Helicon modes in uniform plasmas. I. Low m modes

J. M. Urrutia and R. L. StenzelDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA

(Received 2 June 2015; accepted 14 August 2015; published online 15 September 2015)

Helicons are whistler modes with azimuthal wave numbers. They arise in bounded gaseous and

solid state plasmas, but the present work shows that very similar modes also exist in unbounded

uniform plasmas. The antenna properties determine the mode structure. A simple antenna is a

magnetic loop with dipole moment aligned either along or across the ambient background

magnetic field B0. For such configurations, the wave magnetic field has been measured in space

and time in a large and uniform laboratory plasma. The observed wave topology for a dipole along

B0 is similar to that of an m¼ 0 helicon mode. It consists of a sequence of alternating whistler

vortices. For a dipole across B0, an m¼ 1 mode is excited which can be considered as a transverse

vortex which rotates around B0. In m¼ 0 modes, the field lines are confined to each half-

wavelength vortex while for m¼ 1 modes they pass through the entire wave train. A subset of

m¼ 1 field lines forms two nested helices which rotate in space and time like corkscrews.

Depending on the type of the antenna, both m ¼ þ1 and m¼�1 modes can be excited. Helicons in

unbounded plasmas also propagate transverse to B0. The transverse and parallel wave numbers are

about equal and form oblique phase fronts as in whistler Gendrin modes. By superimposing small

amplitude fields of several loop antennas, various antenna combinations have been created. These

include rotating field antennas, helical antennas, and directional antennas. The radiation efficiency

is quantified by the radiation resistance. Since helicons exist in unbounded laboratory plasmas,

they can also arise in space plasmas. VC 2015 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4930105]

I. INTRODUCTION

Whistler waves, first discovered in the ionosphere,1 are

electromagnetic waves in dense magnetoplasmas. Helicons

are usually referred to as whistler modes in bounded plasmas

and solids. Their salient features are azimuthal eigenmodes.2

Helicon research in solids started in the 1960s,3,4 and soon

thereafter in gaseous plasmas.5 In the 1980s useful plasma

sources were developed6 for various applications such as

plasma processing,7,8 propulsion,9 and current drive in toroi-

dal plasmas.10 In space plasmas, the present focus of whistler

research is on the interaction of whistler modes with resonant

particles, which are usually energetic electrons. Whistler

modes have also been proposed to enhance magnetic recon-

nection.11 Whistlers are useful for diagnostic purposes in

solid state plasmas. But neither in solids nor in space plas-

mas, in situ measurements of the wave field lines have ever

been performed. In the present work, a large uniform labora-

tory plasma is used to diagnose the waves emitted from mag-

netic loop antennas immersed in the plasma. The observed

waves have similarities to, but also differences from tradi-

tional helicon modes. In bounded plasmas, the radius of the

plasma column sets the perpendicular wavenumber. In a uni-

form plasma, an antenna launches wave packets with phase

fronts near 45�, implying nearly equal parallel and perpen-

dicular wave numbers. Although the phase velocity is highly

oblique, the group velocity is nearly field aligned as in

Gendrin modes.12,13 The oblique phase velocity produces

conical screw phase surfaces. We focus on the field topology

and present three-dimensional (3D) displays of helicon field

line spirals. Their unique locations and properties are

explained. Different antenna orientations and combinations

are used to study m¼ 0 and m ¼ 61 helicon modes.

Frequently, the m¼�1 mode is found to be much weaker

than the m ¼ þ1 mode. We demonstrate that it is the type of

antenna and not the property of the plasma which creates this

imbalance. Directional radiation of twisted antennas and

small phased arrays are demonstrated. The radiation effi-

ciency of different loops has been quantified through their

radiation resistance. Helicons with higher m-modes are

described in Paper II.14

The paper is organized as follows: After briefly describ-

ing the experimental setup in Section II, the measurements

and evaluations are shown in Section III for single loops with

different orientations, for multiple loops to produce rotating

modes, and the radiation efficiencies of loop antennas. The

findings are summarized in the Section 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, 0.4 mTorr Ar, and uniform axial magnetic field

B0¼ 5 G in a large device (1 m diameter, 2.5 length) shown

schematically in Ref. 15. Figure 1 displays a photograph of

the interior of the plasma device, showing the heated red

glowing cathode, the purple argon plasma, and the loop

antenna for exciting whistler modes. The electrically insu-

lated loop of 4 cm diameter and 4 turns can be rotated with

respect to B0 and translated vertically. For two configurations

1070-664X/2015/22(9)/092111/12/$30.00 VC 2015 AIP Publishing LLC22, 092111-1

PHYSICS OF PLASMAS 22, 092111 (2015)

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of the dipole moment along and across B0, the wave fields

Brf have been measured with a triple magnetic probe (three

orthogonal loops with 6 mm diameter <c=xp ’ 1:6 cm) in

y–z and x–y planes. The frequency is chosen at f¼ 5 MHz or

f/fc¼ 0.357 when normalized to the electron cyclotron fre-

quency. The wave amplitudes are small (B< 0.1 G) so that

nonlinear effects do not arise. The rf waveform consists of

repeated phase-locked tone bursts so as to observe wave

energy flow and subsequent continuous wave propagation at

different afterglow times. The pulsed discharge is repeated at

a rate of 1 Hz, the rf waveform is triggered at the same after-

glow time and averaged over 10 shots so as to improve the

signal-to-noise ratio. One can obtain the multipoint field to-

pology with a single probe with minimal plasma perturba-

tions since the discharge pulses are highly reproducible. The

analog probe signals are digitized with a four-channel digital

oscilloscope with 10 ns time resolution. Since we are inter-

ested in the field produced by plasma currents, the vacuum

field of the antenna is measured on alternate shots and sub-

tracted from the total field measured in the presence of

plasma.

The linearity between antenna current and wave field

has been established. It allows the superposition of fields

from two or more antennas which has been earlier verified

for counter-propagating whistler vortices.16 Due to the uni-

formity in density and ambient magnetic field, it is also valid

to rotate or shift the measured wave pattern from the single

loop to a different position and then add the two fields so as

to obtain the field of two antennas. This concept can be

extended to multiple antennas, e.g., rotating dipoles, multi-

pole antennas, and antenna arrays, to predict their radiation

properties. Antenna arrays are typically aligned perpendicu-

lar to the dc magnetic field. When the loops are closely

spaced, the wave fields do not vary along the array, suggest-

ing that plane waves can be excited by a two-dimensional

array in a plane transverse to B0. An actual construction of

an array is more involved than the superposition but is rou-

tinely accomplished for electromagnetic arrays in free

space.17 Phasing orthogonal loops produces circularly polar-

ized fields, while phasing an array controls the angular direc-

tion of the radiation pattern.

