Page 1. ARTICLES. The following article was originally published in Physics of Plasmas 6, 2989 (1999). It is reprinted here in its entirety to correct Figs.
PHYSICS OF PLASMAS
VOLUME 6, NUMBER 12
DECEMBER 1999
ARTICLES
The following article was originally published in Physics of Plasmas 6, 2989 (1999). It is reprinted here in its entirety to correct Figs. 1–6 and 9, which were incorrectly processed during the original production process.
Laboratory studies of magnetic vortices. I. Directional radiation of whistler waves based on helicity injection R. L. Stenzel and J. M. Urrutia Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547
共Received 18 February 1999; accepted 23 April 1999兲 A novel principle for the directional excitation of whistler waves is demonstrated in a laboratory experiment. It is based on helicity conservation of electron magnetohydrodynamic fields in plasmas. Whistler wave packets propagating in opposite directions to a static magnetic field have opposite signs of helicity. Injection of helicity of one sign produces radiation in one direction. This is accomplished with an antenna consisting of a loop linked through a torus. Directionality of 20 dB is readily achieved. The direction of radiation is electronically reversible. Transmission between two antennas is unidirectional, hence nonreciprocal. Possible applications include secure communication, direction finding, and efficient power deposition in radio frequency 共rf兲 heating. © 1999 American Institute of Physics. 关S1070-664X共99兲01308-7兴 waves in free-space32 or plasmas,33 the present directional antenna is short compared to the wavelength and achieves its directionality by matching the helicity of a wave packet. The paper is organized as follows: After describing in Sec. II the plasma device and measurement techniques, the experimental results are presented in Sec. III, divided into various subsections. The conclusion, Sec. IV, points out the relevance of the present findings to related observations and applications.
I. INTRODUCTION
The helicity of magnetic fields plays a fundamental role in many fields of physics. It has been studied in cosmology in connection with the creation of the universe1 and the formation of stars.2 In astrophysics, it plays a role in the dynamics of magnetic clouds3,4 and in the creation of magnetic fields by dynamos.5,6 It has been extensively studied in solar physics in connection with flares,7,8 the solar wind,9,10 and the physics of magnetic reconnection.11–13 Related problems have also been studied in laboratory plasmas, such as magnetic reconnection using spheromaks,14,15 turbulent dynamo action in reverse field pinches,16,17 current drive in tokamaks,18,19 the helicity properties of whistler waves,20–24 helicon waves,25,26 and Alfve´n waves.27,28 Finally, magnetic helicity is also of interest in solid state physics when twisted magnetic fields of different helicities exist in different domains of antiferromagnets.29 In the present work we consider the basic helicity properties of whistler wave packets emitted from magnetic loop antennas in a laboratory plasma. These waves possess antisymmetric helicity, i.e., they carry positive helicity when propagating along a background magnetic field and negative helicity when propagating in the opposite direction. Based on these properties, the injection of helicity from a suitable antenna structure can excite one propagation direction preferentially, i.e., produce a directional radiation pattern. This concept has first been tested with knotted antennas in a computer simulation.30 Here we present experimental results confirming highly directional radiation from an antenna with linked fields, a loop-torus antenna.31 In contrast to phased arrays used to produce directionality of electromagnetic 1070-664X/99/6(12)/4450/8/$15.00
II. EXPERIMENTAL ARRANGEMENT
The experiments are performed in a large laboratory plasma device schematically shown in Fig. 1共a兲. A 1 m diam ⫻2.5 m long plasma column of density n e ⯝6⫻1011 cm⫺3, electron temperature kT e ⯝2 eV, and argon gas pressure p n ⯝0.26 mTorr, are produced in a uniform axial magnetic field B 0 ⯝5 G with a pulsed dc discharge (V dis⯝50 V, I dis⯝600 A, t pulse⯝5 ms, t rep⯝1 s兲 with the help of a large oxidecoated cathode.34 In the quiescent, uniform, current-free afterglow plasma, pulsed currents or tone bursts are applied to magnetic loop and torus antennas. The time-varying magnetic fields associated with the plasma currents are measured with a triple magnetic probe, recording (B x ,B y ,B z ) versus time at a given position. By repeating the highly reproducible discharges and moving the probe to many positions in a three-dimensional volume, the vector field B(r,t) is obtained with high resolution (⌬r⯝1 cm, ⌬t⯝50 ns兲. At any instant of time the spatial field distribution can be constructed from the digitally stored temporal traces. Plasma parameters are obtained from a small Langmuir probe ( r 2 ⯝2.6 mm2兲 which is also movable in three dimensions. 4450
© 1999 American Institute of Physics
Phys. Plasmas, Vol. 6, No. 12, December 1999
Laboratory studies of magnetic vortices. I. Directional . . .
