Helicons in Uniform Plasmas

The name "helicon" was suggested in 1961 by Agrain for whistler modes in solid state plasmas. Shortly thereafter it was used for whistler modes in gaseous plasma columns. Whistler wave theory, earlier formulated for plane waves in space plasmas (Storey, 1953), was adapted for wave propagation in cylindrical geometry (Klozenberg et al, 1965) which is still the same today (Chen, 2015). Helicon waves were found to be useful for producing dense plasmas for applications like plasma processing and propulsion (Boswell, 1984). A large community developed which understood helicons as whistler modes in radially bounded plasma column propagating axially along the ambient field Bo. The dispersion relation in cylindrical coordinates implies that the waves have axial, radial and azimuthal wavenumbers, the latter being quantized by integers m. Radial boundary conditions are assumed to produce radial standing waves described by Bessel functions. Radial density gradients also affect the mode structure and suggested the existence of other modes (Trivelpiece-Gould). The actual helicon physics is complicated by unexplained wave damping, different mode excitations, ionization nonlinearities, neutral gas depletion, double layer formation, diagnostic difficulties, etc., as summarized in a recent review by Chen, 2015.

In parallel the research on electromagnetic waves with angular momentum developed rapidly (Twisted Photons: Applications of Light with Orbital Angular Momentum, Juan P. Torres (Editor), Lluis Torner (Editor), 2011, ISBN: 978-3-527-40907-5). These are waves with helical phase fronts produced by axial and azimuthal phase propagation. The latter forms azimuthal eigenmodes of mode number m and are pursued for enhancing information flow m-fold. Lasers with angular momentum are used in atomic and molecular physics, photon and electrons with angular momentum are produced, astronomy employs helical waves. Some call helical electromagnetic waves in free space also helicons (Rop et al, 2012). Thus the common feature of these waves is the orbital angular momentum, which applies to electromagnetic waves in plasma and free space alike. But so far, there seems to be little connection between the different research communities.

Helicon modes in plasmas can also have helical phase fronts, hence angular momentum. The fact that they are bounded column modes is not the cause for their azimuthal propagation. It is the wave excitation with special antennas which rotates the wave phase. In helicon devices the antennas are usually external to the plasma and non-rotating. In order to study the waves without the complications of helicon sources we have used a large afterglow plasma of uniform density and magnetic field, absence of boundaries, internal antennas for excitation of linear waves which are measured and resolved in 3D space and time. Under these conditions we can employ linear superposition principles for measured waves and construct the radiation patterns of antenna arrays used to excite high order helicons of both positive and negative modes. First measurements of the phase fronts and wave magnetic field lines have been done. These observations show that helicons, defined as whistler modes with helical phase fronts ( Stenzel and Urrutia, 2015), exist in unbounded plasmas just like helical electromagnetic waves in free space. It also implies that plasma helicons can exist in space plasmas. Although whistler waves were discovered in space plasmas there seems to be no awareness of whistler modes with angular momentum in space plasma. A few publications mention helicons in space plasmas, but refer to whistler flux ropes rather than waves with angular momentum (Papadopoulos et al, 1994, Lakhina and Tsurutani, 1997). Thus the field of helicons is wide open in several areas of plasma physics.

The present research describes the properties of helicons in uniform and unbounded plasmas. The waves are excited with magnetic loop antennas in a large (1 m x 2.5 m) afterglow plasma of density 10^11 cm^-3, temperature 2 eV, magnetic field 5 G, 0.4 mTorr Ar, frequency 5 MHz, with low amplitudes (<0.1 G) so as to justify linear superposition of waves from multiple antennas forming arrays. Using a small triple diagnostic probe, movable in 3D space, the wave field components are measured for two different orientations of the single loop. When the dipole moment is parallel to Bo an m=0 mode is excited which has no azimuthal mode number. When the dipole moment is perpendicular to Bo an m=1 mode is excited. In both cases the finite size antennas result in a radial phase and amplitude profile with significant spread in the k-spectrum. The phase fronts are oblique at about the Gendrin angle, implying that the energy flow is field aligned and there is no energy flow in the direction of the phase velocity.