III. RESULTS OF EXPERIMENTS AND DATASUPERPOSITIONS

A. Whistler modes excited by single loop antennas

1. Magnetic vs electric dipoles

In low frequency whistler modes, the ratio of electric

energy density to magnetic energy density is given by

�0E2=ðB2=l0Þ ¼ ðv=cÞ2 ¼ 1=n2 � 1. Thus, in order to effi-

ciently excite whistler modes, it is better to induce an oscil-

lating magnetic field using a loop antenna than to excite

wave electric fields with an electric dipole antenna. All heli-

con sources use magnetic antennas. In space plasmas, active

experiments with loop antennas had problems deploying a

large loop;18 hence, electric dipoles are still preferred. Small

amplitude waves can be excited in laboratory experiments

with either electric or magnetic antennas, but large amplitude

waves with wave magnetic fields comparable to the ambient

field require magnetic loops. Magnetic antennas also avoid

nonlinear sheath effects and near-zone heating and instabil-

ities which arise when high voltages are applied to electric

dipoles. The present work considers whistler modes excited

by magnetic antennas.

2. Loop size

The radiation pattern of antennas changes significantly

with antenna size compared to the wavelength. Small electric

and magnetic antennas radiate mainly along the group veloc-

ity cone.19 As the antenna size increases, the radiation pat-

tern becomes more field-aligned.20–22 For large loops with

dipole moment along B0, the perpendicular field penetration

into the loop center is delayed, which can create an inverted

conical phase front.23 When the loop dipole moment is ori-

ented across B0, there is wave interference of radiation from

opposite sides of the loop. When large loops are used as

receiving antennas phase mixing reduces the received signal.

In order to avoid interference effects, the antenna size is

kept to a fraction of the wavelength. However, by stacking

small antennas in the form of arrays, one can produce uni-

form source fields over large surface areas. It will be shown

that approximate plane parallel whistlers are excited with

identical array elements. Introducing a phase shift between

array elements produces oblique plane waves. When an

antenna consists of phased circular arrays, one can generate

azimuthal wave propagation as in helicons.

B. Properties of the m 5 0 mode

The simplest antenna configuration is a single loop with

dipole moment along B0, as shown schematically in Fig.

2(a). In vacuum, the field is a dipolar field with (Br, Bz) com-

ponents which do not vary with /; hence, m¼ 0. This

antenna does not produce helical phase fronts. By our defini-

tion, it is not a helicon mode, but the helicon community

accepts it as a helicon since it also produces dense plasmas.

We include it since phased arrays of m¼ 0 loops excite heli-

cons with m> 0.

The time varying loop current produces an azimuthal

inductive electric field E/. In plasma, the inductive field

FIG. 1. Photograph of the experimental device.

092111-2 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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creates a radial electron E/ � B0 drift but no ion drift. The

resulting space charge imbalance24 produces a radial space

charge electric field. It drives an azimuthal Hall current

which creates a dipolar magnetic field. These are the proc-

esses near the antenna.

The plasma response to the applied field convects to

both sides away from the antenna. It forms a self-consistent

field which is maintained by fields and currents sketched in

Fig. 2(b). The poloidal current J/ maintains the toroidal

(dipolar) magnetic field (Br, Bz). Due to the axial propaga-

tion, the flux change induces azimuthal electric fields 6E/

near the front and end of the wave packet where the flux

change maximizes. The associated E/ � B0 drifts drive ra-

dial currents which close with field aligned currents parallel

to the toroidal magnetic field lines. The toroidal currents cre-

ate a poloidal magnetic field which links with the toroidal

field to form a vortex with positive linkage or helicity. The

convective derivative of the poloidal flux creates two oppos-

ing inductive electric field loops (Er, Ez) in the leading and

trailing half of the vortex. A space charge field cancels the

parallel inductive electric field but adds to the perpendicular

field, thereby maintaining Er and J/ in the center of the vor-

tex. The wave magnetic field and current density are parallel

for wave propagation along B0 and antiparallel for the oppo-

site propagation direction. Likewise the magnetic helicity

changes sign with wave propagation direction. The field link-

age or magnetic helicity is positive for wave propagation

along B0.

For sinusoidal continuous wave antenna currents, the

wave train consists of repeated vortices for each half-

wavelength with alternating polarities. The spiraling field

lines close within each half-wavelength section since the

field lines must turn back at the nulls of Bz. The field lines

are straight on axis, forming the spine of a 3D null point due

to opposing Bz fields between each half-wavelength. The

field lines flare radially out as a spiral fan. In the outer region

of the toroidal field, they spiral back to the fan at the trailing

end of the half-wavelength section. In a fixed z-plane, the

transverse magnetic field ðBr; J/Þ rotates counter clockwise

(ccw) in time, as plane parallel whistlers do. However, the

vortex is a wave packet with a broad k-spectrum. This sim-

plified picture is modified by the observed conical phase

fronts which are addressed further below.

The schematic pictures of the field lines are confirmed

by observations shown in Figs. 3(a) and 3(b). Contours of (a)

the axial and (b) the azimuthal field component are shown in

the central y-z plane at an instant of time during the continu-

ous wave propagation. In the x¼ 0 plane, Bx ¼ B/ while

By ’ Br form the dipole field. The inserted black lines show

the dipolar (poloidal) and toroidal fields, stretched due to the

oblique wave propagation.

The field components have also been measured in or-

thogonal x-y planes. These data are used to extrapolate to a

three-dimensional (3D) vector field as follows: It has been

FIG. 2. Schematic picture of a loop antenna exciting an m¼ 0 whistler

mode. (a) Loop with dipole axis in z direction (B0). (b) Fields and currents

of the excited wave packet for a half-wavelength propagating along B0.

