Whistler wave propagation and whistler wave antenna radiation resistance measurements

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 33, NO. 2, APRIL 2005 637

Whistler Wave Propagation and Whistler WaveAntenna Radiation Resistance MeasurementsWilliam E. Amatucci, David D. Blackwell, David N. Walker, George Gatling, and Gurudas Ganguli

Abstract—Whistler waves are a common feature of ionosphericand magnetospheric plasmas. While the linear behavior of thesewaves is generally well understood, a number of interesting ob-servations indicate that much remains to be learned about thenonlinear characteristics of the mode. For example, in space,very low frequency (VLF) emissions triggered by whistler modeslaunched from ground-based transmitters have been observed.Emission is assumed to come from transverse currents formedby counterstreaming electrons that are phase bunched by thetriggering signal. In the laboratory, it has been shown that withincreasing amplitude of the driving signal applied to an antenna,the whistler mode radiation pattern forms a duct with diameterof the order of the parallel wavelength. The ducted waves wereobserved to propagate virtually undamped along the length ofthe plasma column. These observations have prompted an NavalResearch Laboratory’s (NRL) Space Physics Simulation Chamberstudy of whistler wave dynamics. The goals are to investigatewhistler wave ducting, self-focusing, and amplification, and tostudy nonlinear whistler-plasma interactions.

Index Terms—Antenna impedance, laboratory plasma, waveducting, wave propagation, whistler waves.

I. INTRODUCTION

WHISTLER waves are a ubiquitous feature of iono-spheric and magnetospheric plasmas [1], [2]. Natural

occurrences of these electromagnetic signals often propagatedown to the surface of the earth, where they can be detected inthe very low frequency (VLF) range ( 1-10 kHz). Naturallyoccurring modes are typically initiated by broadband bursts ofradio noise released during lightning discharges. The whistlermode propagates predominantly along the magnetic field as aright-hand-circularly polarized wave with frequency below theelectron cyclotron frequency, [1], [3]. Here, isthe electron charge, is the magnetic field strength, and isthe electron mass.

There have been significant advances in understanding thebasic physics of whistler wave propagation since the earliestrecorded observations [1]. The waves have been extensivelystudied in situ using sounding rockets and satellites, via ground-based transmitters and receivers, through theoretical investiga-tions, and they have also been studied in laboratory experiments.Several of these investigations revealed interesting nonlinear

Manuscript received October 28, 2004; revised December 6, 2004. This workwas supported in part by the Office of Naval Research and in part by the DefenseAdvanced Research Projects Agency (DARPA).

W. E. Amatucci, D. N. Walker, and G. Ganguli are with the Plasma PhysicsDivision, Naval Research Laboratory, Washington, DC 20375 USA (e-mail: [email protected]; [email protected]; [email protected]).

D. D. Blackwell and G. Gatling are with SFA Inc., Largo, MD 20774 USA(e-mail: [email protected]; e-mail: [email protected]).

Digital Object Identifier 10.1109/TPS.2005.844607

Fig. 1. Whistler wave dispersion relation. Frequency normalized by theplasma frequency is plotted along the horizontal axis while the reciprocal of theindex of refraction (= v =c) is plotted on the vertical axis. This dispersionrelation was calculated for a plasma density of 10 cm and a magnetic fieldstrength of 20 G in argon plasma. Under these conditions, ! = � 5, whichscales to typical magnetospheric conditions at L = 2.

effects associated with whistlers. For example, Stenzel [4] re-ported on the self-ducting of large amplitude whistler waves ina laboratory plasma. Those experiments showed that with in-creasing amplitude, the radiation pattern from a small dipoleantenna becomes increasingly narrow, ultimately forming a ductwith diameter of the order of the parallel wavelength. The ductedwaves were observed to propagate virtually undamped alongthe length of the plasma column. In subsequent experiments byKostrov et al., [5] whistler wave ducting within spatially lo-calized magnetic-field-aligned density enhancements was ob-served. In the space environment, observations of artificiallystimulated VLF emissions triggered in the magnetosphere bywhistler modes from VLF transmitters as reported by Stiles andHelliwell [6] are quite well known. Emission radiation is as-sumed to come from the transverse currents formed by electronsthat have been temporarily phase bunched by the constant fre-quency triggering signal. However, many important questionsregarding the nonlinear behavior of these waves, including is-sues regarding the self-ducting of large amplitude whistlers, am-plification and secondary emission of whistlers in the presenceof energetic electrons, and whistler–plasma interactions are yetto be fully understood.