Discharge schematic and plasma picture

Fig. 1. Picture of the plasma produced by a 1 m diam hot cathode. Also seen is a 4 cm diam loop antenna inside the plasma, a diagnostic magnetic probe at the right, and a schematic picture of an array antenna which can excite helicons with positive or negative mode numbers.

The experimental studies are performed in a large laboratory plasma shown schematically in Fig. 1. A picture of the device interior conveys the fact that the plasma is large compared to the antenna, hence boundary effects are negligible. The waves are detected with a movable magnetic probe which fully resolves the wave field in 3D space and time.

It is useful to start understanding first the field topology from a single loop antenna. Two different antenna orientations will be considered. When the loop dipole moment is aligned with respect to Bo the dipole field has no azimuthal dependence and excites an m=0 mode. It has been studied extensively and described as a whistler "vortex" for a single pulse or a string of vortices for cw excitation. The vortex field is easier to visualize by two linked field, a poloidal or dipolar field, linked by a toroidal field. The 3D field consists of nested spirals which reverse in the outer part of the vortex. The vortex is usually shaped conically which complicates the field lines since they can meander from one vortex to the next. The conical structure implies a radial phase propagation. The cone of the wave phase front should not be confused with the group velocity cone excited by very small antennas which forms cones along the group velocity ( Stenzel, 1976). It should also be pointed out that an m=0 vortex field does not have a phase rotation, hence is not a helicon defined by a helical phase rotation.

Whistler wavepacket

Fig. 2. Whistler wavepacket excited by a single loop antenna with dipole moment parallel to the uniform axial magnetic field Bo. (a) Contours of Bz(x=0,y,z) at a fixed time. In 2D the waves have V-shaped phase fronts and propagate in z-direction along B0. (b) Isosurfaces of Bz in 3D showing that the phase fronts are conical. Field lines are tangential to the phase fronts and confined to each half wavelength section due to the closed isosurfaces with Bz=0.

We start by describing the field topology of the m=0 mode. Strictly speaking it is not a helicon mode since it has no azimuthal mode or phi dependence. Fig. 2a shows a 2D contour plot of the field component Bz(x=0,y,z) and Fig. 2b shows in 3D the isosurfaces Bz(x,y,z)=const. Contours or surfaces can be considered as the phase fronts of the wave packet which propagates in time along z or Bo. The V-shaped contours or conical phase front arises from a transverse wave number which equals the axial wavenumber since the phase front is inclined by 45 degrees. The wave pattern is much larger than the dipolar antenna field in vacuum sketched in (a). Field lines are tangential to the phase fronts, hence cannot cross the Bz=0 surface between half-wavelength sections. The field topology can be described by a vortex which is axially inclined. The dipolar (poloidal) field is linked by an azimuthal field Bphi, resulting in spiraling field lines, in particular near the 3D null points between the half-wavelength sections. The axial propagation of the vortex field lines yields a right-hand circular polarization of the wave vector in time which is the property of whistler modes.

When the antenna axis is perpendicular to Bo the antenna imposes a dipole field across Bo. Near the z-axis the wave field is perpendicular to Bo with a right hand circular polarization of parallel whistlers. Thus the entire wave field rotates in +phi direction which forms an m=+1 helicon mode. Due to the finite radial size of the antenna the wave amplitude drops off radially and the perpendicular field lines turn axial to form the spirals characteristic of a helicon mode. The antenna field has no twisted field lines. These arise from the circular polarization. The helicon fields on either side of the antenna have opposite spatial twists but equal temporal rotations. The helical field carries a field angular momentum which is a conserved quantity. Since the wave twist defines the field angular momentum L the total momentum is zero and conserved since the antenna does not inject a momentum.

