PHYSICS OF PLASMAS 15, 072114 共2008兲
Quasielectrostatic whistler wave radiation from the hot electron emission
of a laser-produced plasma
Stephen Vincena,1,a兲 Walter Gekelman,1 M. A. Van Zeeland,2 James Maggs,1 and
Andrew Collette1
1
Department of Physics and Astronomy, University of California at Los Angeles,
Los Angeles, California 90095, USA
2
General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA
共Received 30 January 2008; accepted 20 June 2008; published online 21 July 2008兲
Measurements are presented of radiated wave electric fields which result from the creation of a
dense, laser-produced plasma within a large, uniform background magnetoplasma. The radiated field
patterns are consistent for waves propagating along the quasielectrostatic branch of the whistler
wave dispersion curve calculated from the background plasma parameters. The energy source of
these waves coincides with an observed energetic tail electron population escaping the
laser-produced plasma. A prominent feature of the radiated electric fields is a bipolar spike in both
time and space, with a cross-field size near that of the initial escaping electron burst and a duration
equivalent to one oscillation at the lower hybrid frequency within the background plasma.
Additionally, time-windowed snapshots of the whistler wave radiation patterns are shown to provide
a remote diagnostic of the cross-field speed of the laser-produced plasma. © 2008 American
Institute of Physics. 关DOI: 10.1063/1.2956994兴
I. INTRODUCTION
The pervasive extent of ionized matter in the universe
guarantees the interaction of relatively dense, energetic plasmas with their distinct surrounding plasma environments.
Examples include coronal mass ejections,1 galactic jets,2
cometary surface ionization,3 and supernovas.4 Man-made
versions include fuel-pellet injection in tokamaks,5 chemical
releases in the ionosphere,6 and high-altitude nuclear
detonations.7 The interaction of two disparate plasmas is thus
of general interest to many subfields of plasma physics.
As a method for the creation of localized, dense, and
energetic plasmas, solid-target irradiation by high-power lasers is becoming commonplace. Detailed studies of the creation of laser-produced plasmas 共LPPs兲 have been made for
plasma expansion into vacuum and neutral gases, both with
and without a background magnetic field present.8–12 Although the studies of the LPP in isolation or with neutral
gases are quite mature, the presence of a second, uniform
plasma which surrounds the LPP opens up the study of how
the impulsive LPP event couples its mass and energy to the
surrounding plasma environment. One example of such coupling is the Coulomb shielding effect of the background
charge carriers which permits the spatial separation of LPP
ions and electrons. The escaping field-aligned burst of energetic LPP electrons and subsequent return current system
have been shown, for example, to excite Alfvén waves in the
background plasma.13,14 Such a coupling was possible since
the majority of escaping field-aligned electrons had a beam
velocity close to the phase velocity of the Alfvén waves.
Escaping electrons with greater energies may, of course, give
rise to other types of plasma waves with higher phase velocia兲
Electronic mail: vincena@physics.ucla.edu; http://plasma.physics.ucla.edu.
1070-664X/2008/15共7兲/072114/14/$23.00
ties, and this hypothesis is the motivation for the present
investigation.
Electron beams have long been associated with the production of quasielectrostatic whistler waves 共also known as
lower hybrid waves兲 in both laboratory15,16 and space
plasmas.17–20 In fact, these waves were among the first radiofrequency waves detected by satellites 共see Ref. 21, and references therein兲. Their characteristic “V” or hyperbolic shape
in spacecraft electric field spectrograms earned them the
moniker, very low frequency 共VLF兲 saucers. The shape is a
consequence of the wave propagation along ray paths from
localized, broadband sources in the lower ionosphere. These
VLF saucers continue to be studied today22 since the origin
of the electron bursts themselves is not well understood.
However, given the existence of the electron bursts, analytical models19 and simulations23 predict the generation of
whistler waves, both electromagnetic and electrostatic.
The inverse process, that of electron acceleration by
quasielecrostatic whistlers, has been invoked to explain the
observations of x-ray emissions from comets and supernova
remnants.24–26 These waves have also been observed in the
spontaneously generated magnetic fields during nonuniform
laser irradiation of solid targets;27 as such they remain confined to the local magnetic fields.
The present and unique study is an investigation of
quasielectrostatic whistler waves generated by fast electrons
escaping a laser-produced plasma; these waves are then free
to propagate into a large, uniform background magnetoplasma.
The remainder of this manuscript is organized into the
following sections: a review of the salient properties of LPPs
and of quasielectrostatic whistler waves in the background
plasma are, respectively, presented in Secs. II A and II B; the
details of the experimental setup and diagnostics are pro-
15, 072114-1
© 2008 American Institute of Physics
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-2
Phys. Plasmas 15, 072114 共2008兲
Vincena et al.
t=0
t>0
In order to interpret the data, a brief review of the properties of quasielectrostatic whistler waves is helpful. For
more detailed information, the reader may wish to refer to
some of the pioneering experimental38–42 and theoretical43–45
works on this subject. For this discussion, we will use notation appropriate for propagation in the xz-plane, though it
should be kept in mind that these concepts apply equally
well to cases involving cylindrical symmetry. Quasielectrostatic whistler waves are, as their name implies, primarily
electrostatic plasma oscillations. They propagate at angular
frequencies above the lower hybrid frequency ␻LH and below the smaller of the electron plasma ␻ pe or electron cyclotron frequency ␻ce. The lower hybrid frequency is defined by
2
2
␻LH
⬅ ␻ce␻ci / 共1 + ␻ce
/ ␻2pe兲 with ␻ci being the angular ion cyclotron frequency. Their cold-plasma dispersion relation46
may be written as
laser beam
Target
LPP Ions
E
LPP elec.
rb
Background elec.
B. Quasielectrostatic whistler waves
A. Properties of the laser-produced plasma
Background elec.
II. REVIEW
Background
Plasma
LPP elec.
vided in Sec. III; the experimental results and commentary
are given in Sec. IV; finally, concluding remarks are in Sec.
V. Throughout this manuscript, Gaussian units are used for
all equations, although specific numbers may be quoted in
other units for convenience.
lated with an average perpendicular expansion speed of v⬜
= 1.4⫻ 107 cm/ s which corresponds to a kinetic energy
about half ELPP. These numbers are consistent with experimentally derived scaling laws and previous experiments citing laser-to-plasma kinetic energy conversion efficiencies as
large as 90% at these laser fluences.8,32,33 Both the diamagnetic bubble scaling and ambipolar field have been observed
in previous experiments where the expansion occurred into
vacuum11 and a background gas.9,12,28 For example, in the
present experiment, with a laser energy of 1.5 J, we take
ELPP = 0.75 J, and with a magnetic field of 1.0 kG, the bubble
radius is 3.6 cm.
Cross-field motion of the LPP can be maintained10 in a
magnetic field at the velocity of the initial plasma expansion
because of LPP polarization and subsequent E ⫻ B drift;
cross-field motion by compact, but nonlaser-produced plasmas, or “plasmoids” has also been well established.34–36
On a much grander scale, Litwin and Rosner37 have
theorized that in the process of planetoid impacts onto magnetized neutron stars, the planetoid becomes ionized and also
undergoes an E ⫻ B drift. Although the mechanism for generating the electric field is different, one result is similar:
outside of the plasmoid, there is a stray electrostatic field
with a component directed parallel to the ambient magnetic
field which can accelerate particles along those field lines. In
the astrophysical case, the authors conclude that this process
may explain the origin of some the highest energy
共⬎1020 eV兲 cosmic rays.
Target
x
z
Magnetic field
FIG. 1. Simple cartoon of the initial cross-field, laser-plasma expansion into
a background plasma. Shown is a cross section of a cylindrical, solid target.