A new experimental program at the Naval Research Labora-tory’s (NRL) Space Physics Simulation Chamber (SPSC) hasbegun in conjunction with an NRL theoretical effort in orderto investigate the nonlinear characteristics of whistler waves.The objective of the experimental program is to examine the

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638 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 33, NO. 2, APRIL 2005

Fig. 2. Photograph and schematic diagram of magnetic loop antennas.

key physics associated with formation of ducts, propagationof whistler waves with minimal loss, amplification of whistlerwaves, and whistler–plasma interactions. In parallel, experi-ments are being performed to study the radiation impedance ofvarious antenna configurations. This paper reports on the initialexperimental progress toward these goals.

II. WHISTLER WAVE GENERATION AND MODE VERIFICATION

Fundamental to each of the experimental goals listed aboveis the ability to launch and detect whistler waves in the SpaceChamber plasma. If we consider wave propagation along thebackground magnetic field direction (i.e., ) with the waveelectric field oriented perpendicular to , two solutions to thedispersion relation exist corresponding to right- and left-handcircularly polarized modes [3]. The parallel propagating whistlerwave corresponds to the right-hand circularly polarized modeand its cold plasma dispersion relation is given by

where is the index of refraction, is theplasma frequency, is the plasma density, and is the permit-tivity of free space [3]. The dispersion relation is shown in Fig. 1,where the reciprocal of the refractive index ( )is plotted as a function of wave frequency (normalized by theplasma frequency). For frequencies less than half of the elec-tron cyclotron frequency, we see that the wave phase velocityincreases with frequency, a feature which produces the familiarfalling tone of lightning-produced whistlers. When viewed overa sufficiently wide frequency range, this mode exhibits both aresonance and a cutoff. A resonance in the dispersion relationoccurs where the wavenumber becomes infinite (i.e., wave-length ) and a cutoff occurs at frequencies where van-ishes (i.e., wavelength ). The mode exhibits a res-onance at the electron cyclotron frequency and a cutoff at

. The resonance at resultsfrom the right-hand circularly polarized wave resonating with the

Fig. 3. Schematic diagram of basic experimental setup.

electron gyromotion. At the location in the plasma where thisoccurs, significant electron energization may be observed dueto resonant absorption of the waves.

In the Space Chamber experiments, electromagnetic wavesare launched using a magnetic loop antenna that is electricallyinsulated from, but immersed in, the plasma. An identical probeis used as one of the receiving antennas. Fig. 2 shows a pho-tograph and schematic diagram of the magnetic loop antenna.The loops consist of four turns of wire, with a loop diameter of1.25 cm. The leads are tightly twisted and brought through thegrounded probe shaft to a 1:1 center-tapped transformer. Thetransformer is used to eliminate any electrostatic pickup at thecoil, which can cause unwanted noise signals.

A schematic diagram of the experimental setup is shown inFig. 3. The main section of the NRL Space Chamber is 1.8-min diameter by 5-m long. An additional 0.55-m diameter, 2-m-long chamber section is also available for the experiment, pro-viding up to 7 m of total plasma column length. The steady-statemagnetic field strength in the main chamber section plasmacolumn is variable up to 50 G, while the direct current (dc) mag-netic field strength in the smaller chamber section is variable

AMATUCCI et al.: WHISTLER WAVE PROPAGATION AND WHISTLER WAVE ANTENNA RADIATION RESISTANCE MEASUREMENTS 639

TABLE ITYPICAL BACKGROUND PLASMA PARAMETER RANGES FOR THE SPSC

AND RADIATION BELT

up to 1.1 kG. Space Chamber plasma is produced using a largearea thermionic discharge plasma source, producing a plasmacolumn that is approximately 75 cm in diameter. Argon gas isused for all experiments described in this report. Typical plasmaparameters ranges are summarized in Table I.