2D helicon rotation

Fig. 3. This movie shows the characteristic rotation of an m=1 helicon field in 2D. The wave field is similar to the field topology of the loop antenna in vacuum. In an x-y plane the vector field shows the flaring field lines of a dipole field. The field lines also flare out in an y-z plane which explains the axial field component Bz, displayed as a contour plot. The main difference of the vacuum field and the wave field is its propagation. The wave field translates along B0 and rotates around B0. The combination of phase translation and rotation results in spiraling phase surfaces. Isosurfaces of the Bz component represent the phase surface which are shown in the next movie.

Click on the image to open the movie of the helicon rotation. Click on the back arrow to return to the presentation.


3D helicon spiral

Fig. 4. Displayed are isosurfaces of the axial field component Bz which represent the phase fronts of the wave in 3D space and time. The phase forms a helix, which is the reason of calling it a "helicon" wave. In contrast to a corkscrew the spiral does not rotate in space but just translates axially along B0. When the wave passes through a transverse z=const plane the phase rotates in phi direction. The field lines are tangential to the phase surface. They spiral outwards on the orange surface (Bz>0) and return along the blue surface (Bz<0), closing across B0 at both ends of the wave packet.

Click on the image to open the movie of the helicon rotation. Click on the back arrow to return to the presentation.

The polarization of the wave magnetic field can differ from the vector rotation which for whistler modes is right hand circular in a plane perpendicular to the direction of the wave vector k. The angular momentum of the wave can be determined with a phased antenna. When phased azimuthally the wave will rotate azimuthally, too. This is demonstrated in Fig. 5 where the antenna has a pase rotation of 3 periods in either +phi and -phi direction. The wave rotation follows the antenna field rotation, resulting in right or left handed spirals or m=+3 and m=-3 helicons.

In order to impose a field rotation one has to provide a set of antennas with angular delay, i.e. a traveling wave antenna in phi direction. Fig. 1 insert shows a sketch of a circular array of 16 m=0 loops. Its radiation pattern is obtained by the superposition of the measured field from the single loop, which is then displaced on a circle of 8 cm radius. For a constant density and magnetic field the radiation pattern of a loop is not changed when the antenna is moved. In addition a time lag is introduced from loop to loop which creates a circularly traveling wave. It becomes an eigenmode when there are m wavelengths per circumference, m*lambda=2pi*r. The amount of phase shift determines the mode number and the sign of the phase shift determines the direction of wave propagation in phi. Modes with m=0 to m=8 have been investigated. Helicons with positive and negative m-modes propagate equally well. The shape of the phase front does mot determine the polarization which is always right-hand circular for whistler modes.

3D field lines

Fig. 5. Topology of an m=3 helicon mode. (a) 2D contour plot of the field component Bz(x=0,y,z) and 3 D field lines passing through the peaks of Bz. The field lines form six nested, left-handed helical screws of alternating directions on a cylindrical surface along Bo. (b) Added to (a) are isosurfaces of Bz(x,y,z) which describe the helical phase fronts of a helicon wave. For m=3 there are three azimuthal wavelengths per circumference in phi and each helix has an axial pitch of three wavelengths, describing the phase relation cos(m phi+kzz-omega t).

As an example we show the properties of m=+3 modes. Fig. 5a displays a snapshot in time of contours Bz in an x-y plane at a distance z=18 cm from the antenna array. Field lines in 3D are traced from each peak of Bz. The field lines form 6 left-handed spirals with alternating field directions along a cylindrical surface. In addition Fig. 5b shows isosurfaces of Bz each of which can be considered a surface of constant phase. The phase also forms helices and the field lines are tangential to the phase front such that k*B=0. The axial pitch of each spiral is 3 wavelengths, since it takes 3 periods for a complete azimuthal rotation.

Time-of-flight

Fig. 6. Three consecutive snapshots of Bz(x,y,z=18cm) demonstrate the phase rotation in -phi direction for an m=-3 helicon mode. The phase fronts have the shape of an Archimedian spiral due to radial outward propagation at a constant speed dr/dphi with dphi/dt=const. In 3D the phase surface forms a right-handed Archimedian screw. White circle indicates location of the antenna array.

