2. THE EFFICIENT METHODS FOR PLASMA WAVE
EXCITATION
Plasma waveguide accelerator (PWGA) has been defined as: (a) electron
beam–plasma interaction in magnetized plasma waveguide
(beam–plasma instability) and (b) external ultra-high oscillator
(Fainberg, 2000) and co-workers (since
1956).
Plasma beat–wave accelerator (PBWA) has been defined by Tajima
and Dawson (1979) as:
where fi is the radiation frequency,
fp is the Langmuir frequency (see also
Litvak (1964), Rosenbluth and Liu (1972), for detailed information concerning this and
the next reference can be seen in Balakirev et al. (2002)). PBWA, an electric field of 1.8 ×
109 V/cm and energy of accelerated particles of 20 MeV
were obtained by Clayton et al. (1993a,b),
and Ebrahim (1994). Amiranoff et al.
(1995) observed acceleration of electrons that
were injected at an energy of 3.3 MeV and accelerated up to 4.7 MeV after
plasma, at an energy gain of 1.2 GV/mV by a beat-wave generated in
a plasma by two Nd-YAG and Nd-YLF laser wavelengths. Kitagawa et
al. (2003a,b) presented the following results from
the Institute of Laser Engineering, Osaka University, Japan:
- Table—Top Beat—Wave Accelerator is under construction.
- Two Ti: Saphire lasers drive a 500-fs pulse Beat-wave.
- Will accelerate electrons to > 20 MeV in 2004.
Self–modulated laser wake–field accelerator (SmLWFA) is a
self-modulation of laser pulse as defined by Andreev et al.
(1992), Krall et al. (1993), Antonsen and Mora (1992), and Esarey et al. (1993); see also Andreev et al. (2000). The most impressive results as seen by
Nakajima et al. (1994), Modena
et al. (1995), Coverdale et
al. (1995), Le Blanc et al. (1996), Umstadter et al. (1996a,b),
and Ting et al. (1998) on plasma
acceleration of charged particles were obtained on the SmLWFA, that is,
an electric field amplitude of 1.5 × 108 V/cm to 2
× 108 V/cm, an energy of accelerated particles of
100 MeV to 300 MeV (see also Faure et al. 2000). The extremely large acceleration gradients
generated by laser pulses propagating in plasmas can be used to
accelerate electrons.
In the standard laser wake–field accelerator (LWFA), a short
laser pulse, on the order of a plasma wavelength long, excites a
trailing plasma wave that can trap and accelerate electrons to high
energy. There are a number of issues that must be resolved before a
viable, practical high energy accelerator can be developed. These
include Raman, modulation, and hose instabilities that can disrupt the
acceleration process. In addition, extended propagation of the laser
pulse is necessary to achieve high-electron energy. In the absence of
optical guiding, the acceleration distance is limited to a few Raleigh
ranges, which is far below that necessary to reach GeV electron
energies.
Laser pulse shaping (LPSh) as proposed by Bulanov et al.
(1996), the particle in cell (PIC) simulation
results have been demonstrated that regular wake electric fields may be
obtained by a properly shaped laser pulse (sharp steeping of its
leading front).
Resonant laser-plasma accelerator (RLPA) is a train of laser pulses
with independently adjustable pulse widths and interpulse spacing as
proposed by Dalla and Lontano (1994) and
Umstadter et al. (1994).
Laser wake–fields accelerator (LWFA) is a short laser pulse as
defined by Tajima and Dawson (1979), see also
Gorbunov and Kirsanov (1987); for
relativistic strong pulse see Bulanov et al. (1990). To achieve multi GeV electron energies in
the laser wake-field accelerator, it is necessary to propagate an
intense laser pulse over long distances in plasma without disruption.