FIG. 3. Measured fields of an m¼ 0

whistler mode. (a) Axial field compo-

nent Bzðy; zÞ and (b) azimuthal field

Bxðy; zÞ which links with the dipolar

field to produce positive magnetic hel-

icity. (c) 3D view of the (semitranspar-

ent) conical isosurfaces of Bz with

embedded field lines. The field lines

are confined to half-wave sections of

opposite field directions. On the central

axis, spiral null points are formed

where Bz ’ 0. (d) 3D field lines near

the spiral null points separating oppos-

ing Bz fields. Reprinted with permis-

sion from R. L. Stenzel and J. M.

Urrutia, Phys. Rev. Lett. 114, 205005

(2015). Copyright 2015 American

Physical Society.

092111-3 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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found from the y–z plane that the waves propagate with con-

stant axial velocity vk ’ 70 cm/ls. The data from the trans-

verse x-y plane can be translated to a different z position by

Dz ¼ vDt. The data have also been scaled by the axial ampli-

tude decay but found to have no effect on the field topology

since all components decrease by the same ratio.

The 3D fields are displayed in Fig. 3(c) as isosurfaces of

Bz and field lines of the total wave field. The isosurfaces con-

vey the conical shape of the phase fronts which indicate

oblique wave propagation. When projected onto a plane, the

phase fronts are V-shaped [Fig. 3(a)]. As discussed below,

the same oblique propagation is observed when the loop is

rotated by 90�. The field lines form vortices in each half-

wavelength section. The field lines reverse in adjacent sec-

tions where the opposing Bz components form nulls.

Figure 3(d) presents a view of the spiraling field lines on

either side of a null point. The blue fan lines rotate right-

handed inward and continue outward along the axial spine

with Bz < 0. The yellow fan lines rotate right-handed out-

ward, originating from an axial spine with Bz > 0. As the

vortex propagates along B0 the local field vectors rotate

around B0 but the phase fronts do not.

Whistler vortices have been studied earlier both in the

linear and nonlinear regime.25 The latter includes vortices

with magnetic fields exceeding the ambient field, thereby cre-

ating propagating null points,26 strong electron heating and

excitation of whistler instabilities by temperature anisotro-

pies,27 and whistler modes propagating without background

field in their strong self-fields.28 The present work deals only

with linear waves for which superposition is applicable.

C. V-shaped phase fronts

An obvious feature of the excited whistler modes is their

V-shaped or conical phase fronts. This has not been observed

in most helicon devices where the plasma column is smaller

than the surrounding antenna and there are no helicon waves

outside the plasma cylinder. However, inverted V-shaped

phase fronts have been seen in large diameter devices which

arise from the phase delay by radial inward propagation.23,29

In conventional helicon sources, the radial field dependence

is a standing wave profile with wavenumber determined by

the column radius (k? � kk). In an unbounded plasma, the

transverse wave number develops self consistently.

In order to explain the shape of the phase fronts, we first

examine its space-time evolution near the antenna. A phase-

locked rf tone burst is applied, and the field is measured as

the wave begins to form. Figure 4(a) displays contours of

Bxðy; zÞ at different times during the turn-on of the rf oscilla-

tion. The V-shaped phase front arises during the first cycle in

the near zone of the antenna. Since the antenna field extends

beyond the antenna radius, the wave field also expands in y-

direction, although with a steep amplitude drop since the

group velocity is nearly axial.

The wave’s inclined phase front does not change as it

propagates away from the antenna. The phase normal or

phase velocity makes an angle of h ’ 45� with respect to B0

which for our parameters equals the Gendrin angle, defined

by cos h ¼ 2x=xc. Plane wave theory predicts that under

these conditions the group velocity is parallel to B0 and equal

to the parallel phase velocity.12,13 However, the present

wave packet has conical and not plane phase fronts which

may be the reason why the amplitude peaks indicate that the

group velocity is slightly oblique to B0. In any case, finite-

size antennas cannot produce plane phase fronts since it

would imply an unphysical infinite phase velocity across B0.

Plane phase fronts (z¼ const) exist in waveguides with trans-

verse standing waves which can be decomposed into reflect-

ing oblique plane waves. Whistler “waveguide” modes have

also been produced in unbounded plasmas by interference of

oblique plane waves.30 While plane waves cannot be pro-

duced with single antennas, they can be excited with large

plane antenna arrays.

In order to produce interference of plane waves, two lin-

ear antenna arrays in y-direction are placed next to each

other. One is phased so as to produce þky and the other �ky

along the entire array. Since the energy flow for adjacent

arrays overlaps, standing waves in y-direction are produced

while kz provides propagation along B0. This whistler wave-

guide mode is equivalent to helicon modes with a uniform

density profile and reflecting boundaries at the standing

wave maxima or minima. Helicon theory assumes radial

standing waves due to boundary reflections, such that waves

propagate only axially and azimuthally (m> 0), leading to

helical phase fronts. In the absence of boundaries, the radial

propagation produces conical phase fronts for m¼ 0 modes

and radially expanding screw-like phase surfaces for m> 0.

The radial phase propagation does not imply wave energy

spread. The group velocity of Gendrin-like modes is still

mainly field aligned.

V-shaped phase fronts can also be generated without am-

plitude decay. The field becomes uniform along the loop

array when loop antennas are stacked in a line across B0 and

each loop is driven in phase by the same current.30 When

each loop current is delayed by a time Dt and spaced by a

distance Dy, the rf signal travels along the array with phase

velocity vy ¼ Dy=Dt or wavenumber ky ¼ x=vy. The excited

plane wave propagates obliquely with a wavevector k ¼(ky, kz). When, as shown in Fig. 4(b), the upper half of the

array is phased with þky ¼ kz and the lower half with �ky

the phase front has the same V-shape as that from a single

loop. Conversely, the single loop pattern can be explained by

a wave propagation across and along B0 with equal k-vector

components when h ¼ 45� ¼ arctanðk?=kkÞ. While the par-

allel and perpendicular phase velocities are similar, the

group velocity is mainly parallel to B0. Across B0, the ampli-

tude drops off steeply along the V-shaped wings. The wave

packet can be described by an envelope propagating along

B0 and phase fronts propagating obliquely outward of the

wave packet.