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FIG. 1. 共a兲 Experimental setup and basic plasma parameters. Schematic picture of the linked loop-torus antenna in a cross-sectional view 共b兲 and a side view 共c兲.
Figures 1共b兲 and 1共c兲 show a schematic picture of a linked loop-torus antenna which exhibits directionality due to helicity injection. It consists of a single-turn loop (⬃8 cm diam兲 located on the axis of a torus (r major⯝r minor⯝4 cm兲. The latter consists of many identical thin wire turns 共1 mm diam Cu兲 spaced evenly on a toroidal surface with symmetry axis along B0 . The toroidal turns are connected in parallel, which avoids poloidal fields and minimizes the inductance. The wire antenna is highly transparent to the plasma, electrically insulated from it, and produces little density loss. The loop and torus antennas have separate electrical feeds so that each antenna can be energized separately or both jointly with the desired polarity. III. EXPERIMENTAL RESULTS A. Excitation of whistler vortices from magnetic antennas
We start with the radiated fields from a simple loop antenna with dipole axis along a uniform dc magnetic field B0 . Theoretically, the penetration of a magnetic field into a uniform magnetoplasma is governed by Faraday’s law and Ohm’s law, which in an ideal plasma yields B/ t⫽ⵜ⫻(v ⫻B). It describes magnetic fields frozen into the fluid of velocity v, which, in electron magnetohydrodynamics 共EMHD兲,20 is the electron drift velocity producing currents and fields via v⫽⫺J/ne⫽⫺ⵜ⫻B/(ne 0 ). For small field perturbations 关 ˜B (r,t)ⰆB 0 兴 propagating with wave velocity z/ t⫽⫾ v 储 along the dc magnetic field, the linearized solu˜ /B 0 . These fields have a positive tion yields J/ne⫽⫾ v 储 B ˜ for propagation along 共oppo共negative兲 helicity density J•B site to兲 the dc magnetic field B0 . The same holds for the
FIG. 2. Snapshots of the topology and helicity of EMHD vortices excited by ˜ tor a loop antenna. 共a兲 Vector magnetic field of the toroidal field component B in an x⫺y plane through the center of the vortex. 共b兲 Linked poloidal field ˜ pol in an x⫺z plane through the symmetry axis. 共c兲 Magnetic component B ˜ •B ˜ of the two oppositely propagating vortices excited self-helicity density A by the loop. The helicity is positive 共negative兲 for propagating along 共against兲 B0 .
