High-Frequency Fluctuations of a Modulated, Helical Electron Beam

Transcription

1 Publiatios 1996 High-Frequey Flutuatios of a Modulated, Helial Eletro Beam Mark Athoy Reyolds Embry-Riddle Aeroautial Uiversity, reyoldb@erau.edu Follow this ad additioal works at: Part of the Plasma ad Beam Physis Commos Sholarly Commos Citatio Reyolds, M. A. (1996). High-Frequey Flutuatios of a Modulated, Helial Eletro Beam. Physis of Plasmas, 3(). Retrieved from This Artile is brought to you for free ad ope aess by Sholarly Commos. It has bee aepted for ilusio i Publiatios by a authorized admiistrator of Sholarly Commos. For more iformatio, please otat ommos@erau.edu.

2 High-frequey flutuatios of a modulated, helial eletro beam M. A. Reyolds a) ad G. J. Morales Departmet of Physis ad Astroomy, Uiversity of Califoria, Los Ageles, Califoria Reeived 17 Jue 1996; aepted 7 August 1996 The high-frequey eletromageti field geerated by a desity-modulated, helial eletro beam propagatig i a magetized plasma is alulated. The mageti flutuatios are foud to exhibit spatially loalized evaeset resoaes at harmois of the eletro-ylotro frequey, whose width is determied by the pith agle of the beam, ad whose existee is a osequee of the helial geometry. I additio, eletrostati modes are radiated ear the hybrid frequeies, ad eletromageti modes are radiated above the upper-hybrid frequey. The predited frequey spetrum ad mode struture i ofiguratio spae are i good agreemet with experimetal observatios of disrete emissio lies at the eletro-ylotro harmois Phys. Fluids B 5, Ameria Istitute of Physis. S X I. INTRODUCTION jeted ito the ioosphere. They iluded realisti beam effets, suh as a beam frot ad the fat that the beam has a fiite thikess o the order of the Brilloui radius. Harker ad Baks 10,11 alulated the power radiated by a helial eletro beam through the far field, ad also the expliit form of the ear field. However, they restrited their study to the whistler ad lower-hybrid frequey regimes, ad also to a square-wave desity modulatio. This type of modulatio allows ertai itegrals to be evaluated aalytially, but hiders the uderstadig of the itrisi frequey spetrum of the beam whih must be multiplied by the spetrum of the modulatio to obtai the radiated spetrum. Aother motivatio for the study of loalized beams is the explaatio of laboratory experimets like those of Ladauer 5 where may harmois of e are observed i the frequey spetrum of the fields measured outside of the plasma hamber. Caobbio ad Croi 1 were amog the first to suggest sigle-partile radiatio as a possible explaatio for Ladauer s experimet. They osidered the radiatio of eletro Berstei modes by a sigle eletro i a hot plasma ad foud ehaed emissio at the upper-hybrid frequey, but did ot determie how these quasi-eletrostati waves ould overt to the eletromageti radiatio observed by Ladauer. I additio to treatig all frequey regimes equally, the preset study osiders the desity modulatio to be arbitrary, ad separates out those effets that are due to the geometry of the beam from those that are due to the details of the modulatio. Also, while the geeral formulatio allows for either io or eletro beams, this study oetrates o eletro beams ad frequeies above e. It is foud that the resoaes observed by Stezel ad Golubyatikov 4 are due to the helial geometry of the beam, whih fores variatios i z to be mapped ito variatios i, ad whih produes large fields oly for the beam modes, e k U. I the frequey regime e pe the plasma modes are evaeset i the perpediular diretio so that the fields are loalized ear the beam. Near the hybrid frequeies, however, the plasma modes propagate ad are eletrostati i harater, i agreemet with the fidigs of Caobbio. 