Whistler modes in highly nonuniform magnetic fields. I. Propagation in two-dimensions

The refraction of whistler modes in nonuniform plasmas has been studied for a long time both theoretically¹ and experimentally.^2–4 Since the refractive index of low frequency whistler modes depends on the density and magnetic field, refraction is not only caused by density gradients but also by magnetic field nonuniformities. The refraction in nonuniform magnetic fields has received less attention in both theory^5–8 and observations.^9,10 Refraction problems are usually solved by ray tracing which assumes the validity of the Wentzel-Kramers-Brillouin (WKB) condition as satisfied.¹¹ However, if the refractive index changes on a scale length shorter than the wavelength, ray tracing is not justified and numerical methods are required.¹² Experiments in this regime are scarce and difficult to interpret.

The present work presents experimental results of whistler mode propagation in highly nonuniform magnetic fields. These include magnetic fields with X- and O-type null points, mirror and cusp fields, and different separatrix surfaces. The main findings are the following: Plane waves are not formed in nonuniform magnetic fields. Wave packets excited by antennas change topology in nonuniform fields. Waves do not refract in the direction of the increasing refractive index. Wavelengths do not change as predicted by the refractive index dependence of plane waves in uniform media. Waves can propagate at angles not allowed by oblique cyclotron resonance. Polarization can range from linear to circular. These findings are of intrinsic value and of relevance to space plasmas and helicon plasma applications.

This paper is organized as follows: After describing the experimental setup and measurement procedure in Sec. II, the observations and evaluations are presented in Sec. III in several subsections. The findings are summarized in Conclusion, Sec. IV.

The experiments are performed in a pulsed dc discharge plasma of density $ne≃ 1010−1012$ cm⁻³, electron temperature $kTe≃$ 0.5–3 eV, neutral pressure p_n = 0.4 mTorr Ar, and uniform axial magnetic field B₀ = 3–6 G, in a large vacuum chamber (1.5 m diam and 2.5 m length), shown schematically in Fig. 1(a). A range of densities is obtained by working in the afterglow of the pulsed discharge (5 ms on and 1 s off). The discharge uses a 1 m diam oxide coated hot cathode, a picture of which is shown in Fig. 1(b).

Whistler modes are excited with magnetic loop antennas (4 cm diam). The loops excite m = 0 helicon modes when their dipole moment is aligned with B₀. When the dipole moment is perpendicular to B₀, the loop excites an m = +1 helicon mode. The antennas are driven by 5 MHz rf bursts (20 rf periods duration, 5 μs repetition time) whose turn-on reveals phase and group velocity during the wave growth.

The wave magnetic field is received by a small magnetic probe with three orthogonal loops (6 mm diam) which can be moved together in three orthogonal directions. The spatial field distribution is obtained from repeated pulses, averaged over 10 shots by moving the probe through orthogonal planes. Three-dimensional (3D) data are obtained from a multitude of adjacent planes. The field in vacuum is also measured and can be subtracted from the total measured field, so as to obtain only the field produced by plasma currents. The vacuum field drops off rapidly on the scale of the loop radius (⁠$Bvac∝rloop−3$⁠) so that distant fields (>λ) are due to propagating waves. Plasma parameters are measured with Langmuir probes also attached to the movable probe. In order to obtain the local plasma parameters vs time, the probe current is recorded at a dc voltage which is incremented in small steps so as to obtain I(V) at any time. All signals are acquired with a 4-channel digital oscilloscope.

In order to produce nonuniform magnetic fields, we use a straight line current in the x-direction across B₀ which is in the z-direction. The line current is closed by a circular wire along the chamber wall, such that the current loop has the shape of a half circle with the diameter of the chamber (1.5 m). The 3D magnetic field of the half-circular current loop is calculated from Biot-Savart's law. The finite length of the line current and its return current loop modify the field from that of an infinite line current. The relevant region is the center of the plasma column where the field is essentially two dimensional $(Bx=0,By,Bz)$⁠.

The line current is pulsed with a waveform of a single-oscillation sine wave produced by a switching transistor, a charged capacitor, and the circuit's inductance. The field period (T = 300 μs) is long compared to the rf period (0.2 μs). Pulsing the wire current avoids overheating of the wire by large dc currents, minimizing density perturbations and providing the measurement of wave properties for different field topologies within one wire pulse. Whistler wave bursts (f = 5 MHz and t_rep = 8 μs) are excited before, during, and after I_wire(t). The waves are excited on closed field lines inside the separatrix, on open field lines outside the separatrix, or directly on the X-type null point. The X-point is a true null point without a guide field along the separator. The measurements are performed in the afterglow of a highly repetitive pulsed discharge plasma (t_rep = 1 s). The density decay provides different plasma frequencies selected by the afterglow time.

The superposition of a uniform axial magnetic field and the field from the wire current is displayed in Fig. 2 by field lines and contours of the field strength for different times of the current waveform shown below each panel. For I_wire = 0, the total field is axial and uniform, B₀ = 3 G (not shown). For $Iwire,max≃$ 90 A × 8 turns, the total field exhibits a null point well above the wire at $y≃$ 26 cm, x = z = 0, above the measurement plane [Fig. 2(a)]. Since $∂/∂x≃0$⁠, the null point forms a 2D X-type null line along x. After the current reverses (I_wire < 0), the null line reappears below the wire [Fig. 2(b)] and finally splits into two null lines separating in the ±z direction far below the measurement plane [Fig. 2(c)]. However, the field lines become essentially circular near the wire. For very large negative currents, the field strength near the wire is an order of magnitude larger than the smallest uniform field B₀ = 3 G. The field strength peaks at the wire and decreases radially as $B≃μ0I/(2πr)$⁠. The gradient scale length varies with the radial distance from the wire as $L=B/|∇B|≃r$⁠.

