2D PIC
simulations . The simulations were performed ina range
of plasma densities and laser pulse parameters close to the
experimental ones. In the following, lengths are in units of
the laser wavelength ?, times in units of the laser period
TL¼ ?=c ¼ 2?=!, densities in units of the critical den-
sity nc¼ me!2=4?e2, and field amplitudes are expressed
in terms of the dimensionless parameter a ¼ eE=me!c
(see  for conversions to standard units). The largest
grid employed in the simulations was a 7750 ? 2400
mesh with a spatial resolution of ?=10, and the simulations
were running up to a time t ¼ 1500TL. The plasma density
was linearly ramping from zero at x ¼ 25? to its peak
value ne¼ 0:1ncat x ¼ 425?, remaining constant after
that. Ions with Z=A ¼ 1=2 were assumed. The laser pulse
(propagating along x) had a duration of 330TL, it was S
polarized (electric field in the z direction, normal to the
(x;y) simulation plane), and its field peak amplitude was
a ¼ 2:7.
In the simulation a low-density channel is ponderomo-
tively bored into the plasma by the laser pulse, with the
channel breaking into a number of secondary narrower
channels in the higher background density region .
Following the initial evolution [9,10] and after the laser
pulse has exited the simulation box, both the ion and
FIG. 1 (color online).
laser shot). The probing times, relative to the arrival of the peak
of the interaction pulse at x ¼ y ¼ 0, are (a) t ¼ 3:5 ps,
t ¼ 7 ps,
?12 MeV (a), ?11 MeV (b) and ?8 MeV (c)].
Typical PPI data (acquired in a single
and(c) 17ps [probeprotonenergies
FIG. 2 (color online).
structures at late times (t ? 140 ps, probe proton energy
?8 MeV). (b) Detail of a secondary filament exhibiting a mul-
tiple bubble pattern (t ? 37 ps, probe proton energy ?11 MeV).
(a) Detail of PPI data showing bubble
PRL 105, 175002 (2010)
22 OCTOBER 2010
electron densities are left with a depression on the channel
axis while peaking at the channel edges, giving rise to a
space-charge separation electric field, mainly in the y
direction, Ey. At this stage (t > 650TL) Eyhas evolved
into two ambipolar fronts on each side of the channel; i.e.,
it points outwards outside the channel and inwards inside
it. An electron current is generated along the channel axis
propagation direction) and it is compensated by two cur-
rent sheaths in the opposite direction along the channel
edges. Such currents persist until the latest simulation
times and produce a quasistatic magnetic field Bz, with
Bz< 0 (i.e., entering the simulation plane) above the
channel axis and Bz> 0 below it.
Ateven later times(t > 750TL)the simulationsshow the
onset of localized modulated patterns in the particle, cur-
rent,and fields’distributionsinsidethemainand secondary
channels. A detail of the EM field distribution in a single
localized structure is shown in Fig. 3. The frequency-
resolved analysis of the fields reveals an oscillating EM
field component (Bx, By, Ez) which has a frequency just
below the plasma frequency of the surrounding plasma and
is therefore trappedinside the structure [3(b)and 3(c)].The
rise to a double lobe of quasistatic magnetic field Bz, with
Bz< 0 above the channel axis and Bz> 0 below it [3(d)].
We stress here that this is a constant field, associated with a
stationary electron current, differently from the oscillating
field observed in other works  and also here. The com-
bined effect of the magnetic pressure associated with the
static Bzcomponent and of the radiation pressure associ-
ated with the oscillating fields drills a density depression in
correspondence with each localized structure. The result-
ing space-charge separation gives rise to a quasistatic
electric field ðEx;EyÞ in the radial direction from each
structure center [3(a)]. Such field is less pronounced near
the channel axis (due to the fact that the plasma density is
lower here), and at the latest times (t > 875TL) it tends to
overlap with the ambipolar field at the channel edges.
It should be noted here that the presence of a trapped
oscillatingfieldistypical ofEMsolitonsandPSs, whilethe
current and quasistatic magnetic field distributions re-
semble those of MDVs. As these features coexist and the
static and oscillating fields are of similar strength, it is
difficult to identify the observed structures with either
solitons or vortices. However, their position in the plasma
channels provides a hint of the mechanism leading to their
formation and hence of their primary nature. The localiza-
tion of the structures towards the end of the channels (both
in the experiment and PIC simulations) is qualitatively
consistent with estimations of the laser-pulse-depletion
length (ldepl? cTLðnc=neÞ ? mm range) , indicating
that frequency downshift of the laser pulse and the conse-
quent trapping of EM radiation is likely to be the relevant
should be regarded as PSs. A likely scenario is that the
electron currents readjust according to the density distribu-
tion associated with the preformed soliton structure, there-
fore giving rise to the vortexlike pattern. As these currents
an unavoidable feature of laser-excited solitons.