13 Fially, for frequeies above pe, the plasma is essetially trasparet ear the eletro-ylotro harmoa Preset address: Beam Physis Brah, Plasma Physis Divisio, Naval Researh Laboratory, Washigto, D.C There are several areas of otemporary plasma physis i whih radiatio from loalized harge distributios plays a importat role. Curret ad temperature filamets i tokamaks, 1 auroral eletro beams i the ioosphere, ad artifiial partile beams ijeted ito spae 3 are some of these areas. This aalytial study is motivated by a laboratory experimet performed by Stezel ad Golubyatikov 4 at the Uiversity of Califoria, Los Ageles UCLA, i whih the properties of the flutuatig mageti field ear a helial eletro beam i a high-desity magetized plasma were measured. This is a fudametal experimet whih explores the soure of flutuatios ear the harmois of the eletroylotro frequey, ad a theoretial uderstadig is eessary to lay the foudatio for more ivolved studies ad appliatios. Whe the plasma is weakly magetized ( e pe, where e ad pe are the eletro-ylotro ad the eletro-plasma frequeies, respetively, the observed frequey spetrum of the axial ompoet of the mageti field is haraterized by sharp resoaes at the harmois of e. 4 Resoaes of this type have bee observed previously, e.g., remotely by Ladauer, 5 but ot i situ as i the UCLA experimet. These loal measuremets show that o plasma istability or oliearity is ivolved, but that the fields are geerated by the free-streamig eletros. That is, large fields are produed oly whe the oditio e k U is satisfied, where U is the veloity of the beam parallel to the mageti field ad is the azimuthal mode umber. Previous theoretial work o liear fields has foused o propagatig modes ad has ot examied the evaeset fields whih haraterize the parameter regime of the UCLA experimet. For example, there has bee log-stadig iterest i the spae-physis ommuity i the problem of modulated eletro beams atig as soures of whistler waves. 3 Lavergat et al. 6 9 studied Berstei modes, plasma waves ad eletromageti whistler waves geerated by beams i- Phys. Plasmas 3 (1), Deember X/96/3(1)/4717/8/$ Ameria Istitute of Physis 4717 Dowloaded 8 Sep 000 to Redistributio subjet to AIP opyright, see

3 FIG. 1. Geometry of the helial beam osidered i the aalysis. is, ad the beam radiates eletromageti fields similar to those i a vauum. Setio II formulates the geeral helial beam problem, ad skethes the alulatio of the fields produed by that beam. Setio III examies i detail the physis of the highfrequey regime ( e ) ad atalogues the harater of the fields, iludig the sharp resoaes at the eletroylotro harmois i the frequey spetrum. The aalytial preditios are ompared with experimetal observatios i Se. IV, ad Se. V is the olusio. II. CALCULATION OF FIELDS The soure of the flutuatios is modeled as a desitymodulated, harged-partile beam of ifiite extet i the z diretio, immersed i a uiform plasma with a bakgroud mageti field B 0 B 0 ẑ. The role of this beam is that of a urret soure; o plasma feedbak is iluded i this formulatio, ad hee o istabilities. Effets due to the ijetio poit ad beam frot are ot ruial to the form of the steady-state fields, ad hee are ot iluded. However, the beam is defied by its properties at the z0 plae, as if it were ijeted at that poit. To failitate future study of differet types of beams withi oe theoretial desriptio, we osider beams whose urret desity j b (x,t) is proportioal to a radial -futio j b x,tr, 1 where is the usual ylidrial radial oordiate ad r is the ylotro radius of the beam partiles. This formulatio is able to desribe sigle partiles, helial beams, ad aular beams. All three types have the property that the partiles remai at a fixed radius r. The experimetal realizatio of this model osists of a beam ijeted otiuously ito the plasma at a speifi loatio, ad at a give agle to the ambiet mageti field, as skethed i Fig. 1. This beam spirals alog the mageti field ad, beause the modulatio affets oly the desity, it retais its helial shape; there is o veloity spread. Let be the umber of partiles per-uit-legth i the z diretio at time t ad positio z; beause the partiles simply free stream i the z diretio, is a futio oly of tz/u. The total urret passig through the z0 plae i.e., ijeted per-uit-time is give by q(t)u, where q is the sigle-partile harge. I the ylidrial oordiates of ofiguratio spae (,,z) the beam urret desity has oly ˆ ad ẑ ompoets j b x,t q U r ẑ e r ˆr e z/u, where e qb 0 /m e ad m e is the mass of the beam eletros. A relativisti orretio may be iluded i e if the beam s veloity is large eough; however, i this paper we restrit our study to beam veloities whih are orelativisti. Alteratively, the perpediular veloity, U e r, a be used to parametrize the beam. The Fourier trasform i.e., assumig a depedee of exp(ik ziti)] of the helial urret i Eq. is j b,,k, qñ r U ẑ er ˆ where ñ d e i k e U r, 3 is the spetrum of the modulatio at z0 the ijetio poit. The formalism a be exteded to the other two types of beams metioed previously: sigle partiles ad aular beams. The oly modifiatios appear i the expressios for ad ñ. For a sigle partile, the desity beomes a -futio ad its trasform is a ostat 4 zu t, ñ 1 U. 5 For a aular beam, the beam desity must be itegrated over the agle at whih the beam rosses the z0 plae. That is, is a futio of both ad, ad ñ is a futio of both frequey ad azimuthal mode umber ñ, d e i d e i,. 