The phase velocity angle does not depend on the orien-

tation of the antenna with respect to B0. The cone angle is in-

dependent of density which has been found when varying the

afterglow time. Figures 5(a)–5(c) show contours of Bxðy; zÞfor different afterglow times where the parallel wavelength

differs by a factor of 2.5 yet the cone angle remains the same

(indicated by black lines). Thus, ky varies with density as

does kz and both are not determined by the fixed antenna size

092111-4 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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or field geometry. In helicon devices, the density varies non-

linearly with wave properties and assumes an optimum when

the antenna length corresponds to half an axial wavelength.8

D. Properties of the m 5 1 mode

The topology of an m¼ 1 helicon mode can be excited

by two types of antennas, shown in Fig. 6. The simplest

configuration employs a single loop antenna with dipole

moment across B0 [Fig. 6(a)]. It excites whistler modes with

a rotating dipolar field across B0 linked by another dipole

field which is shifted axially by k=4 and rotated by 908around B0. The axial field components can also be excited

with two adjacent and opposing current loops with dipole

moment along B0. In this case the plasma produces the trans-

verse dipolar field. Both the m¼ 0 and the m¼ 1 antenna

FIG. 5. Contour plots of Bxðy; zÞ for m¼ 1 modes at different afterglow times ta. The drop in density with ta increases the parallel wavelength but does not

change the angle of wave propagation. With ky=kz ¼ const, the perpendicular wave number varies with density and is not determined by the fixed antenna size.

Density gradients at early times may cause the asymmetries in (a).

FIG. 4. V-shaped phase fronts of m¼ 1

whistler modes. (a) Contours of

Bzðy; zÞ at different times after turn-on

of a rf tone burst. The oblique phase

fronts develop near the antenna within

the first rf cycle and then remain con-

stant during propagation. The normal

to the phase fronts shows oblique

phase velocity. The amplitude peaks

propagate within a narrow angle along

B0. (b) V-shaped plane waves excited

by a phased array. The upper half of

the array imposes a wavenumber þky,

the lower half-array creates �ky.

Choosing jkyj ’ kz produces the same

oblique wave propagation as the single

loop, implying that the loop-excited

waves also have jkyj ’ kz.

092111-5 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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fields are oscillating but are not propagating azimuthally.

They can be considered azimuthal standing modes formed

by oppositely rotating m ¼ þ1 and m¼�1 modes.

In order to rotate the antenna fields in a specific /-direc-

tion, two identical fields with a 90� rotation in angle / and a

690� delay in phase are added [Fig. 6(b)]. The crossed

m¼ 1 loops produce either a right or left-handed circularly

polarized dipole field in vacuum. The four m¼ 0 loops pro-

duce rotating axial field components and the plasma pro-

duces rotating transverse fields required to close the total

field lines.

We start by explaining the formation of the helicon

modes from the applied antenna fields of a non-rotating

m¼ 1 dipole. Although not directly measured, the fields in

the near zone of the loop are shown schematically in Fig.

7(a). The time-varying current in the loop gives rise to an in-

ductive electric field around the loop. Along B0, the induc-

tive field is opposed by a space charge field leaving a small

net parallel electric field. Across B0, the space charge field

adds to the inductive field which gives rise to an electron

Hall current Jx of opposite signs on the right and left sides of

the loop. The opposing currents form an out-of-plane current

loop which produces a By component. This induced current

does not shield the antenna field but rotates it. Its amplitude

peaks when the loop current passes through zero. Thus, the

field at the antenna rotates from Bx to By. At the same time,

the fields shift axially as expressed by the convection equa-

tion @B=@t ¼ r� ðv� BÞ. For example, the x-component

states that @Bx=@t ¼ �ð@=@zÞðvx � B0Þ ¼ @Ey=@z, implying

that with growing Bx the field Ey propagates to larger z with

velocity @z=@t ¼ Ey=Bx.

FIG. 6. Schematic of antennas used to excite m ¼ 61 whistler modes. (a) A

simple loop antenna with an oscillating current I and a dipole moment across

B0 produces an m¼ 1 oscillating standing wave profile [/ cos / cosðxtÞ].Two adjacent m¼ 0 loops with opposing dipole moments along B0 excite

whistler modes, whose transverse field lines are dipolar similar to the current

loops. (b) Rotating dipole fields are produced with crossed dipoles and a

690� phase shift for m¼ 1 or m¼�1 rotations [/ cosð6/� xt)].Likewise, four m¼ 0 loops produce rotating dipole fields when one loop pair

is delayed by 690� with respect to the other one.

FIG. 7. Properties of m¼ 1 whistler modes. (a) Schematic picture of fields and currents near a loop antenna with dipole moment across B0. (b) Schematic pic-

ture of the wave fields decomposed into linked loops for purpose of simplification. (c) Contours of the three measured field components in the y-z plane of the

loop (x¼ 0). The V-shaped phase fronts indicate oblique propagation in the Gendrin mode. The Bx and By components are shifted by k=4 indicating circular

polarization. (d) Contours of the axial field component Bz and vector field of (Bx, By) in the x-y plane at different times within one rf period showing a clock-

wise (cw) rotation. (e) 3D field lines in an m ¼ þ1 helicon mode forming a left-handed spiral. (f) Isosurfaces of Bz ¼ const which form helices with the same

pitch as the field line. (g) Multiple 3D field lines showing field line closure across B0 forming circular polarized whistlers near the axis. The color of the field

lines indicates the field strength which peaks near the axis of the spirals. Reprinted with permission from R. L. Stenzel and J. M. Urrutia, Phys. Rev. Lett. 114,

205005 (2015). Copyright 2015 American Physical Society.

092111-6 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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Figure 7(b) shows the induced plasma fields which propa-

gate out of the antenna region where they are measured

directly. The schematic picture decomposes the 3D fields into

linked field lines which form transverse dipole fields whose

dipole moments rotate in /-direction around B0. An axial field

Bz is needed in order to close the transverse dipolar field lines.

It peaks off the z-axis and has opposite directions on either

side of the transverse dipole field. The dipole fields are linked

so as to produce positive magnetic helicity for the wave field

when it propagates along B0. The phase of the fields varies

approximately as exp iðm/þ kz� xtÞ which describes heli-

cal phase fronts, m/þ kz ¼ const. The magnetic field lines

are parallel to the phase fronts since k � B ¼ 0. For m¼ 1 and

t¼ const, the spiral /þ kz ¼ const is left-handed and for

z¼ const the field’s rotation /� xt ¼ const is in þ/ direc-

tion as in all whistler modes. It is thought that for m¼�1 the

reverse field line rotation would reverse the temporal field line

polarization so that negative m-modes cannot be supported by

whistler modes. This neglects the fact that the helical field

lines are a subset of field lines and most fields are not right-

handed helices. Observations show that the field polarization

of negative m-modes remains right-hand circularly polarized

in time, otherwise the mode would not have been observed.

The three vector components of the wave magnetic field

have been measured in orthogonal planes. Figure 7(c) shows

contour maps in the y–z plane of the antenna (x¼ 0). The

phase fronts are inclined at an angle h ’ 45�, which is close

to that of the Gendrin mode, hG ¼ arccosð2x=xcÞ ¼ 44�.The peaks of Bx and By are shifted by k=4 which produces

circular polarization (see dashed line). The Bz component

vanishes on axis, is an odd function in y and z, and peaks off

axis along y where Bx peaks. The wave propagates at a con-

stant speed along z such that z and t are proportional.

Figure 7(d) displays the fields in a transverse x-y plane at

z¼ 18 cm from the antenna for different times within one rf

period. An inserted white arrow is a guide to show the dipole

axis for (Bx, By). It rotates counter clockwise in time together

with the axial field Bz. The contours have spiral arms which

are caused by the V-shaped phase contours where the field is

delayed with increasing radial distance from the axis. The azi-

muthal dependence of all components is that of an m ¼ þ1

mode. This mode is already formed at the antenna because

Hall’s Ohms law rotates the perpendicular dipole field in þ/-

direction together with Bz and the phase front. The antenna

cannot produce an m¼�1 mode since it does not control the

field rotation. Phased antennas can do it.

The fields measured in orthogonal planes have been ex-

trapolated to 3D fields so as to visualize the actual field lines,

displayed in Figs. 7(e)–7(g). Contours of Bz in a transverse

x-y plane are also shown for reference. The most prominent

field line consists of two left-handed helices of opposing

B-directions spiraling along B0 [Fig. 7(e)]. Their diameter is

given by the spacing of the Bz peaks which is comparable to

the loop diameter. The pitch length is the axial wavelength.

Figure 7(f) shows isosurfaces of Bz which are also helices.

Spirals in the axial field component of m¼ 1 helicons have

also been observed in bounded helicon devices.31

For axial propagation, the phase of helicons, exp iðm/þ kz� xtÞ, predicts a helical phase surface, z ¼ �/=kþ

const. An isosurface of Bz can be considered a phase front

analogous to maxima or zeroes of Bz contours. Thus, the

phase fronts are helical conical surfaces which are left-

handed in space and rotate and propagate axially in time.

The k-vector is normal to the phase front of plane waves;

hence, the B-field must lie in the phase front such that k �B ¼ 0 or r � B ¼ 0. Thus, the field line helix is parallel to

the isosurfaces. The sign of Bz determines the axial direction

of the field. The field line propagates like a rotating cork-

screw. The radial phase propagation is evident from the

V-shaped phase fronts in the 2D y-z plane and from radially

outward spirals in Bz contours in the x-y planes of Fig. 7(d).

Extrapolating to 3D, the phase surface would be a radially

expanding helical surface, a bit difficult to visualize from

isosurfaces of Bzðx; y; zÞ ¼ 0 or Bz;maxðx; y; zÞ in 3D space.

It should be pointed out before further analyzing the spi-

raling field line that it is only one out of many other field

lines within a helicon mode. Since the Bz component has a

null surface, the field lines are transverse to B0. Near the

axis, Bz ’ 0 and field line closure occurs by lines which

cross the z-axis. When the originating plane is shifted along

z, both Bz ’ 0 and the transverse field lines rotate, hence

form a circularly polarized field near the axis similar to that

of a parallel whistler. When following these field lines, they

all turn eventually into the discrete helical field lines. Thus,

one may consider the helix as the field line closure of the

transverse field of helicon modes with ðkr; k/; kzÞ.The spiraling field line parallel to the helical phase front

satisfies Bz=B/ ¼ dz=rd/ ¼ �m=ðrkkÞ. For m¼ 1, r ’ 2 cm,

and kk ¼ 2p=ð12 cm), the spiral is located where Bz ’ B/.

An additional condition is that the helical field line lies, to

first order, on a cylindrical surface which implies no radial

field component. Figure 8 shows that these constraints leave

only a unique position and diameter for field line spirals to

exist. There are only two spiraling field lines on this surface.

Figure 8(a) reveals that 3D field lines traced from many

original positions converge into two spirals of diameter D ’4 cm. Figures 8(b)–8(d) show contour plots of the cylindrical

field components Bz, Br, and B/, respectively. The locations

where Bz ’ B/ and Br ’ 0 is indicated by dots which also

define the diameter of the spiral indicated by a black circle.

In time all patterns rotate ccw consistent with a left-handed

spatial spiral. The vector field ðBr;B/Þ indicates also the dif-

ferent field lines inside and outside the helix. However, all of

them eventually connect into the helical spirals when fol-

lowed. The field line closure in the direction of wave propa-

gation is not within the data volume and may involve radial

spread, damping, axial boundaries, etc.

Helicity is an important property of 3D field topologies.

Whistler mode wave packets have positive helicity when

propagating along B0 and negative helicity when k � B0 < 0.

The left-handed field line spiral in Fig. 7(c) may give the

impression that the wave with k � B0 > 0 has a negative hel-

icity. In order to clarify the sign of helicity, an expanded pic-

ture of the fields in the x-y plane is shown in Fig. 9(a). The

dipolar or poloidal field with axis in x-direction is linked by

an orthogonal toroidal field formed by Bz and By out of the

plane. The linkage is right-handed, i.e., the helicity is posi-

tive throughout the volume. This topology rotates left-

092111-7 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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handed in /-direction as it propagates axially, maintaining

its right-handed linkage or positive helicity. The helicity is

reversed for wave propagation in �z-direction because Bz is

an odd function of z while (Bx, By) are even in z.

The current density has similar helicity properties which

are shown in Fig. 9(b). The contours of the normalized helic-

ity J � B=ðJBÞ ¼ cos a show that the angle a between J and B

is close to zero and the sign of the helicity depends on the

direction of wave propagation relative to B0. The latter also

determines the sense of rotation of the 3D helical field lines.

Note that the rotating helicon fields were excited by a non-

rotating m¼ 1 loop antenna.

