whistler modes in highly nonuniform magnetic fields. iii
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Whistler modes in highly nonuniform magnetic fields. III. Propagation near mirror andcusp fieldsR. L. Stenzel, and J. M. Urrutia
Citation: Physics of Plasmas 25, 082110 (2018); doi: 10.1063/1.5039355View online: https://doi.org/10.1063/1.5039355View Table of Contents: http://aip.scitation.org/toc/php/25/8Published by the American Institute of Physics
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Whistler modes in highly nonuniform magnetic fields. III. Propagation nearmirror and cusp fields
R. L. Stenzel and J. M. UrrutiaDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA
(Received 7 May 2018; accepted 26 June 2018; published online 14 August 2018)
The properties of helicon modes in highly nonuniform magnetic fields are studied experimentally.
The waves propagate in an essentially unbounded uniform laboratory plasma. Helicons with mode
number m¼ 1 are excited with a magnetic loop with dipole moment across the dc magnetic field.
The wave fields are measured with a three-component magnetic probe movable in three orthogonal
directions so as to resolve the spatial and temporal wave properties. The ambient magnetic field has
the topology of a mirror or a cusp, produced by the superposition of a uniform axial field B0 and
the field of a current-carrying loop with the axis along B0. The novel finding is the reflection of
whistlers by a strong mirror magnetic field. The reflection arises when the magnetic field changes
on a scale length shorter than the whistler wavelength. The simplest explanation for the reflection
mechanism is the strong gradient of the refractive index which depends on the density and mag-
netic field. More detailed observations show that the incident wave splits when the k vector makes
an angle larger than 90� with respect to B0 which produces a parallel phase velocity component
opposite to that of the incident wave. The reflection coefficient has been estimated to be close to
unity. Interference between reflected and incident waves creates nodes in which the whistler mode
becomes linearly polarized. When the magnetic field topology is that of a reversed field configura-
tion (FRC), the incident wave is absorbed near the three-dimensional (3D) magnetic null point
which prevents wave reflections. However, waves outside the separatrix are not absorbed and con-
tinue to propagate around the null point. When waves are excited inside the FRC, their polarization
and helicon mode are reversed. Implications of these observations on research in space plasmas
and helicon sources are pointed out. Published by AIP Publishing.https://doi.org/10.1063/1.5039355
I. INTRODUCTION
Helicon modes are low frequency whistler modes with
parallel, perpendicular, and azimuthal wavenumbers. Their
phase fronts and field lines are helical. Historically, they
have been associated with waves in bounded solid state slabs
where boundary reflections lead to radial standing waves.1
This model has been adapted for whistler wave studies in
gaseous plasma columns.2 The interest in helicons grew by
their application to produce dense rf plasma sources.3
Discrepancies with linear wave theory arose: The perpendic-
ular wavenumber was not due to boundary reflections but
radial density gradients.4 Helicon waves have also been
observed in unbounded plasmas without density gradients5
where the antenna determines the perpendicular wavenum-
ber. Radial wave propagation has been frequently
observed4,6,7 which contradicted the theoretical prediction of
“paraxial” wave propagation, f ðkrrÞ exp iðm/þ kzz� xtÞ,where f ðkrrÞ describes a radial amplitude profile of a stand-
ing wave with radial wavenumber kr, m ¼ k/r is the azi-
muthal eigenmode number, kz is the axial wavenumber, and
x is the radian frequency.
Waves excited from antennas resemble more closely
helicon modes than plane waves. When the antenna imposes
a perpendicular magnetic or electric field, it usually excites a
rotating m¼ 1 helicon mode.8 The significant, yet unex-
plored, property of helicons is their angular field momentum,
which can interact with electrons or waves. Landau and
Doppler-shifted cyclotron resonance exist not only for the
linear field momentum but also for the angular orbital
momentum.9 The transverse Doppler shift has so far received
little attention.
The propagation of whistler modes in nonuniform plas-
mas has been studied extensively. For density gradient scales
which are large compared to the wavelength, ray tracing has
been applied to whistler wave trapping in density ducts in
space plasmas.10 The guiding of whistler modes in narrow
density troughs has been observed experimentally.11 Since
the refractive index depends not only on density but also on
the magnetic field, the refraction of whistlers also occurs in
nonuniform magnetic fields. It usually assumes large gradi-
ent scale lengths where the Wentzel-Kramers-Brillouin
(WKB) approximation is valid.12 Ray tracing studies have
been done for wave propagation in the Earth’s magnetic
dipole field13 and for wave injection along expanding field
lines of plasma thrusters.14 Experiments on cyclotron absorp-
tion have been done in several laboratory experiments.15–17
However, the topic of wave propagation in highly non-
uniform magnetic fields (B=jrBj < k) has received little
attention since it is not amenable to ray tracing. It has been
addressed in the present work and its preceding Papers I and
II.18,19 In the present paper, the nonuniform field topologies
are an axially symmetric mirror field and a field reversed
configuration (FRC), both with short gradient scale lengths.
1070-664X/2018/25(8)/082110/14/$30.00 Published by AIP Publishing.25, 082110-1
PHYSICS OF PLASMAS 25, 082110 (2018)
The major new finding is the reflection of whistler modes
from a mirror field. It can be explained by the abruptly
changing refractive index. But a more detailed explanation is
the splitting of the phase front when the incident wave vector
turns beyond 90� with respect to B0, which produces a
reverse phase and group velocity along B0. Since the latter is
more field aligned than the phase velocity, a small kk can
cause a significant deflection of the wave amplitude. The
interference between incident and reflected waves is
observed to produce locally a linear polarization of whistler
modes. This leads to a significant temporal amplitude modu-
lation which is absent for circularly polarized whistlers.
As we report in this paper, when a whistler mode propa-
gates against a null point of an FRC, the wave is absorbed in
regions where x=xc � 1. Absorption eliminates the chance
for wave reflection. For an FRC smaller than the lateral
wave dimension, the outer part of the wave does not encoun-
ter cyclotron resonance and keeps propagating, typically on
the separatrix and open field lines. When the wave is excited
inside and on the axis of an FRC, the waves become trapped
on closed field lines. The field reversal changes the rotation
of the B vectors inside the separatrix vs that on open field
lines. The wave topology also rotates differently inside the
separatrix compared to outside since azimuthal phase rota-
tion and polarization rotate together in electron Hall physics.
The direction of wave propagation remains the same with
and without field reversal; hence, the reversal of the phase
rotation changes the spatial twist of the helical phase surface
to a right-handed one.
