High-Frequency Fluctuations of a Modulated, Helical Electron Beam

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1996

Mark Anthony Reynolds Embry-Riddle Aeronautical University, [email protected]

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Scholarly Commons Citation Reynolds, M. A. (1996). High-Frequency Fluctuations of a Modulated, Helical Electron Beam. Physics of Plasmas, 3(). Retrieved from https://commons.erau.edu/publication/394

This Article is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Publications by an authorized administrator of Scholarly Commons. For more information, please contact [email protected]. High-frequency fluctuations of a modulated, helical electron beam M. A. Reynoldsa) and G. J. Morales Department of Physics and Astronomy, University of California, Los Angeles, California 90095 ͑Received 17 June 1996; accepted 27 August 1996͒ The high-frequency electromagnetic field generated by a density-modulated, helical electron beam propagating in a magnetized plasma is calculated. The magnetic fluctuations are found to exhibit spatially localized ͑evanescent͒ resonances at harmonics of the electron-cyclotron frequency, whose width is determined by the pitch angle of the beam, and whose existence is a consequence of the helical geometry. In addition, electrostatic modes are radiated near the hybrid frequencies, and electromagnetic modes are radiated above the upper-hybrid frequency. The predicted frequency spectrum and mode structure in configuration space are in good agreement with experimental observations of discrete emission lines at the electron-cyclotron harmonics ͓Phys. Fluids B 5, 3789 ͑1993͔͒.©1996 American Institute of Physics. ͓S1070-664X͑96͒01312-2͔

I. INTRODUCTION jected into the ionosphere. They included realistic beam effects, such as a beam front and the fact that the beam has a There are several areas of contemporary plasma physics finite thickness on the order of the Brillouin radius. Harker in which radiation from localized charge distributions plays and Banks10,11 calculated the power radiated by a helical an important role. Current and temperature filaments in electron beam through the far field, and also the explicit form tokamaks,1 auroral electron beams in the ionosphere,2 and of the near field. However, they restricted their study to the artificial particle beams injected into space3 are some of whistler and lower-hybrid frequency regimes, and also to a these areas. square-wave density modulation. This type of modulation This analytical study is motivated by a laboratory experi- allows certain integrals to be evaluated analytically, but hin- ment performed by Stenzel and Golubyatnikov4 at the Uni- ders the understanding of the intrinsic frequency spectrum of versity of California, Los Angeles ͑UCLA͒, in which the the beam ͑which must be multiplied by the spectrum of the properties of the fluctuating magnetic field near a helical modulation to obtain the radiated spectrum͒. electron beam in a high-density magnetized plasma were Another motivation for the study of localized beams is measured. This is a fundamental experiment which explores the explanation of laboratory experiments like those of 5 the source of fluctuations near the harmonics of the electron- Landauer where many harmonics of ⍀e are observed in the cyclotron frequency, and a theoretical understanding is nec- frequency spectrum of the fields measured outside of the essary to lay the foundation for more involved studies and plasma chamber. Canobbio and Croci12 were among the first applications. When the plasma is weakly magnetized to suggest single-particle radiation as a possible explanation (⍀eӶ␻ pe , where ⍀e and ␻ pe are the electron-cyclotron and for Landauer’s experiment. They considered the radiation of the electron-plasma frequencies, respectively͒, the observed electron Bernstein modes by a single electron in a hot plasma frequency spectrum of the axial component of the magnetic and found enhanced emission at the upper-hybrid frequency, field is characterized by sharp resonances at the harmonics of but did not determine how these quasi-electrostatic waves 4 ⍀e . Resonances of this type have been observed previ- could convert to the electromagnetic radiation observed by ously, e.g., remotely by Landauer,5 but not in situ as in the Landauer. UCLA experiment. These local measurements show that no In addition to treating all frequency regimes equally, the plasma instability or nonlinearity is involved, but that the present study considers the density modulation to be arbi- fields are generated by the free-streaming electrons. That is, trary, and separates out those effects that are due to the ge- large fields are produced only when the condition ometry of the beam from those that are due to the details of ␻Ϫn⍀eϭkʈUʈ is satisfied, where Uʈ is the velocity of the the modulation. Also, while the general formulation allows beam parallel to the magnetic field and n is the azimuthal for either ion or electron beams, this study concentrates on mode number. electron beams and frequencies above ⍀e . It is found that Previous theoretical work on linear fields has focused on the resonances observed by Stenzel and Golubyatnikov4 are propagating modes and has not examined the evanescent due to the helical geometry of the beam, which forces varia- fields which characterize the parameter regime of the UCLA tions in z to be mapped into variations in ␪, and which experiment. For example, there has been long-standing inter- produces large fields only for the beam modes, est in the space-physics community in the problem of modu- ␻Ϫn⍀eϭkʈUʈ . In the frequency regime ⍀eϽ␻Ͻ␻ pe the lated electron beams acting as sources of whistler waves.3 plasma modes are evanescent in the perpendicular direction Lavergnat et al.6–9 studied Bernstein modes, plasma waves so that the fields are localized near the beam. Near the hybrid and electromagnetic whistler waves generated by beams in- frequencies, however, the plasma modes propagate and are electrostatic in character, in agreement with the findings of 13 a͒Present address: Beam Physics Branch, Plasma Physics Division, Naval Canobbio. Finally, for frequencies above ␻ pe , the plasma Research Laboratory, Washington, D.C. 20375. is essentially transparent near the electron-cyclotron harmon-