As shown in Fig. 6(a), m¼ 1 helicons can also be cre-

ated with two adjacent but opposing m¼ 0 loops. The vac-

uum field of the antenna does not rotate. In plasma a single

m¼ 0 loop excites a dipolar field (Bz, Br) linked by an azi-

muthal field B/, a vortex with m¼ 0. When two opposing

vortices are linearly superimposed the two ðBr;B/Þ fields

produces a dipole field across B0. It is linked by the (Bz, Br)

field of the two loops. Since the latter does not rotate, the

entire field cannot rotate in / direction. As shown in Fig.

10(a), it propagates axially but remains a standing wave in /direction [/ cos / cosðkz� xtÞ] which can be decomposed

into a superposition of an m¼þ1 helicon and an m¼�1

helicon of equal amplitudes.

In comparison, the wave excitation from a single loop with

dipole moment across B0 gave a different result. Although the

vacuum field of this antenna neither rotates, it excites a rotating

plasma field. The reason is that the induced plasma current cre-

ates a magnetic field which rotates the wave field only in one /direction, forming an m¼þ1 helicon. Hall physics does not

allow a field rotation in the opposite direction. The induced cur-

rents for m¼ 0 loops are azimuthal, creating a (Bz, Br) field

without / rotation, forming m ¼ 61 standing modes. Thus,

the mode excitation depends on the type of antenna.

Returning to Fig. 10(a), field lines have been launched

in the extrema of Bz and found to meander throughout the

wave train, guided axially by Bz and radially by the peak

transverse field components which arise where Bz ’ 0. The

transverse fields reverse direction at a spacing of a half-

wavelength. Since Bzð/Þ is a standing wave, the meandering

field lines represent a superposition of oppositely rotating

circular spirals. It is worth noting that these azimuthal stand-

ing waves still propagate axially, that is, the entire structure

observed in the figure moves forward in time.

Figure 10(b) shows that closed figure-eight field lines

are found when the line tracing starts on axis between two Bz

maxima (indicated by a black dot). These lines are confined

within each half-wavelength section, only one of which is

displayed. The line differs from a dipole field line because

FIG. 8. Location and diameter of the helical spiral spirals. (a) Bzðx; yÞ contours and 3D field lines for an m¼ 1 helicon mode. The field lines are launched at

multiple points in the x-y plane and all converge into two axial helices of diameter D. (b) Bzðx; yÞ contours and approximate location of the helicon spirals (see

dots). They pass through the Bz peaks and rotate in time ccw around a circle of diameter D ’ 4 cm (see circles). (c) Contours of the radial field component Br

which vanishes near the spirals hence they are laying on a cylindrical surface. (d) Contours of the azimuthal field component B/ which is comparable to Bz.

FIG. 9. Helicity properties of m¼ 1 helicon modes. (a) Contours of Bz and vectors of (Bx, By) in an x-y plane at z ¼ 18 cm from the loop antenna. Solid lines

are schematic displays of the dipolar in-plane field and the toroidal out-of-plane field closing the 6Bz field. The right-handed linkage implies positive magnetic

helicity. (b) 3D field lines on either side of the antenna. Their helical twist changes sign like the magnetic helicity. Contour plots of the normalized J � B show-

ing that J ’ jj6B.

092111-8 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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the opposing Bz fields prevent field line crossing in the x-yplane. Adding figure-eight loops in each section to the mean-

dering field lines produces on axis right-hand circularly

polarized fields as in plane parallel whistlers.

E. Rotating antenna fields

In order to selectively excite an m-mode, a rotating

antenna field is required. This can be accomplished with

phased crossed dipoles or phased pairs of m¼ 0 mode loops,

as shown in Fig. 6(b).

Figure 11 shows a comparison of the wave modes

excited by rotating antenna fields from two crossed and

phased m¼ 1 loops. A ccw rotating antenna field excites

a strong whistler mode whose fields rotate in the m¼ 1

mode [Figs. 11(a) and 7(d)]. A cw rotating antenna field

excites smaller local field peaks without spatial resemblance

of a rotating m¼�1 mode. In order to quantify the

two cases, we determine the power radiated Prad ¼Ð

S � da

¼Ð

vgroupðB2=2l0Þ � da where S is the Poyntings vector and

the integration is taken over an area normal to the group ve-

locity, i.e., the x-y plane at Dz ¼ 18 cm. The power ratio for

the two modes is found to be Pm¼þ1=Pm¼�1 ’ 5.

A different result is obtained with eight phased m¼ 0

loops as demonstrated in Figs. 12(a) and 12(b). Each loop is

delayed by T=10 with respect to its neighbor. A ccw rotation

of the antenna field produces the rotating Bz components

shown in the contour plot. The radially expanding spiral

arms in the contour plots are due to the V-shaped phase

fronts or radial wave propagation. The 3D field lines form

left-handed spirals, indicating an m ¼ þ1 helicon mode. For

cw rotation, an m¼�1 helicon mode is observed, whose Bz

contours rotate cw and the field lines form right-handed spi-

rals. Both m ¼ þ1 and m¼�1 modes are excited equally

strong.

This shows that the plasma can support a negative

m-mode but its excitation depends on the type of the antenna.

The m¼ 1 loops couple to the transverse field components.

No wave is excited if the antenna field rotates in the opposite

direction to those of whistler modes. The m¼ 0 loops couple

to the axial field component while the transverse field devel-

ops self-consistently. A whistler mode is excited irrespective

of whether Bz rotates or not. Rotation twists the helical field

line in either direction, analogous to the foot point rotation

of frozen-in solar field lines. The whistler mode remains

right-hand circularly polarized irrespective of the sense of

rotation of the helical field line. Similar results will be shown

for the m¼ 2 and higher order modes. Negative m-modes

can always be excited with m¼ 0 loops but not with m¼ 1

FIG. 10. Field lines and Bz components of a helicon field generated by two adjacent m¼ 0 loop antennas. (a) Bzðx; yÞ contours and 3D isosurfaces and a field

line launched at the peak of Bz. The non-rotating isosurfaces are from equal m ¼ þ1 and m¼�1 helicon modes. The field line meanders through the isosurfa-

ces along the whole wavetrain. (b) Bz isosurface for a half-wavelength section. A field line launched on axis between the Bz peaks (see black dot) forms a

closed Figure 8. Each half-wavelength section contains closed and meandering field lines. Reprinted with permission from R. L. Stenzel and J. M. Urrutia,

Phys. Rev. Lett. 114, 205005 (2015). Copyright 2015 American Physical Society.