The space is filled with a background plasma. Left panel, the laser pulse is
incident on the target from the left. Right panel, the LPP expands outward
with unmagnetized ions initially dragging electrons across the field. Simultaneously electrons are escaping down field lines 共escaping current兲 and
being replaced by background electrons 共return current兲. Cross-field neutralization is then accomplished by background ions via polarization drift or
electron-ion collisions. The magnetic bubble radius is rb and E is an ambipolar electric field.
When a laser-produced plasma expands across a background magnetic field in vacuum it goes through several
phases. A cartoon version is shown in Fig. 1. The initial laser
impact results in the immediate ionization of surface atoms
and a burst of fast electrons which rip ions from the target
surface due to a large ambipolar field. Generally, however,
the more massive ions hold back the electrons and eventually
overshoot them due to their relatively unmagnetized state.
This creates a radially inward directed ambipolar field which
in turn causes an electron E ⫻ B drift and, in conjunction
with ⵜP ⫻ B currents, the observed laser-plasma
diamagnetism.13 The LPP expands as a diamagnetic
cavity11,12,28 until, roughly, the kinetic energy of the LPP,
ELPP, equals the expelled background magnetic field energy,
ELPP = 34 ␲r3bB20 / 8␲, where rb is commonly referred to as the
“bubble” radius which upon rearrangement is
rb =
冉 冊
6ELPP
B20
1/3
.
When a background plasma is present, the situation becomes
more complicated because the background electrons can
shuttle along field lines and partially short out the initial
large ambipolar field, allowing laser-plasma electrons to escape. Prompt electrons have been measured in unmagnetized
laser-target experiments in vacuum29,30 and partial vacuum.31
In this experiment, Faraday cup measurements in
vacuum have shown approximately 2.5⫻ 1015 ions are ab-
␻ 共kx,kz兲 =
2
2
␻LH
+
2
␻2pe␻ce
共␻2pe
+
2
␻ce
兲
冉冊
kz
kx
2
,
where the wave vector k = kk̂ makes an angle ␪ with respect
to the background magnetic field vector B0 = B0ẑ so that the
parallel component is kz = k · ẑ = k cos ␪ and the perpendicular
component is kx = k · x̂ = k sin ␪. This dispersion relation is appropriate for the frequency regime sampled by the present
experiments, ␻LH 艋 ␻ Ⰶ ␻ce ⬍ ␻ pe. For brevity, we will refer
to the quasielectrostatic whistler waves propagating in the
above frequency range as lower hybrid waves. Using the
dispersion relation, the group velocities parallel vgz and per-
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-3
Phys. Plasmas 15, 072114 共2008兲
Quasielectrostatic whistler wave radiation…
Field-aligned
electrons
ex
B0 , z^
x^
vg
z
kx
α
kz
k
θ
Source
vgx =
冏 冏 冉
冏 冏 冉
⳵␻
⳵kx
=
kx
冊
2
␻ ␻2 − ␻LH
,
kz
␻2
=−
kz
共1兲
冊
2
␻ ␻2 − ␻LH
.
kx
␻2
共2兲
The minus sign in vgx indicates that the wave is backward
propagating 共with respect to the phase velocity兲 in the perpendicular direction.
For a point source of lower hybrid waves located at the
origin 共see Fig. 2兲, a measurement made at the location 共x , z兲
will detect those waves whose group velocity vector makes
an angle defined by tan ␣ = x / z with respect to B0. For propagation in a uniform plasma after a time ⌬t, a ray will travel
from the source a distance x = vgx⌬t across the field and z
= vgz⌬t along the field. Using the above expressions for the
group velocities, the time ⌬t cancels out, and the group velocity angle is then given by tan ␣ = −kz / kx. Equating the two
expressions for tan ␣, and inserting into the dispersion relation, the expected frequency as a function of the observation
point is
2
␻2共x,z兲 = ␻LH
+
2
␻2pe␻ce
冉冊
x
2
2
共␻ pe + ␻ce兲 z
2
.
f ce
冑1 + 共f ce/f pe兲2
冋 冉 冊册
me
x
+
mi
z
2 1/2
,
共4兲
III. EXPERIMENTAL SETUP
pendicular vgx to the background magnetic field are
⳵␻
⳵kz
f共x,z兲 =
where the positive root has been taken, me / mi is the electronto-ion mass ratio, and the frequencies are now expressed in
Hz. Except near x = 0, the frequency may be approximated as
f ⬇ f ce 兩 x / z兩.
FIG. 2. Illustration of the wave and group velocity vectors. For simplicity,
the propagation is drawn towards positive x and positive z, whereas in the
experiment the propagation is towards negative z and both positive and
negative values of x.
vgz =
fixed z, a probe measuring the wave electric field as a function of x will detect different frequencies according to Eq.
共3兲, with the minimum frequency of ␻LH at x = 0 and increasing frequency with increasing 兩x兩. Strictly speaking, the
group velocities calculated using cold plasma theory go to
zero at the lower hybrid frequency, though kinetic effects
may smooth out a true resonance and instead set a minimum
group velocity rather than zero.
In terms more transparent to the present experimental
observables and variables, the dispersion relation 关Eq. 共3兲兴
can be rewritten as
共3兲
Note that if the source is monochromatic, then there is only
one angle along which the waves will be observed; these are
the well-known lower hybrid resonance cones, so called
since the propagation is actually along cones in three dimensions. If, however, the source is broadband, then there are a
continuum of nested cones, each with its own frequency. For
A. Large plasma device
These experiments are performed at the Basic Plasma
Science Facility, in the upgraded Large Plasma Device
共LaPD兲,47 located at the University of California, Los Angeles. The device 共shown schematically in Fig. 3兲 is a stainless
steel cylindrical vacuum chamber 共total length= 20.7 m and
main diameter= 1 m兲. The main section of the chamber is
surrounded by solenoidal electromagnets which produce a
confining axial magnetic field, variable to 3 kG. The device
is backfilled with noble gases 共here both neon and argon as
noted兲 at pressures of approximately 10−5 Torr. The plasma
is produced by means of a pulsed 共t ⬇ 10 ms兲 electron discharge between a heated, barium-oxide-coated nickel cathode and a molybdenum mesh anode. The cathode-anode
separation is 52 cm, leaving the remainder of the device with
zero net current and quiescent compared to the source region.
The highly reproducible discharge is repeated once per second which allows for the collection of ensemble datasets.
Table I summarizes key parameters for the four experimental
setups where the magnetic field, background plasma species,
and target material are varied.
B. Laser-target system
The laser is a commercially available 共Spectra Physics,
Quanta Ray Pro兲 Q-switched Nd:YAG infrared 共1064 nm兲
laser with 1.5 J/pulse and 8 ns duration 共Gaussian, full width
at half maximum兲. The laser target is a solid, high-purity rod
共30 cm long by 1.9 cm diameter兲 composed of either aluminum or graphite, as noted. The rods are inserted through
vacuum interlocks on top of the device 共see Fig. 3兲. Only one
type of rod at a time is placed in the chamber. When different
targets are used, the origin of the experimental coordinate
system always corresponds to the laser impact site for that
target. The pulsed beam is steered with mirrors and focused
using a plano-convex lens. The beam then enters the device
through a side window. All optical surfaces are coated with
antireflective coatings. When the beam reaches the target, the
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-4
Phys. Plasmas 15, 072114 共2008兲
Vincena et al.