6 0 The Fourier trasforms of the urret desity of all three types of beams have the same struture due to the fat that they are made up of free-streamig partiles. The partile ature of the soure urret is maifested i Eq. 3, whih holds for all three types; oly the orret expressio for ñ must be iserted Eq. 4, 5 or 6. The alulatio of the liear eletri ad mageti fields due to a urret soure with ylidrial symmetry i a old plasma is similar to the derivatio of the fields i a uiform waveguide, 14 but must be modified for a tesor dieletri. 15 Beause the tehique is well-kow ad straightforward, oly a sketh is give. The plasma is represeted by the old dieletri tesor i H 0 i H , where the elemets of are futios of oly. The separatio of the fields ito parallel ad perpediular ompoets with respet to ẑ, the diretio of the stati mageti 4718 Phys. Plasmas, Vol. 3, No. 1, Deember 1996 M. A. Reyolds ad G. J. Morales Dowloaded 8 Sep 000 to Redistributio subjet to AIP opyright, see

4 field deouples Maxwell s equatios, resultig i differetial equatios for the parallel field ompoets, E z ad B z. The solutios away from the beam are writte i terms of Bessel futios, ad the itegrated over the soure to obtai the overall magitude of the fields. The perpediular field ompoets a the be determied i terms of the parallel ompoets if desired. This method works whe the medium is uiform ad a be desribed by a salar dieletri. 14 It is also appliable to tesor media whe, as i the preset ase, the dieletri tesor is both Hermitea ad blok diagoal. 15 The omplete solutio for the parallel eletri ad mageti fields is E z r, qñ r e iz/u B z r, qñ r e iz/u where h j e i e z/u h J k r j H 1 k r, 8a e i e z/u h J k r j H 1 k r, 8b i U k r H 1 k r J k r k 0 r N H i N U H i N U H 1 k r J k r. Here, J ad H (1) are Bessel futios ad Hakel futios of the first kid, respetively, the prime deotes differetiatio with respet to their argumet, N ( e )/k 0 U is the saled parallel wave umber, k 0 /, ad k /k 0 N i H N i H N N H k /k The quatity k is the perpediular wave umber, ad is give by the two solutios of the dispersio relatio N N N H N N N 0, 11 where N N N ad N k /k 0. The fields may alteratively be expressed i terms of the modified Bessel futios I ad K by makig the replaemets H 1 k r J k i K r I, 1a J k r H 1 k i I r K, 1b where k. Whe the harater of the fields is primarily evaeset, the modified Bessel futios provide a more oveiet form for both iterpretatio ad umerial evaluatio. The perpediular ompoets of the eletri ad mageti fields are liear ombiatios of the parallel ompoets, ad a be determied by applyig the operator ẑ to Ampère s law ad Faraday s law to obtai B z ik B ik 0 ẑ E 0, 13a E z ik E ik 0 ẑb 0, 13b where ẑ/z. Equatios 13 a be solved algebraially for E ad B i k spae, ad the resultig expressios iverse-trasformed if ofiguratio-spae depedee is desired. III. GENERAL FIELD BEHAVIOR Equatios 8 are ow evaluated for a eletro beam ad for frequeies above e. For weak mageti fields ( e pe ) this rage of frequeies osists of three regimes: the evaeset regime ( e pe ) where the fields are radially loalized, the upper-hybrid regime ( pe uh ) where the beam exites propagatig upperhybrid waves, ad the vauum regime ( uh ) where the beam exites propagatig eletromageti modes ear the harmois of e. Whe the mageti field is strog ( pe e ) oly the last two of these regimes exist above e. Before evaluatig the fields umerially, it is helpful to ivestigate the physis qualitatively by examiig the struture of Eqs. 8. All of the field ompoets, for example B z, may be writte i the dimesioless form B z B z r, qñ/r b z,e iz e /U e i, 14 where b z is the stregth of the th term. A