To verify that the field distributions observed in the 2D
PIC simulation, extrapolated to a 3D geometry, may pro-
duce the observed proton images, PT simulations  have
been carried out. In PTs the deflections of test protons
crossing a given 3D EM field distribution are numerically
computed and the particle density in the proton detector is
calculated. Only static fields have been considered, as for
our experimental conditions the contribution of the oscil-
lating components to the proton deflection is canceled out
by integration along the particle trajectories. The input 3D
field distribution is extrapolated from the 2D PIC outputs
by assuming an azimuthal symmetry around the channel
axis, and a reference framewith cylindrical coordinates (?,
?, x) (x being the channel axis) is considered. The electric
field is chosen to be oriented along the radial ^ ? direction
and the magnetic field to form closed loops in the azimu-
We first simulated the deflections given by the fields
associated with the plasma channel before the longitudinal
modulations appear. In the simulation the electric field
is takenof theform
f?ð?Þ ¼ 2:33E0½ð? ? ?Þ=??expf?½ð? ? ?Þ=??2g,
the magnetic field of the form B?¼ 2:33B0ð?=?Þ?
exp½?ð?=?Þ2?, where the numerical factor 2.33 is chosen
in order for E0and B0to represent the peak fields’ ampli-
tudes. The parameters ?, ?, and ? can be related to the
E?¼ f?ð?Þ þ fþð?Þ, with
FIG. 3 (color online).
single PS from PIC simulations. (a) Quasistatic electric field (Ex,
Ey) at t ¼ 875TL, (b) oscillating electric field Ez, (c) oscillating
magnetic field (Bx, By), and (d) quasistatic magnetic field Bzat
t ¼ 800TL.
Details of the fields’ distribution in a
PRL 105, 175002 (2010)
22 OCTOBER 2010
spatial characteristics of the PPI image of the channel, Download full-text
yielding ? ’ ? ’ ð15–20Þ ?m and ? ’ ð5–10Þ ?m. The
PPIdata were best reproduced
109Vm?1and B0’ ð150–300Þ T [Fig. 4(a)], in fair agree-
ment with the values obtained from the PIC simulations.
Inspection of the PT results clarifies how the experimental
proton images form. The ambipolar Eyfield tends to pile
up the protons on the channel axis and at the channel edges
[see deflection map 4(b)]. The vpxBzcomponent of the
vp? B (vpbeing the proton velocity, with vpxmainly
arising from the proton beam divergence) force tends to
focus the protons on the axis for x < 0 (where vpx< 0,
while Bz< 0 for y > 0 and Bz> 0 for y < 0), and at the
channel edges for x > 0 (where vpx> 0) [4(c)]. Hence Ey
gives rise to the dark lines delimiting the channel and
contributes to the central dark line visible for x < 0. Bz
contributes to the central dark line for x < 0, while for
x > 0 it cancels the piling up on the channel axis given by
Eyand contributes to the external dark lines.
We next introduce a modulation [described by a sin2
ð2?x=lÞ weight function, where l ’ 60 ?m] of B?along
the x direction. Such a field distribution describes a row of
tori and represents the simplest possible extrapolation to a
3D geometry of the magnetic field associated with the
vortexlike patterns observed in the 2D PIC simulations.
tative agreementwith the experimentalones [Figs. 1(c)and
2]. The PT also indicates that the proton deflection in the
y direction arises from the same effects described above,
with additional longitudinal modulationsin the proton den-
sity introduced by the B?dependence on x. We stress here
that the presence of a modulated magnetic field, introduced
E0’ ð1–2Þ ?
for consistency with the 2D PIC simulations, is critical in
the experimental results.
In conclusion, we have shown that stable modulated
patterns observed via PPI inside laser-plasma channels
observed in PIC simulations. PIC simulations also reveal
the simultaneous presence of a quasistatic magnetic field
associated with a vortexlike current distribution, and PT
simulations evidence that such a magnetic field is essential
for their experimental detection. The peculiar features of
these structures, such as their magnetic nature, their orga-
nization into periodical patterns, and their detailed 3D
topology, should be stimulating for further theoretical and
We acknowledge the support of the RAL/CLF staff. This
work has been supported by EPSRC Grants No. EP/
E035728/1 (LIBRA consortium) and No. EP/C003586/1,
by British-Council-MURST-CRUI, ESF-COST, TR18, and
GRK1203 networks. Part of the simulations were per-
formed at CINECA (Bologna, Italy) sponsored by CNR/
Polytechnique, 91128 Palaiseau, France, and LULI,
Universite ´ Paris VI, 3 rue Galile ´e, 94200 Ivry-sur-Seine,
†Present address: Institut fu ¨r Physik, Universita ¨t Rostock,
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address: CEA/DSM/IRAMIS/LSI, E´cole
(a) Simulated proton image and (b),(c) maps of the proton de-
flection ?? ? ?vpy=vparising from (b) the E field alone (y scale
enlarged for x > ?0:5 mm to highlight the Eyfocusing effect
insidethechannel)and from(c)theB fieldalonefortheunmodu-
lated channel. (d) Simulated proton image for the modulated
(color online).Particletracing simulations.
PRL 105, 175002 (2010)
22 OCTOBER 2010