The physics of laser beams propagating in plasmas has been studied in
great detail and there exists sample experimental confirmation of
extended guided propagation in plasmas and plasma channels. In addition
to these issues, dephasing of electrons in the wake-field can limit the
energy gain. Spatially tapering the plasma density may be useful for
overcoming electron dephasing in the wake-field. Sprangle et
al. (2000, 2002) proposed and studied guiding and stability of
an intense laser pulse in a uniform plasma channel, and analyzed the
wake-field acceleration process in an inhomogeneous channel. The
coupled electromagnetic and plasma wave equations were derived for
laser pulses propagating in a plasma channel with a parabolic radial
density profile, and arbitrary axial density variation. For a uniform
channel, Raman and modulation instabilities were analyzed. For a
nonuniform channel the axial and radial electric fields associated with
the plasma wave were obtained inside and behind the laser pulse. It was
shown that by optimally tapering the plasma density, the wake-field
phase velocity several plasma wavelengths behind the laser pulse can be
equal to the speed of light in vacuum. A three-dimensional (3D)
envelope equation for the laser field has been derived that includes
non-paraxial effects, wake-fields, and relativistic nonlinearities. In
the broad beam, short pulse limits the nonlinear terms in the wave
equation that leads to Raman and modulation instabilities cancellation.
Long pulses (several plasma λp wave lengths)
experience substantial modification due to these instabilities. The
short pulse LWFA, although having smaller accelerating fields, can
provide acceleration for longer distances in a plasma channel. By
allowing the plasma density to increase along the propagation path,
electron dephasing can be deferred, increasing the energy gain. A
simulation example of a GeV channel guided LWFA accelerator is
presented. Simulations of Sprangle et al. (2002) also show that multi-GeV energies can be
achieved by optimally tapering the plasma channel. Kitagawa et
al. (2003a,b) presented the following results from
the Institute of Laser Engineering, Osaka University, Japan:
- Laser of 15 J at 1 μm in 0.5 ps was injected into glass
capillary of 1 cm in length and 60 μm in diameter and accelerated
the plasma electrons to 100 MeV.
- The electrons are trapped in the wake-field, showing the bump at
the maximum energy.
- 1-D and 2-D PIC codes speculated that a laser wake-field, whose
wavelength is close to the laser pulse length, was excited with a
gradient of 10 GV/m, accelerating electrons in the
capillary.
Plasma wake–fields accelerator (PWFA) defined by Chen et
al. (1985), Katsouleas (1986), Amatuni et al. (1986), Keinigs and Jones (1987) as short rectangular relativistic electron
bunch or periodic train of relativistic electron bunches. Rosenzweig
et al. (1991) sees it as the Blow
out regime of PWFA. In the PWFA, an electric field of 6 ×
104 V/cm and energy of accelerated particles of 6 MeV
were obtained by Rosenzweig et al. (1988), see also Rosenzweig (1990) and Schoessow et al. (1993). In blow out regime of PWFA, energy gradients
of 700 MeV/m were measured in the experiment E–157 by Lee
et al. (2000) and Assmann et
al. (1999). An intense, high-energy
electron or positron beam can have focused intensities rivaling those
of today's most powerful laser beams. For example, the 5 ps
(full-width, half-maximum), 50 GeV beam at the Stanford Linear
Accelerator Center (SLAC) at 1 kA and focused to a 3 micron rms spot
size yields intensities of 1020 W/cm2 at a
repetition rate of 10 Hz. Unlike a ps or fs laser pulse which interacts
with the surface of a solid target, the particle beam can readily
tunnel through tens of cm of steel. However, as it is shown, the same
particle beam can be manipulated quite effectively by the plasma that
is a million times less dense than air! This is because of the very
strong collective fields induced in the plasma by the Coulomb force of
the beam. The collective fields in turn react back onto the beam
leading to many clearly observable phenomena. The beam particles can
be:
(1) deflected leading to focusing, defocusing, or even steering of
the beam;
(2) undulated causing the emission of spontaneous betatron X-ray
radiation;
(3) accelerated or decelerated by the plasma
fields.