Measurements of three components of the wave magnetic field are the primary data evaluated to understand the wave propagation.

The standard wave analysis is interferometry. A signal from the magnetic probe [$B cos (k·r−ωt)$_] is multiplied with a reference signal (⁠$cos ωt$⁠), time averaged to yield amplitude and phase information [$B cos (k·r)]$⁠. Amplitude and phase are spatially resolved but without time resolution. This approach is mostly used in helicon research, but often impractical in space plasmas due to the lack of multipoint measurements.

Modern data acquisition allows time resolution by recording simultaneously the waveform of three field components with a digital oscilloscope triggered by a phase-locked reference signal. In the present experiment, wave bursts are used to study the growth, steady state, and decay of wave packets. The space-time dependence indicates the direction of phase and group velocities and hodogram evaluations produce alternate diagnostics.¹³ 

The phase velocity direction is obtained from the normal of magnetic hodograms. It is based on the plane wave property that the hodogram plane lies on the phase front and its normal must be in the direction of the wave vector such that ∇ ⋅ B = 0 and its equivalent in Fourier space, k ⋅ B = 0, is satisfied. One can trace streamlines of the phase velocity by plotting a vector field of hodogram normals. However, additional information about the direction of wave propagation is required to resolve ta sign ambiguity in k.

The hodogram polarization can be quantified by the ellipticity ϵ = B_min/B_max, where B_min and B_max are the minimum and maximum values of B for a given hodogram. Linear polarization yields ϵ = 0, circular polarization has ϵ = 1, and elliptical polarization falls in between. Interference of whistler modes can produce locally linear polarization when two field components have a node and the third component has an antinode. Linear polarization can arise near the antenna when the vacuum field becomes dominant.

The group velocity can theoretically be obtained from the slope of the dispersion curve, $vgroup=∂ω/∂k$⁠. Alternatively, it can also be theoretically obtained from the propagation of the envelope of a wave packet,¹⁴ but this is not practical for wave propagation in highly nonuniform media. In this case, one can obtain the energy flow from the direction of the Poynting vector, $S=vgroupB2/2μ0$⁠. The Poynting vector $S=E×H$⁠, where $H=B/μ0$⁠, requires the electric field which can be obtained from Ohm's law dominated by the Hall effect in low frequency whistlers, $E=J×B0/ne$⁠. If B data are only taken in 2D, one can approximately use the property $J∥B$ which is valid for plane waves as derived below. 3D measurements show that this prediction also holds fairly well for wave packets such as helicons.¹⁵ 

In ideal EMHD, the convection of magnetic fields is given by $∂B/∂t=−∇×E=∇×(v×B)$⁠. When the electron velocity is replaced by a current density, $J=−nev$⁠, a useful relation between the current density and magnetic field is obtained, $∂B/∂t=−∇×(J×B)/ne$⁠. Starting with the simplest case of a uniform guide field, B₀, and a small wave field B, one finds $∇×(J×B0)=J(∇·B0−B0(∇·J)+(B0·∇)J−(J·∇)B0≃(B0·∇)J$ since $∇·B=∇·J=∇B0=0$⁠. If $ẑ$ is the direction of the straight magnetic field, then $B0·∇=B0∂/∂z$⁠, which yields the convection equation $∂B/∂t=−B0(∂/∂z)J/ne$⁠. For a plane parallel whistler mode [$∝ exp i(kz−ωt)$], one has $ωB=B0k∥J/ne$ or $J=v∥σHB$⁠, where $v∥=ω/k∥$ is the phase velocity parallel to B₀ and $σH=ne/B0$ is the Hall conductivity. Thus, one has $J∥B$ when the propagation direction is along B₀, and, conversely, $J∥−B$ when $k·B0<0$⁠.

This result also shows that $E=J×B0/ne∝B×B0$⁠, i.e., orthogonal to B and B₀. The Poynting vector has a parallel component $S∥=E⊥×H⊥$ and a perpendicular component $S⊥=E⊥×H∥$⁠. One obtains the group velocity direction by plotting streamlines of a vector field of S. This is the proper method in highly nonuniform media where ray tracing is not applicable.

The case of a nonuniform magnetic field B₀ complicates the wave physics. The phase velocity and the Hall conductivity become nonuniform and $J∥B$ may be modified since the term $(J·∇)B0$ cannot be neglected in the above derivation. The first term in $∂B/∂t$ predicts that $∂B1st/∂t=ωB1st=(B0·∇)J/ne=B0k∥J/ne$⁠. The second term yields $∂B2nd/∂t=ωB2nd=(J·∇)B0/ne=Jr(∂/∂r)B0(r)/ne=−Jr(B0/r)/neêϕ$⁠, where $êϕ$ is a unit vector along the nearly circular B₀ lines and $B0=μ0I/2πr$⁠. One can now estimate $|B2nd/B1st|=(Jr/r)/(Jk∥) ≃ Jr/J$ since $k∥r ≃ 2πr/λ∥ ≃ 1$⁠. 3D measurements¹⁵ have shown that $J$ and B are still parallel; hence, both lie in the circular phase fronts, implying that $Jr ≪ J$ and $B2 ≪ B1$⁠.

We also address the difference between time resolved and time averaged wave data which involve square terms such as energy density and Poynting vectors. The time resolved square terms have a mean value and an oscillation at twice the wave frequency. For propagating waves, the temporal oscillations also appear as spatial oscillations. The mean value for a single field component equals the oscillation amplitude [$cos2ωt=(1+cos 2ωt)/2]$⁠. For two components with circular polarization, there are no oscillations [$sin2ωt+cos2ωt=1$]. Thus, one can infer the polarization from the amplitude oscillation which will be defined further below.