Figure 6 shows how the phase propagates in time for an m=-3 helicon mode. It displays a sequence of three snapshots of Bz(x,y) at about one third rf period apart. The hexapole wave pattern rotates clockwise by about 1/3 rotation. The contours exhibit a spiral pattern which is due to the radial outward wave propagation, which was also evident from the V-shaped phase fronts in the y-z plane. Closer examination of the contours shows that they form Archimedian spirals (r/phi=const) in 2D. The combined azimuthal, radial and axial phase propagation forms in 3D a phase surface which resembles an Archimedian screw surface. All three cylindrical field components Br, Bphi and Bz have the same topology of phase surfaces.

Whistler standing waves

Fig. 7. Field line closure in an m=2 helicon mode. (a) Field lines are started at many locations along the negative contours of Bz(x,y). All field lines converge to the right into the characteristic helices. To the left the field lines cross B0 in regions of Bz=0. They return through the plane and join the spirals with opposite field line direction. Thus, cross field closure is possible due to angular clefts in the Bz isosurfaces. Near the axis the transverse field have the circular rotation of parallel whistlers.

Figure 7 shows selected field lines which form spirals. An obvious question is the field line closure of spirals and the field topology inside the helices. Figure 5 shows lines of an m=2 helicon which are launched at many points along the phase front, i.e. the crest of the negative Bz contours. To the right side they converge into the characteristic helicon spirals, to the left they diverge, cross the center z-axis where Bz vanishes, then return through a positive Bz contour to form another spiral. The field line near the axis rotate every quarter wavelength consistent with circular polarization of parallel whistler modes. Thus the volume is filled with predominantly rotating transverse field lines which close along spiraling field lines where Bz is large. The location of the spiral is uniquely defined by the the condition that the B-field (Bphi,Bz) has to be tangential to the phase front, whose normal is (kphi,kz). The field line closure to the right end of the spiral is outside the measurement volume. In spite of the radial wave propagation the m pairs of field line spirals remain on a cylindrical surface. Since they form a subset of all field lines they do not determine the helicity or polarization of the bulk wave field. It remains right-hand circularly polarized in time irrespective of the rotation of the field line spirals.

The literature on helicon devices frequently states that helicons with negative m-modes cannot be supported by the plasma (Chen, 2015). This is demonstrably not the case and arises from a confusion between the polarization of the magnetic field and the direction of the azimuthal wave propagation. The latter does not determine the former or vice versa. Furthermore the antenna coupling is not correctly understood. A transverse m=1 type loop couples to the perpendicular magnetic field of helicons. When the antenna is twisted it couples better or worse to the spiral of the helicon surface. The injection of angular momentum creates directional propagation. The directionality can give the impression that a negative mode number may not be "supported" by the plasma. In order to impose an azimuthal field rotation the antenna has to produce a kphi. This is accomplished with a circular phased array or slow-wave structure. It is analogous to a spiral phase plate in optical science or an azimuthally variable delay structure which generates helical electromagnetic waves with orbital angular momentum (Bekshaev, 2009, ISBN 13: 9781604561142, Tamburini, 2012).

Directional Antennas

Fig. 8. Contours of Bz(x,y,z=18 cm) and spiraling field lines for (a) m=+1 and (b) m=-1 helicon modes, excited by a phased array of four m=0 loops. The phase shift determines the direction of rotation in time. The pronounced spiral structure of the contours is due to the outward radial propagation of the phase. Positive and negative m-modes propagate equally well.

A linear phased antenna array has been shown to excite oblique plane waves (see Fig. 8 in WhistlerWaveAntennas). The waves can propagate to either side of the antenna in +-B0 direction. When the linear array is bent into a circle oblique waves with kphi and kz, i.e. helicons, are excited. The sign of kphi is controlled by the phase shift such that helicons with m>0 and m<0 can be excited. These propagate equally well as shown for m=+-1 modes in Fig. 8. The contours of Bz(x,y) rotate in different directions causing a different sense of rotation in the helical field lines. But the wave amplitudes of the m=1 and m=-1 helicons are essentially the same. Likewise the field polarization is right hand circular since both helicons are whistler modes.