Using the 28.5 GeV electron beam from the SLAC linac, a series of
experiments have been carried out that demonstrated clearly many of the
above mentioned effects. The results were compared with theoretical
predictions and with 2D and 3D, one-to-one, particle-in-cell code
simulations. These phenomena may have practical applications in future
technologies including optical elements in particle beam lines,
synchrotron light sources, and ultrahigh gradient accelerators. As can
be seen from spatial distribution of excited wake-field, the electric
field can attain high values only over very short distances. Therefore,
we think that the energy doubles for a linear SLAC collier problem is
not very realistic. Blue et al. (2003) and Hogan et al. (2003) shown for the first time that a beam of
positrons can drive and be used to probe the longitudinal electric
field component of the plasma wake-field. When a 28.5 GeV and 2.4 ps
long positron beam at the Stanford Linear Center (SLAC) containing 1.2
× 1010 particles propagates through a lithium plasma
of electron density 1.8 × 1014 cm−3,
the main body of the beam is decelerated at a rate of approximately 49
MeV/m, while a beam slice containing 5 × 108
positrons in the back of the same bunch gains energy at an average rate
of ∼56 MeV/m over 1.4 m. With such a field structure, the beam
energy is transferred from a large number of particles in the core of
the bunch to a fewer number of particles in the back of the same bunch.
The wake-field thus acts like a transformer with a ratio accelerating
electric field to decelerated electric field of ∼1.3. The
acceleration gradient of 56 MeV/m measured in this proof-of
principal experiment can be increased to the GeV/m level in future
experiments by a combination of in the drive beam charge, a decrease in
the drive beam pulse width (with a corresponding increase in the plasma
density), and by employing a plasma channel rather than a uniform
plasma. Furthermore, in real application of such a plasma wake-field
accelerator, the drive positron bunch will be followed by an optimally
placed trailing witness bunch with a sufficient current to both realize
high-gradient acceleration and a reasonable beam load and narrow energy
spread. These results are critical to the development of future
plasma-based linear colliers.
An interesting result has been established by us (Karas‘ et
al., 1997, 2000), Balakirev et al. (2001, 2002), that for a
certain relation among the parameters of the plasma-bunch-magnetic
field system, the hybrid nature of the wake waves (which are excited by
a relativistic electron bunch in a magnetized plasma and are a
superposition of the surface and spatial modes) makes it possible to
increase the electron energy of the accelerated bunch to a value that
is significantly higher than the initial electron energy of the
accelerating bunch (even when the bunch is initially unmodulated in the
longitudinal direction). We have discussed 2.5-D numerical modeling on
the formation of an ion channel as a result of the radial ion motion in
self-consistent electromagnetic fields excited by a train of
relativistic electron bunches (Karas‘ et
al., 1997). The parameters of the fully developed channel
are determined by the plasma-to-bunch density ratio and the ratio of
the bunch radius to the skin depth. The effective dimensions of the
channel and its “depth” (i.e., the high ion density at the
channel axis) increase monotonically both in time and in the direction
opposite to the propagation direction of the bunches. The formed ion
channel stabilizes the propagation of relativistic electron bunches,
which generate stronger accelerating fields. The results of the
wake-field excitation during the self-modulation of a long relativistic
electron bunch (Balakirev et al. 1998) has shown that the maximum electron density
in the bunch becomes comparable to the plasma density and the amplitude
of the plasma density perturbations becomes larger than the initial
plasma density by a factor of 4.5. This indicates a very strong
modulation of both the bunch density and the plasma density. That is
why, even in the above case of a low-density bunch (in which the
unperturbed electron density is about two orders of magnitude lower
than the plasma density), it is incorrect to describe the plasma in the
linear approximation. The amplitude of the longitudinal field is about
0.8 of the maximum electric field that can be generated in the plasma,
and the amplitude of the radial field is about 0.4 of the maximum
possible field. This shows that the driven bunch needs to be placed in
the acceleration stability region. An important point is that, the
field amplitude increases only over a certain distance along a
relativistic electron bunch; hence, it would be of no use to operate
with bunches whose length exceeds the distance over which the
longitudinal field amplitude is maximum, doing so would provide no
additional increase in the excited wake field. The results obtained
with allowance for all possible nonlinearities give a better insight
into the 3D behavior of relativistic electron bunch in a plasma and may
help to ensure the optimum conditions for the wake-field generation
during the dynamic self-modulation of the bunches. The results of
investigations of the excitation of accelerating fields by an
individual relativistic electron bunch or by a train of such bunches in
a plasma (in particular, in the presence of an external magnetic field)
make it possible to evaluate the potentialities of the wake-field
acceleration method.