Before beginning experimentation on the nonlinear aspectsof the whistler mode or on the radiation resistance of whistlerantennas, experiments were performed in order to confirm thatthe waves being generated in the laboratory plasma are whistlerwaves. By driving the transmitting magnetic loop antenna withan oscillator and an radio frequency (RF) amplifier with con-trollable output power, electromagnetic waves were launched inthe chamber. The waves were detected with receiving antennasand observed either with oscilloscopes, a spectrum analyzer, orwith a network analyzer. Three preliminary experiments wereperformed in order to verify the identity of the waves.

A. Experiment 1

The transmitting antenna was driven in vacuum in order todetermine if there were any “machine resonances” near thefrequencies of interest. The lowest order “cavity” mode in theSpace Chamber without plasma is the mode, which hasa frequency of approximately 100 MHz [7]. Whistler modefrequencies of interest, on the other hand, correspond to levelsmostly less than . Since even for the maximum magneticfield strength of 50 G in the main Space Chamber section,

, it is clear that we are well below any relevantcavity resonances.

The response of a receiving antenna to a transmitting fre-quency of 455 kHz is shown as the black circles in Fig. 4. In thisinstance, the receiver was located approximately 70 cm axiallyfrom the transmitting antenna, which was radially centered onthe chamber axis. Since this frequency is well below any sig-nificant electromagnetic radiation frequency from an antennaof this size and since the driving frequency is well below thefrequency of the lowest “chamber” mode, the response is es-sentially zero as expected. The grey circles in Fig. 4 show thereceived wave power at the same frequency and driving am-plitude in a plasma with a density of . As in theprevious experiment, the transmitting antenna was located at

, and a symmetric double-humped profile of the waveamplitude is visible. This double-peaked structure is character-istic of low-power waves that lie along the whistler-lower hybrid

Fig. 4. Received wave amplitude as a function of radial position in vacuum(black circles) and in plasma (grey circles) with density of 10 cm .

Fig. 5. Received whistler wave power (left) and phase (right) versus radialposition and driver frequency. Data show that for driving low-amplitudes, thewave power spreads along a cone with origin at the driving antenna.

wave branch, which propagate at an angle to the backgroundmagnetic field determined by the ratio of the perpendicular toparallel wavenumbers ( ), forming a coneabout the magnetic field lines. This characteristic is further il-lustrated in Fig. 5, which shows the radial profiles of mode am-plitude [Fig. 5(a)] and phase [Fig. 5(b)] of the received waves


Fig. 6. Schematic diagram of the experimental setup for the whistler polarization measurement.

Fig. 7. Time series of received wave amplitude on the E (black) and E (red) dipole probes. Data indicate a phase shift of approximately 120 .

as a function of driver frequency for low driving power. Thesedata were obtained by driving the transmitting loop antenna lo-cated on the plasma column axis at the end of the plasma columnopposite the plasma source. The receiving loop antenna waslocated approximately 2 m from the transmitting antenna. Foreach driving frequency, the receiving antenna was stepped radi-ally across the plasma column and a time series of the receivedpower was obtained at each radial position. The data indicatethat as the driving frequency is increased, the waves propagateat an increasing angle with respect to the background magneticfield.

For these experiments, unless otherwise noted, the neutralpressure torr and the magnetic field strength

. Under such conditions, the plasma is essentiallycollisionless since the wave frequencies of interest range from

500 kHz to 80 MHz while the ion-neutral collision frequency( ) and the electron neutral collision

frequency ( ). Further-more, for a typical wave frequency of 10 MHz, the cold plasma

dispersion relation indicates that the whistler group velocity is. Since the probes are separated by m, the wave

transit time is of the order of 0.1 . This is much shorter thaneither the electron or ion collision times, which are and220 , respectively.

B. Experiment 2

The dispersion relation indicates that whistler waves trav-eling along the magnetic field direction have right-hand-circularpolarization. While the waves launched in the Space Chamberplasma may not be exactly magnetic field aligned, in orderfor the mode to be a whistler mode, right-hand polarization(possibly elliptical) should be observed. In order to test forthis condition, a pair of crossed electric field dipole probeswas constructed to measure the wave electric field in the planeperpendicular to the background magnetic field ( ). Aschematic diagram of the experimental setup is shown in Fig. 6.Fig. 7 shows a time series of the received wave amplitude in the

and directions. These data indicate that the mode launched


Fig. 8. Polar plot showing the wave amplitude as a function of phase angle. Data indicate the elliptical polarization of the mode launched in the plasma column.

in the Space Chamber plasma does propagate with right-handpolarization. In the example illustrated in Fig. 7, the two com-ponents have a relative phase of approximately 120 . Fig. 8 isa polar plot of the wave amplitude as a function of phase angle,illustrating the elliptical polarization of the mode. This behavioris consistent with that expected for whistler waves propagatingat an oblique angle with respect to the magnetic field. In thisview, the magnetic field would be directed out of the page.

C. Experiment 3

The whistler mode dispersion relation indicates that a reso-nance should exist at the electron cyclotron frequency. To testfor this condition, the frequency of the transmitting magneticloop antenna was swept from well below to well above andthe amplitude of the signal detected by the receiving loop an-tenna was monitored using an HP8753D network analyzer. Thenetwork analyzer is a useful device for the characterization oftwo-port networks, providing a driving signal of variable fre-quency (30 kHz–3 GHz) and measurements of the forward andreflected wave amplitude and phase. Both probes were locatedon the plasma column cylindrical axis. The experiment was re-peated for three different magnetic field strengths (15, 26.5, and36.6 G), corresponding to three different electron cyclotron fre-quencies (42.0, 74.2, and 102.5 MHz). The data in Fig. 9 illus-trate that for each different value of magnetic field strength, thereceived wave power abruptly goes to zero at a frequency corre-sponding to the electron cyclotron frequency. Based on the goodagreement between the characteristics of the waves observed in

the laboratory and those predicted for the whistler mode, weconclude that the observed waves are whistlers.

Once the identity of the mode was established, the investiga-tion into the nonlinear aspects of the waves was initiated, be-ginning with a study of whistler wave ducting. The first ex-periments were aimed at benchmarking our experiments withthose of Stenzel[4], [8], [9], Sugai et al. [10], and Kostrov et al.[5], and extending the results to include propagation over longerdistances. In the experiments described by Stenzel [8], whistlerwaves were launched by driving a 4.4-cm-long dipole antenna(oriented perpendicular to ) at a frequency of 150 MHz in aplasma with a density of 5 ( )and a magnetic field strength of 65 G ( ). Thewaves were detected downstream by a 1-cm-long wire-tippedantenna which could be scanned radially. For low values of ap-plied RF power, the observed waves diverge from the dipole andthe amplitude was found to decay with distance from the trans-mitter. Our results on wave propagation within a cone are con-sistent with these findings. However, for high values of appliedRF power, Stenzel found that the waves could be confined to anarrow channel and propagate along the column axis undamped.The width of the channel was found to be approximately equalto the axial wavelength of the whistler mode.

By the controlled increase of the RF power delivered tothe magnetic loop antenna by an RF amplifier, whistler waveducting is being investigated in the Space Chamber experi-ments. Fig. 10 shows radial profiles of wave amplitude for caseswhere the transmitting antenna is located on axis (solid line)and 10 cm off axis (dashed line). The plasma density for both


Fig. 9. Observed whistler wave cutoff at the electron cyclotron frequency.

Fig. 10. Whistler wave ducting. Solid line indicates the received wave power� 2 m downstream from the transmitting antenna, with the transmitter locatedon the cylindrical axis of the plasma column. Dashed line shows the receivedwave power when the transmitting antenna is located at r = 10 cm.

these trials was ( ) andthe magnetic field strength was 36.6 G ( ).The magnetic loop antenna was driven with 20 W at a fre-quency of 80 MHz, which corresponds to an axial wavelengthof for the given plasma conditions. The axial distanceto the receiving antenna was approximately 2 m. Fig. 10 showsthat unlike the earlier observations of the wave power spreadingout over broader distances as the mode propagates axially, inthese cases the wave power remained localized within channelshaving width on the order of 10 cm. This observation is similar

to the findings of Stenzel, but over a longer axial distance. Inthe Space Chamber experiment, the distance between the trans-mitting and receiving antennas was approximately . Theseare preliminary results, however, and further investigation isrequired.

III. WHISTLER WAVE ANTENNA RADIATION

RESISTANCE MEASUREMENTS

In order for an antenna to be effective at driving whistlerwaves in the plasma, the coupling of the antenna to the mediummust be considered. An antenna usually has complex impedancewhen driven at its operating frequency, with nonzero resistancein series with a complex reactance, . The powerdelivered by a complex current (at some phase angle to theimpedance ) is split between a resistive portion ( ) and areactive portion ( ). The energy transferred by the reactiveportion is stored mostly in the reactive near field. The energytransferred to the antenna by the resistive power flow generallyis split between resistive loss (heating of the antenna structure)and energy radiated to the surrounding medium. Although weinclude both of these terms in our characterization of resistanceas , it must be kept in mind that resistive heating of the antennaelement itself and radiation resistance (defined aswhere is the radiated electromagnetic power and is the an-tenna current) are fundamentally distinct quantities.

The radiation resistance experiments currently underway inthe Space Chamber have proceeded along two main directions.In the first effort, the radiation resistance of the loop antennadescribed above has been measured for a wide range of drivingfrequencies and compared to theoretically predicted values. A


Fig. 11. (a) Schematic diagram of experimental setup and the resulting equivalent circuit. (b) Example of antenna radiation resistance (R ) data showing the:vacuum theory prediction (red dashed line) and the experiment measurements (blue solid line) for a small loop antenna.

simplified schematic diagram of the experimental setup and theequivalent circuit is shown in Fig. 11(a) and an example of thedata is shown in Fig. 11(b). The experimental data shown inFig. 11(b) (solid line) were obtained using the network analyzerto sweep the driving frequency while measuring the load resis-tance . The theoretically predicted values are shown as thedashed line. The data show reasonable agreement with the the-oretical predictions for antenna radiation into a vacuum envi-ronment. The additional peak found in the experimental data isdue to reflections in the vacuum vessel.

In a plasma, the impedance of an antenna can be expressedby

where is the driving frequency, is the plasma dielec-tric response function (i.e., the dielectric tensor), is the wavevector, and is the antenna surface current density. The disper-sion relation of the plasma is given by ,which describes the various modes “preferred” by the plasma.These modes depend on plasma parameters such as density, tem-perature, species, etc. The total impedance of the antenna atsome frequency can be thought of as a sum of contributionsfrom individual modes. If a component of the antenna surfacecurrent density is large (i.e., when the numerator of the integralis large), then that mode will contribute significantly to the totalimpedance.

In order to theoretically predict the frequencies at which theantenna energy will be deposited, it is necessary to determinethe appropriate dispersion relation, which, in turn, requires anaccurate description of the dielectric properties of the plasma.The dispersion relation can be determined by beginning fromMaxwell’s equations [11]. Combining with

, we obtain

. Linearizing and assumingplane wave solutions, we obtain

(1)

where the subscript 1 denotes the perturbed quantities and theequilibrium velocity of each species is assumed to be zero.

In an anisotropic medium, like a plasma immersed in a mag-netic field, Ohm’s Law is given by where isthe conductivity tensor. Using this expression, (1) can be reex-pressed as

(2)

where is the unit tensor. The plasma dielectric tensor is givenby . The plasma dispersion relation isgiven by setting the determinant of the dielectric tensor to zero.

Obviously, finding solutions to the plasma dispersion relationcan become quite complicated when all background quantitiesand plasma effects are incorporated. Therefore, in order to testantenna responses in the laboratory and make meaningful com-parisons with theory, it is useful to first begin with a very sim-plified model of the plasma. Using such a model and a simpleprobe geometry, we can make quantitative theoretical predic-tions that can be tested in the laboratory.

The simplified model of the plasma that we begin with con-sists of uniform, cold, collisionless plasma with no backgroundmagnetic field. We will assume that the large ion mass preventsion motion, thus the ions form a neutralizing background. If aperturbation is applied to the electrons, their motion away fromtheir equilibrium position creates a perturbation electric fieldthat acts as a restoring force. This, in turn, creates an oscillatorymotion of the electrons about their equilibrium position. Theequation of motion for the electrons (in the absence of a mag-netic field) is . Again, linearizing andassuming plane wave solutions and solving for the perturbedelectron velocity, we obtain

(3)


Since the perturbed current is given by the sum of theperturbed particle velocities over the different species ,

, (1) yields

Substituting (3), we obtain

This expression can be simplified to

Finally, using the definition of the electron plasma frequency,we obtain

(4)

Written in this form, we see that the dielectric constant of theplasma is given by

(5)

The solution to this dispersion relation represents electro-static oscillations of the electrons at the plasma frequency,which is one of the fundamental resonances in a plasma.

The need to test the applicability of this simplified plasmamodel before use in interpretation of the loop antenna ex-periments led to the second phase of the antenna radiationresistance measurement experiments. In this phase, a simplerspherical probe geometry was selected and measurements ofthe impedance of a driven spherical capacitive probe wereperformed. The capacitance of such a probe in vacuum is wellknown and is given by , where is the radius of thesphere. With the addition of plasma, the capacitance changes bythe relative permittivity of the medium: .Comparison of the measurement of the probe’s capacitancein the plasma with its vacuum value can therefore provideinformation on plasma parameters such as the density andcollision frequency.

The symbols in Fig. 12(a) show the measured value of theprobe impedance as the frequency is swept, while the dashedline shows the value of impedance predicted by the simplifiedplasma model. These measurements were performed in an argonplasma with density of 4.7 with a background mag-netic field strength of . The radius of the sphere was 1 cm.At frequencies near and above the plasma frequency, reasonableagreement is found. This is expected since this model is validprimarily for high frequency plasma oscillations.

In Fig. 12(b), the phase is observed to switch from to, which is indicative of the load switching from being pri-

Fig. 12. (a) Measured (symbols) and predicted (dashed and solid lines) valuesof capacitive reactance and (b) measured (symbols) and predicted (solid line)phase as a function of driver frequency for a 19-mm diameter spherical probe.Plasma density n = 4:7 � 10 cm and the background magnetic fieldB = 3G. Vertical dashed line marks the plasma frequency as determined fromplasma density measurements with a nearby Langmuir probe.

marily inductive to primarily capacitive. This sharp phase tran-sition is in excellent indicator of the plasma frequency reso-nance. Strictly speaking, this cutoff is expected to occur at theupper hybrid frequency . However, at a

density of 4.7 and a magnetic field strength of 3 G,, so the upper hybrid frequency is essentially

equal to the plasma frequency.Below approximately half of the plasma frequency, the

agreement between the simplified plasma model and the ex-perimental data breaks down. To more accurately model theseeffects, other physical aspects must be accounted for. Thesolid line illustrates a model that incorporates more of thephysics relevant at lower frequency. For the conditions of thisexperiment, at lower frequencies, the effects of the magneticfield can become important. To account for these effects,the dispersion relation must be solved for finite magneticfield. If we assume that the waves propagate parallel to themagnetic field, this solution yields the dispersion relation forright-hand circularly polarized electromagnetic waves ( -modeor whistler waves). Thus, the plasma dielectric constant be-comes .