Stacking antennas

Fig. 9. Field topology of a non-rotating m=4 helicon. It can be decomposed into couter rotating m-modes which form a standing wave in phi direction. (a) Contours of Bz(x,y) which oscillate in time without propagating in space. The contours exhibit no spirals but radial oscillations. (b) Isosurfaces of Bz(x,y,z) form discrete shapes which propagate axially but not azimuthally. Field lines traced do not form simple continuous spirals but meander through different Bz surfaces.

Since both positive and negative m-modes can propagate their superposition will propagate too. The superposition of anti-rotating fields start at the antenna where an oscillatory but not rotating field is applied. This is simply done by applying opposite polarities to adjacent loops or groups of loops so as to specify the m-number. As an an example Figure 9a shows Bz(x,y) for an oscillating octopole which excites an azimuthal m=4 standing wave propagating in z-direction. For standing waves the contours show no spirals but radial phase reversals. Figure 9b shows isosurfaces of Bz in 3D space forming 8 peaks in phi direction with alternating polarities. All surfaces propagate in z-direction with alternating polarities every half wavelength. Field lines traced through this helicon does not form simple spirals but meandering lines through different Bz surfaces. At Bz=0 surfaces the field lines are perpendicular to B0. Field lines are parallel to B0 where Bperp=0 with nonzero Bz.

Stacking antennas

Fig. 10. Comparison of helicons with low and high mode numbers, m=0 to m=8. (a) Bz(x,y) for m=0 helicon shows a peak amplitude on axis even though the wave source is at r=8 cm. Wave energy is focused radially inward and outward, as for a linear array of m=0 loops. (b) Bz(x,y) for an non-rotating m=8 helicon. Its mode structure vanishes radially inward when the azimuthal wavelength becomes comparable to the electron inertial scale, c/omegap=1.6 cm.

Figure 10 shows a comparison of some properties for helicon of low and high mode numbers, m=0 to m=8. For m=0 (Fig. 10a) the contours of Bz(x,y) at z=18 cm from the loop array show a radial oscillation and a peak amplitude in the center, even though the wave source is at r=8 cm. Thus, the waves propagate both radially outward and inward, the latter creating a focusing effect on axis, somewhat reminiscent of a zero order Bessel beam in free space (McLeod, 2008). The focal spot would be of interest for creating nonlinear effects. Bessel beams do not spread and they reconstitute behing an obstacle on axis. In contrast the high order helicons have an amplitude minimum on axis. Figure 10b shows an m=+-8 standing wave octopole which is the highest mode number excitable with our 16 loop array. The m=8 mode number is easily identifiable at the antenna radius of r=8 cm but the wave structure vanishes toward the center. The azimuthal wavelength lambdaphi=2 pi r /m decreases with radius reaching the electron inertial scale c/omegap=1.6 cm at r=2 cm. Cold plasma whistler modes vanish in the electron inertial regime or change character to electrostatic modes, such that the wave B-field vanishes on axis. In optical science this effect is used to pass a laser beam over objects located on axis. In astronomy a bright object on axis is eliminated when the received light is converted to a high order helical wave ("helicoid").

In Fig. 11 a new Doppler shift effect in helicons is shown. The familiar Doppler shift arises by an observer moving along or against the axial direction of wave propagation. In helicons there is another Doppler shift due to the rotation of the wave. It arises when an observer moves azimuthally or radially, i.e. transverse to the axial wave propagation. The figure caption explains the details of its verification. Here we explain the significance of this effect. When an electron spirals around the B-field against the direction of wave rotation it experiences an upshifted frequency which can be the electron cyclotron resonance. The transverse Doppler-shifted cyclotron resonance gives rise to wave-particle interactions just like the parallel Doppler shift.