We would like to note recent and interesting experimental results
from Yakimenko et al. (2003) on the
generation of plasma wake-fields by relativistic electron bunch and on
phasing between the longitudinal and transverse fields in the wake. The
leading edge of the electron bunch excites a high-amplitude plasma wake
inside the over dense plasma column, and the acceleration and focusing
wake-fields are probed by the bunch tail. By monitoring the dependence
of the acceleration upon the plasma's density, the authors
approached the beam-matching condition and average acceleration
gradient of 35 MeV/m. These results confirmed that the maximum
acceleration gradient in the over dense regime
ne >>
nb, where nb
= 1015 cm−3 is the average electron density
in the bunch, and can be estimated by the following formula (Lee et al., 2000):
where Q and σz are charge and
longitudinal size of electron bunch, respectively.
3. PHYSICAL MECHANISM FOR GENERATION OF
VERY HIGH “QUASI-STATIC” MAGNETIC FIELDS
We will now discuss the physical mechanism for generation of very
high “quasi-static” magnetic fields in the interaction of
an ultra-intense short laser pulse, with a plasma target owing to the
spatial gradients and non-stationary character of the ponderomotive
force. The interaction between intense laser radiation and matter is
known to produce a wealth of nonlinear effects. Those include fast
electron and ion generation indicating that ultra-strong electric
fields are produced in the course of the laser–plasma
interaction. An equally ubiquitous, although less studied, effect
accompanying laser–matter interaction is the generation of
ultra-strong magnetic fields in the plasma. Magnetic fields can have a
significant effect on the overall nonlinear plasma dynamics. Extremely
high (few megagauss) azimuthal magnetic fields play an essential role
in the particle transport, propagation of laser pulses, laser beam
self-focusing, penetration of laser radiation into the overdense plasma
and the plasma electron and ion acceleration. By means of a 2.5-D
numerical simulation on the macroparticles method, it is possible to
find the magnetic field spatial and temporal distribution without usage
of an adapted parameter unlike the conventional
∇nx∇T mechanism (see for example, Haines, 1997). On the other hand, theoretical model
for the generation of a magnetic field proposed by Sudan (1993) is not appropriate, this model has a very
large ratio of plasma density to critical density and when the
∇nx∇T contribution is not relevant.
3.1. Generation of an axial magnetic field in plasma
The generation of an axial magnetic field in plasma by a circularly
(or elliptically) polarized laser is often referred to as the inverse
Faraday effect (IFE). Theoretically described by Steiger and Woods
(1972), results from the features of the
electron motion in a circularly polarized electromagnetic wave. During
the interaction of the plasma electrons with the circularly polarized
laser pulse, electrons absorb both the laser energy and the angular
momentum of the laser pulse. In particular, the angular momentum
absorption leads to the electron rotation and generation of the axial
magnetic field by the azimuthal electron current. Naturally, IFE does
not occur for a linearly polarized laser pulse since it does not
possess any angular momentum. IFE has since been measured in several
experiments (Horovitz et al., 1997;
Lehner, 2000: Najmudin et al.,
2001). The conditions under which IFE is possible are still not
fully explored. What is theoretically known (Abdullaev & Frolov,
1981) is that there is no magnetic-field generation during the
interaction of the inhomogeneous circularly polarized electromagnetic
waves with the homogeneous plasma. Magnetic field can be produced in
the presence the strong plasma inhomogeneity either preformed or
developed self-consistently during the interaction. Recently, a
mechanism has been proposed for the generation of an axial magnetic
field through the transfer of the spin of the photons during the
absorption of a transversely nonuniform circularly polarized radiation
(Haines, 1997). The magnetic field thus
generated has a magnitude proportional to the transverse gradient of
the absorbed intensity and inverse proportional to the electron
density, the latter sealing being in contrast to earlier theories of
IFE (Haines, 1997).
Along the same lines as Haines (1997) and
Davies et al. (1997), it has been
demonstrated that in laser-plasma interactions, strong axial magnetic
fields can be generated through angular (spin) momentum absorption by
either electron-ion collisions or ionization.
Another mechanism of magnetic field generation has been proposed by
Kostyukov et al. (2002), in
framework of classical electrodynamics and Karas‘ et al.