˜ •B ˜ , where A ˜ is the vector magnetic self-helicity density A potential of the field perturbation defined in the Coulomb˜ ⫽ⵜ⫻A ˜ . Fourier analysis of the frozen-in equation gauge, B yields eigenmodes with dispersion of low-frequency whistlers, ⯝ ce (kc/ pe ) 2 , where ce and pe are the cyclotron and plasma frequency for the electrons, respectively. Wave packets with a spectrum of oblique whistler modes form three-dimensional 共3D兲 vortices as predicted by theory20,21 and observed in experiments.22–24 Physically, the spheromak-like vortex in the perturbed field can be decomposed into linked toroidal fields ˜B formed by axial currents, ˜ r , ˜B z ) and an axially oriented poloidal 共dipolar兲 field (B
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formed by toroidal Hall currents. The currents are driven by both inductive and space charge electric fields.35 Figure 2 presents a measurement example of an EMHD vortex excited by a loop antenna. The topology is displayed by vector fields in orthogonal planes at one instant of time. ˜ tor in an x Figure 2共a兲 shows the toroidal field component B ⫺y plane through the center of the vortex. The linked poloi˜ pol in an x⫺z plane through the symdal field component B metry axis is displayed in Fig. 2共b兲. Selected field lines have been traced to emphasize the field topology. One can notice a V-shape in the poloidal field which arises from the propagation properties of oblique whistlers. Oblique modes propagate slower than parallel whistlers, causing the sides of the wave packet to lag behind the center. The vortex propagates along B0 and has a right-handed linkage or positive helicity. Helicity properties are indicated in Fig. 2共c兲 by showing sur˜ •B ˜ in 3D faces of constant magnetic self-helicity density A ˜ is determined from space at t⫽const. The vector potential A ˜ the measured B by 3D Fast-Fourier transformation 共FFT兲 ˜ ⫽ik⫻B ˜ /k 2 , then inverse transformation into into k-space, A real space. The fields are adequately sampled and contained within the measurement volume. The latter avoids discontinuities at the periodic boundaries when calculating the FFT, which would produce high wave numbers and aliasing errors. The loop excites two oppositely propagating vortices, one with positive helicity propagating along B0 , the other with negative helicity propagating against B0 . Consistent with helicity conservation, the total helicity is zero since the antenna injects no helicity. The loop radiates symmetrically along B0 , hence exhibits no directionality. Vortices can be excited by either coupling the dipole field of a loop antenna to the poloidal vortex field or by exciting the toroidal vortex field with a toroidal antenna. Figure 3 demonstrates that a torus antenna excites the same vortices as a loop antenna. While the toroidal field is imposed by the antenna, the poloidal field develops selfconsistently. The reverse holds true for the loop antenna. Right and left propagating vortices have the same signs in ˜ for torus, ˜B z for loop兲, but opposite the imposed fields (B signs in the linked fields. Thus, the sign of the magnetic self-helicity density depends on the direction of wave propagation along B0 . Neither loop nor torus alone injects any net magnetic self-helicity but, as shown below, a linked looptorus antenna does. It should be pointed out that helicity and polarization of whistlers describe very different field properties which should not be mixed. Helicity describes the field topology at an instant of time; polarization describes the trajectory a field vector traces out in time at a fixed position 共or vice versa兲. For example, at a fixed point off-axis (r⯝4 cm兲 where ˜B⬜ ˜ ⬜ rotates in time from maximizes and ˜B z ⯝0, the vector B ˜ ˜ ˜ ⫹B r to ⫹B to ⫺B r , i.e., in a right-handed sense with respect to B0 like the electron rotation. Plane whistlers have right-handed circular polarization irrespective of wave propagation direction along B0 , yet no helicity since the field lines are straight. In a wave packet, the self-helicity depends on propagation direction with respect to B0 and the polariza-
R. L. Stenzel and J. M. Urrutia
FIG. 3. Vortex excited by a toroidal antenna. 共a兲 Vector plot of the toroidal ˜ tor⫽(B ˜ x , ˜B y ) in an x⫺y plane at the center of the vortex field component B ˜ y , ˜B z ) in a y⫺z plane (z⫽⫺10 cm兲. 共b兲 Poloidal field components ˜B pol⫽(B along the axis of the vortex (x⫽0). Note the left-handed linkage of ˜B tor and ˜B pol due to vortex propagation against B0 .