Phys. Plasmas 3 (12), December 1996 1070-664X/96/3(12)/4717/8/$10.00 © 1996 American Institute of Physics 4717 Downloaded¬28¬Sep¬2000¬to¬138.238.124.47.Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/pop/popcpyrts.html. nq ˆ jb͑x,t͒ϭ ͑Uʈzˆϩ⍀erc␪͒␦͑␳Ϫrc͒␦͑␪Ϫ⍀ez/Uʈ͒, ͑2͒ rc

where ⍀eϭqB0 /mec and me is the mass of the beam electrons. A relativistic correction may be included in ⍀e if the beam’s velocity is large enough; however, in this paper we restrict our study to beam velocities which are nonrelativis- tic. Alternatively, the perpendicular velocity, UЌϵ⍀erc , can be used to parametrize the beam. The Fourier transform

͓i.e., assuming a dependence of exp(ikʈzϪi␻tϩin␪)] of the FIG. 1. Geometry of the helical beam considered in the analysis. helical current in Eq. ͑2͒ is ˜ 2␲cqn Uʈ ⍀erc ˜j ͑␳,n,k ,␻͒ϭ zˆϩ ␪ˆ ics, and the beam radiates electromagnetic fields similar to b ʈ r ͫ c c ͬ c those in a vacuum. ␻Ϫn⍀e Section II formulates the general helical beam problem, ϫ␦ k Ϫ ␦͑␳Ϫr ͒, ͑3͒ ͩ ʈ U ͪ c and sketches the calculation of the fields produced by that ʈ beam. Section III examines in detail the physics of the high- where frequency regime (␻Ͼ⍀e) and catalogues the character of the fields, including the sharp resonances at the electron- ϱ ñ ϭ d␶ ei␻␶n͑␶͒ ͑4͒ cyclotron harmonics in the frequency spectrum. The analyti- ͵Ϫϱ cal predictions are compared with experimental observations is the spectrum of the modulation at zϭ0 ͑the ‘‘injection’’ in Sec. IV, and Sec. V is the conclusion. point͒. The formalism can be extended to the other two types of II. CALCULATION OF FIELDS beams mentioned previously: single particles and annular The source of the fluctuations is modeled as a density- beams. The only modifications appear in the expressions for modulated, charged-particle beam of infinite extent in the z n and ñ . For a single particle, the density becomes a direction, immersed in a uniform plasma with a background ␦-function and its transform is a constant ˆ magnetic field B0ϭB0z. The role of this beam is that of a 1 current source; no plasma feedback is included in this for- nϭ␦͑zϪUʈt͒, ñϭ . ͑5͒ mulation, and hence no instabilities. Effects due to the injec- Uʈ tion point and beam front are not crucial to the form of the For an annular beam, the beam density must be integrated steady-state fields, and hence are not included. However, the over the angle ␪Ј at which the beam crosses the zϭ0 plane. beam is defined by its properties at the zϭ0 plane, as if it That is, n is a function of both ␶ and ␪Ј, and ñ is a function were injected at that point. To facilitate future study of dif- of both frequency ␻ and azimuthal mode number n ferent types of beams within one theoretical description, we ϱ 2␲ consider beams whose current density jb(x,t) is proportional ñ͑␻,n͒ϭ d␶ ei␻␶ d␪Ј ein␪Јn͑␶,␪Ј͒. ͑6͒ to a radial ␦-function ͵Ϫϱ ͵0

jb͑x,t͒ϰ␦͑␳Ϫrc͒, ͑1͒ The Fourier transforms of the current density of all three types of beams have the same structure due to the fact that where ␳ is the usual cylindrical radial coordinate and r is c they are made up of free-streaming particles. The particle the cyclotron radius of the beam particles. This formulation nature of the source current is manifested in Eq. ͑3͒, which is able to describe single particles, helical beams, and annu- ˜ lar beams. All three types have the property that the particles holds for all three types; only the correct expression for n must be inserted ͓Eq. ͑4͒, ͑5͒ or ͑6͔͒. remain at a fixed radius ␳ϭrc . The experimental realization of this model consists of a The calculation of the linear electric and magnetic fields beam injected continuously into the plasma at a specific lo- due to a current source with cylindrical symmetry in a cold plasma is similar to the derivation of the fields in a uniform cation, and at a given angle to the ambient magnetic field, as 14 15 sketched in Fig. 1. This beam spirals along the magnetic field waveguide, but must be modified for a tensor dielectric. and, because the modulation affects only the density, it re- Because the technique is well-known and straightforward, tains its helical shape; there is no velocity spread. Let n be only a sketch is given. the number of particles per-unit-length in the z direction at The plasma is represented by the cold dielectric tensor time t and position z; because the particles simply free ⑀Ќ Ϫi⑀H 0 stream in the z direction, n is a function only of ␧ϭ i⑀H ⑀Ќ 0 , ͑7͒ ␶ϵtϪz/Uʈ . The total current passing through the zϭ0 ͫ 00⑀ͬ plane ͑i.e., ‘‘injected’’͒ per-unit-time is given by qn(t)Uʈ , ʈ where q is the single-particle charge. In the cylindrical coor- where the elements of ␧ are functions of ␻ only. The sepa- dinates of configuration space (␳,␪,z) the beam current den- ration of the fields into parallel and perpendicular compo- sity has only ␪ˆ and zˆ components nents ͑with respect to zˆ, the direction of the static magnetic

4718 Phys. Plasmas, Vol. 3, No. 12, December 1996 M. A. Reynolds and G. J. Morales Downloaded¬28¬Sep¬2000¬to¬138.238.124.47.Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/pop/popcpyrts.html. field͒ decouples Maxwell’s equations, resulting in differen- ͑1͒ 2i tial equations for the parallel field components, E and B . Jn͑kϮrc͒Hn ͑kϮ␳͒ In͑␬Ϯrc͒Kn͑␬Ϯ␳͒, ͑12b͒ z z → ␲ The solutions away from the beam are written in terms of 2 2 Bessel functions, and then integrated over the source to ob- where ␬ϮϭϪkϮ. When the character of the fields is prima- tain the overall magnitude of the fields. The perpendicular rily evanescent, the modified Bessel functions provide a field components can then be determined in terms of the more convenient form for both interpretation and numerical parallel components if desired. This method works when the evaluation. medium is uniform and can be described by a scalar The perpendicular components of the electric and mag- dielectric.14 It is also applicable to tensor media when, as in netic fields are linear combinations of the parallel compo- the present case, the dielectric tensor is both Hermitean and nents, and can be determined by applying the operator zˆ؋ to block diagonal.15 Ampe`re’s law and Faraday’s law to obtain The complete solution for the parallel electric and mag- ˆ ЌBzϪikʈBЌϩik0z؋␧•EЌϭ0, ͑13a͒“ netic fields is ˆ ЌEzϪikʈEЌϪik0z؋BЌϭ0, ͑13b͒“ qn˜ -z. Equations ͑13͒ can be solved algeץ/ץî␻z/Uʈ in͑␪Ϫ⍀ez/Uʈ͒ where ϭ Ϫz Ez͑r,␻͒ϭ e ͚ e “Ќ “ rc n braically for EЌ and BЌ in kʈ space, and the resulting expressions inverse-transformed if configuration-space dependence h J ͑k ␳͒ ␳Ͻr Ϯ n Ϯ c is desired. ϫ͚ ͑1͒ , ͑8a͒ Ϯ ͭjϮHn ͑kϮ␳͒ ␳Ͼrcͮ III. GENERAL FIELD BEHAVIOR qn˜ i␻z/U in͑␪Ϫ⍀ez/U ͒ Bz͑r,␻͒ϭ e ʈ͚ e ʈ Equations ͑8͒ are now evaluated for an electron beam rc n and for frequencies above ⍀e . For weak magnetic fields hϮJn͑kϮ␳͒ ␳Ͻrc (⍀eϽ␻ pe) this range of frequencies consists of three re- ϫ ␩Ϯ ͑1͒ , ͑8b͒ gimes: the ‘‘evanescent’’ regime (⍀eϽ␻Ͻ␻ pe) where the ͚Ϯ j H ͑k ␳͒ ␳Ͼr ͭ Ϯ n Ϯ cͮ fields are radially localized, the ‘‘upper-hybrid’’ regime where (␻ peϽ␻Ͻ␻uh) where the beam excites propagating upper- hybrid waves, and the ‘‘vacuum’’ regime (␻uhϽ␻) where ͑1͒Ј hϮ Ϫ␲i UЌ Hn ͑kϮrc͒ the beam excites propagating electromagnetic modes near ϭ k r ⑀ Ϫ Ϯ c Ќ c the harmonics of ⍀e . When the magnetic field is strong jϮ ⑀Ќ͑␩Ϯ ␩ϯ͒ ͫ JЈ͑kϮrc͒ ͭ ͮ ͭ n ͮ (␻ peϽ⍀e) only the last two of these regimes exist above ⍀e . 2 Uʈ Ϫ k0rc͓Nʈ⑀HϪi␩ϯ͑⑀ЌϪNʈ ͔͒ Before evaluating the fields numerically, it is helpful to ͩ c investigate the physics qualitatively by examining the struc-

͑1͒ ture of Eqs. ͑8͒. All of the field components, for example UЌ Hn ͑kϮrc͒ Bz , may be written in the dimensionless form ϩn͓⑀Hϩi␩ϯNʈ͔ . ͑9͒ c ͪͭ Jn͑kϮrc͒ ͮ ͬ Bz iz͑␻Ϫn⍀e͒/Uʈ in␪ (1) Bz͑r,␻͒ϵ ϭ͚ bzn͑␳,␻͒e e , Here, Jn and Hn are Bessel functions and Hankel functions ͑qn˜/rc͒ n of the first kind, respectively, the prime denotes differentia- ͑14͒ tion with respect to their argument, Nʈϭ(␻Ϫn⍀e)/k0Uʈ is where bzn is the strength of the nth term. An investigation of the scaled parallel wave number, k0ϭ␻/c, and Bz ͑rather than Bz) has the advantage of separating the ef- 2 2 2 fects of the density modulation, ñ , from those inherent to the ⑀Ќ͑kϮ/k0͒Ϫ⑀ʈ͑⑀ЌϪNʈ ͒ ␩ ϭ Ϯ i⑀ N beam. The dependence of the nth term on ␪ and z is oscil- H ʈ latory i⑀HNʈ⑀ʈ ϭ . ͑10͒ ␻Ϫn⍀e ⑀ ͑⑀ ϪN2͒Ϫ⑀2 Ϫ⑀ ͑k2 /k2͒ exp͑in␪͒exp iz . ͑15͒ Ќ Ќ ʈ H Ќ Ϯ 0 ͫ ͩ U ͪͬ ʈ The quantity kϮ is the perpendicular wave number, and is This structure arises as a consequence of the beam geometry, given by the two solutions of the dispersion relation which demands that if the azimuthal dependence is harmonic, exp(in␪), then the axial dependence is a phase oscil- ͑⑀ ϪN2 ͓͒͑⑀ ϪN2͒͑⑀ ϪN2͒Ϫ⑀2 ͔ϪN2 N2͑⑀ ϪN2͒ϭ0, ʈ Ќ Ќ Ќ ʈ H Ќ ʈ Ќ lation due to the free streaming, i.e., only those waves which ͑11͒ satisfy the Doppler-shifted cyclotron condition where N2ϭN2 ϩN2 and N2 ϭk2 /k2 . The fields may alter- Ќ ʈ Ќ Ϯ 0 ␻Ϫn⍀ ϭk U , ͑16͒ natively be expressed in terms of the modified Bessel func- e ʈ ʈ tions In and Kn by making the replacements are excited. The strength of the nth term is governed by bzn(␳,␻), whose evaluation requires a detailed knowledge ͑1͒ 2i of the dispersion relation because it depends sensitively on Hn ͑kϮrc͒Jn͑kϮ␳͒ Kn͑␬Ϯrc͒In͑␬Ϯ␳͒, ͑12a͒ → ␲ kϮ .