Finally, we have also measured the rf field of the antenna in vacuum. These data allow us to separate the fields due to antenna currents and plasma currents. The latter “plasma” field shows the evolution and topology of the whistler mode. Furthermore, it clarifies the polarization and the energy flow of the two field contributions. The vacuum field is curl-free and linearly polarized, while the plasma field approximately satisfies $∇×B∝$ B and the polarization is close to circular.

Whistler modes are excited by magnetic loop antennas as described earlier. Helicon mode theory^16–18 assumes boundary reflections resulting in radial standing waves which forms “eigenmodes” with axial and azimuthal (“paraxial”) propagation. The twisted phase front is the salient feature of helicon waves which distinguishes them from plane waves. This aspect broadens the notion of helicons to unbounded plasmas.¹⁹ The radial amplitude profile of unbounded helicons is determined by the antenna field. The highly parallel group velocity of low frequency whistler modes collimates the helicon wave packets even when the phase velocity is oblique. Helicons carry orbital angular momentum which can lead to yet unexplored perpendicular wave-particle interactions.²⁰ 

Although the m = 0 mode has no azimuthal phase rotation, it is useful for comparison purposes. Its wave packets can approximately be compared with a sequence of vortices with axis along B₀ of alternating polarities. On axis, the field is axial, off axis, the field rotates, and, in general, the field lines are helical with different pitches and close in each vortex. The phase fronts are cones due to the outward radial wave propagation. The wave has a linear polarization on axis where the transverse field components vanish while the parallel component peaks.

The m = 1 mode is excited by a magnetic dipole field with axis across B₀. Since the induced currents and fields obey EMHD physics, the field rotates around B₀ just like the electron cyclotron rotation but at a different frequency. This results in a right-handed polarization of the field since the field lines are nearly parallel to the phase front to satisfy k ⋅ B = 0. The field topology is approximately a rotating dipole field but tilted due to the radial phase propagation. The phase fronts are helices. The axial field component vanishes on axis, while the transverse fields peak. On axis, the polarization is nearly right-hand circular with $k∥B0$⁠, yet it is not a plane parallel whistler due to the finite dimensions across B₀ and the field rotation around B₀. Although on axis it looks like a plane wave, it has axial field components off axis required for field line closure. In helicon wave bursts, the field lines spiral parallel to the phase front and return to the source. In m = 0 modes, or plane waves, the field lines close within each half wavelength section. Helicon waves can be excited with electric²¹ or magnetic dipoles²² across B₀. Helicon plasma sources use variations such as double loops with or without a twist to match the helical field rotation of the wave.²³ 

Figure 3(a) shows the propagation of an m = 1 helicon mode toward a nonuniform background magnetic field. The wave is launched from a loop antenna 30 cm away from the line current where the local field strength increases by a factor 5. Except for the direction of wave propagation, there is little change from the field topology in a uniform axial field. The axial wavelength is not observed to scale as predicted by plane wave theory (⁠$λ∝B01/2$⁠). Refraction should bend the wave normal toward the region of high refractive index, i.e., upward toward lower B₀-field regions. This does not happen since the V-shaped phase fronts remain unchanged. The axis of the m = 1 helicon wave packet, indicated by the zeroes of the $B∥$ component, follows the local B₀ indicated by the superimposed field lines. Thus, the m = 1 field topology is maintained in the nonuniform field. The wave is not damped by oblique cyclotron resonance. Hence, to first order, the nonuniformity of the magnetic field does not refract the wave as predicted by plane wave theory. However, the observed propagation can be understood by the predominantly field aligned group velocity. After all, the energy flow determines where the wave amplitude is large.

The flow of the phase velocity is obtained from B-field hodograms.¹³ The hodogram normal n indicates the direction of the wave vector or phase velocity, which for plane waves satisfies $∇·B∝k·B=0$⁠. Figure 3(b) shows contours of the parallel component $n∥$ and streamlines through the vector field of components (n_y, n_z) in the central y–z plane (x = 0). The wave propagates nearly along B₀ along the axis of the wave packet, but the wave becomes highly oblique off axis.

The energy flow is described by the Poynting vector, $S=E×B/μ0=vgroupB2/2μ0$⁠. The electric field is obtained from Hall Ohm's law, $E=J×B0/ne$⁠, where $J=v∥σHB$ and $v∥=ω/k∥$ is the parallel phase velocity and $σH=ne/B0$ is the Hall conductivity. The Poynting vector can be plotted as a hodogram, but its conical shape lends no meaning to its normal. Instead, the Poynting vector has been averaged over one cycle, thus describing the time-averaged direction of the energy flow [see Fig. 7(b) in Ref. 15)].

Figure 3(c) shows streamlines of S, equivalent to the group velocity flow, and contours of the parallel Poynting vector. The energy flow is highly field aligned in the central region of the wave packet. Thus, the propagation of the wave packet is mainly controlled by the field-aligned group velocity. The field lines converge in the direction of wave propagation helping to collimate the wave packet. The oblique phase velocity also helps to collimate the wave. At the Gendrin angle, the group velocity of plane waves is field aligned. For steeper propagation angles, a diverging phase flow creates a converging energy flow. Thus, the direction of wave propagation is mainly determined by the group velocity rather than the refraction of the phase velocity by the nonuniform field strength.

Finally, the properties of the ambient field B₀ are displayed in Fig. 3(d). The field strength changes locally from B₀ = 9 G to 12 G within the first wavelength (⁠$λ∥≃15$ cm) and to 20 G in the next, such that WKB theory²⁴ is not applicable. Evidently, plane wave theory is neither applicable since the wavelength does not vary as predicted (⁠$λ∝B01/2$⁠).