tion is not particularly meaningful since it varies with position due to the superposition of many k-modes. From singlepoint measurements, such as performed with satellites in space, one can neither determine the helicity nor the polarization unambiguously without making assumptions about the spatial field distribution 共e.g., plane waves, flux ropes, etc.兲. B. Helicity injection from a linked loop-torus antenna
The principle of the directional helicity antenna is explained with the help of Fig. 4. A loop antenna alone excites two vortices, one of positive helicity propagating along B0 , the other of negative helicity propagating opposite to B0 . No net helicity is produced and the antenna radiates symmetrically. The same holds for the torus antenna alone. However, when the linked loop and torus antennas are both driven together, the applied antenna field exhibits helicity. The sign of the helicity is determined by the relative direction of the current flow in both antennas, i.e., the current linkage. Loop and torus are typically connected in series. When the applied magnetic helicity matches that of a naturally propagating
Phys. Plasmas, Vol. 6, No. 12, December 1999
Laboratory studies of magnetic vortices. I. Directional . . .
FIG. 4. Schematic figure explaining the principle of antenna directionality due to helicity injection. 共a兲 A simple loop with axis along B0 excites two vortices of opposite helicity propagating in opposite directions along B0 . No helicity is injected, the net helicity of both vortices is zero, and the radiation pattern is symmetric. 共b兲 Injecting positive helicity with a linked loop-torus antenna excites only a positive-helicity vortex which propagates to the right ( 储 B0 ). In the receiving mode the directional antenna detects positivehelicity vortices propagating from the left ( 储 B0 ). Reversing the sign of the injected helicity switches the direction of preferred wave excitation/ reception.
FIG. 6. Directional radiation of a whistler tone burst from a loop-torus ˜ z (z,t) on axis (x⫽y⫽0) indicating approxiantenna. Contour plots of B mately 20 dB stronger signals propagating in the direction selected by the antenna helicity 共along B0 for H⬎0). Applied current wave form sketched in gap occupied by antenna ( 兩 z 兩 ⬍4 cm兲.
vortex, the latter is efficiently excited. For positive helicity injection, a vortex propagating only along B0 is excited, for negative helicity injection a vortex propagating only against B0 is excited. Thus, helicity injection produces an asymmetric radiation pattern, i.e., antenna directionality. This concept
FIG. 5. Measured fields of a directional loop-torus antenna with positive ˜ (x,y) at a time when the center of the vortex helicity. 共a兲 Vector plot of B is at z⫽15 cm. 共b兲 Contour plot of ˜B z (x,y,z⫽15 cm兲 indicating right˜ z in the y handed linkage or positive helicity. 共c兲 Axial field component B ⫺z plane along the axis of the vortex (x⫽0) on either side of the antenna at z⫽0. A strong vortex is emitted along B0 , no vortex is excited opposite to B0 .
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has been first confirmed in computer simulations for a knotted antenna30 and will now be demonstrated experimentally. Figure 5 displays field components excited by a looptorus antenna with positive helicity. A single current pulse (I⯝12 A, ⌬t⯝1 s兲 is applied to the antenna and the displayed plasma fields are induced by the rise of the antenna current. In the transverse x⫺y plane at ⌬z⫽15 cm from the antenna, Fig. 5共a兲 displays a vector field of the toroidal field ˜B and Fig. 5共b兲 a contour plot of the poloidal field ˜B z . The linkage is right-handed, i.e., the helicity density is positive. Figure 5共c兲 shows the poloidal field component ˜B z in the y ⫺z plane on axis (x⫽0). A sketch of the antenna explains the data gap near the origin. The field distribution is axially highly asymmetric. In the direction of B0 a large-amplitude vortex is excited, while in the opposite direction essentially no vortex is discernible and the stray fields are about an order of magnitude smaller than ˜B z,max . In order to show the propagation of the vortex, the poloidal field component is plotted in Fig. 6 versus axial position and time, ˜B z (x⫽y⫽0,z,t). Instead of a single current pulse, an oscillating current is applied at t⫽0 so as to demonstrate the transition from transient to cw wave excitation. The tone burst I ant(t) is sketched in the data gap near the origin. The inclination of the contours is a measure for the axial propagation speed, v z ⯝9⫻107 cm/s. Again, largeamplitude waves are excited by a positive-helicity antenna only in the direction along B0 . CW excitation simply produces a sequence of vortices of alternating polarity but same positive helicity. C. Transmission between loop-torus antennas
The antenna directionality based on helicity injection does not only apply to wave excitation but also to wave detection. From transmission experiments between two iden-
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FIG. 7. Transmission between two identical loop-torus antennas separated along B0 by ⌬z⫽50 cm. Received voltage V rec(t)⬀⫺ ˜B / t⬀ I ant / t shows high transmission for positive helicity of both antennas and poor transmission (⫺20 dB兲 for negative helicities. Transmission is unidirectional and nonreciprocal.