One possibility to explain the minimum in the impedancelocated at is a sheath-plasma resonance [12]–[15],which results from the capacitance associated with the Debyesheath surrounding the probe. In the case of the sphericalprobe, the sheath will also be roughly spherical and will beseparated from the probe surface by a few Debye lengths( ). This forms an additional capacitance

, where is the radius of the spher-ical probe, and is the radius of the sheath. The capacitancedue to the sheath surrounding the probe basically adds anotherimpedance in series with the probe, so the total impedance is


given by . The solution to thisexpression, shown as the solid line in Fig. 12, gives a moreaccurate representation of the actual antenna impedance overthe entire frequency range. The data indicate that the effectivethickness of the sheath surrounding the spherical probe isapproximately 4– .

IV. SUMMARY

An experimental program to test the feasibility of whistlerwave-induced particle scattering for removal of energetic ra-diation belt electrons has been designed and is now un-derway in the NRL Space Physics Simulation Chamber Lab-oratory. Transmitting and receiving magnetic loop antennasand crossed electric field dipole receiving antennas have beenfabricated and tested. Sensitive detection electronics and RFamplifiers have been assembled, tested, and calibrated. Elec-tromagnetic modes launched in the Space Chamber plasmahave been identified as whistler waves due to the correspon-dence of the wave properties with theoretical predictions. Theinitial investigations into the nonlinear properties of whistlerhave begun.

In parallel to the whistler wave experimentation, an experi-mental effort to characterize the efficiency of various antennaconfigurations for projecting whistler wave power is underway.Initial results from these experiments confirm the accuracy andperformance of the plasma diagnostics by identification of theelectron plasma frequency. Simplified plasma models have beenemployed to help guide the experimental effort.

ACKNOWLEDGMENT

The authors gratefully acknowledge the useful discussionswith Dr. C. Swenson, Dr. D. Papadopoulos, Dr. P. Schuck, andDr. O. Storey.

REFERENCES

[1] R. A. Helliwell, Whistlers and Related Ionospheric Phenomena. Stan-ford, CA: Stanford Univ. Press, 1965.

[2] L. R. O. Storey, “An investigation of whistling atmospherics,” Phil.Trans. Roy. Soc. Lond., vol. A246, pp. 113–141, 1953.

[3] F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 2nded. New York: Plenum Press, 1984.

[4] R. L. Stenzel, “Self-ducting of large-amplitude whistler waves,” Phys.Rev. Lett., vol. 35, p. 574, 1975.

[5] A. V. Kostrov et al., “Whistlers in thermally generated ducts with en-hances plasma density: Excitation and propagation,” Physica Scripta,vol. 62, p. 51, 2000.

[6] G. S. Stiles and R. A. Helliwell, “Frequency-time behavior of artifi-cially stimulated VLF emissions,” J. Geophys. Res., vol. 80, p. 608,1975.

[7] J. D. Jackson, Electrodynamics. New York: Wiley, 1975.[8] R. L. Stenzel, “Whistler wave propagation in a large magnetoplasma,”

Phys. Fluids, vol. 19, p. 857, 1976.[9] , “Filamentation instability of a large amplitude whistler wave,”

Phys. Fluids, vol. 19, p. 865, 1976.[10] H. Sugai et al., “Whistler wave ducting caused by antenna actions,”

Phys. Fluids, vol. 21, p. 690, 1978.[11] G. K. Parks, Physics of Space Plasmas. Redwood City, CA: Addison-

Wesley, 1991.[12] F. W. Crawford and K. J. Harker, “Energy absorption in cold inhomoge-

neous plasmas: The Herlofson paradox,” J. Plasma Phys., vol. 8, p. 261,1972.