The wave propagation in a nonuniform magnetic field depends on several parameters: the ambient field topology, the antenna location, the helicon mode number, and the electron density and temperature. As shown earlier, the ambient magnetic field is slowly varied by a sinusoidal line current while repeated rf bursts are excited and their field topologies are measured as shown in Fig. 4. The simplest configuration is that of an antenna placed above the line current, in either an m = 0 or m = 1 orientation. By varying the current, the topology of the ambient field will change at the antenna such that launched waves will encounter a variety of conditions. These conditions include the presence of a null point at or near the antenna region, as well as open and closed lines, allowing us to test for the possibility of trapped or standing waves which could lead to the establishment of whistler eigenmodes.

Figure 4(a) displays the slow sinusoidal waveform of the z component of the ambient magnetic field at the center of the m = 0 antenna, located at y = 6 cm. The field is uniform along z with a magnitude of 3 G prior to turn off of the line current. The ambient field topology is displayed in Fig. 4(b) at the maximum negative current by contours showing the field strength and streamlines indicating its direction. The ambient field has no x component nor x variation in the plane of measurement. The x component of the wave field topologies generated by the antenna at selected points during the line current evolution [see black circles in (b)] is presented in Figs. 4(c)–4(i). The contours are shown at the same time during the continuous wave (cw) stage for each selected burst. Streamlines of the ambient magnetic field indicating its own topology are also included in each figure. The first picture recalls the wave propagation in uniform magnetic fields [Fig. 4(c)]. The waves propagate axially even though the phase fronts are oblique to B₀. For m = 0 modes, the transverse B_x component has an odd dependence in the radial direction and hence a null point on axis.

For a strong negative field, $B0,z$ = 20 G, a null point lies well above the antenna [Fig. 4(d)]. The waves propagate both on closed field lines diagonally upward in the general direction of the diverging open lines, i.e., toward regions of decreasing magnetic field strength. For a smaller field [Fig. 4(e)], the null point appears in the plane of measurement. The antenna is located inside the separatrix on closed field lines with small circumference compared to the parallel wavelength. The field near the antenna remains large, and the emitted waves are weak, i.e., the antenna-wave coupling is poor. When the null point coincides with the antenna position, the wave excitation is the weakest; nevertheless, some waves are emitted since the antenna vacuum field extends well beyond the evanescent null point region [Fig. 4(f)].

When $B0,z$ reverses sign, the null point moves below the wire and the antenna. Waves are excited directly by the antenna with characteristics of m = 0 modes, i.e., V-shaped wings and a B_x null on the axis which follows along the open field lines [Figs. 4(g) and 4(h)]. The wave amplitude is as large as in a uniform field.

At the strongest positive $B0,z$ field, the phase fronts and amplitude levels of the waves are stretched along B₀ [Fig. 4(i)]. Since B_x is also odd in the z-direction, the x component has a null at z = 0. It could be thought of an m = 0 mode on circular fields with axis along the y-axis and wings which are nearly field aligned. Alternatively, counter-propagating waves on closed field lines interfere and form a node at the antenna and its conjugate point and hence could be considered standing waves around the B₀ lines with radially outward propagation across B₀. The complementary case is counter propagating m = 0 modes on straight field lines where standing waves are formed axially while radial propagation continues (see Fig. 5 in Ref. 13). When the straight field lines are bent into a circle, the wings are elongated, and B_x is odd in parallel and perpendicular directions to B₀.

The general feature which emerges is that waves propagate through the separatrix and follow the diverging open field lines since the group velocity is preferentially along B₀. When the antenna is located well away from the wire, the waves also propagated along open field lines toward increasing B₀ (see Fig. 3). Thus, the phase refraction which should lead to weak field regions is less important than the nearly parallel group velocity direction. However, this general rule fails in the special case of circular field lines where a parallel group velocity would lead to wave trapping. However, the observed cross-field wave propagation requires a small perpendicular group velocity component. This special case will be presented in Sec. IV D.

Increasing the background field from 3 to 6 Gauss improves the coupling and permits a more careful exploration of the wave propagation. Again, the antenna has been oriented to excite an m = 0 mode so that the effect of the curved field on the polarization of the wave packet can be examined. The results are summarized in Fig. 5 where the magnetic topologies observed at both positive and negative maxima are presented. The background magnetic field B₀ is presented in Figs. 5(a) and 5(e) by contours showing the field strength and streamlines indicating the field direction. It is worth noting that improving the coupling comes at a cost: the increase in background field changes the location of the separatrix: it lowers the null point from 26 cm to 16 cm above the wire for maximum negative line current. Similarly, the separatrix crossing point of the y axis is lowered from y ≈ 24 cm to 10 cm for the maximum positive current. Hence, no circular field lines can be produced. Nevertheless, the antenna is still situated inside the separatrix for both configurations.

Figures 5(b) and 5(f) show contours of the transverse wave field B_x, which display phase fronts (⁠$|Bx|$ = const) and wave amplitudes. The waves are excited on closed field lines inside the separatrix for both cases but, in contrast to (f), the wave packet “wings” cannot propagate vertically past the null point since whistlers have no group velocity across B₀.

Vertically downward, the packet is trapped, but unfortunately, much of the area of interest is inaccessible due to the presence of the wire and associated support structure. However, the wings of the downward packet are able to cross the separatrix and give rise to a new wave packet that follows along the open field lines. This transmission process causes the new packet to lose its m = 0 topology as it no longer has a null in the perpendicular components.

Figure 5(f) demonstrates that even though the antenna is close to the separatrix, the majority of the excited packet is able to escape because most of it initially spans open field lines. This allows the packet to retain the alternating polarities observed when the field is uniform that is unique to the m = 0 mode. The axis is not as well aligned with the ambient field as seen in Fig. 3, but it still roughly follows its direction leading to oblique propagation across the separatrix onto open field lines.