1660 cm
32cm
laser
laser
B0 ,z
y
x
probes
100 cm
chamber
Ni
cathode
Al
target
(not to scale)
Mo
mesh
anode
256cm
60 cm
plasma
C
256cm
target
(not to scale)
Cu
terminating
mesh
FIG. 3. 共Upper兲 mechanical drawing of the large plasma device showing the vacuum chamber, magnets, and access ports. 共Lower兲 schematic of the
experimental arrangement of lasers, targets, probes, and plasma boundary conditions. The chamber is grounded, while the copper terminating mesh and
cathode-anode circuit are independently electrically floating.
spot size has a diameter of 0.5 mm. The rods are held on one
end with an electrically insulated adapter which joins them to
a 50 cm long, 0.95 cm diameter Inconel rod. Outside of the
vacuum chamber, the Inconel rod is mated to a two-axis
stepper motor system which allows the target rod to be rotated about its axis and translated in the y-direction. Except
as noted, the laser hits a fresh face of the target every five
plasma discharges; after five laser firings, a hole begins to
form on the target surface and the amplitude of measured
magnetic and electric field oscillations in the plasma begins
to be noticeably reduced.
C. Probes and data acquisition
The primary diagnostics for this experiment are a Langmuir probe and an electric dipole probe. The probes are connected to separate, two-axis stepper motor systems which
allow motion of the probe in the xy-plane under the control
of a custom, computerized data acquisition system.
A single-sided, planar Langmuir probe is used to measure ion saturation current. The collection area is
1.5⫻ 10−2 cm2 and is directed with the normal vector of the
face along the direction of the magnetic field. The probe is
biased −70 V relative to the grounded vacuum chamber and
the cathode-anode voltage drops to typically −40 V during
the main plasma discharge; the probe thus excludes all primary electrons and serves as a diagnostic of both ion saturation current and very energetic electrons produced by the
laser. The probe is also separately used in a swept-voltage
mode to determine the background electron temperature of
6 eV. The probe density measurements are calibrated using a
56 GHz microwave interferometer.
Fluctuating electric fields are measured using a dipole
probe with tip separation of 3 mm radially along the probe
shaft. Such probes have previously been used to measure
lower hybrid wave patterns in LaPD launched from an antenna at fixed frequencies.48 Along the line y = 0, the probe
measures the x-component of the electric field. The dipole is
displaced by 0.5 cm in the +ŷ direction to avoid overlap with
the probe support structure. This support structure is a ceramic tube with length 25 cm and diameter 0.64 cm. The
ceramic tube 共which is also used on the Langmuir probe兲 is
mated to a 0.95 cm diameter, nonmagnetic, stainless steel
shaft. Probes are introduced into the plasma chamber using a
system of vacuum interlock valves and cyrogenic pumps.
The dipole probe time-series data are amplified using a Stanford Research Systems model SR445 amplifier, then subtracted using a Minicircuits ZSCJ-2-1 inverting combiner,
and sampled at a rate of 1 GHz with a Tektronix TVS645A
digitizer. The bandwidth of the system is roughly from
1 to 200 MHz. Since the probe tip separation is much larger
than the Debye length 共⬃20 ␮m兲 plasma shielding prevents
a true calibration of the electric field measurements and the
reported magnitudes should be treated as a lower limit.
TABLE I. Experimental parameters for the several configurations used in these experiments. Plasma parameters
are those of the background plasma. The initial laser energy for each is equal to 1.5 J and the conversion
efficiency to plasma kinetic energy is 50%. The size of the diamagnetic cavity is given in terms of the bubble
radius rb as described in the text.
Setup
No.
Target
element
Background
ion species
B0
共Gauss兲
rb
共cm兲
ne
共⫻1012 cm−3兲
␻ pe / 2␲
共GHz兲
␻ce / 2␲
共GHz兲
␻LH / 2␲
共MHz兲
1
2
3
4
Al
Al
Al
C
Ne
Ne
Ne
Ar
750⫾ 15
1000⫾ 20
1500⫾ 30
400⫾ 8
4.3
3.6
2.7
6.6
2.0⫾ 0.2
2.0⫾ 0.2
2.0⫾ 0.2
1.0⫾ 0.25
12.7⫾ 0.6
12.7⫾ 0.6
12.7⫾ 0.6
9.0⫾ 1.1
2.1⫾ 0.04
2.8⫾ 0.06
4.2⫾ 0.08
1.12⫾ 0.02
10.8⫾ 0.2
14.3⫾ 0.3
20.8⫾ 0.4
4.1⫾ 0.1
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-5
Phys. Plasmas 15, 072114 共2008兲
Quasielectrostatic whistler wave radiation…
0.4
( A / cm 2 )
0.2
0.0
-0.2
z = -352 cm
z = -576 cm
-0.4
-0.6
0
1
2
3
4
5
6
time (μs )
FIG. 4. Energetic electron detection during ion saturation current collection.
For time-of-flight measurements, the vertical lines mark the centroids of the
negative pulses with the same line style. The data are acquired under the
conditions of setup #3.
IV. EXPERIMENTAL RESULTS
Using the same type of laser as in the present experiments,
Issac et al.31 have measured initial bursts of electrons with
energies of 60⫾ 5 eV. In previous experiments,14,50 it is established that bursts of electrons which last for times on the
order of the diamagnetic cavity lifetime 共⬃1 ␮s兲 and have
mean speeds above the background Alfvén speed vA
= B0 / 冑4␲nimi = 5.2⫻ 107 cm/ s are responsible for the production of Alfvén waves via Cherenkov radiation. The
prompt detection of much higher frequency waves suggests a
similar process may occur for an energetic tail electron population which couples to a plasma wave with a much higher
phase velocity.
Figure 5 shows the raw, nonaveraged time series for five
plasma discharges from the ion saturation-biased probe for
early times. The measurements are made at a location close
to the target 共z = −34 cm兲 and field-aligned with the laser
impact site 共x , y兲 = 共0.0, 0.0兲 cm, again for setup #3. Any jitter in the experimental triggering is less than 10 ns. Although
there is shot-to-shot variation, there are three common features:
1.
A. Energetic electrons
One of the key features of the interaction of an energetic
LPP with another background plasma is the ability of the
LPP ions and electrons to become physically separated from
each other due to the shielding effect of the background
plasma. In the present case, magnetized LPP electrons can
escape along the background magnetic field as they are replaced by thermal, ambient plasma electrons.
Figure 4 shows time series of the ion saturation current
measured at two axial distances from the aluminum target
with B0 = 1500 G 共setup #3 of Table I兲. The solid curve is
recorded at z1 = −352 cm downstream from the target and the
dashed curve at z2 = −576 cm. In the plane transverse to the
background magnetic field, the probe was located at 共x , y兲
= 共−1.0, 0.0兲 cm which lies within the field-line projection of
the diamagnetic cavity. For any setup, the origin in space is
the laser focal point, and t = 0 marks the arrival time of the
peak of the 8 ns Gaussian laser pulse at the target surface,
corrected for all cable delays. High-frequency oscillations 共to
be discussed later兲 are evident at early times 共t ⬍ 1 ␮s兲 and
much lower frequency density fluctuations are observed at
later times 共t ⬎ 2 ␮s兲. The latter have been presented
previously49 and are attributed to evanescent compressional
Alfvén waves. At intermediate times, the large negative signals indicate the presence of electrons with energies sufficient to overcome the 70 V probe bias, which corresponds to
an electron streaming velocity ue of 3.5⫻ 108 cm/ s. The bias
is with respect to the anode of the discharge circuit which is
at plasma potential. The background electron temperature Te
is 6 ⫾ 1 eV with a thermal velocity vth,e = 冑Te / me of 1
⫻ 108 cm/ s. The time-of-flight velocity 共using the centroid
of the negative pulse兲 from the target to z1 is 3.2
⫻ 108 cm/ s and for z2 is 3.4⫻ 108 cm/ s. The centroids for
each curve are marked by vertical lines having the same line
style. The propagation speed between the two locations is
consistent at 3.7⫻ 108 cm/ s. It is also consistent with ue.
2.
3.
A negative peak at 300⫾ 50 ns which we attribute to
energetic electrons. The electron velocity from time of
flight for the peak at 300 ns is 1.1⫻ 108 ⫾ 1.8
⫻ 107 cm/ s. This peak has roughly one-third the velocity of the peaks shown in Fig. 4, but here, the probe is
field-aligned with the laser impact site, whereas for Fig.