[13] P. Meyer, N. Vernet, and P. Lassudrie-Duchnese, “Theoretical and ex-perimental study of the effect of the sheath on an antenna immersed ina warm isotropic plasma,” J. Appl. Phys., vol. 45, p. 700, 1974.

[14] V. P. T. Ku, B. M. Annaratone, and J. E. Allen, “Plasma-sheathresonances and energy absorption phenomena in capacitively coupledradio frequency plasmas. Part I,” J. Appl. Phys., vol. 84, p. 6536,1998.

[15] , “Plasma-sheath resonances and energy absorption phenomena incapacitively coupled radio frequency plasmas. Part II. The Herlofsonparadox,” J. Appl. Phys., vol. 84, p. 6546, 1998.

William E. Amatucci received the B.S. degrees inphysics and mathematics from Saint Vincent Col-lege, Latrobe, PA, and the M.S. and Ph.D. degreesin plasma physics from West Virginia University,Morgantown.

After receiving the Ph.D. degree in 1994, he wasawarded a National Research Council PostdoctoralFellowship at the Naval Research Laboratory (NRL),Washington, DC. Following completion of thepostdoctoral fellowship, he served one year as acontractor in NRL’s Plasma Physics Division before

becoming a full-time Staff Member in 1997. He is currently Head of the SpaceExperiments Section at NRL. His areas of research interest include plasmawaves and instabilities, wave-particle interactions, plasma diagnostics, and thephysics of dusty plasmas.

Dr. Amatucci is a Member of the American Physical Society, the Amer-ican Geophysical Union, the International Union of Radio Science (URSI), andSigma Xi.

David D. Blackwell received the Ph.D. degreein electrical engineering from the University ofCalifornia, Los Angeles, in 1999.

From 1999 to 2001, he was a National ResearchCouncil Postdoctoral Associate at the Naval Re-search Laboratory, Washington, DC. Since 2001,has been employed as a Research Physicist in theAdvanced Technology Division of SFA, Inc., Largo,MD. While at SFA, he has worked with the large areaplasma processing system (LAPPS) group and theSpace Experiments Section of the Charged Particle

Physics Branch at the Naval Research Laboratory, Washington, DC.. Hisresearch interests include radiofrequency plasma phenomona, plasma sourcesand applications, and plasma diagnostics.

David N. Walker received the B.S. degree in physicsfrom the University of Maryland, College Park, andthe M.S. and Ph.D. degrees also in physics from theUniversity of New Hampshire, Durham.

He is currently a Research Physicist in the PlasmaPhysics Division at the Naval Research Labora-tory, Washington, DC, where he has been sincehe received the Ph.D. degree in 1975. Since thebeginning of his work, he has been concerned withbasic research in both space-related and laboratoryplasma physics. Areas of research interest include

chaotic particle orbits in the magnetosphere, space vehicle charging, plasmaprobe experimental techniques, dusty plasmas, and, recently, investigations ofphenomena associated with plasma sheath resonance in collisionless plasmas.He has published a number of articles in each of these areas.


George Gatling received the B.Sc. degree in elec-trical engineering from Brigham Young University,Provo, UT, in 2002.

He is currently a Contractor for the Plasma PhysicsDivision of the Naval Research Laboratory, Wash-ington, DC. He is actively involved with research onplasma diagnostic development for both laboratoryand space plasmas.

Gurudas Ganguli is a Research Physicist in theBeam Physics Branch at the Naval Research Lab-oratory, Washington, DC. He has a broad range ofexperience in the study of plasma processes in themagnetosphere and ionosphere, and in laboratoryplasma devices. His areas of expertise includenonlocal effects associated with the generationand propagation of waves in plasma and stabilitytheory of both kinetic and fluid plasmas. His currentresearch interests concern the study of turbulentprocesses in collisional and collisionless plasmas,

including negative ion and dusty plasmas.Dr. Ganguli is a Member of the American Physical Society and the American

Geophysical Union and a Fellow of the American Physical Society.

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Whistler wave propagation and whistler wave antenna radiation resistance measurements