The direction of parallel wave propagation is identified from the sign of the helicity $B·(∇×B)$⁠, which is that of the current density, J ⋅ B. It has been found from 3D measurements (not shown here) that each component of the dot product has the same sign properties such that it is sufficient to show $JxBx$ for 2D data in the plane aligned with the antenna. Figures 5(c) and 5(g) show that for z < 0, the negative sign of $JxBx$ implies $k·B0<0$ and the reverse holds for z > 0. This result may not be so obvious when the wave fronts in the measurement plane appear to propagate perpendicular to B₀. The contour spacing is $λ∥/2$⁠, and the amplitude is proportional to the wave energy density; hence, the crest describes the energy flow. No energy flows along the y-axis, i.e., perpendicular to B₀. As noted before, the wavelength does not change significantly in spite of the rapidly varying field strength B₀.

Another characteristic of a magnetic wave is the field polarization. By tracing out magnetic hodograms, the ellipticity $ϵ=Bmin/Bmax$ determines whether the polarization is linear (ϵ = 0), circular (ϵ = 1), or elliptical (0 < ϵ < 1). Figures 5(d) and 5(h) present contours of the field ellipticity for negative and positive line currents, respectively. The loss of the m = 0 topology is confirmed as there is no zero polarization region in the traveling wave and the mode is nearly circular as is characteristic of whistler modes. Instead, polarization is found to be nearly linear for the wave trapped within the separatrix. The low polarization in the bottom and the central top of the figure is due to the wave being non-existent in those regions. The linear polarization observed on the axis of the m = 0 helicon wave packet confirms that the topology has not been lost since there is only one field component (⁠$B⊥=0$⁠). The field is elliptical to nearly circular off-axis confirming that the mode is an oblique whistler. The null line is not only topologically interesting but also energetically. With $B⊥=0$⁠, there is no axial energy flow, but $B∥$ provides a stored energy. It is worth noting that the helicon mode is retained in this nonuniform magnetic field which is not a general feature.

We have shown the dependence of the radiation patterns for different B₀ topologies and a fixed antenna location near the current-carrying wire. Now, we show the effect of placing the antenna at different places within the separatrix for a fixed B₀ topology. To ensure that all locations chosen are well within the separatrix, we will return to the initially lower background magnetic field case as in Fig. 4 when the null point is at 26 cm above the wire at the maximum of the negative line current. We have explored several locations but we will show in Fig. 6 the two extremes: antenna just above the wire (center at y = 2 cm, 30 G < B < 100 G across the antenna) and y = 10 cm (6 G < B < 11 G). Contours of the magnitude of B are plotted at the same instant of time in the cw of the burst with all other parameters being identical.

When the antenna center is close to the origin [Δy = 2 cm, Fig. 6(a)], the circumference along the closed field line intersecting the antenna is smaller than the typical parallel wavelength (⁠$λ≃$ 15 cm) in a uniform background field. Given that the antenna is placed inside a very large gradient, there is absolutely no possibility of any whistler wave emission. Nevertheless, there is a coupling between the oscillations at the antenna and the plasma since a weak whistler oscillation is present outside the separatrix and propagates along curved lines of the ambient magnetic field similarly to what is observed in Fig. 4 where the antenna center is at 6 cm above the wire. When the antenna is moved to Δy = 10 cm, the waves can propagate on closed field lines inside the separatrix [Fig. 6(b)]. The greater radial distance from the wire enables these stronger waves to propagate closer to the separatrix. This in turn allows their “wings” to extend past the separatrix and themselves propagate into open field lines, as the B contours clearly demonstrate. A more detailed discussion of this propagation process is presented in Sec. IV E.

The temporal oscillation of the wave amplitude provides a useful diagnostics tool for the polarization of waves. The wave amplitude involves the square of the field components each of which oscillates sinusoidally but with a different phase. In general, the amplitude has a time-average value and a small oscillation, which can be quantified by a modulation degree $Bmod=(Brms−⟨B⟩)/⟨B⟩$⁠, where $Brms=⟨(Bx2+By2+Bz2)⟩$⁠, B denotes the magnitude of $B(t)$⁠, and $⟨⟩$ denotes the time average $1/T∫dt$⁠. It is worth noting that the magnitude of the modulation has no dependence on the wave amplitude and, hence, can be obtained even for cases where the coupling is poor. For a plane parallel whistler mode with circular polarization, the modulation degree vanishes (⁠$cos2ωt+sin2ωt=1$⁠), while for a linearly polarized field, such as the antenna field in vacuum, the modulation degree has a maximum value $Bmod=π/(22)−1≃0.1$⁠.

Figure 6(c) shows strong modulations near the separatrix for the loop at y = 2 cm, i.e., no propagating waves. When the antenna is located at y = 10 cm, the modulation degree is small both near and far from the antenna which implies propagating whistler modes. The observations indicate that the definition of “near” and “far” zones for antennas in plasmas is not as simple as that given for vacuum. That is, the ability of the antenna to couple to the plasma is determined by the local ambient field magnitude and topology and not by the characteristics of the antenna field. While there is no wave observed near the antenna [as in (a)], the non-propagating oscillations imposed on the plasma do couple at some distance from the antenna to whistler modes. The location at which the waves are generated is therefore not determined by the antenna but by the local conditions and what can be considered “near-zone” in one case does not apply to another.

As shown by Fig. 3, whistler packets easily propagate along curved magnetic fields. As the subsequent figures demonstrate, this simple propagation can be complicated if the packets are launched while the antenna is within strong field gradients. To explore these effects, we have placed a slightly larger (5 cm diameter) m = 1 antenna in a region where the magnetic field gradient is not strong at both negative and positive maxima of the line current. The results are displayed in Fig. 7, where the loop is placed in the left hemisphere at $(y,z)≃(2,−8)$ cm). The measurement plane is actually in front of the antenna (⁠$Δx≃−3$ cm) which allows scanning the diagnostic probe across the antenna so as to gain some insight into what should be its near-zone.

For reference, the magnetic field topology is shown in Fig. 7(a) and is obtained when the background magnetic field is 3 G. Figure 7(b) displays contours of B_x at a time in the burst where cw has been established. Waves are observed to leave the antenna region in both directions of the local B₀. They are strong in the opposite hemisphere (z > 0), proving that waves propagate from one hemisphere into the other. For z > 0, the antenna excites waves upward and across the separatrix. In the opposite direction, the waves first travel along closed lines downward and then reflect and travel diagonally upward along open field lines. The reflection process will be explained below in Fig. 8. The reflected wave is weak and delayed. The phase shift leads to $Bx≃0$ at the interface of the two waves. An m = 1 helicon mode has a maximum in B_x but not a minimum on the symmetry axis. In the right hemisphere, the waves refract from oblique to nearly parallel whistlers.

The waves shown in the lower panels of Figs. 7(b) and 7(c) travel initially on almost circular B₀ lines. No reflection process occurs. The antenna excites waves to both sides with equal polarity of B_x. The waves fill the left (z < 0) and right (z > 0) hemispheres with equal amplitudes. The J_xB_x contours show that the waves propagate away from the antenna along and against the curved B₀ lines. Of course, the dominant wave vector component is normal to the phase front, i.e., across B₀. The maxima in the right hemisphere occur on nearly the conjugate point of the antenna, suggesting a constructive interference between oppositely propagating waves on circular field lines. With the increasing radius, the field lines deviate from circles. Then, the waves begin to refract radially outward and do not match along closed field lines. Thus, refraction dominates over interference. We will return to interference of waves propagating on truly circular field lines in a separate chapter.

As presented through Fig. 6(b), an almost plane whistler wave packet can be generated from inside a separatrix. We investigate the growth into steady-state of this whistler packet in Fig. 8. This is done by presenting a sequence of snapshots of B_x with time intervals much shorter than an rf period (Δt = T_rf/10). B_x is displayed because it is the dominant field component of the m = 1 loop in vacuum. The ambient magnetic field lines are superimposed on the first and other frames and the separatrix (dashed black line over thick white line) is also shown.

The first signal is a strong negative B_x field (blue contours with white dot) which rapidly expands along closed field lines inside the separatrix. Ahead of the first wave is a “precursor” wave which is needed to close the wave field line. The wave reaches the bottom left boundary close to the conjugate point of the antenna within less than half an rf period (0.4T). As made evident by Fig. 4, waves are symmetrically launched from both sides of the antenna and it is expected that any signal that travels into the opposite hemisphere will constructively interfere with the oppositely traveling wave. Unfortunately, most of the area of interference is inaccessible to our diagnostic probe.

The antenna field reverses at Δt = 0.5 T. The new field (yellow contours with white diamond) expands like the first one, expanding radially to the separatrix while traveling down the closed field lines. The first wave decays, while the second wave grows. At Δt = T, a third wave starts to form which at Δt = 1.2 T splits the second wave: one of which travels downward (marked by white diamond) and the other (marked by a black-filled circle) propagates upward across the separatrix. The first wave stagnates in the y-direction, but by Δt = 1.3 T, it begins to rise again in the y-direction and z-direction. It merges with the growing third wave at Δt = 1.7 T to form a long curved phase front almost normal to the open field lines.

The merging of the first and third waves splits the second wave into two oppositely propagating waves. The split could be caused by strongly curved phase front advances against a less curved field line. The bulk of the wave, as represented by the higher amplitude contours, behaves as a plane wave maintaining its phase front perpendicular to the ambient field. The schematic picture, shown in the last panel, explains that the phase normal, or k-vector, develops opposing $k∥$ components which causes opposite directions of wave propagation as seen at Δt = 1.2 T. The refraction is mainly due to the change in the direction of the B₀ lines and to a lesser degree due to the bending of the phase front normal, k. Well outside the separatrix, the waves and precursors propagate as nearly parallel whistler modes. The nonuniform magnetic field has completely modified the helicon mode as produced in a uniform field.

Next, we compare the wave properties during the cw phase and the turn-off phase of the same rf burst as shown in Fig. 8. Figure 9(a) shows contours of B_x at two times well after cw has been established but separated by T_rf/2 = 100 ns. As expected, the contours differ only by the sign of B_x. The oblique wavelength is readily determined, $λ≃$ 8 cm, from the contours. Combining this observation with the known frequency, the phase velocity is $vphase=fλ$ = 40 cm/μs.

Figure 9(b) shows the same data as Fig. 9(a) but taken at the end of the rf burst. At the first instant of time, $Δt=0$⁠, one can see no waves inside the separatrix, one last contour of $Bx(y,z)$ at the separatrix, and the parallel whistler modes on open field lines. Extrapolating backwards, these waves come from the middle of the separatrix but not from the antenna. An interesting observation is that the rf period increases at the end of the rf burst, $Trf≃150$ ns, possibly because of the inductance of the antenna circuit. The lower frequency wave creates a longer wavelength, $λ/2=5$ cm. The effect is useful to determine the group velocity from first principles, $vgroup=∂ω/∂k≃ Δω/Δk$⁠. For $Δω=2π(5−3.33)$ MHz and $Δk=2π(1/8−1/10)$ cm⁻¹, one finds v_group = 67 cm/μs. The direction of the group velocity is along B₀. The phase velocity is also readily obtained, $vphase=fλ$ = 33 cm/μs. The ratio $vgroup/vphase=67/33≈$ 2 is in rough agreement with plane wave theory, $vgroup/vphase=2(1−f/fc)=2(1−3.3/14)=1.6$ since $B≈$ 5 G in this region. The reason for the agreement is that in the right upper corner, the wave is essentially a parallel whistler mode in a uniform plasma and ambient field.