4, the probe is displaced 1 cm back along the laser path.
A discussion of the spatial distribution of the energetic
electrons is presented later in conjunction with its impact on the lower hybrid radiation patterns.
A broad distribution of faster electrons with the earliest
deviation from ion collection at 25⫾ 15 ns, which requires electrons with velocities of 1.3⫻ 109 ⫾ 2.3
⫻ 108 cm/ s, albeit with very low current densities.
High-frequency oscillations riding on top of this fast-tail
electron distribution. The period of the earliest such
cycle 共which occurs within the first 100 ns兲 is
50⫾ 10 ns, which yields a frequency of 20⫾ 4 MHz.
This matches the lower hybrid frequency in the background plasma which is 20.8⫾ 0.4 MHz.
Similar results are obtained using the carbon target in setup
#4: field-aligned with the laser impact site, the bulk of electrons are emitted with a beam velocity of 1.3⫻ 108 cm/ s
with tail electrons observed in excess of 1 ⫻ 109 cm/ s. For
both targets, the signature of the fastest electrons was difficult to measure beyond an axial distance of 32 cm due to
large electrostatic fluctuations.
B. Spatial amplitude distributions
of the wave electric field
We next present the amplitude of electric field fluctuations, measured using the dipole probe, as a function of x at
a fixed axial displacement 共z = −256 cm兲 from the laser impact site of the aluminum target. The arrangement is illustrated in Fig. 3 and corresponds to setup #3 in Table I. The
amplitude spectra of the fast-Fourier-transformed signals
Ex共f兲 are presented in Fig. 6 for three spatial locations: x
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-6
Phys. Plasmas 15, 072114 共2008兲
Vincena et al.
A/cm 2
s hot 1
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-200
x=7.45 cm
x=15.3 cm
x=23.1 cm
0
200
400
t (ns )
600
800
1000
A/cm 2
s hot 2
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-200
0
200
400
t (ns )
600
800
1000
A/cm 2
s hot 3
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-200
0
200
400
t (ns )
600
800
1000
A/cm 2
s hot 4
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-200
0
200
400
t (ns )
600
800
1000
A/cm 2
s hot 5
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-200
0
200
400
t (ns )
600
800
1000
FIG. 5. Earliest energetic electron detection during ion saturation current
collection. Taken at x = 0, y = 0, z = −32 cm. The data are acquired under the
conditions of setup #3.
= 关7.45, 15.3, 23.1兴 cm. Each spectrum is formed from an ensemble of 25 plasma discharges. A 128-point, Hanningwindowed section of the full time series is used. After signal
propagation delays through the measurement cables are removed, the window starts 150 ns after the target is struck by
the laser. The effects of sliding the window in time are discussed later. The full time series contains 8192 points and is
digitized at 1.0 GHz. The amplitudes have been normalized
to the largest value of all three spectra, excluding the dc
level, before taking the common logarithm. For each spatial
location, a vertical line of the same corresponding line style
is drawn at the frequency expected to be received from a
FIG. 6. Spectra of electric field fluctuations 共logarithmic scale兲 for three x
locations at z = −256 cm. Solid, dashed, and dotted lines are, respectively, at
x = 7.5 cm, 15.3 cm, and 23.1 cm. Vertical lines indicate the predicted frequency 共for spectra with the same corresponding line style兲 for a point
source of lower hybrid waves located at the laser impact site. The data are
acquired under the conditions of setup #3.
point source of lower hybrid waves 关Eq. 共4兲兴. For all three x
locations, the local peak in the spectrum occurs at a higher
frequency than predicted for a broadband point source at the
origin. This observation is made clearer by examining the
spectra obtained during a high-resolution spatial scan in x, as
presented next.
In Fig. 7 we present data showing the received frequency
spectra measured at z = −256 cm for 128 x locations and for
three values of the magnetic field 共750 G, 1000 G, and
1500 G兲. These correspond to setup numbers 1–3 of Table I.
The data are processed in the same manner as for Fig. 6. In
the plot for each magnetic field, there is a hyperbolic black
line drawn to indicate the expected frequency from the dispersion relation, Eq. 共4兲. The distributions of measured frequencies agree very well with the predictions as a function of
both position and magnetic field. Based on this agreement,
the electric field oscillations are identified as quasielectrostatic whistler waves. The spread of frequencies away from
the point source predictions will be discussed next.
When plotted in this manner, the spectra resemble the
frequency versus time spectrograms of VLF saucers21 commonly observed in the Earth’s ionosphere. Such measurements are well correlated with upward-going field-aligned
bursts of energetic electrons from below the spacecraft. As
with spacecraft data, the propagation characteristics of lower
hybrid waves can be used as a remote measurement of the
size of the wave source region. First, consider a source with
a finite extent in z of length 2zs, as shown in Fig. 8. There are
a range of group velocity vectors which point from the extended source to the measurement location, and so there will
be a range of frequencies observed at the measurement location according to the dispersion relation 共here rewritten in
terms of the propagation angle ␣兲,
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-7
Phys. Plasmas 15, 072114 共2008兲
Quasielectrostatic whistler wave radiation…
FIG. 7. 共Color兲 Frequency spectra as a function of x for three magnetic fields at fixed axial location z = −256 cm. The black parabolic curves show the expected
frequency observable from a point source of lower hybrid waves at the origin 关Eq. 共4兲兴. Note that these curves generally define a minimum observed frequency
indicating that the growth region for the waves is closer to the observation point than the laser impact site. This is discussed further in the text.
f共␣兲 =
f ce
冑1 + 共f ce/f pe兲2
冋
me
+ tan2 ␣
mi
册
1/2
,
共5兲
where have used tan2 ␣ = tan2共␲ − ␣兲 for propagation to positive x and negative z. Compared with the point source, higher
共lower兲 frequencies will be detected if the source extends
closer to 共farther from兲 the observation point. Note that for
the line source extended in z the received spectrum should be
symmetric about x = 0.
Figure 9共a兲 shows the spectra for the 1000 G case 共setup
#2兲 from Fig. 7, together with the bounds in frequency predicted for the finite z line source. A source size of zs
= 30 cm was used, which qualitatively best contained the observed spectra. In the background plasma, this source size
corresponds to 80 ion gyroradii, 65 electron inertial lengths,
and 2.3⫻ 104 Debye lengths. In the figure, the red line shows
the limit of expected frequencies from the end of the source
at z = −zs 共closer to the observation locations兲 and the white
line is the limit for the opposite end of the source at z = + zs.
The black line as in the previous figure is for the point source
at the laser impact site. From this, it is evident that the source
region is well confined to negative values of z and to within
30 cm of the laser impact site. Similar line scans with the
probe in the region of positive z show the reverse trend: there
the source appears confined to regions of positive z. This
bidirectional nature of the source is consistent with the lower
hybrid wave source being the outflow of energetic electrons
which we have also measured on both axial sides of the laser
target.
Figure 9共b兲 shows the effect of a source extended in the
y direction, and also of measurements made out of the plane
of y = 0. Here, the white curve indicates the spectrum from a
point source at y = + rb and the red curve for y = + 2rb. The
hyperbolic shape of the curve is more evident at greater separation and is the form most easily recognized in spacecraft
electric field spectrograms, where the satellite or rocket usually passes near but not through magnetic field lines which
map back to the source region. The red curve does seem to fit
the spectra better below 100 MHz, but it fails to track the
spectra at large 兩x兩 since it and the white curve asymptotically
approach the black curve for large 兩x兩. Additionally, the enhanced amplitude of the low-frequency portion of the spectrum extends to all negative values of x and may be caused
by a charging of the probe shaft by the electron burst since
the probe enters from the positive x axis. However, broadband electrostatic noise is a feature commonly observed
when satellites pass through field lines which do map back to
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-8
Phys. Plasmas 15, 072114 共2008兲
Vincena et al.
x
α
ma x
|z|
αmin
2z s
x
B0 , z
FIG. 8. Example of a finite line source 共extending ⫾zs in the z-direction兲
showing the minimum and maximum propagation angles to the observation
point 共x ⬎ 0 , z兲. For this source geometry, the angles and therefore the measured frequency spectra are symmetric about x = 0 with tan ␣max
= 兩x兩 / 共兩z 兩 −zs兲 and tan ␣min = 兩x兩 / 共兩z兩 + zs兲.