The transmission between two antennas is shown and
discussed. Wave propagation along the symmetry axis
through an FRC is not possible due to the two null points;
likewise, propagation into and out of an FRC is cut off as
previously reported.20 Incident waves with large lateral
dimensions propagate on and beyond the separatrix around
the FRC, which complicates the comparison of cyclotron
damping experiments with plane wave theory. Inside the
FRC, waves can propagate but with reversed polarization
and wave rotation compared to outside the separatrix. The
difference between polarization and phase rotation is shown
here, which is relevant to helicon mode theory and experi-
ments. The importance of orbital angular field momentum to
new wave-particle interactions is also pointed out.
Hodograms are a standard tool in space plasma to deter-
mine the direction of wave propagation.21,22 A hodogram is
the curve which the tip of a three-component wave electric
or magnetic field vector traces out in time. For plane parallel
whistlers, the hodogram rotates anti-clockwise or forms a
right-handed rotation in time around B0. For plane but obli-
que whistlers, the field vector rotates around the k-vector but
not around B023,24 which is confirmed by observations.
The wave field lies in the plane of the hodogram which indi-
cates the local phase front of the plane wave. The normal to
the hodogram is parallel or antiparallel to the wave vector k,
depending upon the direction of propagation with respect to
B0 and the sign of the field polarization around B0. In either
case, the normal satisfies Maxwell’s equation in real or
wavenumber space, r � BðrÞ / k � BðxÞ ¼ 0.
Single spacecraft data do not provide sufficient spatial
resolution of phase fronts which prevents the distinction of
wave packets from plane waves or the direct verification of
wave refraction and reflection. Since the refractive index
depends on both the density and the magnetic field, a nonuni-
formity in either parameter leads to wave refraction.
Propagation in density nonuniformities such as ducts has
been studied in laboratory experiments,11,25–27 by theo-
ries28–31 and by numerical ray tracing models.10,28,32
Comparatively little is known on how whistler modes propa-
gate in strongly nonuniform magnetic fields.33 The Earth’s
magnetic field is weakly nonuniform, such that ray tracing
theory is justified.13,34 It predicts that multiple reflected
whistlers gradually refract away from the Earth. Such predic-
tions cannot be made for whistler modes in lunar crustal
magnetic fields35 or for low frequency whistlers at the bow-
shock36 or near reconnection regions.37,38
Nonuniform magnetic fields also exist in the exhaust
region of helicon thrusters14,16,39 or toroidal plasma devi-
ces39–42 with the focus on plasma production rather than
wave refraction. Refraction in nonuniform magnetic fields
destroys the cylindrical geometry of helicons. Wave propa-
gation in nonuniform and time-varying magnetic fields has
also been studied in a large laboratory device.43–45 In the
present laboratory experiment, we establish controlled non-
uniform magnetic fields of known topology and strength and
measure the wave propagation under various conditions.
This paper is organized as follows: After describing the
experimental setup and measurement procedure in Sec. II, the
observations and evaluations are presented in Sec. III in sev-
eral subsections. The findings are summarized in Conclusion,
Sec. IV.
II. EXPERIMENTAL SETUP AND DATA EVALUATIONS
The experiments are performed in a pulsed dc discharge
plasma of density ne ’ 1011 cm�3, electron temperature
kTe ’ 2 eV, neutral pressure pn ¼ 0.4 mTorr Ar, and uniform
axial magnetic field B0 ¼ 3 G in a large vacuum chamber
(1.5 m diam, 2.5 m length), shown schematically in Fig.
1(a). A range of densities is obtained by working in the after-
glow of the pulsed discharge (5 ms on, 1 s off). The after-
glow plasma has Maxwellian distributions and is free of
instabilities. The discharge uses a 1 m diam oxide coated hot
cathode, a photograph of which is shown in Fig. 1(b).
Whistler modes are excited with magnetic loop antennas
(4 or 5 cm diam, 4 turns). The axis of the dipole is perpendic-
ular to B0 which excites an m ¼ þ1 helicon mode.18 The
antenna is driven by 5 MHz rf bursts (20 rf periods duration,
5 ls repetition time) whose turn-on reveals phase and group
velocity during the wave growth and decay. A small field
amplitude (B< 0.1 G) is applied in order to avoid nonlinear
effects.
The wave magnetic field is received by a small magnetic
probe with three orthogonal loops (6 mm diam) which can be
moved together in three orthogonal directions. The spatial
field distribution is obtained by moving the probe through
orthogonal planes and acquiring at each position an average
082110-2 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
over 10 shots which improves the digital resolution and
reduces spurious noise. Plasma parameters are measured
with Langmuir probes also attached to the movable probe.
All signals are acquired with a 4-channel digital oscilloscope
with an 8 bit amplitude resolution and a 10 ns time
resolution.
In order to produce nonuniform magnetic fields, we
apply a current through a circular loop (16 cm diam, 4 turns)
whose dipole moment is aligned along B0. The applied cur-
rent (Iloop, max ¼ 640 A-turns) has a sinusoidal waveform of a
single period (T ’ 175 ls) which is long compared to the rf
period (0.2 ls). Thus, for each burst, the magnetic field is
nearly constant but it varies from burst to burst. Figure 1(c)
shows the waveforms of the loop current and the rf current
bursts. Since the magnetic field perturbation is applied in the
afterglow, it does not modify the plasma production during
the discharge, i.e., the density profile remains uniform.
The superposition of a uniform axial magnetic field B0
and the field from the current loop is displayed in Fig. 2.
When the loop field opposes the weaker axial magnetic field,
an FRC field is produced [Fig. 2(a)]. It has two 3D null
points on axis where the axial loop field cancels the uniform
field. A separatrix divides field lines closing around the loop
from the “open” field lines produced by an external solenoid.
The total field strength is calculated from Biot-Savart’s law
and displayed by contours of Btotal for a y–z plane at x¼ 0.
The field is symmetric around the axis (0, 0, z).
When the loop field adds to the axial field, a mirror-type
magnetic field is produced [Fig. 2(b)]. The field lines con-
strict toward the coil with a large mirror ratio,
B0ð0; 0; 0Þ=B0ð0; 0; 40 cmÞ ’ 10. There also exists a null
point located in the plane of the loop at r > rloop. It occurs
all around the loop and hence is a 2D null line. The topology
varies during the slowly varying loop current from uniform
to cusp to mirror and back to uniform magnetic field. The
wave propagation depends not only on the field topology but
also on the gradient scale length of the field compared to a
whistler wavelength. Refraction explains wave propagation
in the regime of L ¼ B0=jrB0j � k, and reflection may arise
for short gradient scale lengths. Figures 2(c) and 2(d) show
contour plots of the gradient scale length for both cases. The
smallest scale lengths arise at the null points and near the
loop wires. Waves approaching the null point will encounter
cyclotron resonance (x/xc ¼ 1), while there is no resonance
at the mirror point (0, 0, 0). It is worth nothing that refraction
and ray tracing concepts are no longer valid when L < k.