Stacking antennas

Fig. 11. Demonstration of the transverse Doppler shift. (Aa) Field lines or phase fronts of an m=-3 helicon mode in the rest frame. A observer rotates and sees a different frequency in its moving frame. (Ab) The field is observed with an array antenna whose radiation pattern looks only at the transverse field propagation. (Ba) Doppler shift for different types of helicons. Top row displays the helicon field, bottom row (Bc) the observed frequency vs rotation velocity, normalized to the wave's rotational velocity. The frequency is upshifted when the observer rotates against the wave. It is downshifted to zero when the observer rotates with the wave. (Bf) No frequency shift arises for the m=0 mode since its field is independent of rotation angle phi. (Bc) The Doppler shift reverses sign with the sign of the mode number. (Bd). For an azimuthal standing wave there is the same Doppler shift for right and left rotation of the observer. A realistic "observer" is a rotating electron. It can experience a transverse Doppler shifted cyclotron resonance. Electron scattering, wave damping and growth are possible due to the transverse Doppler shift.

Wave-wave interactions play an important role in creating turbulence. The interactions involve nonlinear processes which lead to cascading in k-space such as observed in solar wind turbulence. However, small amplitude turbulence in the inertial range may not arise from linear processes but may just be a superposition of linear waves with random amplitudes and and phases. Certainly, the observed whistler turbulence excited by thermal electrons in a Maxwellian afterglow plasma ( Golubyatnikov and Stenzel, 1993) are not produced by nonlinear effects. In order to study the basic elements of linear superposition of differently propagating helicons we examine three cases of wave collisions. These yield an amazing variety of field topologies. The figure captions are self-explanatory.

Collisions of helical fields is also of interest in high energy physics. Wave-particle duality describes electrons as fields which can carry orbital angular momentum. Such twisted electron-electron collisions have been explained in terms of counter propagating helicons (Ivanov, 2011).

Stacking antennas

Fig. 12. Collision of two oppositely propagating m=0 helicon modes. The wave is excited by an antenna array of 16 identical m=0 modes on a circle of 8 cm radius. (a) Axial field amplitude in the transverse x-y plane at z=15 cm from the antenna at the time of a standing wave maximum. In time the contours move radially outward and inward implying cross-field propagation. (c) Isosurfaces of Bz(x,y,z) showing axial standing waves, no azimuthal propagation but radial waves. The field maxima/minima are located on axis.

Stacking antennas

Fig. 13. Axial collision of two oppositely propagating m=-3 helicons. (a) The field topology shows two helical phase surfaces of opposite twist but rotating in the same sense and propagating axially against each other. The resultant interference produces (b) axial standing waves. Although the waves propagate in the same azimuthal direction their opposite helical phase fronts interfere and form rotating maxima and minima. (c,d,e) Contours, isosurfaces and field lines in 3D near one axial standing wave maximum. The field can be described by a spinning wheel with rotational angular momentum.

Stacking antennas

Fig. 14. Axial collision of m=-3 and an m=+3 helicons. (a) The field topology consists of helical phase surfaces with the same twist since the helicons propagate in opposite directions with respect to B0. The opposite direction of wave propagation in axial and azimuthal direction produces standing waves in both directions. The helical phase surfaces are maintained and do not rotate but just oscillate between amplitude maxima (b) and minima (nodes, c).

Field polarization

Fig.15. Polarization of the wave magnetic fields during helicon collision (m=+3 and m=-3). (a) Vector fields (Bx, By) in the transverse (x,y) plane of an axial standing wave maximum, displayed at 4 consecutive times separated by a quarter rf period. The vector length and rotation distinguishes linear, elliptical and circular polarization, all of which can be found in the vector field. Whistler modes are supposed to be circularly polarized. Elliptical and linear polarizations arises from a projection of a circle on a plane. Thus, the vectors with three components are investigated by plotting hodograms in 3D space for an rf period. These differ from point to point between (b) circular polarization to (c) nearly linear polarizations. The spatial variation of polarization can be described by the ratio of vector length during one rf period. A contour map of |B|min/|B|max shows that circular polarization (ratio=1) dominates in the field maxima while destructive interference areas have elliptical/linear polarizations (ratio<<1). No left-handed circular polarization is observed. Note that the field polarization, the phase propagation and field lines are all different wave properties.

References