VLF saucer source regions.22 In space, this noise can extend
to much higher frequencies 共scaled to the lower hybrid frequency兲 than can be observed with the present laboratory
setup.
Figure 9共c兲 shows the effect of a point source displaced
in the x direction. The white curve is for a source at x =
+ rb and the red curve for x = −rb. Again, the black curve is
for the point source at the origin. For x ⬍ 0, the observed
spectra are well bracketed by the black and white curves;
however, for x ⬎ 0, it is the black and red curves which best
contain the pattern. To be consistent, the curves must be used
on both sides of x = 0. The conclusion, therefore, is that the
extent of the source in the x direction is less than the bubble
radius. In the case of VLF saucers in the ionosphere,
Temerin51 concisely demonstrates that some features of the
observed spectrograms are best described not by a point
source, but by sources which are “extensive in longitude but
narrowly confined in the other two dimensions.”
In summary of the laboratory observations, the impact of
the laser initially results in a bidirectional line source 共along
the magnetic field兲 of lower hybrid waves. The initial size of
the source region is approximately one bubble radius crossfield and 30 cm along the magnetic field on both sides of the
target. These characteristics are consistent with the source of
the waves being the observed energetic electrons.
Next, the lower hybrid waves may also be used as a
remote diagnostic of the LPP velocity. This technique uses a
sliding, Hanning-window function on the time series Ex共t兲
before taking the Fourier transform to produce a timeresolved picture of the radiated lower hybrid waves. The
results from this analysis are presented in Fig. 10 for experimental setup #3 of Table I. The time indicated in each frame
indicates the start of the 128-point 共128 ns兲 windowed transform. The amplitudes of the Fourier transforms are shown on
a log scale and in each frame, the magnitudes are first scaled
to the maximum amplitude for that frame before taking the
logarithm. At t = 100 ns, the pattern is roughly symmetric
about the laser impact site 共x = 0兲 but as time advances, the
pattern becomes smeared out, develops asymmetries, and
shifts in the direction of LPP expansion. The black hyperbolic curves show the expected distribution of received frequencies as a function of x for a point source 共also the minimum frequency for a z-extended line source兲 of lower hybrid
waves located at the origin, but moving to the left with a
velocity of 1.6⫻ 107 cm/ s. This value was selected as a
qualitative single value which best tracked the moving radiation pattern for all times, and is consistent with the velocity
of the expanding LPP. The forward-directed velocity of LPP
ions in vacuum and without a magnetic field was previously
measured using a Faraday cup and found to be
1.4⫻ 107 cm/ s. This directed expansion velocity can be
maintained in the presence of a magnetic field; the LPP polarizes due to the difference in the gyroradius of the ions and
electrons and the LPP then undergoes E ⫻ B motion across
the magnetic field.9–11,52,53 As it does so, it continually emits
energetic electrons which are replaced by colder, background
plasma electrons.14 Additionally, the higher-frequency portions of the spectrum are diminished at later times. This may
indicate that fewer fast electrons are present to couple to the
higher frequency waves which have higher phase velocities.
The broadening of the frequency spectrum may be understood in terms of the evolution of the spatial distribution
of fast electrons. Figure 11 shows three snapshots in time of
the xy-distribution of fast electrons near the target 共z
= −32 cm兲. This measurement was made during an ion saturation current diagnostic of the background plasma for setup
#3. Again, we attribute the negative current values to the
impact of energetic electrons on the probe surface.
The fastest velocity associated with the expansion is that
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-9
Quasielectrostatic whistler wave radiation…
Phys. Plasmas 15, 072114 共2008兲
FIG. 9. 共Color兲 Wave source region reconstruction using the measured spectra vs. x at 1 kG from Fig. 7. Black curves indicate the expected distribution from
a point source at the origin, using Eq. 共4兲. 共a兲 Displacement of source in z; the red curve is for a point source 30 cm axially closer to the measurement line,
and the white curve is for a point source 30 cm farther away. 共b兲 Displacement of source in y; the red 共white兲 curve is for the point source moved +2rb 共+rb兲
along the y axis. 共c兲 Displacement of source in x; the red 共white兲 curve is for the point source moved −rb 共+rb兲 along the x axis.
of the leading edge which moves from x = 0 cm at t = 0 ns to
x = −8.5⫾ 1 cm at t = 720 ns, or at −1.2⫻ 107 ⫾ 1.4
⫻ 106 cm/ s. This velocity is somewhat slower than the measurements in vacuum or from the lower hybrid radiation pattern. It may be that the density of the most energetic electrons at the leading edge is insufficient to offset the level of
ion collection, but this has not been studied in detail. Additionally, the expansion velocity of the leading edge is
roughly constant: −1.3⫻ 107 ⫾ 4.6⫻ 106 cm/ s at t = 120 ns,
and −1.4⫻ 107 ⫾ 1.2⫻ 106 cm/ s at t = 420 ns.
C. Phase-coherent and solitary electric fields
We next present data on the space-time radiation pattern
at a fixed axial displacement of z = −256 cm without the use
of Fourier transforms. These data correspond to the experimental setup #4 of Table I. Figure 12 shows the x-component
of the electric field as a function of time and distance perpendicular to the background magnetic field along the line
y = 0. The expansion of the laser-produced plasma is from the
origin towards negative x. The data are averaged over 20
plasma discharges and so elucidate phenomena that maintain
a high degree of phase coherency from shot-to-shot. Figures
12共a兲 and 12共b兲 are identical except that different scales have
been used for the color bar to emphasize different features.
In Fig. 12共a兲, the color bar levels are saturated at approximately one-tenth of the maximum data values; this reveals
information on the phase of the electric field fluctuations.
Coherent oscillations of relatively high frequency are observed first, followed by several oscillations with spatial
scales as large as the measurement region and temporal period on the order of the diamagnetic cavity lifetime 共1 ␮s兲.
Previous experiments by this group13 have conjectured that
these slower oscillations may be due to compressional
Alfvén waves. The pattern of all oscillations shows some
symmetry about x = 0 which corresponds to the magnetic
field line connecting back to the laser impact site. However,
the pattern is clearly more washed out for x ⬍ 0 where the
LPP forms and moves, indicating a lack of phase coherency
during the ensemble averaging. For x ⬎ 0, the lower hybrid
waves are still evident after 5 ␮s near x = 5 cm. For reference, the ion gyroradius is 1.4 cm and the electron inertial
length is 0.53 cm; the ion temperature of 0.8 eV was measured using laser-induced fluorescence in argon under similar
discharge conditions.54
In Fig. 12共b兲 the limits of the color bar are set by the
maximum amplitude observed 共2.3 V / m兲. Here the asymme-
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-10
Vincena et al.
Phys. Plasmas 15, 072114 共2008兲
FIG. 10. 共Color兲 Sliding, Hanning-windowed amplitude spectra 共log scale兲
of the x-component of the electric field at 1500 G. The width of the window,
centered on the times shown, is 128 ns and the digitization period is 1 ns.