(2003), based on the resonant absorption by
energetic electrons of a circularly polarized laser pulse. The
resonance occurs between the fast electrons, executing transverse
(betatron) oscillations in a fully or partially evacuated plasma
channel, and the electric field of the laser pulse. The betatron
oscillations are caused by the action of the electrostatic force of the
channel ions and self-generated magnetic field. This type of resonant
interaction was recently suggested as a mechanism for accelerating
electrons to highly relativistic energies (Kostyukov
et al., 2002). When a circularly polarized laser pulse
is employed, its angular momentum can be transferred to fast resonant
electrons along with its energy. The resulting electron beam spirals
around the direction of the laser propagation, generating the axial
magnetic field. The intensity of the magnetic field generated in
relativistic laser channel was calculated taking into account
self-generated static fields, which are neglected in known IFE theories
(Kostyukov et al., 2002).
These calculations are in agreement with the recent experiments at
the Rutherford Appleton Laborato-ry (RAL) which exhibited very large
(several megagauss) axial magnetic fields during the propagation of a
sub-picosecond laser pulse in a tenuous plasma (Borghesi et
al., 1998a,b). The relevant aspect of the RAL
experiment is that both fast electrons and the strong magnetic field
were measured in the same experiment.
3.2. Nonrelativistic two-dimensional treatment of
self-generated magnetic fields
In an under dense longitudinal inhomogeneous plasma, the
nonrelativistic 2D treatment of self-generated magnetic fields was
presented by Tripathi and Liu (1994), showing
that a laser beam propagating along a plasma gradient produces a
rotational current which gives rise to a quasi-static magnetic field.
An analogous mechanism was considered earlier, whereas the nonlinear
mixing of two electromagnetic waves in nonuniform plasma was discussed
by Bychenkov et al. (1994). They
investigated a circularly polarized pulse for which the generation of
low-frequency electromagnetic field is due to the inverse Faraday
effect. In the extremely strong relativistic regime, the magnetic field
generated by the laser beams in under dense plasma was recently studied
numerically by Gorbunov et al. (1996). The main objective of the work by Gorbunov
et al. (1996) was to investigate
self-generated quasi-static magnetic fields both in the laser pulse
body and behind the pulse in the region of the wake-field. These
authors treated the laser radiation as linearly polarized and the
plasma as uniform and under dense. The analytical work was based on a
perturbation theory applied to a set of relativistic cold electron
fluid equations and Maxwell's equations. The quasi-static magnetic
field generated by a short laser pulse in uniform rarefied plasma is
found analytically and compared to 2D particle-in-cell simulations.
It is shown that a self-generation of quasi-static magnetic fields
takes place in fourth order with respect to the parameter
ve /c, where
ve and c are the electron quiver
velocity and the light of speed, respectively. In the wake region, the
magnetic field possesses a component which is homogeneous in the
longitudinal direction, and is due to the steady current produced by
the plasma wake-field and a component which is oscillating at the wave
number 2kp, where
kp is the wave number of the plasma wake,
a known property of nonlinear plasma waves. Numerical particle
simulations confirm the analytical results and are also used to treat
the case of high intense laser pulses with
ve /c > 1. The resultant
magnetic field has a focusing effect on relativistic electrons in the
plasma wake-field accelerator context.
3.3. Magnetic fields in the interaction of an ultra
intense short laser pulse
We will now discuss the physical mechanism for generation of very
high “quasi-static” magnetic fields in the interaction of
an ultra intense short laser pulse, with an over dense plasma target
owing to the spatial gradients and non-stationary character of the
ponderomotive force. Numerical (PIC) simulations by Wilks et
al. (1992) of the interaction of an
ultra intense laser pulse with an over dense plasma target have
revealed non-oscillatory self-generated magnetic fields up to 250 MG in
the over dense plasma, that this nonoscillatory magnetic field around
the heated spot in the center of the plasma, the magnetic field
generation being attributed to the electron heating at the
radiation-plasma interface.
The spatial and temporal evolution of spontaneous megagauss magnetic
fields, generated during the interaction of a picosecond pulse with
solid targets at irradiances above 5 × 1018
W/cm2 have been measured using Faraday rotation with
picosecond resolution, the observations being limited to the region of
under dense plasma and after a laser pulse (Fig.
1), see Borghesi et al.,
1998a. A high density plasma jet has been observed
simultaneously with the magnetic fields by interferometer and optical
emission. Because of the high temporal resolution of the probe
diagnostic, a quantitative measurement of the transient nature of the
fields has been obtained for the first time. Interestingly, no Faraday
rotation was detected immediately after the interaction, a possible
reason being that the fields were still limited to regions not
accessible for probing. After 5 ps the typical signatures corresponding
to a toroidal field surrounding the laser axis (i.e., a dark and a
bright pattern on opposing sides of the axis, in the proximity of the
target surface) began to appear. In papers by Clark et al.
(2000) and McKenna et al. (2003), the first direct measurements of high energy
proton generation (up to 18 MeV and 100 MeV, respectively) and
propagation into a solid target during such intensive laser-plasma
interactions were reported.
Magnetic field distribution extracted from the polarigram. The
magnitude is in units of megagauss. The plasma region either obscured
by self-emission or not accessible for probing is shown (From Borghesi et al., 1998a).
The sense of rotation is the same as observed in previous
measurements in longer pulse regimes and is consistent with fields
generated by the thermoelectric mechanism (see for example Haines
(1997)).
Measurements of the deflection of these energetic protons were
carried out which imply that magnetic fields in excess of 30 MG exist
inside the target. The structure of these fields is consistent with
those produced by a beam of hot electrons which has also been measured
in these experiments. Such observations are the first evidence of the
large magnetic fields which have been predicted to occur during
laser-target interactions in dense plasma (Najmudin
et al., 2001). The intensity on target of laser
radiation was up to 5 × 1019 W/cm2 and
was determined by simultaneous measurements of the laser pulse energy,
duration, and focal spot size. Borghesi et al. (2002) proposed that due to their particular
properties, the beams of the multi-MeV protons generated during the
interaction of ultra intense (I > 1019
W/cm2) short pulses with thin solid targets are most
suited for use as a particle probe in laser-plasma experiments. The
recently developed proton imaging technique employs the beams
in a point-projection imaging scheme as a diagnostic tool for the
detection of electric fields in laser-plasma interaction experiments.
In recent investigations carried out at the RAL, a wide range of
laser-plasma interaction conditions of relevance for inertial
confinement fusion (ICF)/fast ignition has been explored. Among the
results obtained will be discussed: the electric field distribution in
laser-produced long-scale plasmas of ICF interest; the measurement of
highly transient electric fields related to the generation and dynamics
of hot electron currents following ultra-intense laser irradiation of
targets; the observation in under dense plasmas, after the propagation
of ultra-intense laser pulses, of structures identified as the remnants
of solitons produced in the wake of the pulse.
Numerical (PIC) simulations by Wilks et al. (1992), the interaction of an ultra intense laser
pulse with an over dense plasma target have revealed non-oscillatory
self-generated magnetic fields up to 250 MG in the over dense plasma,
that this non-oscillatory magnetic field is generated around the heated
spot in the center of the plasma, the magnetic field generation being
attributed to the electron heating at the radiation-plasma interface.