One can continue to evaluate the theoretical dispersion relation to find the plasma density. From the observed wavelength λ = 10 cm, the cyclotron frequency f_c = 14 MHz, and signal frequency f = 3.33 MHz, one can determine the electron density from $(kc/ω)2≃ ωp2/(ω(ωc−ω))$ which results in a plasma frequency of 5.4 GHz and an electron density $ne=3.6×1011$ cm^–3.

We will now show wave propagation in fields with true circular field lines. These are produced by a larger line current (⁠$Iwire,max$ = 752 A vs 544 A in all other cases) and the lowest axial field (⁠$B0,z=3$ G) required to confine the electrons in a long discharge column. The wave is similarly excited with the same antenna but to ensure that any generated oscillation is trapped, the loop will be placed just above the line current in the m = 0 configuration.

Figure 10(a) shows a time sequence of $Bx(y,z)$ for the first two rf periods as $Δt=0$ is actually one quarter period after turn on of the rf burst for only the right-side of the measurement plane. The field topology is therefore shown from wave start to essentially cw waves. The circular field lines of B₀ are superimposed in various snapshots by white lines for reference. Within $Δt=T/4$⁠, the waves have been established around the circular field lines to the conjugate point of the antenna. This implies a high parallel phase and group velocity [$πr/(T/4)≃$ 31.4 cm/50 ns = 628 cm/μs] compared to the radial expansion (10 cm/50 ns = 200 cm/μs). The plasma response is like that shown in Fig. 4(c): B_x is an odd function in z. However, it is worth noting that a true m = 0 whistler packet topology will never develop because the magnetic field at the center of the antenna is 75 G and that sets the parallel wavelength to more than 120 cm. Nevertheless, the plasma response still shows that the amplitude of B_x must vanish along the y-axis where z = 0. This amplitude null is produced by interference but not by wave damping. The interference is only in the local z-direction since there are no opposing radial waves.

The wave propagation is similar to the collision of two oppositely propagating helicon modes on linear B₀ lines [see Fig. 5 in Ref. 13]. When the two wave packets interfere axially, they form standing waves with nodes (nulls) if the component is odd and anti-nodes (maxima) if the component is even about the center of the antenna. Since parallel and perpendicular field components are phase shifted, they have different locations for nodes and antinodes, and hence, there is no true null for the wave magnitude. The even parallel component B_z has an antinode at the symmetry point. Thus, the wave is a standing wave in the axial direction with radial propagation across B₀.

By analogy, the interference on circular field lines can be understood by bending the slightly curved field lines shown in Fig. 5(f) into circles. A single antenna launches waves which propagate in opposite directions at the conjugate point (180° from the antenna). While the interference produces standing waves along B₀, the radial propagation still remains. By causality, the wave energy and phase propagate radially outward and this is what generates the observed radially outward propagating shells. Since the parallel group velocity is faster than the radial velocity, the wave is rapidly distributed axially and gradually leaks out radially.

On a circular field line, B_x has one maximum, one minimum, and two null points, which characterizes a single mode standing wave, $k∥r=1$⁠. If there were several wavelengths around the circular field lines, there could be several interference minima, but this does not apply here where the antenna is close to the origin and the field line circumference is far less than a whistler wavelength. Once a circular phase front is created near the antenna, it retains its shape since $k∥−∇|B0|$⁠. Every reversal of the antenna field produces another concentric shell which leads to the cw pattern. However, when the waves eventually reach the separatrix, they will refract into parallel whistlers [see Fig. 5(b)].

Azimuthal and radial k vector components define the propagation angle with respect to B₀. At the outer radius (r = 20 cm), the propagation angle with respect to B₀ is given by $θ=arctan(λ∥/λ⊥)=arctan(2π20/9)≈ 86°$⁠. This far exceeds the resonance cone angle $θ=arccos(ω/ωc)=arccos(5/24)≃ 78°$ where the local $B0≃8.5$ G. According to plane wave theory, this highly oblique whistler mode should not propagate.

Standing waves and eigenmodes are usually associated with boundaries causing wave reflections. In the present case, there are no boundaries but the interference arises from opposing waves generated from a single antenna. The evolution of the waves after the antenna is switched off can be observed at the end of the rf burst shown in Fig. 10(b). As is natural, the waves near the antenna vanish, causing the waves above the antenna to disappear first as the driving current is switched off. The waves in the lower part of the measurement plane are still moving radially away across B₀. The turn off of the wave source no longer permits the wave to continue to move along the field lines and this ends the wave interference.

When the field lines are not circular, the interference effects diminish. This arises when the counter propagating waves do not meet at the conjugate point due to unequal propagation paths or refraction across the separatrix. An example of this situation is shown in Fig. 7(b, bottom). The even B_x contours of the m = 1 mode do not properly connect from the left to the right z-hemisphere. Although an antinode can be seen, its location may not be at the conjugate point. At large radii, the waves do not stay on closed field lines, and hence, no interference can arise.

Finally, we address the near-zone field concept in the context of the observations presented in this work. In contrast to waves propagating in vacuum, the establishment of a wave structure in a plasma is dependent on the ability of the antenna to couple to waves that are supported by the plasma. As is observed in Fig. 4, the magnitude of the propagating wave varies with the ambient field topology. Rather than examining each of these cases, we have chosen to present only the m = 0-like mode on circular B₀ lines as in Fig. 10.

We present in Fig. 11 snapshots of contours of the wave magnetic field for all three components under cw conditions. The time at which the rf current is a maximum within a cycle has been selected. In addition, the field of the antenna has been measured without plasma and then subtracted from the measured total field in plasma. The difference is the “plasma” field, produced by plasma currents, in contrast to the “vacuum” field, produced by the antenna current. Additional wave properties are also shown.