Phys. Plasmas, Vol. 3, No. 12, December 1996 M. A. Reynolds and G. J. Morales 4719 Downloaded¬28¬Sep¬2000¬to¬138.238.124.47.Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/pop/popcpyrts.html. The parallel index of refraction Nʈ is a measure of the frequency separation from the nth harmonic

Nʈ ϰ ␻Ϫn⍀e . ͑17͒

Two limits are useful: NʈϷ0 ͑near the nth harmonic͒ and Nʈ ϱ͑far from the nth harmonic͒. In the small Nʈ limit, the two→ solutions for the perpendicular wave number are approximated by 2 Ϫ 2 2 ⑀Ќ ⑀H NЌϷ⑀ʈ , , ͑18͒ ⑀Ќ which are the well-known dispersion relations for the ordinary and extraordinary modes, respectively. For large Nʈ , 2 NЌ approaches FIG. 2. Frequency spectrum of the scaled magnetic ﬁeld B . The spatial ⑀ z 2 2 ʈ 2 location is ␳/rcϭ0.7, ␪ϭzϭ0, the beam parameters are ␪ pϭ88°, NЌ ϪNʈ , Ϫ Nʈ , ͑19͒ 2 2 4 → ⑀Ќ ␤ϭ0.01, and ␻ pe/⍀eϵ␣ϭ10 . The choice of zϭ0 is a convenience, and is made in all of the ﬁgures. The dashed line is Eq. ͑23͒ for nϭ4 and the which are the dispersion relations for the electromagnetic dotted line is Eq. ͑26͒. and electrostatic modes, respectively. In Eq. ͑19͒, Nʈ has been taken to be larger than all elements of the dielectric, which means the frequency does not correspond to a hybrid resonance. UЌ rc b exp͑Ϫk ͉␳Ϫr ͉͒, ͑23͒ There are three important dimensionless parameters: the zn→ c ͱ␳ ʈ c pitch angle where the exponential behavior stems from the large argu- Ϫ1 ␪ pϭtan ͑UЌ /Uʈ͒, ͑20͒ ment expansion of the radial Bessel functions. The approximation in Eq. ͑23͒ is shown as a dashed line in Fig. 2 for the the scaled velocity nϭ4 term. To determine the frequency scale, the argument 2 2 of the exponential can be written in the form ͱUʈ ϩUЌ ␤ϭ , ͑21͒ c ͉␻Ϫn⍀e͉ k ͉␳Ϫr ͉ϭ , ͑24͒ and the effective magnetization ʈ c ⌬

2 ␻ pe where ␣ϭ . ͑22͒ ⍀2 e ⌬ ͑Uʈ /UЌ͒ cot ␪ p ϵ ϭ , ͑25͒ ⍀ 1Ϫ␳/r 1Ϫ␳/r A. Evanescent regime: ⍀e<␻<␻pe e ͉ c͉ ͉ c͉ 2 In this frequency regime, the real part of NЌ is negative is defined as the width of the resonance, corresponding to for all values of Nʈ , hence the fields are radially evanescent one e-folding. For the values of the parameters in Fig. 2, Eq. and are localized near the beam. We will show that each term ͑25͒ predicts a half-width of ⌬/⍀eϭ0.12, in good agreement in Eq. ͑14͒ decays exponentially as the frequency is varied with Fig. 2. Sharp resonances, therefore, are obtained at ra- away from cyclotron harmonics. Figure 2 shows a typical dial locations far from the beam (␳/r ϱ or ␳/r Ϸ0), and c→ c spectrum, Bz(␻), for a beam with a pitch angle of ␪ pϭ88° for large pitch angles. For large pitch angles, Eq. ͑25͒ im- and a scaled velocity of ␤ϭ0.01, immersed in a plasma of plies that the resonance width depends sensitively on ␪ p . ␣ϭ104. This spectrum is taken at the spatial location of This dependence is depicted in Fig. 3 which shows the fre- ␳/rcϭ0.7 and ␪ϭzϭ0. It is convenient to choose zϭ0 be- quency spectrum for three values of the pitch angle, cause, except for the phase factor due to the free-streaming ␪ pϭ87°, 88°, and 89°, and at the same spatial location as ͓see Eq. ͑15͔͒, the field is periodic in z with a period of Fig. 2 (␳/rcϭ0.7 and ␪ϭzϭ0). The determination of the resonance width as a function of is complicated by the fact 2␲Uʈ /⍀e . All figures in this paper will therefore evaluate ␳ the fields at zϭ0. Peaks occur at the cyclotron harmonics, that the magnitude of the field is also a function of radial with a maximum amplitude that decreases with increasing position, as is investigated next. harmonic number. To understand this spectrum, each term in When ␻ is near a harmonic, NʈϷ0, and Eq. ͑18͒ can be Eq. ͑14͒ needs to be considered in more detail. used to approximate the perpendicular wave number. If, in Near the nth harmonic, the nth term in the sum is larger addition, ␳ is not too large, the Bessel functions may be than the other terms and has the shape of a resonance, with expanded for small argument to approximate the magnitude exponential wings. To illustrate this feature, consider ␻ to be of the field at the peak of the nth harmonic different from n⍀ . In this case, N is large, the asymptotic e ʈ U 1 ͑⑀ ϩ⑀ ͒͑␳/r ͒n ␳Ͻr solutions in Eq. ͑19͒ may be used, and the strength of the Ќ H Ќ c c bznϷϪ n . ͑26͒ nth term approaches the value c ⑀Ќ ͭ͑⑀HϪ⑀Ќ͒͑rc /␳͒ ␳Ͼrcͮ

4720 Phys. Plasmas, Vol. 3, No. 12, December 1996 M. A. Reynolds and G. J. Morales Downloaded¬28¬Sep¬2000¬to¬138.238.124.47.Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/pop/popcpyrts.html. ͑1͒ Each term in the sum has axial and azimuthal dependence given by Eq. ͑15͒, and a radial dependence given by Eq. ͑26͒ at the harmonic, and by Eq. ͑23͒ far from the harmonic. ͑2͒ For large pitch angles and far from the radial location of the beam, the frequency spectrum consists of sharp resonances at the cyclotron harmonics. In this parameter regime, these resonances are well-approximated by one term in the infinite sum — the other terms are exponentially small. Hence, the field has the same spatial dependence as the single, dominant, term. ͑3͒ Close to the radial location of the beam, ␳Ϸrc , and for small pitch angles, many terms contribute significantly to the magnitude of the field. In this case, the field magnitude at a given frequency is a superposition of those FIG. 3. Frequency spectrum of the scaled magnetic field B for three dif- z terms that are ‘‘closest,’’ in the sense of Eq. ͑17͒. ferent values of the pitch angle ␪ pϭ87° ͑dashed line͒, 88° ͑solid line͒, and 89° ͑dotted line͒. The spatial location is ␳/rcϭ0.7, ␪ϭzϭ0, the beam ve- These properties are due to the Bessel functions which ap- locity is ␤ϭ0.01, and ␣ϭ104. pear in the expression for the field, and hence are a consequence of the helical geometry. Because all components of This is due solely to the extraordinary mode, for the polar- E and B depend on this same product of Bessel functions, they all exhibit similar structure. ization of the ordinary mode is BzϭE␪ϭ0. This expression contains both the decrease of peak amplitude with harmonic number n for a given radial position, shown as the dotted B. Upper-hybrid regime: ␻pe<␻<␻uh line in Fig. 2, as well as the dependence of peak amplitude This frequency regime is characterized by electrostatic on ␳/r for a given harmonic number. The radial dependence c wave propagation. As long as ␻ and ␻ are far from the of B for ␻ϭ2⍀ and for three different values of the azi- pe uh z e harmonics of ⍀ , the value of N is large for all terms in the muthal angle, ␪ϭ0°, 90°, and 180°, is shown in Fig. 4. Far e ʈ sum, the approximation in Eq. ͑19͒ can be used, and because from ␳ϭr , the fields are well-approximated by a power law c the electrostatic mode has a real and positive N2 , the fields in ␳, as predicted by Eq. ͑26͒. Of course, as ␳ becomes large, Ќ propagate. Although the dispersion relation is well approxi- the small-argument expansion is no longer valid, and the mated by the electrostatic dispersion relation, fields become evanescent. Near the cyclotron radius, the k2 ⑀ ϩk2⑀ ϭ0, the fields have a significant magnetic com- width of the harmonic resonances increases, implying that Ќ Ќ ʈ ʈ ponent, hence we continue to use B as characteristic of the many terms contribute significantly to the sum, and the radial z field structure. The electromagnetic mode may be ignored dependence departs from the simple power law. This is the because it is evanescent with a short decay length mathematical manifestation of the fact that the induced LϵkϪ1Ӷr , where the inequality is due to the fact that plasma current near the beam is strong and has a complex Ќ c k ϳN . spatial structure. Ќ ʈ To illustrate the behavior in the upper-hybrid regime, we The following picture emerges: choose the value of ␣ϭ20.25, which implies ␻ peϭ4.5⍀e and ␻uhϷ4.61⍀e . This upper-hybrid regime corresponds to the frequency interval between the nϭ4 and nϭ5 harmonics

so that the large Nʈ approximation is appropriate for all terms in the infinite sum. Figure 5 shows the magnetic field due to the nϭ5 and nϭ6 terms at a radial position of ␳ϭ2rc , outside of the beam. The oscillations in the field are due to the Bessel functions J5(kϪrc) and J6(kϪrc), where kϪϷk0NʈͱϪ⑀ʈ /⑀Ќ.As␻approaches ␻uh , ⑀Ќ approaches zero, and the oscillations are more closely spaced. The field in this limit approaches the value

Uʈ rc ⑀ʈ 1 1 ͉b ͉ 2N2 cos k r Ϫ n␲Ϫ ␲ . zn ʈ c ͱ␳ͱϪ⑀ ͯ Ϫ c 2 4 ͯ → Ќ ͩ ͪ ͑27͒ Because these waves propagate away from the beam, it is useful to discuss the power radiated by the beam and its radiation resistance. The total power radiated per-unit-length FIG. 4. Radial dependence of the scaled magnetic ﬁeld B for three different z P may be found by integrating the Poynting vector S over a values of azimuthal angle ␪ϭ0° ͑solid line͒, 90° ͑dashed line͒, and 180° 4 cylinder of length L and radial position ӷr that is centered ͑dotted line͒. The parameters are ␻/⍀eϭ2, ␤ϭ0.01, ␣ϭ10 , and ␳ c ␪ pϭ88°. on the beam

Phys. Plasmas, Vol. 3, No. 12, December 1996 M. A. Reynolds and G. J. Morales 4721 Downloaded¬28¬Sep¬2000¬to¬138.238.124.47.Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/pop/popcpyrts.html. FIG. 5. Frequency dependence of bz5 ͑thin line͒ and bz6 ͑thick line͒ for the FIG. 6. Frequency spectrum of the scaled magnetic ﬁeld Bz in the vacuum parameter values of ␣ϭ20.25, ␤ϭ0.01, ␪ pϭ88°, and a spatial location regime. The spatial location is ␳/rcϭ2, ␪ϭzϭ0, and the parameters are ␳/rcϭ2, ␪ϭzϭ0. ␤ϭ0.01, ␪ pϭ88°, and ␣ϭ1. For comparison, the dashed line is the spectrum for ␣ϭ104, with all other parameters the same.

1 L 2␲ Pϭ dz d␪ ␳S . ͑28͒ L ͵ ͵ ␳ 0 0 ever, the beam radiates near the cyclotron harmonics, and the The contributions from the endcaps of the cylinder cancel. A radiation resistance per-unit-length is again a useful quantity. radiation resistance per-unit-length, which is a measure of Unlike the upper-hybrid regime, the regions in frequency the efﬁciency of the beam, may be deﬁned as space corresponding to propagation do not overlap for different n, so that the radiation resistance per-unit-length for 2 ␻ϭn⍀e ͑on the harmonics͒ becomes Rϵ 2 P, ͑29͒ ͉I͉ 2 2 2 ␲␻ Uʈ ⍀e/␻ 2 2 2 2 ˜ 2 Rϭ J ͑kϩr ͒ϩ where ͉I͉ ϭq Uʈ ͉ n ͉ . In this upper-hybrid frequency re- 2 n c 2 2 U ͫͩ c ͪ ⑀Ќ͑⑀ЌϪ⑀ ͒ gime, the resistance is approximately ʈ H 2N 1 1 1 2 ʈ 2 ϫ͕n⑀HJn͑kϪrc͒ϪkϪrc⑀ЌJnЈ͑kϪrc͖͒ , ͑32͒ RϷ͚ cos kϪrcϪ n␲Ϫ ␲ . ͑30͒ ͬ n cr ͩ 2 4 ͪ c ͱϪ⑀ʈ⑀Ќ where For the parameters in Fig. 5, the magnitude of each term is approximately UЌ k r ϭn ⑀ , ͑33a͒ ϩ c c ʈ 2N ohm ʈ Ϫ4 Ϸ1.2ϫ10 n⍀e , ͑31͒ 2 2 crc cm UЌ ⑀ЌϪ⑀H kϪrcϭn . ͑33b͒ Ϫ1 c ⑀Ќ if ⍀e is measured in s . Electrostatic waves are also radiated above the lower-hybrid frequency. The two terms in the square brackets represent the power radiated into the ordinary mode and extraordinary mode, re-

C. Vacuum regime: ␻uh<␻ spectively. As ␻ becomes large compared with ␻ pe , the radiated For frequencies larger than the upper-hybrid frequency, fields closely approximate those that would be radiated by a N2 can be real and positive for small values of N2 , i.e., the Ќ ʈ beam in vacuum. That is, ␧ approaches l in this high- ordinary mode propagates for ␻Ͼ␻ pe , while the extraordi- frequency limit, and the arguments of the Bessel functions nary mode propagates for ␻Ͼ␻uh . This implies that for fre- approach nUЌ /c.IfUЌ/cӶ1 as well, i.e., the beam is non- quencies very near the nth harmonic, the nth term describes relativistic, the radiation resistance takes a simple form a propagating electromagnetic field. The contribution of 2 1 U 2n other terms is exponentially small because for them Nʈ is Ќ n 2 4 large. ␲n⍀ ͩ 2 c ͪ U UЌ e ʈ RϷ ϩn4 , ͑34͒ To illustrate this behavior, we choose ␣ϭ1 which places U2 ͑n!͒2 ͫͩ c ͪ ͩ c ͪ ͬ all harmonics above the upper-hybrid frequency, so that both ʈ which for large n becomes modes will propagate for small Nʈ . A typical near-field spectrum is shown in Fig. 6, for ␳/r ϭ2, ␤ϭ0.01, and 2n c ␲n⍀e e UЌ ␪ ϭ88°. Also shown is the spectrum for ␣ϭ104 ͑evanes- R͑n ϱ͒ , ͑35͒ p → → U2 ͩ 2 c ͪ cent regime͒ for comparison. As in the evanescent regime, ʈ sharp resonances occur because the parameter ⌬/⍀e is small, so that the resistance decreases with harmonic number as and the strength of each term decays exponentially. How- ␤2n, as expected.

4722 Phys. Plasmas, Vol. 3, No. 12, December 1996 M. A. Reynolds and G. J. Morales Downloaded¬28¬Sep¬2000¬to¬138.238.124.47.Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/pop/popcpyrts.html. IV. COMPARISON WITH EXPERIMENT

In the experiment performed at UCLA by Stenzel and Golubyatnikov,4 the parallel component of the fluctuating magnetic field is measured using a B˙ probe, which consists of a circular wire coil in which the induced emf driven by the changing magnetic flux through the loop is detected. The B˙ probe is 2.3 cm in radius, with a thickness of 2.6 mm. This radius is approximately equal to the cyclotron radius of the beam electrons, the exact ratio depending on the beam’s perpendicular velocity. The background plasma in which the beam is immersed is weakly magnetized, ⍀eӶ␻ pe , and its temperature is low enough to warrant the approximation of the dielectric tensor by that of a cold plasma. There are three experimental facts which, when con- FIG. 7. Radial dependence of the scaled magnetic field Bz ͑theoretical pre- trasted with the theoretical idealization presented here, make diction͒ for the first two harmonics ␻/⍀eϭ1.01 ͑solid line͒ and ␻/⍀eϭ2 the quantitative comparison between experiment and theory ͑dashed line͒. The parameters are ␪ pϭ88°, ␤ϭ0.01, zϭ0, and difficult. First, and most important, is the finite size of the ␪ϭ0°, 180°. Compare with Fig. 12 of Ref. 4. B˙ probe. The experimental technique cannot detect variations in the magnetic field on a scale smaller than the size of shown in Fig. 7 is seen in the experiment. However, the the probe. For typical beam energies of 30 V–100 V, the measured field decreases more slowly, a feature expected corresponding cyclotron radii are rcϷ1.7 cm–3 cm. This from the finite size of the probe. At the cyclotron frequency means that the probe is not much smaller than the beam, and (␻ϭ⍀e), however, there is a significant discrepancy. While is sometimes larger. The present theory, however, deter- the experimental measurement does show a sharp drop in mines the field locally. Because of the mathematical form in magnitude for ␳уrc , it remains large for ␳Ͻrc and does not which we have expressed the fields ͑an infinite sum of Bessel show a minimum at ␳ϭ0. The present theory does not pre- functions͒, it is difficult to integrate the field over the area of dict this type of behavior, most likely because it does not the probe, except when the probe is located either completely include any dissipation mechanisms which might be present inside the beam (␳Ͻrc) or completely outside (␳Ͼrc). Sec- in the experiment. ond, while the theory singles out a specific pitch angle and The frequency dependence of the magnetic fluctuations beam velocity, the experimental beam has a finite spread in is perhaps the most striking result of the experiment, and is these quantities produced by the broadband noise inherent in explained by the present theory. The theoretical spectrum in the injection process. Thus, there is no well-defined beam Fig. 8, where the parameters were chosen to approximate the axis. Third, the beam is injected from a specific location and experimental conditions, agrees qualitatively with the experi- does not extend infinitely in z. This fact leads to the experimental spectrum seen in Fig. 6 of Ref. 4. A quantitative mental observation that when the probe is near the injection comparison is difficult to make, for the reasons mentioned point, the field spectrum is broadband, corresponding to the earlier. Due to the dependence of the resonance width on noise of the injection. However, when the probe is far from pitch angle and radial location, and the experimental spread the injection point, those modes that are not coherent are in these quantities, the resonances of the observed spectra filtered out, and spectra more closely resembling those in Fig. 2 are obtained. This implies that the theory can only be used to explain measurements taken far from the disturbing influence of the injector. The simple axial and azimuthal dependence predicted by Eq. ͑15͒ is observed using interferometric techniques. For large pitch angles (␪ pϷ88°) and frequencies near ⍀e and its first harmonic (␻/⍀eϭ1,2), the observed ␪ dependence is clearly ein␪ and the measured z dependence is uniform, i.e., kʈϭ0. The phase change in z is also measured for interme- diate frequencies (⍀eр␻р2⍀e) and kʈ is found to be con- sistent with Eq. ͑16͒, where n is the ‘‘closest’’ harmonic, i.e., due to that term which has the largest magnitude. The present theory agrees well with these observations. The observed radial dependence at the first harmonic (␻/⍀eϭ2), depicted in Fig. 12 of Ref. 4, also is in agreement with the present theory as shown in Fig. 7. This theoretical plot is a cut along x for yϭ0. That is, negative values FIG. 8. Frequency spectrum of the scaled magnetic field B . The spatial of x correspond to ϭ180° while positive values signify z ␪ location is ␳/rcϭ0.7, ␪ϭzϭ0, and the parameters are ␪ pϭ88°, ␪ϭ0°. The minimum at xϭ0 and the decay for ͉x͉Ͼrc ␤ϭ0.0198, and ␣ϭ8711. Compare with Fig. 6 of Ref. 4.

Phys. Plasmas, Vol. 3, No. 12, December 1996 M. A. Reynolds and G. J. Morales 4723 Downloaded¬28¬Sep¬2000¬to¬138.238.124.47.Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/pop/popcpyrts.html. electron-cyclotron harmonics. This behavior is well described by the excitation of beam modes which satisfy the

condition ␻Ϫn⍀eϭkʈUʈ . The width and magnitude of these resonant peaks can be explained by the form of the fields both near the harmonics and far from the harmonics. It is found that the peaks are functions of the pitch angle and the spatial location. The axial and azimuthal dependence of the fields is also described by the model developed in this study. A discrepancy arises between the experimental results and the theoretical prediction for the radial dependence of the field at ⍀e . The finite size of the measuring instrument, a magnetic loop which integrates the flux, as well as the idealization of the theory, which includes neither a velocity modulation nor a spread in pitch angle, are the probable causes for the discrepancy. FIG. 9. Dependence of the frequency spectrum of the scaled magnetic field In addition, the present theoretical formulation is general Bz on the pitch angle ␪ p of the beam. The parameters are ␤ϭ0.0198, 4 ␣ϭ10 , and the spatial location is ␳/rcϭ0.7, ␪ϭzϭ0. Compare with Fig. 8 enough to describe fluctuations generated in other frequency of Ref. 4. regimes as well as those generated by ion beams. This gen- erality allows the calculation of quantities such as the radiated power in a straightforward manner, without resorting to have a minimum width, regardless of the average parameters approximations. Two of these other frequency regimes, near of the beam. A clear dependence of resonance width on pitch and above the upper-hybrid frequency ␻ , have been con- angle is indeed observed in the experiments ͑see Fig. 8 of uh sidered. Near ␻ the beam radiates a broad spectrum of Ref. 4 . The sensitivity of the experimental resonance width uh ͒ electrostatic waves which propagate away from the beam, on , however, is not as strong as predicted by the present ␪ p and propagating electromagnetic modes are emitted near model. Figure 9 shows that the resonances disappear when those harmonics of ⍀e that are above ␻uh . ␪ pϷ80°, while experimentally they are seen for pitch angles as small as 70°. Because experimentally, the resonance width depends strongly on the distance between the probe ACKNOWLEDGMENTS and the injection point, and, since this is not addressed in the We thank Professor R. Stenzel for illuminating discus- present theory, it is difficult to offer an explanation for the sions concerning his experimental results. discrepancy. However, it is worth noting that while a general This work is sponsored by the Office of Naval Research. tendency is for the non-ideal experimental realities to destroy theoretically sharp features, in this case the observed behav- 1N. J. Lopes Cardozo, F. C. Schuller, C. J. Barth, C. C. Chu, F. J. Pijper, J. ior is contrary to this trend. Lok, and A. A. M. Oomens, Phys. Rev. Lett. 73, 256 ͑1994͒. The linear beam modes excited by spiraling electrons, as 2J. E. Maggs, in Artificial Particle Beams in Space Plasma Studies, edited calculated by the present theory, agree qualitatively with the by B. Grandal ͑Plenum, New York, 1982͒, p. 261. 3T. Neubert and P. M. Banks, Planet. Space Sci. 40, 153 ͑1992͒. experimental results. Some of the discrepancies can be un- 4R. L. Stenzel and G. Golubyatnikov, Phys. Fluids B 5, 3789 ͑1993͒. derstood in light of the method of measurement, i.e., a spatial 5G. Landauer, J. Nucl. Energy C ͑Plasma Phys.͒ 4, 395 ͑1962͒. integration of B rather than a localized measurement, and by 6J. Lavergnat and R. Pellat, J. Geophys. Res. 84, 7223 ͑1979͒. z 7 taking into account the experimental limitations and the J. Lavergnat and R. Pellat, in Ref. 2, p. 535. 8J. Lavergnat and T. Lehner, IEEE Trans. Antennas Propag. AP-32, 177 theoretical idealizations. However, the two points of dis- ͑1984͒. 9 agreement are puzzling: the radial structure for ␻ϭ⍀e , and J. Lavergnat, T. Lehner, and G. Matthieussent, Phys. Fluids 27, 1632 the resonance width for small pitch angles. ͑1984͒. 10K. J. Harker and P. M. Banks, Planet. Space Sci. 33, 953 ͑1985͒. 11K. J. Harker and P. M. Banks, Planet. Space Sci. 35,11͑1987͒. V. CONCLUSION 12E. Canobbio and R. Croci, Phys. Fluids 9, 549 ͑1966͒; R. Croci and E. Canobbio, in Proceedings of the Seventh International Conference on The electromagnetic fluctuation spectrum generated by a Phenomena in Ionized Gases, Beograd, 1965, edited by B. Perovic and D. modulated, free-streaming, helical electron beam has been Tosic ͑Gradevinska Knjiga, Beograd, 1966͒, Vol. II, p. 496. calculated and compared to a recent laboratory experiment.4 13E. Canobbio, Nucl. Fusion 1, 172 ͑1961͒. 14 For the frequency regime of the experiment J. D. Jackson, Classical Electrodynamics, 2nd ed. ͑Wiley, New York, 1975͒, Sec. 8.2. (⍀eϽ␻Ӷ␻ pe), the spectrum consists of localized, evanes- 15W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas cent fields which shows a series of sharp resonances at the ͑MIT Press, Cambridge, MA, 1963͒, Chap. 9.

4724 Phys. Plasmas, Vol. 3, No. 12, December 1996 M. A. Reynolds and G. J. Morales Downloaded¬28¬Sep¬2000¬to¬138.238.124.47.Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/pop/popcpyrts.html.