III. EXPERIMENTAL RESULTS
A. Wave reflections of helicon modes at a magneticmirror
An m¼ 1 helicon mode is excited by a loop antenna
located at z ’ 44 cm and launching waves toward a strong
mirror point. Figure 3(a) displays contours of Bxð0; y; z > 0Þwhich show the wave amplitude and phase front (contours of
Bx ¼ const). The wave propagates against B0 with V-shaped
phase fronts which are cuts through an m¼ 1 helical phase
front in 3D (see Fig. 6 in Ref. 46). As the wave propagates
toward the increasing strength of the converging field, the
parallel wavelength increases and the propagation angles of
the oblique wings become highly oblique. The latter is due
to the rapid change of the B0 topology rather than a change
of the wave phase front.
A new feature is observed: the formation of waves with
phase fronts which are inclined opposite to those of the central
wave packet. In time, these waves propagate opposite to the
incident wave packet and hence must be reflected from the
mirror region. There are no density gradients to produce the
reflection. Since there is no reflection in a uniform magnetic
field, the reflection cannot be produced by the conducting
coil.
Counter-propagating waves are known to produce linear
polarization (see Figs. 5 and 6 in Ref. 18). It arises at
FIG. 1. (a) Schematic of the experimental setup with basic parameters. (b)
Photograph of the discharge plasma with an inserted large wire loop to pro-
duce nonuniform magnetic fields, smaller loops to excite whistler modes, and
a very small triple loop on a telescopic shaft to measure the wave magnetic
fields in space and time. (c) Waveforms of the large loop current (red) and the
rf antenna current which consists of a string of 20 cycle, 5 MHz bursts.
082110-3 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
FIG. 2. Magnetic field topologies for studying whistler mode propagation in highly nonuniform fields. (a) Field lines and field strength of a cusp configuration. A
3D null point is formed on axis. (b) A strong mirror field is formed when the coil field adds to the weak axial field B0 ¼ 3 G. Magnetic field gradient scale length,
L ¼ B0=jrB0j, for (c) the magnetic cusp configuration and (d) the magnetic mirror. When L < k, refraction and ray tracing concepts are no longer valid.
FIG. 3. Observation of wave reflection from a magnetic mirror field. (a) Contours of Bx showing waves propagating from a loop antenna toward a rapidly
increasing magnetic field. It creates a reflected wave with reversed V-shaped phase fronts partly visible in the lower half of the plane. (b) Ellipticity of magnetic
hodograms showing periodic locations of nearly linear polarization (� ’ 0) in the interference zone between incident and reflected waves (see black arrows).
The incident wave has a nearly circular polarization with � ’ 1 at the nodes. (c) Amplitude modulation peaks at the interference nodes thereby also identifying
the locations of linear polarization. (d) Examples of hodograms showing circular polarization in the middle of the incident wave (point A) and linear polariza-
tion at an interference node (point B). Views from two orthogonal directions show a circular disk in (d) and a pencil shaped hodogram in (e).
082110-4 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
interference nodes where two field components vanish while
the third remains finite. With a single component left, the
field has a linear polarization. The latter can be shown from
magnetic field hodograms. The hodogram ellipticity, �¼Bmin/Bmax, assumes a minimum for linear polarization,
while circularly polarized waves have an ellipticity of � ¼ 1.
Figure 3(b) shows a contour plot of the ellipticity �. The
incident wave has a high ellipticity, i.e., the wave polariza-
tion is nearly circular. There is a string of ellipticity minima
in the interference region between incident and reflected
waves implying linearly polarized whistler modes. Since the
ellipticity results from a time average, the minima do not
propagate but are standing wave phenomena.
An alternate feature of the wave interference is the ampli-
tude dependence on time. For circularly propagating waves
with two orthogonal components (b cos xt; b sin xtÞ, the total
amplitude is constant [BðtÞ ¼ ðb2cos2xtþ b2sin2xtÞ1=2
¼ b ¼ const]. For linear polarization, the amplitude varies in
time as BðtÞ ¼ ðb2cos2xtÞ1=2 ¼ b½ð1þ cos 2xtÞ=2Þ�1=2. We
define an amplitude modulation by a suitably normalized
difference between the root-mean-square (rms) value and
the time averaged amplitude, DBrms ¼ ðBrms � hBiÞ=hBi¼ Brms=hBi � 1. Here, Brms ¼ ½hðB2
x þ B2y þ B2
z Þi�1=2
is the
root-mean-square value and hBi ¼ T�1ÐjBðtÞjdt is the time
average of the absolute value jBðtÞj. For example, for linear
polarization, Brms ¼ ðb2cos2xtÞ1=2 ¼ 2�1=2b and hBi¼ bhj cos xtji ¼ ð2=pÞb, resulting in DBrms ¼ 2�1=2=ð2=pÞ�1 ’ 0:1. For circular polarization, Brms ¼ hBi ¼ b and
DBrms ¼ 0.
The contour plot DBrms in Fig. 3(c) shows a string of
localized DBrms peaks where � had minima, both describing
locations of linearly polarized whistler modes. These are
spaced apart by k/2 and created by interference of oppositely
propagating waves of comparable amplitudes. Without inter-
ference, the wave amplitude of the circularly polarized inci-
dent wave is nearly constant in time (DBrms ’ 0).
The lower half of Fig. 3 displays selected hodograms
confirming the polarization properties of interfering whistler
modes. In the center of the incident wave packet [location A,
Fig. 3(d)], the B-field rotates with a constant radius in a
right-hand circle (left plot, view along its normal). The side
view (right plot, view at right angle to its normal) shows that
the vector rotates in a plane with normal predominantly in
the z-direction. At this location, the hodogram describes a
slightly oblique whistler mode. At a location of wave inter-
ference [location B, Fig. 3(e)], the hodogram is nearly linear
with � ¼ Bmin=Bmax ’ 0. The views are the same as in (d)
and confirm that the hodogram is a narrow cylinder. The
major axis describes the direction of linear polarization. The
field components normal to the major axis vanish due to
destructive interference of incident and reflected oblique
whistler modes.