Figures 11(a)–11(c) present the total, vacuum, and plasma fields, respectively, for the perpendicular component, B_x. The existence of a vacuum field indicates that the measurement field is not exactly aligned with the antenna center. Nevertheless, the difference between the total and plasma field is small and the wave is clearly dominant outside of r > d, where d = 4 cm, the antenna diameter. The other two components [Figs. 11(e)–11(g) for y and Figs. 11(i)–11(k) for z] show a greater vacuum contribution. The radius of the area where the antenna field dominates is now more than twice as large (⁠$r≤2d$⁠), which is not surprising as the z component is the strongest in this plane. Yet, both components exhibit the same shell structure in the wave region. It is worth noting that the near-zone is not static: as the rf current changes the direction, the area shrinks as the ratio of B_vacuum/B_plasma drops. Moreover, the size of the near-zone for our experimental parameters varies as it is observed that the area also shrinks significantly as the magnitude of the ambient magnetic field at the antenna is lowered.

Figure 11(d) shows the component of the plasma wave which is parallel to the local field. It is an even function in z near the antenna because there its main contributor is B_z. As expected, it displays the same shell structure as B_x, albeit with a difference: B_x has nodes while $B∥$ has anti-nodes at z = 0. Figure 11(f) displays the x-component of the helicity density J ⋅ B. Its sign confirms that the waves in the left half plane propagate downward against the circular B₀ lines, and in the right half plane (z > 0), they propagate also downward but along B₀. Combining the three field components yields the magnitude of $(Bx,By,Bz)$ which is displayed in Fig. 11(l). The wave amplitude is radially spread with little azimuthal variation. The perpendicular field components (B_x, B_y) have nodes (nulls) on the y-axis, while the even B_z component has an antinode (maximum) at z = 0. This results in a minimum in the wave amplitude rather than a total null, which also occurs in helicon collisions (see Fig. 5 in Ref. 13).

The waves on circular field lines have some similarities to m = 0 modes on straight field lines: even $B∥$ components and odd $B⊥$ properties. The waves leave the antenna to both sides along B₀ and have helicity of opposite signs. The waves spread out across B₀ forming wings on straight field lines and circular phase fronts on circular field lines. The oblique wings become a perpendicular wave on circular field lines. The parallel wavelength on straight lines becomes the perpendicular wavelength on circular lines. A major difference is that straight field lines have no “ends” while circular field lines are closed which could result in interference of counter propagating waves.

The properties of helicon modes in non-uniform magnetic fields have been investigated experimentally. New measurement methods have been employed, and new effects have been observed. A reproducible plasma source without boundary effects makes it possible to obtain multipoint wave field measurements with full time resolution using a single probe with minimal perturbations. Hodograms have been constructed to obtain the direction of the k-vector field or phase velocity. The hodogram ellipticity defines the polarization. Regions of linear polarization are produced by wave interference. The sign of the helicity density J ⋅ B defines the direction of wave propagation along B₀. Wave bursts have been used to study the wave growth and transition into the cw regime. The group velocity has been measured at the end of the rf burst which produces a shift in frequency and wavelength.

Whistler wave excitation and propagation have been studied in nonuniform dc magnetic fields with X-type and O-type magnetic null points. Waves launched on closed field lines are not confined by a separatrix but propagate into open and diverging field lines, i.e., regions of low magnetic field strength. Parallel propagating whistlers have curved phase fronts in diverging field lines.

Theoretically, wave excitation at a magnetic null point should not be possible. However, since the antenna field extends well outside the resonance region (ω/ω_c = 1), waves are excited as observed. Waves excited on open field lines are stronger than those on closed lines.

In highly nonuniform magnetic fields, gradual wave refraction is replaced by the reverse process: The phase front changes slower than the ambient magnetic field. The wavelength does not scale as predicted by plane wave theory. The wave penetrates into regions forbidden by cyclotron damping according to plane wave theory. Wave fronts can split and create two waves propagating in opposite directions. However, the refraction processes are only part of the explanation of the wave propagation. The group velocity overrules the phase velocity by determining where the wave energy flows and the amplitude peaks. For oblique whistlers, the group velocity is usually more field-aligned than the phase velocity.

Waves excited inside a separatrix usually assume phase fronts parallel to B₀. It may result from the radial gradient of $B0∝1/r$ which creates a radial increase in the refractive index. Parallel waves refract radially into perpendicular waves. However, this refraction process may not apply when the field line circumference is smaller than the parallel wavelength. Under these conditions, the wave fields are small, i.e., the antenna-wave coupling is poor. Strong waves are excited when the antenna is located on large diameter closed field lines, i.e., well away from the center of the O-point. In this case, waves leave the antenna to both sides and travel along and opposite to the closed B₀-lines. A small radial group velocity component causes a radial wave expansion. Surprisingly, these highly oblique whistler modes are not strongly damped as predicted by plane wave theory. Since they propagate near the oblique cyclotron resonance, helicon theory predicts the excitation of short wavelength Trivelpiece-Gould modes. The present modes have the same wavelength as waves in uniform magnetic fields and hence are whistler modes and not slow T-G modes.

Measurements in two dimensions can reveal the wave propagation in a plane but not the propagation normal to the plane. These can reveal surprises which will be presented in Paper II.²⁵ 

The properties of whistlers in highly nonuniform magnetic fields are relevant to various other plasma configurations such as the exhaust region of small helicon thrusters,^8,26 micro helicon plasma devices,²⁷ and whistler modes in lunar crustal magnetic fields²⁸ and in Hall reconnection.²⁹ In many applications, whistler modes are analyzed by plane wave theory which is shown to break down when the wavelength exceeds the gradient scale length of magnetic fields or densities.

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