Electric fields are normalized to the maximum value within each subplot
which are on the order of 10 V / m. The black curves indicate the expected
frequencies received by a broadband point source of lower hybrid waves
moving perpendicular to the background magnetic field at a velocity of
1.6⫻ 107 cm/ s.
try due to the LPP motion is much clearer. Two main features
can be gleaned: first, a bipolar electric field structure is observed at a time of 0.4 ␮s; the propagation velocity of this
largest high frequency pulse is 6.4⫻ 108 cm/ s. Second, a
broad pattern with transverse size approximately equal to the
bubble radius 共rb = 6.6 cm兲 is visible from roughly
t = 1.5 ␮s to t = 2.5 ␮s with a spread in time of ⫾0.3 ␮s
where it travels a distance of ⌬x = 10⫾ 2 cm. This cross-field
expansion speed 共1.0⫻ 107 ⫾ 4.5⫻ 106 cm/ s兲 of this second
feature corresponds roughly to the E ⫻ B speed of the LPP of
1.4⫻ 107 cm/ s.
Figure 12共c兲 shows the early time behavior: from
0 to 500 ns. The laser strikes the target at t = 0 and care was
taken to remove the time delays for the various cables by
using a fast photodiode 共1 ns rise time兲 at the same axial
location as the target and with the same cable lengths as
those used to record the electric field data. From the arrival
time, the largest group velocity observed in the experiment is
FIG. 11. 共Color兲 Probe-measured ion saturation current snapshots in a plane
perpendicular to the background magnetic field at z = −32 cm for setup #3 of
Table I. Red indicates the true ion saturation current in the background
plasma. The large negative patches indicate energetic electrons escaping the
laser-produced plasma. The data grid size is 51 points in x and 41 points in
y and an ensemble average of 5 plasma discharges was used. The lower
hybrid radiation patterns in Fig. 10 roughly track the leading edge of the
negative current, which moves to the left at approximately 1.2⫻ 107 cm/ s.
2.3⫻ 109 ⫾ 4.7⫻ 108 cm/ s which is in rough agreement with
the fastest 共⬃1 ⫻ 109 cm/ s兲 electrons emitted from the carbon target.
Although a small tail population of electrons may only
be required to give rise to the fast-arriving oscillations at
relatively low amplitude, the velocity of these electrons is
still quite high from what may be expected to be created in
the laser-produced plasma. In the initial, perpendicular expansion of the LPP, the ions and electrons primarily are ex-
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-11
Quasielectrostatic whistler wave radiation…
Phys. Plasmas 15, 072114 共2008兲
FIG. 12. 共Color兲 Twenty-shot ensemble averaged electric field measurements as a function of x and t for experimental setup #4 at z = −256 cm. The
LPP expands towards negative x. 共a兲 Saturated color bar emphasizing the
wave phase. Lower hybrid waves 共f 艌 f LH = 4.1 MHz兲 are evident at early
times. More phase-coherent, lower frequency oscillations can be seen in the
region x ⬎ 0. 共b兲 The same data as 共a兲 but with the color bar range set by the
maximum amplitude of the electric field 共−2.3 V / m at t = 0.41 ␮s兲. 共c兲 Expanded view of the first 500 ns shown in 共a兲 with saturated color bar emphasizing the fine structure of the early coherent radiation pattern.
pelled at the same velocity, so that the massive ions will
inherit the vast majority 共approximately 1 keV per ion兲 of
the incident laser energy. However, at its initial, near-solid
densities the early LPP is highly collisional. If, through these
collisions, a small fraction of electrons were to achieve an
equipartition of energy with the ions, then a 1 keV electron
would have a velocity of 1.3⫻ 109 cm/ s, which is consistent
with our measurements.
Figure 13共a兲 provides a detailed look at the early, largeamplitude structure seen in Fig. 12共b兲 with the same color
bar. Figure 13共b兲 is a temporal slice through the data along
the horizontal line shown in Fig. 13共a兲, at x = −2.7 cm. The
inset to Fig. 13共b兲 is the linear amplitude spectrum of the
pulse, showing a well-defined peak at the lower hybrid frequency in the background plasma. Figure 13共c兲 is a cut of the
data along the vertical line in Fig. 13共a兲, at t = 4.1 ␮s; this
intersects the largest negative value in Fig. 13共b兲, and shows
an isolated bipolar electric field signature with an apparent
perpendicular wavelength of 2 cm. This is comparable to the
ion gyroradius of the background plasma of 1.4 cm.
FIG. 13. 共Color兲 共a兲 Detailed view of the solitarylike electric field structure
of Fig. 12共b兲, shown here with the same color bar. Black lines indicate the
locations where cuts of the data are taken. 共b兲 Temporal signal at x
= −2.7 cm with the inset of a Hanning-windowed fast-Fourier transform amplitude spectrum of the time series showing a peak at the lower hybrid
frequency of the background plasma. 共c兲 Spatial cut of the data at t = 4.1 ␮s.
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-12
Vincena et al.
Phys. Plasmas 15, 072114 共2008兲
Although an analytic expression for the radiated fields
from the moving LPP source is beyond the scope of this
paper, a simple model can be used to reproduce general features of the observed lower hybrid wave phase variations as
a function of space and time for the initial oscillations.
Consider a disturbance of electric potential ⌽ which
propagates from a broadband source located at the origin as a
plane wave. In the xz-plane, the potential may be written as
⌽共x,z,t兲 = ⌽0 exp共ikzz + ikxx − i␻t兲,
where ⌽0 is an arbitrary constant. Since x / z = −kz / kx, the
potential is simply
⌽共␣,t兲 = ⌽0 exp关− i␻共␣兲t兴,
where the phase varies only with time and the propagation
angle, ␣. The real part of this complex potential is plotted for
the experimental parameters of setup #4 in Fig. 14共a兲. The
pattern is computed at the fixed axial measurement location
of z = −256 cm. The fishbone pattern in the potential is reminiscent of the pattern seen in the electric field data. The
temporal signals which begin in phase for all values of x
become increasingly phase mixed as time advances. Before
computing the model electric field, the group-delayed signal
must be computed. Although the wave frequency can be
computed without explicit knowledge of k储 or k⬜, this is not
the case for the group velocities 关refer to Eqs. 共1兲 and 共2兲兴. A
further assumption of the model is that the physical size of
the current channel of fast electrons sets a single perpendicular wavelength ␭⬜ while the parallel wavelengths are determined, for any given frequency, by the dispersion relation. In
reality, a spectrum of ␭⬜ are present, but this model assumes
the pattern can be approximated using a single value which
is determined by examining the group delay curves plotted
using a range of ␭⬜ as shown in Fig. 15. Since the group
delay diverges as x approaches zero, the delay curves are
only plotted to a delay time of 500 ns. For 兩x兩 ⬎ 5 cm, there is
approximately a single group delay for any given perpendicular wavelength. Since Fig. 12共c兲 also shows a reasonably
abrupt arrival time 共⬇100 ns兲, the value of ␭⬜ = 2.5 cm is
selected for the model. The resulting pattern, displaced in
time by the corresponding delay curve is shown in Fig.
14共b兲, and it is with this pattern that the x-component of the
electric field is computed from Ex = −⳵⌽ / ⳵x. The result is
presented in Fig. 14共c兲. The model electric field has been
convolved in the x-direction with a Gaussian of full-width
共2␴兲 size of 3 mm to simulate sampling the pattern with the
dipole electric field probe with 3 mm tip separation. For
comparison, in Fig. 14共d兲, the same electric field data Ex
from Fig. 12共c兲 is shown, but with each time series digitally
filtered to remove frequencies at or below the lower hybrid
frequency; this effectively removes the large-amplitude, solitary electric field structure and emphasizes the highfrequency phase variations.