The direction of wave propagation along B0 can be iden-
tified from the sign of the helicity density J � B. Figures 4(a)
and 4(b) display instantaneous contours of Bx and Jx, respec-
tively, for cw conditions. Both contours are highly similar
although the sign of Bx and Jx differs for the incident wave
(see black dots) but is equal for the reflected wave (see white
dots). Recall that in ideal electron magnetohydrodynamics
(EMHD), J ¼ vkrHB, where vk is the phase velocity parallel
to B0 and rH ¼ ne=B0 is the Hall conductivity. Thus, one
has J � B> 0 for vphase � B0 > 0 and vice versa. The incident
wave propagates against B0, and hence, J �B< 0, while the
reflected wave has J � B> 0, as shown in Fig. 4(c). Since
J / B, the helicity is proportional to the magnetic energy
density and the periodicity is half a wave length.
The reflected wave is best seen at the end of the wave
burst. After the incident wave has propagated beyond the
mirror point, the reflected waves still propagate to the right
away from the mirror point. No interference occurs. Figures
4(d)–4(f) show contours of the three field components Bx, By,
and Bz, respectively. All show V-shaped phase contours of
alternating polarity, spaced k apart. The wave propagation is
oblique and points radially outward, similar to the incident
waves.
B. Space-time evolution and reflection coefficient
Rf bursts allow us to observe incident and reflected
waves separately. Figure 5 displays a time series of contours
of a component of the helicity density, JxBx, at (a) the begin-
ning of the rf burst, (b) the middle of the burst, and (c) the
end of the rf burst. For each case, a sequence of six consecu-
tive snapshots is presented so as to observe the direction of
wave propagation. Superimposed are white lines showing
the direction of the background magnetic field B0.
At the beginning of the rf burst, only an incident wave is
observed [Fig. 5(a)]. No reflected wave is generated until the
incident wave arrives as the mirror field (Dt ’ 2T ¼ 2=f ).
From the axial displacement of the phase fronts, one obtains
a parallel phase velocity of the incident wave, Dz=Dt ’66 cm/ls. Closer to the mirror point, where the field
increases by an order of magnitude [see Fig. 2(b)], the axial
wavelength increases and the k vector becomes highly obli-
que. When the wave reaches the mirror point (Dt ¼ 2 T¼ 2/
f), the reflection starts. Near the axis, the large incident wave
covers up the smaller reflected waves, but radially outward,
the reflected wave begins to dominate.
In the middle of the rf burst, both the reflected and inci-
dent waves are visible [Fig. 5(b)]. The waves appear to be
stationary during the steady state of the rf burst, but the
phase fronts propagate by half a wavelength each half rf
period. With the increasing distance, the reflected wave con-
tours vanish due to wave decay and discrete contour levels.
The reflection coefficient depends on the axial position due
to wave damping and reaches unity near the reflection point,
as is shown below.
At the end of the rf burst, the incident wave moves out
of the measurement plane so that the phase fronts of the
reflected wave become visible [Fig. 5(c)]. The phase fronts
are also V-shaped but wider than those of the incident wave,
as there is no wave energy along the axis of the reflected
wave. Connecting the centers of the wave amplitude peaks
yields the direction of the group velocity which is oblique
and radially outward. The normal to the phase fronts points
in the direction of the phase velocity, which is even more
oblique than the group velocity. Extrapolating the group
velocity backwards might give the impression that the origin
082110-5 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
for the reflection is near the center of the loop, but the lack
of wave energy on axis suggests otherwise. The large waves
appear to originate near the wire of the loop where the larg-
est gradient scale length and the steepest curvature of the
field lines are located [see Fig. 2(c)]. In this region, k � B0
reverses sign causing the wave to reflect.
The separation of incident and reflected waves at the
end of the rf burst allows one to estimate the reflection
coefficient, here defined as the energy ratio R ¼ U0refl=U0inc,
where U0 ¼ dU=dz ¼Ð
B2=ð2l0Þ2prdr is the wave energy
per axial length. Figure 6(a) shows traces of U0ðt; zÞ at the
end of the rf burst for the mirror field. The early large value
corresponds to the incident wave energy, and the second
peak is the reflected wave. The family of curves shows the
dependence on the axial position. Each wave decays in its
direction of propagation, i.e., in the –z direction for the inci-
dent wave and þz direction for the reflected wave. The
propagation delay of the reflected wave confirms that the
wave propagates in the þz-direction. In the absence of
damping, the reflection coefficient would be independent of
position, but to avoid damping effects, the reflection ratio
should be measured at the point of reflection which is near
FIG. 4. Helicity properties and components of the reflected wave magnetic field. (a) Instantaneous contours of Bx during quasi cw conditions (middle of long rf
burst). (b) Contours of the current density Jx which are nearly identical to those of Bx but with reversed sign for the incident wave (see black dots) and equal
sign for the reflected mode (see white stars). (c) Contours of JxBx, one component of the current helicity density. Its sign depends on the direction of wave
propagation along B0. Its magnitude is proportional to the energy density. The contours indicate the phase fronts whose spatial periodicity is k/2. As the wave
propagates into the constricting mirror field, the wavelength increases and the wave becomes highly oblique. The energy decreases since the propagation veloc-
ity increases. The energy density of the reflected wave is much smaller than that of the incident wave. (d), (e), and (f) Contours of Bx, By, and Bz, respectively,
for the reflected wave which is visible after the incident wave has left the measurement plane at the end of the rf burst. The reflected wave has also V-shaped
phase fronts which trace back to near the mirror coil where the reflection originates.
082110-6 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
the loop. As the incident wave travels from 40 cm to 20 cm,
its energy decays from 1 to 10–1. Extrapolating to z¼ 0, its
energy would decay to 10–1. When the reflected wave trav-
els from 40 cm to 20 cm, its energy increases from 10–3 to
10–2. Thus, the extrapolated reflection coefficient is R ’ 1
at the loop which is probably overestimated since the wave
penetrates to the other side of the loop [see Fig. 7(a)].
Nevertheless, the wave reflection from a strong magnetic
field is very significant.
The same evaluation has been done for the cusp field
topology. Figure 6(b) shows no reflected wave. Likewise, no
reflection is expected or seen in a uniform field (not shown).
C. Wave reflection mechanism in highly nonuniformmagnetic fields
Reflection and refraction of plane waves are usually
explained by the properties of the refractive index n
¼ c/vphase. Reflection occurs when the refractive index
changes abruptly, while refraction applies to a gradually
changing index where reflection is negligible. The refractive
index of low frequency whistler modes depends on the den-
sity and magnetic field, n ’ xp=ðxxcÞ1=2 / ðne=B0Þ1=2. It
is well known that density discontinuities reflect waves,47
but much less is known about reflections from magnetic field
FIG. 5. Time-space dependence of the
reflection of a whistler wave packet
from a magnetic field gradient. A wave
burst is used to observe separately the
incident wave and the reflected wave
as well as their interference. Displayed
are consecutive snapshots of the helic-
ity density JxBx whose sign indicates
the direction of wave propagation with
respect to the dc magnetic field B0. (a)
At the turn-on of the rf burst, an m¼ 1
helicon mode propagates toward the
mirror field. A dashed white line
through a constant phase point indi-
cates the parallel phase velocity,
vk ’ 66 cm/ls. Since k � B0 < 0, the
helicity is negative. When the incident
wave reaches the peak mirror field, the
reflected wave with positive helicity
appears. (b) In the middle of the
20 cycle rf burst, the conditions resem-
ble those of a cw wave. It shows the
superposition of incident and reflected
waves. It forms interference nodes
where the counter propagating waves
have equal amplitudes and opposite
signs. (c) The incident wave is turned
off at the end of the rf burst while the
reflected wave still propagates. The
phase fronts are V-shaped similar to
those of the incident wave, but there is
no wave energy along the axis of the
reflected wave, and hence, the reflec-
tion is not specular. Both phase and
group velocities are oblique to B0 and
radially outward, suggesting that the
reflection surface is convex.
082110-7 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
gradients. This effect has been shown and explained in a
recent letter.48
In the present work, we observe in detail the motion of
the incident phase front in an ambient magnetic field whose
curvature changes on a scale smaller than a wavelength. The
off-axial field lines bend faster than the phase front such that
the incident wave changes from oblique to perpendicular and
beyond. When k � B0 < 0, the wave propagates opposite to
B0, i.e., is reflected. This forms a turning point for the phase
and group velocities which can lead to wave reflection from
the sides of the mirror field. Even a small negative kk produ-
ces a significant reflection of the wave energy since for
highly oblique whistlers, the group velocity is closely field
aligned. The model is based on observations of the space-
time evolution of the wave burst shown in Figs. 5 and 7(a).
1. Wave propagation through mirror and cusp fields
In order to better observe the wave propagation at the
mirror point, the measurement plane has been extended to
both sides of the large loop. The source antenna is now posi-
tioned closer to the large loop. There is an unavoidable data
gap of Dz ¼ 6 2 cm width to prevent collisions between the
diagnostic antenna and the large loop. Figure 7(a) shows the
wave propagation as it travels to and through the large loop.
As the wave travels toward the mirror point, the parallel
phase velocity increases predominantly on axis where B0(r)
peaks. This causes the V-shaped phase fronts to become
elongated such that the k vector becomes highly oblique to
the B0 field. Furthermore, the advancing wave encounters a
rapidly changing magnetic field as it travels along the B0
lines. When the field lines become steeper to the V-shaped
phase fronts, the angle between k and B0 exceeds 90�. This
results in wave reflection.
These conditions occur only on the side of the incident
wave. On the left side of the mirror loop, the transmitted
wave fronts gradually return to V-shapes and the helicon
mode continues to propagate in the –z direction. It shows
that the reflection must be smaller than unity.
The reflection mechanism has some similarity to that
proposed for whistler reflections in the magnetosphere.
When a whistler wave propagates toward the poles, the wave
encounters the lower hybrid resonance where k?B0. This
condition creates a turning point for electron whistler
waves.49 The main difference is that in the present experi-
ment, the propagation direction changes within less than a
wavelength, while in space, the reflection is a classical ray
bending model.
We now come to the wave propagation against a cusp
magnetic field obtained by reversing the loop current. The
cusp field has comparable gradient scale lengths to that of a
mirror field [see Figs. 2(a) and 2(c)], such that one might
also expect a wave reflection. However, Fig. 7(d) shows that
no reflection is observed. The incident wave encounters
cyclotron resonance and is strongly absorbed in regions
around the null point where x/xc > 1. A reflected wave can-
not propagate out of the null point region. This has been
tested by placing the antenna on the null point and observing
no wave excitation (not shown). Since the phase front of the
wave is wider than the null region, the V-shaped wings of the
wave packet continue to propagate along the separatrix albeit
with small amplitudes. The phase normal on the separatrix
indicates a nearly parallel whistler mode.
Highly nonuniform fields imply small spatial scales, as
is shown in Figs. 7(b) and 7(e), where the gradient scale
length, L ¼ B0=jrB0j, is shown for the mirror and cusp
cases, respectively. They interact with part of a larger wave
packet such that only a fraction of the wave energy is
reflected or absorbed. The interaction may be better
described as a scattering process rather than a refraction pro-
cess of a plane wave because there, as shown in Figs. 7(c)
and 7(f), the measured wavelength is smaller than the gradi-
ent scale length.
Figure 8 shows the exciter antenna located at x¼ 0,
y¼ 5 cm, and z¼ 18 cm which is in the right hemisphere of
the 16 cm diam loop. At this location, the loop can radiate
from near the cusp null point. Furthermore, wave propaga-
tion toward and away from the nonuniform B0 field can be
compared. The B0 field lines are shown by white lines. The
receiving probe records the three wave components, one of
which Bx is displayed by contours in the central y–z plane
(x¼ 0), except in gaps near the rf antenna and the large loop.
Without loop current, the field is uniform and small, B0
¼ 3 G. The m ¼ þ1 helicon mode propagates symmetrically
to both sides of the rf antenna with equal amplitudes but odd
signs in the current density, magnetic helicity, and phase
FIG. 6. Estimation of the reflection coefficient of whistler modes from a mir-
ror field. Displayed is the field energy per axial length U0 ¼ dU=dz vs time
and z at the end of an rf burst where the reflected and incident waves are sep-
arated in time. The data are taken (a) in a mirror field and (b) in a cusp field.
Several axial positions are shown to demonstrate propagation delays and
energy decay. In the center of the plane (z ’ 30 cm), the reflection coeffi-
cient is R ¼ U0refl=U0inc ’ 10�3, but this is not meaningful because the waves
are damped as they propagate in opposite directions. One can extrapolate
from the observed damping rate the incident and reflected wave energies to
near the loop where the reflection occurs. They are approximately equal,
implying R � 1 near z � 0.
082110-8 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
helicity. Note that this is a relatively high frequency helicon
mode (x/xc ¼ 0.6). It has the same V-shaped phase contours
as other low frequency mode studies (e.g., x/xc ¼ 0.35 for
B0 ¼ 5 G, Fig. 4(a) in Ref. 18).
