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.

1C. R. Leg�endy, Phys. Rev. 135, A1713 (1964).2J. P. Klozenberg, B. McNamara, and P. C. Thonemann, J. Fluid Mech. 21,

545 (1965).3R. W. Boswell, Plasma Phys. Controlled Fusion 26, 1147 (1984).4K. Niemi and M. Kr€amer, Phys. Plasmas 15, 073503 (2008).5R. L. Stenzel and J. M. Urrutia, Phys. Rev. Lett. 114, 205005 (2015).6A. W. Degeling, G. G. Borg, and R. W. Boswell, Phys. Plasmas 11, 2144

(2004).7C. M. Franck, O. Grulke, A. Stark, T. Klinger, E. E. Scime, and G.

Bonhomme, Plasma Sources Sci. Technol. 14, 226 (2005).8R. L. Stenzel and J. M. Urrutia, Phys. Plasmas 23, 082120 (2016).9R. L. Stenzel, Adv. Phys.: X 1, 687 (2016).

10R. A. Helliwell, Whistlers and Related Ionospheric Phenomena (Stanford

University Press, Stanford, CA, 1965).11R. L. Stenzel, Phys. Fluids 19, 865 (1976).12K. G. Budden, The Propagation of Radio Waves (Cambridge University

Press, Cambridge, UK, 1988).13M. G. Rao and H. G. Booker, J. Geophys. Res. 68, 387, https://doi.org/

10.1029/JZ068i002p00387 (1963).14A. Cardinali, D. Melazzi, M. Manente, and D. Pavarin, Plasma Sources

Sci. Technol. 23, 015013 (2014).15B. D. McVey and J. E. Scharer, Phys. Fluids 17, 142 (1974).16S. Shinohara and A. Fujii, Phys. Plasmas 8, 3018 (2001).17T. Kaneko, K. Takahashi, and R. Hatakeyama, J. Plasma Fusion Res. 2,

038 (2007).18J. M. Urrutia and R. L. Stenzel, Phys. Plasmas 25, 082108 (2018).19R. L. Stenzel and J. M. Urrutia, Phys. Plasmas 25, 082109 (2018).20M. C. Griskey and R. L. Stenzel, Phys. Plasmas 8, 4810 (2001).21V. S. Sonwalkar, X. Chen, J. Harikumar, D. Carpenter, and T. Bell,

J. Atmos. Sol.-Terr. Phys. 63, 1199 (2001).22O. Santolik, M. Parrot, and F. Lefeuvre, Radio Sci. 38, 1010, https://

doi.org/10.1029/2000RS002523 (2003).23O. P. Verkhoglyadova and B. T. Tsurutani, Ann. Geophys. 27, 4429

(2009).24P. M. Bellan, Phys. Plasmas 20, 082113 (2013).25R. L. Stenzel, Phys. Fluids 19, 857 (1976).

26H. Sugai, M. Maruyama, M. Sato, and S. Takeda, Phys. Fluids 21, 690

(1978).27A. V. Kostrov, A. V. Kudrin, L. E. Kurina, G. A. Luchinin, A. A. Shaykin,

and T. M. Zaboronkova, Phys. Scr. 62, 51 (2000).28R. L. Smith, R. A. Helliwell, and I. W. Yabroff, J. Geophys. Res. 65, 815,

https://doi.org/10.1029/JZ065i003p00815 (1960).29M. S. Sodha and V. K. Tripathi, J. Appl. Phys. 48, 1078 (1977).30V. I. Karpman, Whistler Solitons, Their Radiation and the Self-Focusing

of Whistler Beams (Springer, Berlin, 1999).31V. A. Koldanov, S. V. Korobkov, M. E. Gushchin, and A. V. Kostrov,

Plasma Phys. Rep. 37, 680 (2011).32A. V. Streltsov, J. R. Woodroffe, and J. D. Huba, J. Geophys. Res. 117,

A08302, https://dx.doi.org/10.1029/2012JA017886 (2012).33M. E. Gushchin, S. V. Korobkov, A. V. Kostrov, A. V. Strikovsky, T. M.

Zaboronkova, C. Krafft, and V. A. Koldanov, Phys. Plasmas 15, 053503

(2008).34J. Chum, O. Santolik, and M. Parrot, J. Geophys. Res. 114, A02307, https://

dx.doi.org/10.1029/2008JA013585 (2009).35Y. Tsugawa, Y. Katoh, N. Terada, H. Tsunakawa, F. Takahashi, H. Shibuya,

H. Shimizu, and M. Matsushima, Earth Planets Space 67, 36 (2015).36B. T. Tsurutani, E. J. Smith, M. E. Burton, J. K. Arballo, C. Galvan, X.-Y.