Along with the early solitary structure, what the model
does not reproduce is the presence of the wave field near x
= 0; this may be due to thermal effects44,45,55 which can cause
interference fringes to develop both inside or outside the
resonance cones. Additionally, no wave growth or damping
is included in the model. Despite the simplicity of the model,
FIG. 14. 共Color兲 共a兲 Model of an oscillating electric potential disturbance,
⌽, propagating as plane waves, where the frequency varies according to the
lower hybrid dispersion relation, Eq. 共4兲 at z = −256 cm. 共b兲 The same electric potential, but delayed in time by the wave group delay as described in
the text. 共c兲 The electrostatic field component Ex = −⳵⌽ / ⳵x, calculated numerically from the pattern in 共b兲 and then convolved with a Gaussian of
full-width 共2␴兲 size of 3 mm to simulate sampling the pattern with the
dipole electric field probe with 3 mm tip separation. 共d兲 The same electric
field data Ex from Fig. 12共c兲, but with each time series digitally filtered to
remove frequencies at or below the lower hybrid frequency. All values are
normalized to the maximum amplitude for each subplot with red being
positive, blue negative, and green is zero.
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-13
Quasielectrostatic whistler wave radiation…
λ⊥=1cm
2cm
3cm 4cm
FIG. 15. Group delay times for several perpendicular wavelengths. The
wavelength used as an input parameter to the model for Fig. 14 is ␭⬜
= 2.5 cm. The curves are computed for z = −256 cm and for the parameters
of setup #4. The perpendicular wavelengths are all smaller than the LPP
bubble radius rb = 6.6 cm. For reference, the horizontal dashed line is at the
speed-of-light delay of 8.5 ns.
it does reproduce the following: a general arrival front at
approximately 100 ns; the faded nature of the front for
兩x兩 ⬎ 10 cm due to relatively little x variation in the phase of
the potential; the general morphology of the wave phase;
and, the fading of the pattern at late times due to rapid perpendicular phase variations which are averaged away due to
the size of the probe.
V. CONCLUSIONS
We have observed lower hybrid waves generated by a
laser-produced plasma within a background magnetized
plasma. The source of free energy for the waves is consistent
with suprathermal electrons emitted from the LPP.
As with localized sources of lower hybrid waves in the
Earth’s ionosphere, the wave group velocities can be used as
a remote diagnostic of the size of the source region. In the
laboratory, the reverse-mapping of frequency-versus-position
data shows the initial growth region of the waves to be a
bidirectional line source with dimensions of a few centimeters cross-field and about 30 cm along the field. The size of
the source region is consistent with two-dimensional ion
saturation profiles and axial time-of-flight measurements of
the emitted energetic electrons near the laser target. The axial
extent of the growth region also corresponds to the distance
in which the tail electrons of the beam can be distinguished
apart from the electrostatic fluctuations. The agreement between direct measurements of the physical size of an energetic electron burst and the reverse mapping of the distant
lower hybrid radiation patterns validates the reverse-
Phys. Plasmas 15, 072114 共2008兲
mapping technique for VLF saucer source regions in the
ionosphere. Until now, the technique was deemed obvious,
yet lacked direct experimental confirmation.
As the LPP E ⫻ B drifts perpendicular to the ambient
magnetic field, it continues to shed suprathermal electrons
which continue to radiate waves, but with decreasing intensity. Owing to the high group velocities of these waves compared with the perpendicular expansion of the LPP, we have
also established the usefulness of the time-windowed lower
hybrid radiation pattern as a remote diagnostic of the leading
edge of the LPP expansion velocity.
Analysis of the ensemble-averaged, perpendicular wave
electric field data reveals two important phase-coherent phenomena. The first is a low-amplitude electric field radiation
pattern with measured parallel group velocities in excess of
2 ⫻ 109 cm/ s. A simple model of a plane wave source of
lower hybrid waves originating at the laser impact site reproduces the gross features of the measured space-time radiation
pattern taken along a line perpendicular to the background
magnetic field. The abrupt arrival time 共100 ns兲 of the wave
fronts allow the use of a single perpendicular wavelength as
an input parameter to the model. This value of 2.5 cm is
smaller than the 6.6 cm diamagnetic cavity size of the LPP,
and it is this cavity size which has been shown in the present
and previous experiments13,14 to roughly determine an upper
bound on the cross-field size of the escaping electron current
channel. Further experiments, however, are required to establish a definite scaling between the cross-field size of the current channel and the cavity size, especially how well the two
are related as a function of time.
The second, phase-coherent phenomenon is an isolated,
spatially bipolar spike in Ex with a duration of 2␲ / ␻LH as it
passes the axially fixed probe. Here ␻LH is calculated using
the background plasma parameters. As to why the frequency
of the largest amplitude structure should be near that of the
lower hybrid frequency, the interpretation is guided by the
theoretical model used by Maggs17 to explain the ground and
satellite measurements of very low frequency 关共VLF兲 at
1 – 500 kHz兴 radio spectra at high latitudes in the Earth’s
ionosphere. From the observations, these VLF emissions are
closely associated with electron precipitation in the auroral
zones,56–59 and in the model, incoherent whistler noise is
amplified by the beam of precipitating auroral electrons.
Characteristics of the observed differential power spectra of
VLF emissions are accurately reproduced by this model—
most importantly that the peak power flux occurs at a frequency near the lower hybrid frequency in the ambient
plasma.56,59,60 As discussed by Maggs, a variety of factors
contribute to the resulting power flux for any frequency including the total length along a ray path from the source to
the observation point, the average growth rate of the wave,
and the average inverse of the wave group velocity. Although
the model assumes a steady state which precludes direct
comparison with our impulsive experimental results, it does
consider a finite perpendicular spatial extent of the electron
beam as is necessary to understand discreet auroral arcs as
the gain media for the VLF emissions. The effects of finite
transverse source size are also relevant to the present experiment. The group velocities for whistler waves with frequen-
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
072114-14
cies near the lower hybrid frequency are nearly parallel to the
background magnetic field, hence these waves spend the
most time within the gain medium of the electron beam. This
would be the case in the present experiment as well, since the
relatively large amplitude of the bipolar electric field structure occurs in the region x ⬍ 0, which is the direction in
which the laser-produced plasma expands and continues to
shed suprathermal electrons as they are replaced by colder
electrons from the background plasma.
The phase of the solitary structure as seen in Fig. 13共a兲
joins together with the phase of the surrounding wave pattern. The main difference therefore is in amplitude, and could
be a nonlinear phenomenon of the wave growth. A good
indicator of a nonlinear process would be if the energy density of the wave electric field were comparable to the plasma
thermal energy; however, an absolute calibration of the electric field probe cannot presently be made since the probe tip
separation 共3 mm兲 is much larger than the Debye length
共⬃2 ⫻ 10−3 cm兲. The amplitude of the solitary structure electric field is, however, the largest of any observed in the
experiment.
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation 共NSF兲 under Grant No. PHY-0408226, and the experiments were performed at the Basic Plasma Science Facility
through a cooperative agreement between the Department of
Energy 共Grant No. DE-FC02-07ER549118兲 and the NSF
共Grant No. PHYÐ0531621兲.
1
Phys. Plasmas 15, 072114 共2008兲
Vincena et al.
B. C. Low, J. Geophys. Res. 106, 25141, DOI: 10.1029/2000JA004015
共2001兲.
2
A. H. Bridle and R. A. Perley, Annu. Rev. Astron. Astrophys. 22, 319
共1984兲.
3
L. Biermann, B. Brosowski, and H. U. Schmidt, Sol. Phys. 1, 254 共1967兲.
4
I. B. Bernstein, and R. M. Kulsrud, Astrophys. J. 142, 479 共1965兲.
5
B. Pégourié, Plasma Phys. Controlled Fusion 49, R87 共2007兲.
6
P. A. Bernhardt, R. A. Roussel-Dupre, M. B. Pongratz, G. Haerendel, and
A. Valenzuela, J. Geophys. Res. 92, 5777, DOI: 10.1029/
JA092iA06p05777 共1987兲.