The spatial and temporal evolution of spontaneous megagauss magnetic
fields, generated during the interaction of a picosecond pulse with
solid targets at irradiances above 5 × 1018
W/cm2 have been measured using Faraday rotation with
picosecond resolution, the observations being limited to the region of
under dense plasma and after a laser pulse (see Borghesi et
al., 1998a,b). A high density plasma jet has been
observed simultaneously with the magnetic fields by interferometry and
optical emission and a field value is consistent with field generated by
the thermoelectric mechanism (see for example, Haines,
1997). In paper by Clark et al. (2000) the first direct measurements of high-energy
proton generation (up to 18 MeV) and propagation into a solid target
during such intense (5 × 1019 W/cm2)
laser plasma interactions were reported. Measurements of the deflection
of these energetic protons were carried out which imply that magnetic
fields in excess of 30 MG exist inside the target. Batishchev et
al. (1994) solved numerically the problem
of high-intensity, linearly polarized electromagnetic pulse incident onto
a collision less plasma layer in a Cartesian coordinate system in a
2.5-D formulation (z is the cyclic coordinate and there are
three components of the momentum) by means of COMPASS (COMputer Plasma
And Surface Simulation) code. The recent review by Balakirev et
al. (2002) and references therein
combine detailed information concerning COMPASS code as well as its
possibilities and applications. A general advantage of the complete
numerical simulation consists of the possibility of obtaining all
necessary information concerning spatial and temporal dynamics of both
particles and self-consistent electromagnetic fields, without requiring
additional data (reflection and absorption coefficients, changes of
either plasma temperature or different plasma parameters) for a given
situation concerning the interaction of an intensive electromagnetic
pulse with plasmas. We give only the external parameters, both the
initial and boundary conditions for particles and fields, and as
results of a numerical simulation we attain all characteristics of the
plasma together with pulsed self-consistent electromagnetic fields. The
most characteristic feature of the action of an intense, normally
incident electromagnetic pulse onto ultrahigh-density plasma consists
in a “well-digging” effect. Worth nothing is the
time-growing sharp nonuniformity of the perturbed plasma layer in the
transverse direction. As for the magnetic field, we do not observe a
change of its direction, but a significant time-variation of its
strength varies significantly in time. Hence, the magnetic field cannot
be considered as quasi-static because it varies by more than an order
of magnitude over a time of
2πωpe−1. Thereby it is
shown that the magnetic field oscillates with the doubled frequency of
a laser radiation, but it has unchanged direction. The magnetic field
oscillations are seen as good at Fig. 2. Now
let us look at Figs. 2a and 2b, which present
two instantaneous magnetic field Bz
distributions separated in the time by
πωpe−1 (a plasma wave half-period).
It will be noted that a maximum of magnetic field (1.1) in the point
(38,31) in the time tωpe = 200 (see Fig. 2a) after a plasma wave half-period (see Fig. 2b) replaced in this point very low value of
magnetic field.
(a) Spatial distribution of magnetic field
Bz at different times:
tωpe = 200 (from Batishchev et al., 1994). (b)
Spatial distribution of magnetic field Bz
at different times: tωpe = 200 +
p (from Batishchev et al.,
1994).
The magnetic field Bz attains its peak
value of (0.64) at the point (36.5,29), cf. Fig.
2b. It is shown that the magnetic field
Bz is consistent with a linearly modulated
current flowing along the line y =
Ly/2. We do not observe this field to
change its direction, but its strength varies significantly in time,
Hence, the magnetic field cannot be considered as quasi-static because
it varies by more than an order of magnitude over a time of
2πωpe−1.
The magnitude of direct current (dc) magnetic field is ten
times as low as the maximum magnetic field. One should note that in the
articles by Wilks et al.(1992) and
Batishchev et al. (1994), the
numerical simulation has been made under very optimal conditions: a
uniform plasma density makes it sure an own plasma oscillation
resonance with a longitudinal modulation density of particles in a wave
as well as a maximum frequency of nonlinear Tomson scattering
spectrum.
It is important to note, that in a recent article by Baton et
al. (2003), have place direct experiment
evidence of accelerated electron bunches separated by half the period
of the laser light at irradiation of thick solid targets by laser beam
at relativistic intensities. In Borghesi et al. (1998a) experiments, instead, a plasma
inhomogeneity is very essential, with the result that resonant
conditions are fulfilled only in a small plasma region. Subsequently to
the interaction pulse, only the “dc” magnetic
field exists, as measured in the underdense plasma region in the
article by Borghesi et al. (1998a). On the basis of the formula:
where I is the intensity of the incident laser radiation;
one obtains a “dc” magnetic field magnitude of few
MG for the experimental parameters of Borghesi et al. (1998a), and a few tens of MG for the
experimental conditions of Clark et al. (2000). A difference still on order of value is
conditioned that at such intensities only 10% of the incident laser
radiation is absorbed.
3.4. 2.5-D numerical simulation on the macroparticles
method
By means of a 2.5-D numerical simulation on the macroparticles
method, it is possible to find the magnetic field spatial and temporal
distribution without making use of an adapted parameter, in contrast
with the conventional ∇nx∇T mechanism
(see for example Haines, 1997). On the other
hand, the theoretical model for the generation of a magnetic field
proposed by Sudan (1993) does not appear to
be appropriate, this model being valid for a very large ratio of plasma
density to critical density and when the
∇nx∇T contribution is not relevant.