Zhou, D. J. Southwood, M. K. Dougherty, K.-H. Glassmeier, F. M.

Neubauer, and J. K. Chao, J. Geophys. Res. 106, 30223, https://doi.org/

10.1029/2001JA900108 (2001).37X. H. Deng, M. Zhou, S. Y. Li, W. Baumjohann, M. Andre, N. Cornilleau,

O. Santolik, D. I. Pontin, H. Reme, E. Lucek, A. N. Fazakerley, P.

Decreau, P. Daly, R. Nakamura, R. X. Tang, Y. H. Hu, Y. Pang, J.

Buechner, H. Zhao, A. Vaivads, J. S. Pickett, C. S. Ng, X. Lin, S. Fu, Z.

G. Yuan, Z. W. Su, and J. F. Wang, J. Geophys. Res. 114, A07216, https://

dx.doi.org/10.1029/2008JA013197 (2009).38K. Fujimoto, Geophys. Res. Lett. 41, 2721, https://dx.doi.org/10.1002/

2014GL059893 (2014).39T. Shoji, H. Kikuchi, Y. Sakawa, C. Suzuki, G. Matsunaga, and K. Toi, in

Proceedings of the 30th International Conference on Plasma Science(ICOPS) (IEEE Conference Publications, 2003), p. 466.

40P. K. Loewenhardtt, B. D. Blackwell, and S. M. Hamberger, Plasma Phys.

Controlled Fusion 37, 229 (1995).41M. K. Paul and D. Bora, J. Appl. Phys. 105, 013305 (2009).42C. G. Jin, T. Yu, Y. Zhao, Y. Bo, C. Ye, J. S. Hu, L. J. Zhuge, S. B. Ge, X.

M. Wu, H. T. Ji, and J. G. Li, IEEE Trans. Plasma Sci. 39, 3103 (2011).43M. E. Gushchin, S. V. Korobkov, A. V. Kostrov, A. V. Strikovsky, T. M.

Zaboronkova, C. Krafft, and V. A. Koldanov, Adv. Space Res. 42, 979986

(2008).44A. V. Kudrin, P. V. Bakharev, T. M. Zaboronkova, and C. Krafft, Plasma

Phys. Controlled Fusion 53, 065005 (2011).45J. M. Urrutia and R. L. Stenzel, IEEE Trans. Plasma Sci. 39, 2458 (2011).46J. M. Urrutia and R. L. Stenzel, Phys. Plasmas 22, 092111 (2015).47J. Vogt and G. Haerendel, Geophys. Res. Lett. 25, 277, https://doi.org/

10.1029/97GL53714 (1998).48R. L. Stenzel and J. M. Urrutia, Geophys. Res. Lett. 44, 2113, http://

dx.doi.org/10.1002/2016GL072446 (2017).49L. Lyons and R. M. Thorne, Planet. Space Sci. 18, 1753 (1970).50P. Aigrain, in Proceedings of the International Conference on

Semiconductor Physics, Prague, 1960 (Academic Press, New York and

London, 1961), p. 224.51T. Shoji, Y. Sakawa, S. Nakazawa, K. Kadota, and T. Sato, Plasma

Sources Sci. Technol. 2, 5 (1993).52A. S. Kingsep, K. V. Chukbar, and V. V. Yan’kov, in Reviews of Plasma

Physics, edited by B. B. Kadomtsev (Consultants Bureau, New York,

1990), Vol. 16, p. 243.53R. L. Stenzel, J. M. Urrutia, and M. C. Griskey, Phys. Plasmas 6, 4450

(1999).54J. M. Urrutia and R. L. Stenzel, Phys. Plasmas 21, 122107 (2014).55J. M. Urrutia and R. L. Stenzel, Phys. Plasmas 23, 052112 (2016).56F. F. Chen, Plasma Sources Sci. Technol. 24, 014001 (2015).57R. L. Stenzel and J. M. Urrutia, Phys. Plasmas 22, 072110 (2015).58A. V. Karavaev, N. A. Gumerov, K. Papadopoulos, X. Shao, A. S.

Sharma, W. Gekelman, A. Gigliotti, P. Pribyl, and S. Vincena, Phys.

Plasmas 17, 012102 (2010).59S. Otsuka, K. Takizawa, Y. Tanida, D. Kuwahara, and S. Shinohara,

Plasma Fusion Res.: Regul. Art. 10, 3401026 (2015).60C. Charles, J. Phys. D: Appl. Phys. 42, 163001 (2009).61X. M. Guo, J. Scharer, Y. Mouzouris, and L. Louis, Phys. Plasmas 6, 3400

(1999).

082110-14 R. L. Stenzel and J. M. Urrutia Phys. Plasmas 25, 082110 (2018)