7
S. A. Colgate, J. Geophys. Res. 70, 3161, DOI: 10.1029/
JZ070i013p03161 共1965兲.
8
B. H. Ripin, R. R. Whitlock, F. C. Young, S. P. Obenschain, E. A.
McLean, and R. Decoste, Phys. Rev. Lett. 43, 350 共1979兲.
9
B. H. Ripin, E. A. McLean, C. K. Manka, C. Pawley, J. A. Stamper, T. A.
Peyser, A. N. Mostovych, J. Grun, A. B. Hassam, and J. Huba, Phys. Rev.
Lett. 59, 2299 共1987兲.
10
A. N. Mostovych, B. H. Ripin, and J. A. Stamper, Phys. Rev. Lett. 62,
2837 共1989兲.
11
G. Dimonte and L. G. Wiley, Phys. Rev. Lett. 67, 1755 共1991兲.
12
B. H. Ripin, J. D. Huba, E. A. McLean, C. K. Manka, T. Peyser, H. R.
Burris, and J. Grun, Phys. Fluids B 5, 3491 共1993兲.
13
M. VanZeeland, W. Gekelman, S. Vincena, and G. Dimonte, Phys. Rev.
Lett. 87, 105001 共2001兲.
14
M. VanZeeland, W. Gekelman, S. Vincena, and J. Maggs, Phys. Plasmas
10, 1243 共2003兲.
15
T. A. Hall, Plasma Phys. 14, 677 共1972兲.
16
M. Yamada and D. K. Owens, Phys. Rev. Lett. 38, 1529 共1977兲.
17
J. E. Maggs, J. Geophys. Res. 81, 1707, DOI: 10.1029/JA081i010p01707
共1976兲.
18
H. G. James, J. Geophys. Res. 81, 501, DOI: 10.1029/JA081i004p00501
共1976兲.
19
J. E. Maggs, J. Geophys. Res. 83, 3173, DOI: 10.1029/JA083iA07p03173
共1978兲.
20
J. R. Winckler, Rev. Geophys. Space Phys. 18, 659, DOI: 10.1029/
RG018i003p00659 共1980兲.
21
S. R. Mosier and D. G. Gurnett, J. Geophys. Res. 74, 5675, DOI: 10.1029/
JA074i024p05675 共1969兲.
22
R. E. Ergun, C. W. Carlson, J. P. McFadden, R. J. Strangeway, M. V.
Goldman, and D. L. Newman, Phys. Plasmas 10, 454 共2003兲.
23
N. Singh, S. M. Loo, B. E. Wells, and G. S. Lakhina, J. Geophys. Res.
106, 21165, DOI: 10.1029/2000JA000335 共2001兲.
24
S. Klimov, S. Savin, Y. Aleksevich, G. Avanesova, V. Balebanov, M.
Balikhin, A. Galeev, B. Gribov, M. Nozdrachev, V. Smirnov et al., Nature
共London兲 321, 292 共1986兲.
25
R. Bingham, J. M. Dawson, V. D. Shapiro, D. A. Mendis, and B. J. Kellett,
Science 275, 49 共1997兲.
26
R. Bingham, B. Kellett, J. Dawson, V. Shapiro, and D. Mendis, Astrophys.
J., Suppl. Ser. 127, 233 共2000兲.
27
M. G. Haines, Phys. Rev. Lett. 78, 254 共1997兲.
28
S. Kacenjar, M. Hausman, M. Keskinen, A. W. Ali, J. Grun, C. K. Manka,
E. A. McLean, and B. H. Ripin, Phys. Fluids 29, 2007 共1986兲.
29
H. Cronberg, M. Reichling, E. Broberg, H. B. Nielsen, E. Matthias, and N.
Tolk, Appl. Phys. B 52, 155 共1991兲.
30
S. Amoruso, M. Armenante, R. Bruzzese, N. Spinelli, R. Velotta, and X.
Wang, Appl. Phys. Lett. 75, 7 共1999兲.
31
R. C. Issac, G. K. Varier, P. Gopinath, S. S. Harilal, V. P. N. Nampoori,
and C. P. G. Vallabhan, Appl. Phys. A: Mater. Sci. Process. 67, 557
共1998兲.
32
J. Grun, R. Decoste, B. H. Ripin, and J. Gardner, Appl. Phys. Lett. 39,
545 共1981兲.
33
B. Meyer and G. Thiell, Phys. Fluids 27, 302 共1984兲.
34
W. H. Bostick, Phys. Rev. 104, 292 共1956兲.
35
G. Schmidt, Phys. Fluids 3, 961 共1960兲.
36
N. Brenning, T. Hurtig, and M. A. Raadu, Phys. Plasmas 12, 012309
共2005兲.
37
C. Litwin and R. Rosner, Phys. Rev. Lett. 86, 4745 共2001兲.
38
A. W. Trivelpiece and R. W. Gould, J. Appl. Phys. 30, 1784 共1959兲.
39
R. K. Fisher and R. W. Gould, Phys. Rev. Lett. 22, 1093 共1969兲.
40
R. L. Stenzel and W. Gekelman, Phys. Rev. A 11, 2057 共1975兲.
41
P. Bellan and M. Porkolab, Phys. Rev. Lett. 34, 124 共1975兲.
42
R. L. Stenzel, J. Geophys. Res. 82, 4805, DOI: 10.1029/
JA082i029p04805 共1977兲.
43
H. H. Kuehl, Phys. Fluids 5, 1095 共1962兲.
44
H. H. Kuehl, Phys. Fluids 16, 1311 共1973兲.
45
J. M. Chasseriaux, Phys. Fluids 18, 866 共1975兲.
46
A. Swanson, Plasma Waves, 2nd ed. 共Academic, San Diego, 1989兲.
47
W. Gekelman, H. Pfister, Z. Lucky, J. Bamber, D. Leneman, and J. Maggs,
Rev. Sci. Instrum. 62, 2875 共1991兲.
48
S. Rosenberg and W. Gekelman, Geophys. Res. Lett. 25, 865, DOI:
10.1029/98GL00382 共1998兲.
49
W. Gekelman, M. Van Zeeland, S. Vincena, and P. Pribyl, J. Geophys.
Res., 关Space Phys.兴 108, 8 共2003兲.
50
B. V. Compernolle, W. Gekelman, and P. Pribyl, Phys. Plasmas 13,
092112 共2006兲.
51
M. Temerin, J. Geophys. Res. 84, 6691, DOI: 10.1029/JA084iA11p06691
共1979兲.
52
T. A. Peyser, C. K. Manka, B. H. Ripin, and G. Ganguli, Phys. Fluids B 4,
2448 共1992兲.
53
A. Neogi, A. Mishra, and R. K. Thareja, J. Appl. Phys. 83, 2831 共1998兲.
54
N. Palmer, W. Gekelman, and S. Vincena, Phys. Plasmas 12, 2102 共2005兲.
55
R. K. Fisher and R. W. Gould, Phys. Fluids 14, 857 共1971兲.
56
R. E. Barrington, T. R. Hartz, and R. W. Harvey, J. Geophys. Res. 76,
5278, DOI: 10.1029/JA076i022p05278 共1971兲.
57
R. A. Hoffman and T. Laaspere, J. Geophys. Res. 77, 640, DOI: 10.1029/
JA077i004p00640 共1972兲.
58
T. Laaspere and W. C. Johnson, J. Geophys. Res. 78, 2926, DOI: 10.1029/
JA078i016p02926 共1973兲.
59
D. A. Gurnett and L. A. Frank, J. Geophys. Res. 77, 172, DOI: 10.1029/
JA077i001p00172 共1972兲.
60
T. Laaspere, W. C. Johnson, and L. C. Semprebon, J. Geophys. Res. 76,
4477, DOI: 10.1029/JA076i019p04477 共1971兲.
Downloaded 04 Aug 2008 to 128.97.43.206. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp