Trivelpiece-Gould modes in a uniform unbounded plasma

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

(Received 15 July 2016; accepted 16 August 2016; published online 2 September 2016)

Trivelpiece-Gould (TG) modes originally described electrostatic surface waves on an axiallymagnetized cylindrical plasma column. Subsequent studies of electromagnetic waves in such plasmacolumns revealed two modes, a predominantly magnetic helicon mode (H) and the mixed magneticand electrostatic Trivelpiece-Gould modes (TG). The latter are similar to whistler modes near theoblique cyclotron resonance in unbounded plasmas. The wave propagation in cylindrical geometry isassumed to be paraxial while the modes exhibit radial standing waves. The present work shows thatTG modes also arise in a uniform plasma without radial standing waves. It is shown experimentallythat oblique cyclotron resonance arises in large mode number helicons. Their azimuthal wave numberfar exceeds the axial wave number which creates whistlers near the oblique cyclotron resonance.Cyclotron damping absorbs the TG mode and can energize electrons in the center of a plasma columnrather than the edge of conventional TG modes. The angular orbital field momentum can producenew perpendicular wave-particle interactions. Published by AIP Publishing.[http://dx.doi.org/10.1063/1.4962182]

I. INTRODUCTION

Whistler modes are important in space and laboratoryplasmas. In nearly unbounded space plasmas, the wavesare assumed to be plane waves whose theory is welldeveloped.1–3 In bounded plasma columns of cylindricalgeometry, a theory of cylindrical eigenmodes has been for-mulated.4,5 It follows the theory of electromagnetic wavesin waveguides, even though plasmas are not isotropicmedia. The excitation of eigenmodes with antennas is still atopic of current research in helicon discharges.6 The wavetheory for helicons is complicated by nonlinear effectsand nonuniformity and boundary effects. Since the wavedamping could not be explained by collisions and Landaudamping,7,8 anomalous resistivity9,10 and coupling toTrivelpiece-Gould (TG) modes11 have been proposed with-out definitive conclusions. TG modes are slow waves whichcan be easily absorbed and heat electrons, albeit near theplasma boundary. In spite of many theoretical investiga-tions, TG modes have not been clearly observed. The onlyevidence is found from current density measurements,12 butno short wavelength magnetic or electric modes have beenobserved so far.

Helical phase fronts are the salient features of heliconmodes. Such modes exist not only in bounded plasma col-umns but also in unbounded plasmas.13 The present workdeals with helicon wave properties in uniform plasmaswhich offer the simplest conditions. Linear whistler modesare excited from antennas inside a preformed, uniform labo-ratory plasma of dimensions large compared to the heliconwavelength. The field topology of the waves is measured in3D space with full time resolution. Wave bursts are gener-ated to observe phase and group velocities. Depending onthe loop orientation with respect to the ambient field B0, theexcited waves have the same radial field dependence asm¼ 0 and m¼ 1 helicons, where m is the azimuthal modenumber. This leads to the conclusion that in uniform

plasmas, the antenna determines the helicon wave topology,not a helicon eigenmode.

Linear superposition of loop antenna fields allowed thestudy of wave excitation by phased antenna arrays.14 Thesecan excite nearly plane waves, conical waves, and heliconwaves of either rotation up to high mode numbers. The prop-erties of orbital angular momentum of vortex waves havebeen pointed out for helicons.15 The present experimentemploys a real physical antenna array which excites thesame helicons as earlier predicted by field superposition.

A circular antenna array is used to excite m¼ 8 heliconmodes. The antenna consists of 16 loops of alternating dipoledirections parallel to B0, forming an azimuthal standingwave. This approach is much simpler than phasing a largearray. The azimuthal distance between two loops defines halfan azimuthal wavelength. The axial wavelength depends onthe plasma density which is variable in the afterglow of apulsed discharge plasma. Thus the propagation angle can bevaried so as to excite helicon modes approaching cyclotronresonance, i.e., TG modes. The difference to TG modes insmall columns is that the dominant perpendicular wavenum-ber is in azimuthal direction, not in radial direction.Furthermore, we do not require an eigenmode theory toexplain the TG mode. It is simply a whistler mode of anunbounded plasma which rotates around B0 with a shortwavelength compared to the parallel wavelength. When thepropagation angle exceeds the resonance cone angle, givenby cos hres ¼ x=xc, the wave becomes highly electrostaticand strongly damped. An m¼ 0 helicon mode does not dampsignificantly under the same conditions.