FIG. 11. Comparison of wave fields for

different rotations of the antenna field

of crossed m¼ 1 loops. Displayed are

Bzðx; yÞ and vectors of B? ¼ ðBx;ByÞ at

z¼ 18 cm from the antenna. (a) Crossed

dipoles (see inset) produce a ccw rotat-

ing antenna field which excites an m¼ þ1 helicon mode. (b) A cw rotating

antenna field produces nonuniform, non-

rotating small amplitude wave fields

which is not an m¼�1 helicon mode.

092111-9 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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loops or their variations like the helical antennas shown

below.

F. Antenna radiation resistance

The radiation resistance, Rrad, of a wire antenna is

defined by RradI2ant ¼ Prad, where Iant is the loop current and

Prad is the radiated power. A quantitative result is obtained as

follows: The antenna current is measured with a Rogowski

coil and its rms value is obtained from the current waveform.

Its value (Iant ’ 0:11 A) shows a negligible difference in vac-

uum and plasma. The magnetic probe has been calibrated

using the measured and calculated value of the axial magnetic

field of a current loop, BzðzÞ ¼ ðl0NI=2rÞ½1þ ðz=rÞ2��3=2,

where r¼ 2 cm is the loop radius and N¼ 4 is the number of

turns [Bzð0Þ ’ 0.11 G]. By matching the measured and calcu-

lated axial dependence, the calibration factor between probe

voltage and magnetic field is obtained (1 V¼ 0.198 G).

Knowing the field strength for all components, the energy

density of the whistler mode is integrated over the x–y plane

which together with the measured group velocity (vgroup

’ 70 cm/ls) yields the radiated power (Prad ’ 3.3 mW). For

the applied rms antenna current of Irms ’ 0.1 A, a radiation

resistance of Rrad ’ 0.33 X is found for the single loop with

dipole moment across B0 (m¼ 1 mode). When the dipole

moment is aligned with B0 (m¼ 0), the radiation resistance

drops to Rrad ’ 0.19 X. The radiation resistance decreases

with increasing m-order. The loop has an inductance L ’1.25 lH which results in an impedance Z ’ 32 X. The react-

ance is not tuned out since it would slow down the rise time

of rf bursts.

To put the numerical value of the radiation resistance

into perspective, we consider the effects of increasing the

loop current. For a loop current of Iant ¼ 10 A, the radiated

power would increase to 33 W which would strongly heat the

electrons and produce a density trough which ducts whistler

modes.32 Furthermore, the oscillating magnetic field would

rise to Bmax ’ 11 G which exceeds the ambient field B0,

producing highly nonlinear “whistler spheromaks” with

magnetic null points that alter the propagation properties of

whistler modes.26 The electron heating may ionize neutrals

and produce a helicon discharge.6 The present experiment

avoids all these nonlinear effects such that linear superposi-

tion remains valid.

It should be pointed out that the present evaluation of

the radiation resistance is more accurate than measuring the

antenna impedance externally and calculating the absorbed

power from its real part, Pin ¼ RinI2rms, since the radiation Rin

includes antenna losses and dissipation in the sheath and

near zone.33

G. Directional radiation from antennas

Twisted helicon antennas are commonly used in helicon

plasma sources.34,35 They are basically an axially elongated

m¼ 1 loop on either side of the plasma column with a 180�

twist to produce a transverse and axially rotating antenna

field. The twist matches the spatial field polarization in only

one direction of wave propagation for half a parallel wave-

length. Directionality of wave propagation is inferred from

different plasma densities produced on different sides of the

antenna. This wave excitation method can be tested in our

uniform plasma with an axial array of gradually rotated

m¼ 1 loops shown schematically in Fig. 13. Six loops are

displaced axially by 1 cm and each is rotated by 30� to

obtain a half-wavelength field reversal matched to that of the

propagating whistler mode. The field vector of the helical

antenna rotates axially and oscillates in time, but does not

FIG. 12. Comparison of wave fields

excited by eight phased m¼ 0 loop

antennas. (a) The antenna field rotates

ccw which produces a left-hand rotation

of the Bz component and creates left-

hand spiraling field lines typical for

m ¼ þ1 helicons. (b) For cw rotation of

the antenna field, the Bz component also

rotates and the field lines form right-

handed spirals indicating an m¼�1

helicon mode. The Archimedian spiral

arms in the contour plots are due to ra-

dial wave propagation.

FIG. 13. Schematic picture of (a) a helical antenna, (b) its equivalent model

of an axial array of rotated loops, and (c) a vector diagram of the spatially

rotating transverse antenna field.

092111-10 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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propagate axially or azimuthally. Its amplitude variation in

axial and azimuthal directions can be decomposed into

standing waves along z and phi.Figure 14 shows the 3D field lines and 2D contours

maps of the wave amplitude on either side of the twisted

antenna. The helical field lines have different sense of rota-

tion and different radii on opposite sides. The antenna does

exhibit directionality. The energy density has been integrated

over the x-y plane to yield the total power radiated to each

side, Ptot ¼ vgroup

Ð ÐB2dxdy. Defining the directionality

by the ratio of power radiated to either side, we find

Pz>0=Pz<0 ’ 6. For z> 0, the twisted antenna excites a

strong m¼þ1 helicon; in the opposite direction, it produces

a weak mixed m¼ 0 and m¼ 1 mode.

We note that the twisted antenna only matches the spatial

polarization of the wave B-field in the middle of the helicon.

This is the reason for its directionality. It does not determine

the direction of angular wave propagation or the shape of the

spiral field lines of helicons. The frequent conclusion that the

plasma cannot “support” an m¼�1 mode is due to mix up

between polarization of the field vectors and the helicity of

the phase front. The twisted antenna couples to the field

polarization but not the direction of the angular phase veloc-

ity. It will be shown in Papers II and III that azimuthally

phased antennas excite helicons with positive and negative

m-modes which the plasma supports equally well.14,36

A simpler and surprisingly effective directional antenna

for whistler modes has been obtained with only two loops

separated axially by k=4 and temporally phase shifted by

690� as shown in Fig. 3 of Ref. 37. It is based on the princi-

ple of an end-fire array or Yagi antenna. In that figure, both

the energy density profiles in real space and polar plots of

the radiation patterns at r ’ 18 cm from the antenna for

phase shifts of þ90� and �90�. The preferred radiation is

along B0 in one case, in the other case opposite to B0. The ra-

tio of the total radiated power is found to be Pz>0=Pz<0 ’ 32

or 15 dB. The high directionality is most easily explained by

the superposition of the contours such as shown in Fig. 7(c).