Waves can transmit through the field constriction in a
mirror field [Fig. 8(b)]. Surprisingly, the constriction does
not enhance the wave amplitude. The wave speed increases
with increasing B0, whereby the conservation of the energy
flux S ¼ vgroupB2=2l0 requires a decrease in field strength.
Conversely, the waves propagating along expanding field
lines have a larger amplitude on the right side of the
antenna.
In a weak FRC [Fig. 8(c)], the cyclotron resonance
region is small compared to the lateral extent of the helicon
wave packet. The center of the wave packet is damped while
the outer wings do not experience cyclotron damping and
continue to propagate around the FRC. This should also
apply to plane waves whose transverse dimensions are
always larger than an FRC. To the right side of the rf loop,
the wave propagates similar to that in a nearly uniform field.
In a strong FRC, the null point shifts close to the rf
antenna [Fig. 8(d)], but the antenna is vertically offset by Dy¼ 5 cm. No waves can be transmitted axially into the FRC,
but weak waves propagate around the FRC along the separa-
trix. On the right-hand side of the rf loop, the waves propa-
gate in a weakly constricting field which slightly reduces the
wave amplitude compared to that in a uniform field.
Finally, we move the rf antenna close to the large coil
(Dz ¼ 2 cm) so as to excite waves inside an FRC or waves
traveling away from a mirror point. Figure 9 indicates the
antenna position and the field topology for the three cases:
uniform, mirror, and cusp B0. The contours of JxBx yield the
direction of wave propagation from the sign of the helicity
and give a measure for the wave energy density from the
contour levels.
In a uniform field [Fig. 9(a)], the wave propagates along
B0 with the largest signals along the z-axis of the antenna
(x¼ y¼ 0).
In a mirror field [Fig. 9(b)], the waves propagate again
along B0 and spread radially outward since the field lines
diverge away from the loop. No wave reflections are seen
when the wave leaves the peak mirror field.
In a strong FRC field [Fig. 9(c)], the waves propagate to
the right side against B0, and hence, JxBx < 0. To the left
side of the rf antenna (not shown), the wave would propagate
along B0 and have a positive helicity density (JxBx > 0).
Waves propagating along the spine are absorbed near the 3D
null point and hence cannot escape axially. Waves have
curved phase fronts, yet propagate nearly parallel along the
diverging B0 lines.
Figures 9(d) and 9(e) show two hodograms on the
antenna axis (y¼ 2 cm and z ’ 10 cm), one in a mirror and
another in a cusp field. In the mirror field (b), k � B0 > 0
such that the polarization is right-handed with respect to B0.
FIG. 7. An rf loop excites m¼ 1 helicons propagating in highly nonuniform background magnetic fields B0. (a) The wave propagates through a strong mirror
magnetic field. Contours of Bx, mapped on both sides of the mirror loop, show that near the mirror throat, the wave vector is perpendicular to B0, which can
become a turning point for wave reflection. The focusing of the wave by the field constriction does not increase the amplitude since the velocity increases with
B0. The latter also explains the axial stretching of the V-shaped phase fronts. (b) Field lines and contours of the gradient scale length L ¼ B=jrBj of the mirror
magnetic field. (c) Gradient scale length of the mirror magnetic field normalized to the theoretical parallel wavelength kk, indicating that the WKB approximation
fails for small L=kk. (e) and (f) The same as (b) and (c) for the field reverse configuration topology. (d) A negative loop current produces an FRC configuration.
The wave stagnates near the 3D null point. No waves penetrate into the FRC but can propagate on and outside the separatrix. No wave reflections are observed.
082110-9 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
The surface normal n̂ points in the direction of the wave vec-
tor k which is nearly parallel to B0. It confirms that the wave
propagates closely in the þz direction with nearly circular
polarization.
When the field is reversed [Fig. 9(e)], the hodogram also
rotates right-handed in time around the normal which now
points in the direction opposite to the k vector. The hodo-
gram is nearly circular when viewed along n̂. In the FRC
topology, k remains unchanged while B0 is reversed. The
right-handed polarization around B0 is preserved, but the
spatial phase helicity becomes right-handed in the z-direc-
tion. Thus, the hodogram normal, n̂, and the k vector point
in opposite directions when k � B0 < 0, which is a well-
known sign ambiguity for hodograms.18 Since a simple loop
excites helicon modes with phase rotation in the same direc-
tion as the polarization, the phase rotation is in the –/ direc-
tion which shows it is an m ¼ –1 mode. With both negative
m and negative kz, the spatial phase surface is a right-handed
helix, –/ –kz ¼ const. The helix is left-handed and the heli-
con is an m ¼ þ1 mode in a uniform field or mirror field.
The three different field topologies occur during each
discharge pulse. It is worth showing the time dependence of
the received probe signal which depends on many parame-
ters such as Iloop, zrf, loop, rprobe, B0, x, and ne. In Figs.
9(f)–9(h), we show just one parameter variation, the position
of the receiving probe along the z-axis, while all other
parameters remain constant. Closest to the antenna [z ’ 2
cm in Fig. 9(f)], the received signal Bx(t) is large and can be
seen for uniform B0 (Iloop ¼ 0), mirror B0 (Iloop > 0), and
cusp B0 (Iloop < 0). Only for small Iloop, when the null point
coincides with the rf loop position, the signal vanishes.
When the receiving probe is located in the middle of the
measurement plane [z¼ 10 cm, Fig. 9(g)], the signal is
received for Iloop � 0, albeit with a smaller amplitude due to
damping and wave spread. However, for Iloop � 0, the signal
reaches the probe only for large FRC fields where both the
exciting and receiving antennas are inside the separatrix.
Finally, when the receiving probe is located at the far
right-hand side of the plane [z ’ 18 cm, Fig. 9(g)], no signal
is observed because the position of the null point coincides
with that of the receiving probe. Since the transmitting
antenna is inside the separatrix (z ’ �2 cm), there are waves
inside the separatrix but they are absorbed before reaching the
receiver probe. Thus, one antenna on a null point is sufficient
for total blackout between the transmitter and the receiver.
2. Polarization and phase helicity
Polarization and helicity of whistler modes are different
quantities.2 Polarization refers to the rotation of a field vector
which is usually described by a hodogram. The polarization
of electron whistlers ranges from linear to circular, generally
right-hand elliptical. The phase surface of helicons is heli-
cal.50 The theoretical model for helicon modes assumes para-
xial wave propagation, B / exp ½iðm/þ kkz� xtÞ�, which
describes axial and azimuthal phase propagation with radial
FIG. 8. Expanded view of waves propagating against a mirror and a cusp magnetic field. Contours of instantaneous Bx show amplitude and phases of the m¼ 1
helicon mode excited by a loop antenna located at z¼ 16 cm from the large loop. (a) Wave propagation in a uniform field is symmetric in the z-direction. (b)
When the wave propagates against a mirror field, the phase fonts nearly align with B0, i.e., k?B0. (c) Wave propagation against a weak cusp field. The center
of the wave packet stops propagating but the wings continue to propagate outside the separatrix as nearly parallel whistler modes. (d) Wave propagation
against a strong cusp field which moved the 3D null point close to the antenna. Weak waves are found well away from the null region.
082110-10 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
standing waves. Paraxial propagation leads to helical phase
surfaces with a constant radius. This theoretical model does
not explain observations where kr is determined by density
profiles or antenna properties but not by reflections from
radial boundaries. Even in unbounded uniform plasmas,
spiraling whistler modes have been observed.5
There is radial phase propagation in an unbounded
plasma but with little radial energy flow, resulting in weakly
FIG. 9. Wave properties in different B0 fields when the waves are excited with an m¼ 1 antenna near the large current carrying loop. [(a)–(c)] Contours of the
current helicity component JxBx whose sign indicates the direction of wave propagation along B0. Superimposed are B0 field lines in white. (a) In a uniform
field and (b) in a mirror field, the waves propagate with wave vector k � B0 > 0, and hence, JxBx > 0. (c) In a field-reversed configuration, the waves propagate
with k � B0 < 0 and cannot travel beyond the 3D null point. Note that the whistler modes have curved phase fronts, yet are propagating parallel to the diverging
field lines. (d) Magnetic hodogram on axis for the m¼ 1 helicon mode in a uniform magnetic field, showing circular polarization and a hodogram normal
n̂ ’k k k B0. (e) B0 is reversed in an FRC while k remains unchanged; hence, the polarization with respect to B0 is still right-hand circular but the field rota-
tion with respect to k is reversed. The phase surface forms a right-handed spiral. The field rotates in the –/ direction when it passes through a z ¼ const plane,
identifying it as an m ¼ –1 mode. (f) Rf bursts (red traces) received on axis at z¼ 2 cm from the loop. Superimposed is the waveform of the loop current (black
trace) which varies from the mirror to cusp configuration. The signal passes for both field configurations. (g) Received signal at z¼ 10 cm passes through the
mirror field but not through an FRC unless the probe is within the separatrix which requires large negative currents. (h) Received signal at z¼ 18 cm is cut off
by an FRC when its null point does not expand beyond the position of the receiving probe.
082110-11 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
conical phase surfaces. At a fixed instant of time, the phase
front of a parallel propagating helicon (k k B0 in the z-direc-
tion) is described by m/þ kkz ¼ const. The phase surface
forms a left-handed helix, d/=dz ¼ �kk=m < 0. If either mor kk, but not both, is reversed, the helix is right-handed or
the phase helicity is positive.
When time increases, the helical phase surface translates
along z but does not rotate, md/þ kkdz� xdt ¼ 0. When
the helix passes through a plane z ¼ const, the phase con-
tours in the plane rotate around the z-axis or B0. The sense of
rotation d/=dt ¼ x=m is positive or ccw for m> 0, irrespec-
tive of kk. Thus, when a loop antenna with the axis across B0
excites waves to both sides, they have the same sense of rota-
tion but opposite spatial phase helicities. This is not always
agreed upon (see Fig. 7 in Ref. 51).
Since the wave field lines lie approximately on the phase
surfaces (see Fig. 7(g) in Ref. 46), a twisted phase front implies
twisted field lines, i.e., magnetic helicity A � B, where A is the
vector potential. Magnetic helicity is a conserved quantity in
ideal electron magnetohydrodynamics (EMHD).52 If the
antenna injects helicity, the wave will acquire the same helicity
and travel only in one direction.53 The radiated wave is much
stronger when the antenna twist matches that of a helicon
mode which can only occur on one side of the antenna. The
directionality switches sides when the B0 field is reversed. This
has sometimes been associated with different properties of m¼ –1 and m ¼ þ1 modes, but the effect is due to the antenna
directionality. A single loop does not inject helicity, and hence,
the oppositely propagating waves must have opposite helicities
and equal amplitudes to conserve zero total helicity.
FIG. 10. Rotation of helicon modes for different field topologies of B0. The rotation can be seen in the sequence of snapshots of Bz(x, y, z ¼8 cm) at one quarter
rf period apart. (a) For propagation along a uniform B0 field, the rotation is in the þ/ direction (see white circle in the last frame). This identifies the wave as
an m ¼ þ1 helicon mode [exp iðm/þ kkz� xtÞ]. The phase surface forms a left-handed helix in space. The polarization, which describes the wave vector
rotation in time around B0, is right-handed [see hodogram in Fig. 9(d)]. (b) When B0 has the topology of a mirror field, the field strength is increased but the
direction is unchanged; hence, the helicon rotation is the same as in (a). (c) When B0 has the topology of an FRC, B0 is reversed. The polarization [Fig. 9(e)]
and the field rotation are reversed, identifying the helicon as an m ¼ –1 mode. Since the wave propagates against B0, the phase surface forms a right-handed
spiral in space (–/ þ kzz ¼ const). When the right-handed spiral passes through a z ¼ const plane, the field rotates in the –/ direction, as observed. [(d)–(f)]
Rotation angles D/(t) with Dt ¼ T/20 (10 ns) time resolution for the three B0 topologies. (d) The rotation rate in a uniform field is d//dt ¼ x/m ¼ const. For
m¼ 1, the angle changes by 2p in one rf period. (e) In a nonuniform magnetic field, the average rotation rate is positive but exhibits small jumps every half rf
period, indicating a contribution from linear field polarization. (f) When B0 is reversed, d//dt < 0 with small jumps as in (e).
082110-12 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
In the present experiment, we can readily change the
direction of B0 with the FRC configuration. The m¼ 1 loop
is placed inside the separatrix and its wave propagation is
measured on axis. For comparison, the wave properties are
also measured in mirror field topologies and uniform fields.
Of particular interest is the helicon field rotation which is
seen by a sequence of successive snapshots separated by Dt¼ T/4.
Figures 10(a) and 10(b) show the field rotation of Bz(x,
y, z¼ 8 cm) when k is in the same direction as B0 which is
either a uniform field or in a mirror configuration. In both
cases, the phase rotates in the þ/ direction. By the above
definition, a temporal rotation in the þ/ direction (d//dt¼x/m> 0) identifies it as an m ¼ þ1 helicon mode.
When B0 is reversed, while the wave propagation direc-
tion remains along z, the sense of phase rotation of Bz in the
x–y plane is found to reverse in the –/ direction defining it
as an m ¼ –1 mode [Fig. 10(c)]. The amplitude also remains
the same since the antenna is not phased. Thus, the m ¼ –1
mode propagates just as well as the m ¼ þ1 mode. The
reversed rotation and the same kz imply that the helical phase
surface has a right-handed twist. The phase rotates in the
same direction as the polarization, i.e., right-handed with
respect to the reversed B0.
The azimuthal phase rotation for the three B0 topologies
is shown with higher temporal resolution in Fig. 10(d). For a
uniform field, the rotation is nearly constant (d//dt ¼ x/m¼ const), while in nonuniform fields, there are jumps in
phase which indicate partially linear polarizations such as
the antenna vacuum field or that the plane is not everywhere
orthogonal to the curved B0 lines.
Since the magnetic field lines are approximately parallel
to the phase fronts (see Fig. 3 in Ref. 19), the twist of the
phase surface is related to the helicity of the wave magnetic
field or the current density since J ’k B. The sign of the hel-
icity is given by that of k � B0, and hence, the helix twist is
reversed in the FRC compared to the mirror field. Likewise,
the twist of the helicon phase surface in a uniform magnetic
field differs on either side of the antenna. However, the tem-
poral phase rotation is the same when the different spirals
propagate in different directions through z ¼ const planes.
Thus, the m-number is the same for helicons emitted to both
sides. Reversal of k with respect to B0 is not reciprocal.
For a single loop, the field rotation is the same as that of
the polarization because EMHD physics rotates the wave
field in the same direction as the polarization (see Fig. 7 in
Ref. 46 or Fig. 8 in Ref. 54). With a circular antenna array,
phased in the / direction, one can impose the m-number of
helicon modes such that field rotation and polarization differ
(see Fig. 7 of Ref. 55). Oppositely propagating waves (6k or
6 m) can produce axial or azimuthal standing waves where
helicons exhibit linear polarization (see Fig. 4 in Ref. 48).
The early antennas for helicon plasma sources were
“double saddle” antennas (also known as “Boswell” or
“Nagoya Type III” antennas, see Fig. 8 in Ref. 56), which
are essentially Helmholtz coils draped over a cylindrical
glass tube to produce an rf magnetic field across B0. These
antennas excite equal m ¼ þ1 modes to both sides just like a
single internal loop (see Fig. 6 in Ref. 3). When such
antennas are elongated (length ’ k=2) and twisted by 180�
around B0 (see Fig. 9 in Ref. 56 or Figs. 11 and 12 in Ref.
54), the radiation pattern becomes directional. The preferred
direction is that where the antenna helicity matches the wave
helicity. Directionality arises from the fact that the phase hel-
icity switches sign on either side of the antenna.
Directionality can also be obtained from axially phased
arrays which match the parallel wave propagation (see Fig. 3
of Ref. 57). The antenna directionality explains the confu-
sion about the existence of m ¼ –1 helicon modes in plas-
mas.56 There is no question that negative m-mode helicons
can propagate in plasmas (see Fig. 7 of Ref. 55).
Rotating magnetic field (RMF) antennas have also been
used for more efficient coupling to whistler waves.58,59 The
rf field is perpendicular to B0 and rotates in the þ/ or –/direction by a 690� phase shift in the currents of the two
orthogonal loops. When the antenna field rotates in the same
direction as the field polarization (m ¼ þ1), the wave excita-
tion is stronger than for the opposite direction of rotation.
But the enhancement over a single loop is modest and can be
explained by the fact that the rotating field isffiffiffi2p
larger than
for the non-rotating field for the same antenna current.
However, field rotation opposite to the electron rotation
decreases the wave amplitude significantly (see Fig. 13 in
Ref. 54). Unlike twisted antennas, the rotating field antennas
impose no kk and hence exhibit no directionality.
IV. CONCLUSION
Experiments on wave propagation in highly nonuniform
mirror and cusp magnetic fields have been described. Wave
reflection from a strong mirror field is observed and
explained. Reflection can arise from strong gradients in the
refractive index which depends on the density and magnetic
field. Although refraction by density gradients is well known,
the complementary effect of reflection by magnetic field gra-
dients has not been demonstrated elsewhere. Strong mag-
netic field gradients can exist in space plasmas at magnetic
shocks or lunar crustal magnetic fields, but the diagnostics
for wave reflection would be very difficult. It may also be
difficult to observe reflections in helicon devices because of
axial density gradients and boundaries which can also pro-
duce wave reflections.
The interference between incident and reflected waves
produces nodes in which two field components vanish. At
these locations, linear polarization arises which is confirmed
by hodograms and by amplitude oscillations.
Wave propagation against a magnetic null point of an
FRC leads to wave absorption by cyclotron resonance. When
the wave is launched inside the separatrix, the wave is
trapped. The polarization and phase rotation inside the FRC
are reversed compared to those outside the FRC. The helicon
phase rotation is that of an m ¼ –1 mode which propagates
as well as an m ¼ þ1 mode.
Propagation against cyclotron resonance can arise in
plasma thrusters with expanding magnetic fields.60 Theories
for cyclotron damping assume plane waves and 1D field
lines which cannot be realized in space or laboratory
082110-13 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)
plasmas. Simulations are required to model wave propaga-
tion in nonuniform magnetic fields.61
This series of papers on whistler modes in nonuniform
plasmas demonstrates the discovery of several other new
effects. Whistler modes propagating radially across circular
magnetic fields are observed. Along circular field lines,
counter propagating waves can form standing waves along
B0 while also propagating across B0.
Whistler wave packets are inherently 3D structures
which require 3D field measurements. The current density
can only then be calculated, the field lines and phase surfaces
resolved, force-free field topologies characterized, and the
orbital angular momentum (OAM) of helicon modes
obtained. Bending of helicon orbital angular momentum
(OAM) on curved field lines produces a precession due to
momentum conservation. This results in phase shifts and
splitting of wave packets. The rotation vanishes in strongly
nonuniform fields. Since whistler modes have force-free
fields, the wave magnetic field lines exhibit writhed flux
tubes rather than straight field lines. The relationship
between angular momentum and helicity has been pointed
out. Novel diagnostics with vector fields of hodograms nor-
mals and of Poynting vectors are used to show the flow of
phase and energy in highly nonuniform fields.
ACKNOWLEDGMENTS
The authors gratefully acknowledge support from NSF/
DOE Grant No. 1414411.
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