tical loop-torus antennas, the receiving and transmitting properties have been explored. A variety of combinations between two loops and tori with different helicities are possible. First, in Fig. 7, we show the transmission signal between two loop-torus antennas, separated by ⌬z⫽50 cm along B0 . When both antennas have positive helicities and the transmission is along B0 , a large-amplitude signal is received. The open-loop voltage on the receiver antenna is
R. L. Stenzel and J. M. Urrutia
given approximately by V rec⬀ ˜B / t. By reversing the polarity of the loop but not that of the torus 共or vice versa兲 the antenna helicity is changed. For identical conditions, the reversal of the antenna helicities produces a negligibly small transmission signal. More specifically, one finds V rec(H ⬎0)/V rec(H⬍0)⭓10 or 20 dB, an excellent directionality for antennas of complete geometric symmetry along B0 . Figure 8 shows a number of combinations of different receiver and transmitter antennas and helicities. The applied current waveform, I exc , to the transmitting antenna (Tx) is a step function shown in Fig. 8共a兲. Starting with a directional loop-torus as an exciter, the positive-helicity vortex propagating along B0 can be detected by a nondirectional receiver antenna (Rx) such as a loop 关Fig. 8共b兲兴 or torus 关Fig. 8共c兲兴. When the receiving antenna is also a directional loop-torus antenna, the received signal depends on the helicity of both antennas. Arrows next to the antennas indicate the optimum direction of wave excitation or reception. When both antennas have positive helicity, the transmission along B0 is optimal 关Fig. 8共d兲兴. When one helicity is reversed, e.g., that of the transmitter, the transmission degrades 关Fig. 8共e兲兴. When both antennas have negative helicity, the transmission along B0 is poorest 共Fig. 7兲 but would be best opposite to B0 . Thus, the transmission is unidirectional and nonreciprocal, i.e., transmitter and receiver cannot be interchanged. Reversing both loop and torus polarity does not change the helicity but simply reverses the polarity of the received voltage or excited fields. The antenna directionality is useful for various applications, one of which is the sensitive detection of reflected vortices. When a vortex propagates against a conducting boundary its helicity reverses.36 In Fig. 8共f兲, the negativehelicity exciter launches a vortex opposite to B0 , which reflects from the metallic plate, propagates with positive helicity along B0 , and is clearly detected with a positive-helicity receiver antenna. If the exciter antenna has a positive helicity 关Fig. 8共d兲兴 or no helicity, the direct vortex propagating along B0 between the antennas is detected while the small reflected signal is difficult to discern. D. Helicity of whistler noise
FIG. 8. Transmission between antennas of different helicities as indicated schematically. Arrows indicate preferred direction of wave excitation (T x ) and reception (R x ). A current step 共a兲 is applied to a loop-torus transmitting antenna of positive helicity. This excites a vortex along B0 which can be either received by a nondirectional loop 共b兲, torus 共c兲, or a directional looptorus antenna of positive helicity 共d兲. Reversing the helicity of the exciter antenna 共e兲 decreases the received signal V rec(t) by an order of magnitude. The delayed signal in 共f兲 is due to reflection from a metal plate, which reverses the sign of helicity.