The second antenna produces the same pattern as the first

one but is spatially shifted by k=4. When temporally delayed

by a quarter period, the contour shifts by another k=4 and

overlays exactly with the contours of the first antenna. On

one side of the array, both antenna fields add with the same

sign, on the other side they add with the opposite sign. An

alternate explanation is that the two-antenna field rotates like

that of the whistler mode. It also matches the axial wave

propagation due to the phase delay between the loops.

The twisted helicon antenna also matches the spatial

field polarization of parallel whistlers. However, the antenna

field does not propagate axially with the wave since there is

no axial phase delay. No helicon wave antenna produces a

traveling wave. This difference may account for a lower di-

rectivity of the twisted antenna vs the phased Yagi antenna.

The directionality arises because the sign of the spatial

polarization, or helicity, depends on the propagation direc-

tion relative to B0. Good directivity is achieved when the

antenna field matches the topology of the propagating wave

field. The transmission between two directional antennas is

not reciprocal.

IV. CONCLUSION

The properties of whistler modes with different azi-

muthal wavenumbers are investigated in a large laboratory

plasma. In contrast to helicon waves in bounded plasma, the

present waves do not involve boundary effects or plasma

nonuniformities or nonlinearities like ionization. These fea-

tures allow linear superpositions of data from single loops to

predict wave properties from multiple antennas located

inside the plasma. First, measurements of 3D wave field lines

have been presented.

Like all finite size sources, a magnetic loop antenna

does not excite plane waves but spatial wave packets.

Theory for helicon plasma sources predicts that the waves

have radial wavenumbers determined by the plasma radius,

azimuthal mode numbers imposed by the antenna, and paral-

lel wavenumbers given by the antenna length and matched

by the dispersion relation.38 Without radial boundaries, a

plausible assumption is that the perpendicular wavenumber

is given by the antenna size.23 However, the present observa-

tions show that the perpendicular wavenumber is comparable

to the parallel wavenumber and not fixed by the antenna

size. The resultant waves propagate like Gendrin modes but

are not plane waves. They have oblique spiral phase fronts,

and their radial propagation should be described by Hankel

functions rather than a Bessel function. The rapid radial am-

plitude drop is caused by evanescence since the group veloc-

ity is nearly parallel to B0.

FIG. 14. Fields from an m¼ 1 antenna

with a half-wavelength 180� twist. The

waves propagating along B0 are much

stronger than those propagating oppo-

site to B0 as indicated by the contours

of the field amplitude in the transverse

x-y plane. The directionality arises

from the spatial twist of the antenna

field which matches the spatial polar-

ization of whistler modes. The antenna

field does not rotate in time and does

not match the helical field lines.

092111-11 J. M. Urrutia and R. L. Stenzel Phys. Plasmas 22, 092111 (2015)

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The azimuthal wavenumber depends on the orientation

of the loop antenna with respect to B0. When its dipole

moment is aligned with B0 an m¼ 0 mode is excited. The

field topology is that of vortices modified by the V-shaped

phase fronts. There is always a radial field component

required to close the axial field lines. Successive vortices

have opposite axial field line directions which are separated

by 3D spiral null points. The fan opens radially and field

lines return within the vortex with dimensions of half an

axial wavelength. Field line spirals are right-handed when

the waves propagate along B0. The magnetic helicity

reverses sign with propagation direction.

When its dipole moment is aligned across B0, an m¼ 1

helicon mode is excited. The topology of the m¼ 1 mode

can be described by a rotating vortex field across B0. The

antenna field is that of an oscillating dipole field. The excited

waves also have a dipolar field, but it rotates right-handed

around B0. The rotation is caused by the Hall effect in

Ohm’s law. The dipole field is linked by an orthogonal field

(Br, Bz) which produces the magnetic helicity of whistler

field lines. The combination of the linked 2D fields produces

3D fields with m pairs of axial field line spirals which are the

characteristics of helicons. The fields are tangential to the

phase fronts which are also spirals. Their rotation is left-

handed in space consistent with the counter clockwise rota-

tion of the vortex fields. Field lines traced in the direction of

wave propagation funnel into the regions of large Bz, where

B? ¼ 0 and spiral together with isosurfaces of Bz. The spirals

are azimuthally offset by 180� (m¼ 1) and carry field lines

in opposite directions. The radial position of the spiraling

fields is unique and the azimuthal origin is determined by the

loop angle. The field lines of opposing spirals close across

B0 in regions of Bz ’ 0. The transverse fields near the axis

are circularly polarized as in parallel whistlers.

The work also addresses the radiation resistance of loop

antennas. The resistance is obtained from first principles by

measuring the radiated power of the waves and loop current.

The m¼ 0 antenna radiates better than the m¼ 1 antenna.

Since the spatial polarization of whistlers depends on

the direction of wave propagation, directional antennas can

be achieved with phased end-fire arrays and with twisted

loop antennas.

The present work shows common and new features of

helicon modes. The 3D spiral field lines have been displayed

which has not yet been done in small helicon devices and

cannot be done in space plasmas or solid state plasmas.

Without boundaries, the helicons propagate obliquely

although the energy flow is nearly field aligned. Trivelpiece-

Gould modes39 are not relevant without boundaries. The

propagation of helicons with negative m-modes has been

demonstrated. Their excitation depends on the type of

antenna but not on the plasma properties. A single loop

antenna does not determine the axial wavelength.

Since helicons exist in an unbounded uniform plasma,

they should also exist in space plasmas. Active experiments

with wave injection from electric or magnetic dipole anten-

nas will undoubtedly excite helicons. They could also be

excited naturally by oscillating and spiraling electron

beams.40,41 However, it would require multi-point field

measurements to identify helicon wavepackets in space.

ACKNOWLEDGMENTS

The authors gratefully acknowledge support from NSF/

DOE grant 1414411.

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