These observations cast a new light on helicon wave the-ory and possibly suggest new heating schemes. It may alsosuggest new active experiments in space plasmas. The injec-tion of waves with high angular momentum causes a signifi-cant transverse Doppler shift which can give rise to newwave-particle interactions.

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PHYSICS OF PLASMAS 23, 092103 (2016)

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The paper is organized as follows. After describing theexperimental setup and measurement procedure in Section II,the observations and evaluations are presented in Section IIIin several subsections. The findings are summarized in theConclusion (Section IV).

II. EXPERIMENTAL SETUP AND DATA EVALUATIONS

The experiments are performed in a pulsed dc dischargeplasma of density ne ’ 1010–1012 cm"3, electron tempera-ture kTe ’ 1–4 eV, neutral pressure pn ¼ 0.4 mTorr Ar, uni-form axial magnetic field B0 ¼ 5 G in a large device (1.5 mdiam, 2.5 m length), shown schematically in Fig. 1. The vari-ation in plasma parameters is due to the choice of time of thepulsed discharge and its afterglow (5 ms on, 1 s off). The dis-charge uses a 1 m diam. oxide coated hot cathode, a photo ofwhich is shown in Fig. 2.

Whistler modes are excited with two types of antennas: asimple loop (4 cm diam) and a circular array of 16 loops(2 cm loop diam, 15 cm ring diam). The antenna axes are par-allel to B0. The dipole moment of the array elements alter-nates between adjacent loops such that an oscillating standingwave of 8 wavelengths per circumference is formed. Thisantenna excites an m ¼ 68 helicon mode, while the singleloop excites an m¼ 0 helicon wave. The bifilar winding ofthe array antenna produces no net azimuthal current, hencedoes not excite an m¼ 0 mode. The antennas are energizedwith 4.5 MHz wave bursts repeated every 50 ls throughoutthe discharge and afterglow. The burst generator is a weaklydamped L-C ringing circuit, triggered with a phase lockedfunction generator. Long wave bursts are also produced witha 5 MHz function generator. This wave burst excitationreveals group and phase velocity properties of helicon modes.

The wave magnetic field is received by a small magneticprobe with three orthogonal loops (6 mm diam) which can bemoved together in three orthogonal directions. The spatialfield distribution is obtained from repeated pulses, averagedover 10 shots by moving the probe through orthogonalplanes. The field in vacuum is also measured and can be sub-tracted from the total measured field, so as to obtain only thewave field produced by plasma currents. The vacuum fielddrops off rapidly on the scale of the loop radius (Bvac /r"3

loop). Plasma parameters are measured with Langmuir

probes also attached to the movable probe. In order to obtainthe local plasma parameters vs time, the probe current isrecorded at a dc voltage which is incremented in small stepsso as to obtain I(V) at any time. All signals are acquired witha 4-channel digital oscilloscope.

We used linear superposition principles in order toinvestigate advanced antenna arrays. In a uniform plasma,waves excited by a single loop placed at different positionscan be superimposed so as to obtain the wave from morethan one antenna. Loops have been arranged in an array ofcircular rings with increasing radii so as to generate a planarcircular array. When the loops are phased azimuthally, arigid-rotor like field rotation is produced which excites heli-con modes of integer m-values. Conical waves are excitedwhen the loops are phased radially. When the amplitudes arevaried radially, Bessel function profiles can be produced tomodel helicon “eigenmodes.” Many of these results havebeen published earlier.16

The present work validates the superposition results bymeasurement from an actual antenna array. A single circulararray is simple to build and to energize without phasing. All16 loops are formed from a single wire with the sense ofrotation varied from loop to loop. All loops are fed withthe same current. The loop reversal is equivalent to a 180#

phase shift between the array elements. The antenna field isan oscillating azimuthal standing wave. It excites whistlermodes with the same azimuthal standing wave topologyFIG. 1. Schematic of the experimental setup with basic plasma parameters.

FIG. 2. Photographs of the interior of the plasma device showing the largehot cathode and circular loop antenna (a) without plasma and (b) during thedischarge. The diagnostic probe enters radially from the right side. The largeloop and the horizontal wire are not used in the present experiment.

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which can be decomposed into equal m ¼ þ8 and m¼"8helicon modes. The latter are modes whose phase rotates azi-muthally in the opposite direction as the polarization of themagnetic field. Unless there is a need for specifying the signof the mode number, this simple antenna eliminates the com-plexity of phasing a multi element array.

III. EXPERIMENTAL RESULTS

A. Helicon excitation by superposition and directmeasurements

First, we show the topology of a high order heliconmode, created by two different approaches. One methodrelies on superposition of waves excited from loop antennasat different positions.17 In this case, the field from a singleloop has been measured. Since density and magnetic fieldare uniform, the same wave pattern is produced when theantenna position is changed. By superposition, new radiationpatterns of the fields of two or more antennas are obtained.Here, we generate the radiation from a circular array of 24phased loops on a 16 cm diam circle indicated by a whitedashed line. The phase of adjacent antennas differs by 120#

which results in a clockwise-rotating m¼" 8 helicon mode.The resultant wave pattern is shown in Fig. 3(a). The snap-

shot of contours of Bzðx; y; z ¼ "15 cmÞ clearly shows thehelicon mode structure. It rotates clockwise in time, i.e., in "/direction. There are 8 wavelengths per circumference, 2pr¼ 8k. With decreasing radius, the local azimuthal wavelengthbecomes shorter or the local k/-vector increases, k/ ¼ m/r.The perpendicular k-vector, k? ¼ ðk/; krÞ, becomes muchlarger than the parallel kz-vector, resulting in a propagationangle h ¼ arctanðk?=kkÞ which reaches and exceeds that ofthe oblique cyclotron resonance, hres ¼ arccosðx=xcÞ.Thus,the helicon mode runs into oblique cyclotron resonance withdecreasing radius where it becomes a TG mode. Such modes

are highly electrostatic and easily damped both of which arereasons for the disappearance of wave magnetic fields in thecenter of the helicon.

The present experiment employs a real antenna array of16 loops as shown in Fig. 2. Since phasing an array of manyloops is rather involved, we chose to apply an azimuthalstanding wave which is much easier to implement than phas-ing. Every other loop is reversed and all loops are connectedin series. An oscillating standing wave can be decomposedinto counter propagating m ¼ þ8 and m¼"8 modes both ofwhich can and do propagate as whistler modes.

Figure 3(b) shows a snapshot of contours for Bzðx; yÞunder similar conditions as in Fig. 3(a). Again, the m¼ 8helicon mode structure is very pronounced at the radius ofthe antenna. Since the measurement is taken at a distance ofDz ¼ 15 cm from the antenna, the field is not the near-zonefield of the antenna but that of a propagating wave. Heliconfields rotate like a rigid rotor; hence, the m¼ 8 mode struc-ture should persist radially since the wave propagates bothradially inward and outward. However, the azimuthal wavestructure is lost radially inward and the wave amplitudedrops rapidly toward the axis of the helicon. The reason isthat the azimuthal wavelength approaches the electron iner-tial length, which is the shortest wavelength of whistlermodes. Alternatively, the propagation angle approaches thatfor oblique cyclotron resonance such that the mode can becalled a TG mode. This direct experiment validates theresults of the earlier superposition method.16

B. Helicon propagation and damping

The snapshots of Figs. 4(a) and 4(b) did not displaythe time dependence of the fields. This is now shown inFigs. 4(c) and 4(d) by time-of-flight diagrams of Bzðz; tÞwhich demonstrate that the fields propagate axially. Theslope of the dashed line indicates the axial phase velocity

FIG. 3. Comparison of two m¼ 8 heli-cons. (a) Superposition of the fields ofproperly phased 24 loops, placed on acircular array indicated by the white-dash circle, producing an m¼"8 heli-con. Only one loop field has been mea-sured, the others are time and spaceshifted copies. Note the empty fieldregion in the center. (b) Measured fieldfrom a physical array antenna of 16alternating loops. It forms an m¼ 8helicon mode with standing waves inazimuthal direction and wave propaga-tion in axial direction. Note the loss ofwaves toward the center where the azi-muthal wavelength decreases and thewave runs into oblique cyclotron reso-nance, i.e., becomes an azimuthal TGmode.

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vz ¼ dz=dt ’ 160 cm/ls in an afterglow plasma. The largeamplitude near the antenna (z ’ 0) for the m¼ 8 modeindicates the oscillating but not propagating antenna field.Since the phase fronts [dashed lines of constant Bzðy; zÞ inFigs. 4(a) and 4(b)] are oblique, the waves also propagateradially. The normal to the phase front forms an angleh ’ 45# implying kr ’ kz ’ 2p=15 cm in a dense dischargeplasma, but much smaller in an afterglow plasma. The radi-ally outward phase propagation does not lead to an energyspread implying that the group velocity is highly fieldaligned. Likewise, the radially inward phase propagation ofthe helicon does not lead to a 1=r amplitude enhancementand focus on axis. Comparing the m¼ 0 mode and the m¼ 8helicon modes, both have comparable speeds but differentamplitudes and damping. It is interesting to see that the firstcontour of the wave burst propagates at the same speed asthe subsequent ones, implying that the group velocity andparallel phase velocity are comparably consistent with theproperties of Gendrin modes.2 The large amplitude near theantenna (z ’ 0) indicates the oscillating but not propagatingantenna field.

The propagation angle of a helicon mode with respectto B0 depends on k?, determined by the antenna, andkk, determined by the density for a given frequency andmagnetic field. The density decays in the afterglow ofthe pulsed discharge. The helicon propagation has beenobserved at different afterglow times which is shownin Fig. 5(a) for the m¼ 0 mode and in Fig. 5(b) for

the m¼ 8 helicon mode. Displayed are contours of thewave energy density, B2 ¼ ðB2

x þ B2y þ B2

z Þ (x¼ 0,y,z), at afixed instant of time for three different afterglowtimes. The single loop antenna excites axially propagatingwaves throughout the measurement plane with little radialenergy spread. As the density decreases, the wave energydrops but the damping remains unaffected. At the verylate afterglow, the large contour values are again due tothe antenna near-zone field rather than the wave fieldintensity.

Under the same experimental conditions, the m¼ 8 arrayantenna also excites axially propagating waves with peakintensities in the flux tubes of the array elements. But thewave damping increases as the density decays. At the lowestdensity, the near-zone field intensity dominates but no travel-ing wave propagates away from the antenna. The change ofwave damping must depend on the difference in field topolo-gies since the plasma parameters are the same. The highm¼ 8 mode has a large k/ while the m¼ 0 mode has k/ ¼ 0both of which are imposed by the antenna. Other low modehelicons (m¼ 1–3) readily penetrate to the axis where Bz

switches sign. The axial wavenumber kz decreases in timeleading to a large propagation angle h ’ arctanðk/=kzÞ,especially radially inward from the antenna array, wherek/ ¼ m=r ' kr ’ kz. Cyclotron resonance occurs when hres

¼ arccosð5 MHz/14 MHz)¼ 69# which implies that tan hres

’ k/=kz ¼ 2:6. At the antenna radius of r ¼ 7:5 cm, onehas k/ ¼ 5:9 cm and cyclotron damping would arise when

FIG. 4. Field topology and propagationof m¼ 0 and m¼ 8 helicon modes. (a)Contours of Bzðx; y; z ¼ 15 cmÞ at afixed time for two different antennas ina high density discharge plasma. A sin-gle loop antenna excites m¼ 0 heliconwith V-shaped phase fronts (see dashedlines). This implies both radial andaxial wave propagation (ky ’ kr). Theoblique wave propagates close to theGendrin angle where the energy flowis highly field aligned. (b) A circulararray of 16 loops with 2 cm diam. andalternating dipole direction excites ahelicon wave. Each loop excites wavessimilar to those of the single loop, butthe superposition of all loop fields pro-duces the m ¼ 68 helicon. (c) A time-of-flight diagram Bzðy ¼ 0; z; tÞ in anafterglow plasma for the single loop,demonstrating that the time dependentmagnetic field propagates axially withlittle damping. (d) The z-t diagram forthe helicon mode which has the sameparallel phase velocity as in (c) butstronger axial damping. The largeamplitude near the antenna is due tothe loop field in vacuum.

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kz ¼ k/ ( tan hres > 15 cm. The values are approximationssince they apply to plane wave theory while helicons are notplane waves. Nevertheless they show that cyclotron dampingcan explain the observed energy decay and the fact that nowaves can be supported inside the high order helicon which isdemonstrated next.

It is interesting to note that at low densities, the 2 cmdiam loops are an order of magnitude smaller than the paral-lel wavelength yet do not exhibit the radiation patterns ofpoint sources, i.e., group velocity resonance cones withopening angle hgroup ¼ arcsinðx=xcÞ ’ 20#.18 Destructiveinterference of nearly overlapping cones with oppositephases may have eliminated the cones which are also pre-dominantly electric field structures.

C. Hollow helicon profiles

Returning to the high order helicon field profiles in thetransverse plane (Fig. 3), we now show their dependenceson density or afterglow time. Figure 6(a) displays contoursof one field component, Bzðx; y; z ¼ "15 cm), at three dif-ferent times in the afterglow. The wave maps the field pat-tern of the antenna array, 16 alternating poles, peaked nearthe location of the antenna array indicated by a white circle.However, with decreasing radius the azimuthal wavelength

should decrease, k/ ¼ 2pr=m, but instead the wavestructure and amplitude vanish. On the other hand, forincreasing radius, the wavelength increases and the wavestructure remains with significant amplitudes. As the den-sity decreases, the field amplitude decreases, the azimuthalwave number remains unchanged, but the central area ofthe helicon becomes nearly field free. These features areconsistent with the earlier interpretation of wave absorptionby oblique cyclotron resonance. The mode number andradial position determine k/, the density determines kz,which leads to TG modes inside a high mode number heli-con at low densities.

Figure 6(b) displays contours of the total waveenergy density, B2 ¼ ðB2

x þ B2y þ B2

z Þðx; y; z ¼ "15 cmÞ. Itshows that all field components have similar spatialproperties as the Bz component. Due to the predomi-nantly axial group velocity, the wave energy axiallyaway from the antenna still peaks on the radial locationof the antenna array. Virtually no energy is observedinside the helicon. As the density decays, the waveenergy decreases, the empty inner region expands, andthe energy density completely vanishes on a constantscale for the contour values. The orthogonal planes ofFigs. 5(b) and 6(b) lead to a three-dimensional pictureof a hollow helicon cylinder.

FIG. 5. Comparison of the propagation of wave energy density in an y-z plane (x¼ 0) for an m¼ 0 mode and an m¼ 8 mode for different densities.(a) Contours of B2ðx ¼ 0; y; zÞ for the single loop at increasing afterglow times or decreasing densities. Although its amplitude decreases with density,the m¼ 0 mode is weakly damped. The large intensity near the antenna at low densities represents the antenna near-zone field. (b) The same display as in(a) but for the m¼ 8 mode. Its strong axial damping is evident and there is negligible wave energy excited at low densities. The interior of the mode isnearly field-free.

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IV. CONCLUSION

High m-number helicons have been excited with acircular antenna array. Depending on mode number andarray radius, the modes can be described as H modes orTG modes. TG modes are formed radially inward, wherethe increasing k/ creates oblique cyclotron resonance.Whether TG modes arise from radial or azimuthal propa-gation does not change the physics of oblique cyclotronresonance. In the present experiment, the TG mode isabsorbed in the central region of the helicon. The hollowhelicon cannot be explained by the dependence of high-order Bessel functions or by the lack of radial energyflow since the field penetration varies with density whichdetermines kk and the resultant oblique propagationangle.

The present observation shows that TG modes (definedas quasi electrostatic modes near the oblique cyclotron reso-nance) are volume modes in uniform unbounded plasmasand not only boundary modes of nonuniform plasma col-umns as originally discovered19 and later adopted to heliconsources.5 In retrospect, our findings may not be surprisingsince it has already been shown that helicon modes exist inunbounded plasmas13 and the difference between heliconand TG modes is just the angle of wave propagation. Ourmethod of generating azimuthal TG modes is simple andstraightforward while radial TG modes are difficult to excitewith standard helicon antennas, hence have rarely beenobserved.

When the circular array is opened into a straight linearray, high k? components can also be imposed along thearray which produce oblique plane wave near cyclotron

FIG. 6. Density dependence of the m¼ 8 helicon mode. (a) Snapshots of Bzðx; y; z ¼ 15 cmÞ at three different afterglow times of successively decreasing den-sity. The parallel wavenumber decreases in time while the perpendicular wavenumber is fixed by the loop spacing. The azimuthal wavenumber increases radi-ally inward. This creates oblique propagation near cyclotron resonance and wave absorption. (b) Snapshots of the total wave energy densityB2ðx; y; z ¼ 15 cmÞ which shows that the inside region of the helicon does not support waves when k? > c=xp.

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resonance (see Fig. 5 in Ref. 17). In spite of its similar phys-ics, the circular version of oblique whistlers near cyclotronresonance is novel and has not been discussed or observed inuniform plasmas and helicon sources.

The present results have several implications.Cyclotron absorption in the center of a helicon mode

produces an absorptive boundary which prevents the forma-tion of radial standing waves which is the underlyingassumption in helicon wave theory. The radial field depen-dence cannot be described by Bessel functions if part of theplasma column is field-free and presents an absorptiveboundary.

The solution of the wave equation for helicon eigenmo-des assumes paraxial wave propagation, i.e., radial standingwaves and axial phase propagation. It is valid for free-spaceelectromagnetic waves in cylindrical waveguides, but not inanisotropic plasmas. Phase and group velocities point in dif-ferent directions. For example, in a Gendrin mode with par-allel group velocity, the radial propagation transports noenergy, hence cannot be described by a Bessel function as inan isotropic medium. In general, oblique phase and groupvelocities reflect differently.2 Many observations of heliconmodes show radial wave propagation.20,21 Helicon wave the-ory may be a crude approximation for very small plasma col-umns, where wave tunneling dominates over wavepropagation effects. It certainly cannot be extended to highmode number helicons, to helicons without boundaries, tononlinear effects in helicon devices, and explain antenna-wave coupling for different antenna configurations.

Although high mode number helicons have not yet beenproduced in helicon devices, the wave absorption in the cen-ter of such modes may be of interest to electron heating andionization. In space plasmas, high-mode number heliconscan produce perpendicular wave-particle interactions notconsidered previously. For example, wave absorption trans-fers the field orbital angular momentum to the electrons,scattering them in velocity space, possibly producing acurrent rings or whistler instabilities by temperatureanisotropies.

An axial density gradient can also lead to TG modes inhigh order helicons. An axial drop in density causes adecrease in kk which for fixed k? brings the wave into cyclo-tron resonance. Alternatively, the density drop increases theelectron inertial scale such that k/c=xp ! 1 which is thelimit for whistler wave propagation. The same can happen ina converging magnetic field where the transverse wavelengthshrinks to the electron inertial scale.

ACKNOWLEDGMENTS

The authors gratefully acknowledge support from NSF/DOE Grant No. 1414411.

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