Having established the directionality of a helicity antenna with test waves, we now use it to study the helicity of natural whistler wave noise in an active discharge plasma. If the noise possesses helicity, its direction of wave propagation can be inferred as well as the possible source of the noise. The latter is clearly associated with a beam of primary electrons emitted by the cathode,37 since in the afterglow plasma the whistler noise is negligibly small.38 Figure 9 summarizes the noise measurements in the pulsed discharge plasma. At t⫽0 the discharge voltage is turned on, the discharge current and density begin to rise and reach steady-state values at t⯝0.5 ms. A single trace of the received voltage V rec(t) on the loop-torus antenna, amplified by a factor of ten, is displayed in Fig. 9共a兲. Strong rf magnetic fluctuations ( f ⭐10 MHz兲 exist as the density rises and then level off to a lower value in the high-density regime ( pe / ce ⬎100). The digitized noise voltage is squared
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FIG. 9. Analysis of whistler wave noise in a pulsed discharge plasma using a directional loop-torus receiving antenna. 共a兲 Single trace of the antenna voltage, V rec(t) 共gain of 10兲, at the beginning of the discharge (t dis⫽3 ms兲. Noise spectrum covers frequency range of whistlers, ci Ⰶ ⭐ ce . 共b兲 Digitally squared 2 2 N 2 antenna voltage, V rec (t), for a single trace. Averaged signal, 具 V rec (t) 典 N ⫽N ⫺1 兺 i⫽1 V rec (t) i , over an ensemble of N⫽500 repeated discharge pulses for 共c兲 negative antenna helicity and 共d兲 positive antenna helicity. Note that noise has a preferential sign of helicity. 共e兲 Difference between negative and positive 2 2 2 2 helicity data, ⌬ 具 V rec 典 and the average 具 V rec 典 ave . 共f兲 Normalized helicity contents ⌬ 具 V rec 典 / 具 V rec 典 ave which show that, on the average, half of the magnetic energy has left-handed helicity, i.e., propagates opposite to B0 away from the cathode which emits a beam of energetic electrons. The latter is in wave-particle resonance with oblique whistlers and has been shown to produce whistler instabilities 共Ref. 37兲.
关Fig. 9共b兲兴 and ensemble averaged over N⫽500 repeated discharges. By reversing the polarity of either the loop or the torus, leaving all other parameters unchanged, the average 2 noise power 具 V rec 典 is observed to be larger for negative antenna helicity 关Fig. 9共c兲兴 than for positive helicity 关Fig. 9共d兲兴. 2 2 The time dependence of the difference, ⌬ 具 V rec 典 ⫽ 具 V rec 典 H⬍0 2 2 2 ⫺ 具 V rec典 H⬎0 , and the average, 具 V rec典 ave⫽ 关 具 V rec 典 H⬍0 2 ⫹ 具 V rec 典 H⬎0 兴 /2, are shown in Fig. 9共e兲 and their ratio, 2 2 ⌬ 具 V rec 典 / 具 V rec 典 ave , in Fig. 9共f兲. These results show that throughout most of the early discharge the magnetic fluctuations in the low frequency whistler branch exhibit more negative than positive helicity, i.e., propagate preferentially against B0 or away from the cathode. Earlier experiments37 have shown that the beam of primary electrons emitted by the cathode drives a Cherenkov instability of oblique whistlers whose parallel phase velocity matches the beam veloc-
ity. The present measurements suggest that the unstable waves evolve into wave packets since plane waves have no helicity. However, a comparison of Figs. 9共c兲 and 9共d兲 with Figs. 8共d兲 and 8共f兲 shows that the relative helicity 共helicity/ energy兲 of the noise is not as large as for the vortices excited by the loop-torus antenna. Note that the loop-torus antenna can also detect plane oblique whistlers which would produce the same amplitude for both signs of antenna helicity. The normalized difference of the received noise power 关Fig. 9共f兲兴 is a measure for the helicity contents in the noise. The larger emission at the beginning of the discharge (20⬍t⬍40 s兲 is due to a higher percentage of beam to background electrons, longer mean free path of the cathode beam, and a higher whistler speed at lower beta values compared to the dense steady-state discharge. However, even the
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2 FIG. 10. Ensemble-averaged loop-torus voltage 具 V rec 典 for both helicity signs vs time in the dense steady-state discharge plasma. The whistler noise exhibits preferentially negative helicity.
lower-level whistler noise during steady state has predominantly negative helicity. This is demonstrated in Fig. 10, which displays the ensemble-averaged loop-torus voltage 2 具 V rec 典 for both helicity signs vs time at 0.6 ms into the discharge.
R. L. Stenzel and J. M. Urrutia
each axial half wavelength a topology which can be described by two vortices of opposite polarity placed radially in juxtaposition. Like all bounded whistler wave packets it has a positive magnetic self-helicity for propagation along B0 . Thus, it is most efficiently excited with a right-handed helical antenna, as indeed observed, although explained by matching the polarization rather than the magnetic helicity. However, it was found surprising that a left-handed helical antenna did not properly excite the m⫽⫺1 mode with opposite polarization.41 This is easily understood by considering the magnetic self-helicity of this mode which is predominantly negative for propagation along B0 . Since this mode violates the basic helicity properties of EMHD vortices, it cannot be excited irrespective of proper phase matching. A left-handed helical antenna excites preferentially helicons propagating opposite to B0 , consistent with matching negative antenna and wave helicity. Thus, the antenna directionality for whistlers is better understood in terms of helicity matching than polarization matching.
IV. CONCLUSIONS
The present laboratory experiments have demonstrated the properties of magnetic helicity of bounded whistler wave packets. The sign of the helicity is uniquely determined by the direction of wave propagation along B0 . Injection of helicity from a suitable antenna results in wave propagation in a preferred direction. Directional radiation from a linked loop-torus antenna has been demonstrated. The antenna also exhibits directionality for receiving EMHD vortices. Transmission between two helicity antennas is unidirectional and nonreciprocal. Interesting applications have been pointed out, such as the sensitive detection of reflected waves or the analysis of helicity of whistler noise in a discharge. In the latter case a preferred sign of helicity has been detected, revealing the direction of wave propagation, and the source of the noise, i.e., a Cherenkov instability driven by the electron beam emitted from the cathode. The present results should also be of interest to the research on helicon waves. Helicons, as defined in plasma processing,25,39 are low frequency whistlers in bounded plasma columns with dispersion / ce ⫽(k 2储 ⫹k⬜2 ) 1/2 ⫻k 储 c 2 / 2pe , where k⬜ is determined by the radial boundaries.40 The waves are used for efficient production of high density, albeit nonuniform plasma columns. The axial dependence of the column eigenmodes is described by ˜B ⬀exp i(m⫹k储z⫺t). The m⫽0 mode has for each half ˜ r , ˜B z ) wavelength a vortex topology with linked (B ˜ ⬀J 1 (k⬜ r) and B ⬀J 0 (k⬜ r), where J 0 , J 1 (k⬜ r) are Bessel functions. The mode is usually excited by a loop antenna wound around a glass tube confining the plasma. Since helicons propagating along B0 have positive magnetic helicity, a positive-helicity antenna at one end of the column would be more efficient than a nondirectional loop antenna. Directionality of helical antennas has been observed for higher order bounded Alfve´n33 and helicon eigenmodes.41 The directionality is usually explained by matching the antenna k-spectrum to that of the column eigenmode, e.g., by proper phase rotation in space and time. The m⫽⫹1 mode has in
ACKNOWLEDGMENT
The authors gratefully acknowledge support for this work from the National Science Foundation under Grant No. PHY 9713240.
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