ivestigatio of B z rather tha B z ) has the advatage of separatig the effets of the desity modulatio, ñ, from those iheret to the beam. The depedee of the th term o ad z is osillatory expiexp iz e U. 15 This struture arises as a osequee of the beam geometry, whih demads that if the azimuthal depedee is harmoi, exp(i), the the axial depedee is a phase osillatio due to the free streamig, i.e., oly those waves whih satisfy the Doppler-shifted ylotro oditio e k U, 16 are exited. The stregth of the th term is govered by b z (,), whose evaluatio requires a detailed kowledge of the dispersio relatio beause it depeds sesitively o k. Phys. Plasmas, Vol. 3, No. 1, Deember 1996 M. A. Reyolds ad G. J. Morales 4719 Dowloaded 8 Sep 000 to Redistributio subjet to AIP opyright, see

5 The parallel idex of refratio N is a measure of the frequey separatio from the th harmoi N e. 17 Two limits are useful: N 0 ear the th harmoi ad N far from the th harmoi. I the small N limit, the two solutios for the perpediular wave umber are approximated by N H,, 18 whih are the well-kow dispersio relatios for the ordiary ad extraordiary modes, respetively. For large N, N approahes N N, N, 19 whih are the dispersio relatios for the eletromageti ad eletrostati modes, respetively. I Eq. 19, N has bee take to be larger tha all elemets of the dieletri, whih meas the frequey does ot orrespod to a hybrid resoae. There are three importat dimesioless parameters: the pith agle p ta 1 U /U, the saled veloity 0 U U, 1 ad the effetive magetizatio pe. e A. Evaeset regime: e << pe I this frequey regime, the real part of N is egative for all values of N, hee the fields are radially evaeset ad are loalized ear the beam. We will show that eah term i Eq. 14 deays expoetially as the frequey is varied away from ylotro harmois. Figure shows a typial spetrum, B z (), for a beam with a pith agle of p 88 ad a saled veloity of 0.01, immersed i a plasma of This spetrum is take at the spatial loatio of /r 0.7 ad z0. It is oveiet to hoose z0 beause, exept for the phase fator due to the free-streamig see Eq. 15, the field is periodi i z with a period of U / e. All figures i this paper will therefore evaluate the fields at z0. Peaks our at the ylotro harmois, with a maximum amplitude that dereases with ireasig harmoi umber. To uderstad this spetrum, eah term i Eq. 14 eeds to be osidered i more detail. Near the th harmoi, the th term i the sum is larger tha the other terms ad has the shape of a resoae, with expoetial wigs. To illustrate this feature, osider to be differet from e. I this ase, N is large, the asymptoti solutios i Eq. 19 may be used, ad the stregth of the th term approahes the value FIG.. Frequey spetrum of the saled mageti field B z. The spatial loatio is /r 0.7, z0, the beam parameters are p 88, 0.01, ad pe / e The hoie of z0 is a oveiee, ad is made i all of the figures. The dashed lie is Eq. 3 for 4 ad the dotted lie is Eq. 6. b z U r expk r, 3 where the expoetial behavior stems from the large argumet expasio of the radial Bessel futios. The approximatio i Eq. 3 is show as a dashed lie i Fig. for the 4 term. To determie the frequey sale, the argumet of the expoetial a be writte i the form k r e, 4 where U /U e 1/r ot p 1/r, 5 is defied as the width of the resoae, orrespodig to oe e-foldig. For the values of the parameters i Fig., Eq. 5 predits a half-width of / e 0.1, i good agreemet with Fig.. Sharp resoaes, therefore, are obtaied at radial loatios far from the beam (/r or /r 0), ad for large pith agles. For large pith agles, Eq. 5 implies that the resoae width depeds sesitively o p. This depedee is depited i Fig. 3 whih shows the frequey spetrum for three values of the pith agle, p 87, 88, ad 89, ad at the same spatial loatio as Fig. (/r 0.7 ad z0). The determiatio of the resoae width as a futio of is ompliated by the fat that the magitude of the field is also a futio of radial positio, as is ivestigated ext. Whe is ear a harmoi, N 0, ad Eq. 18 a be used to approximate the perpediular wave umber. If, i additio, is ot too large, the Bessel futios may be expaded for small argumet to approximate the magitude of the field at the peak of the th harmoi b z U 1 H /r r H r / 6 r. 470 Phys. Plasmas, Vol. 3, No. 1, Deember 1996 M. A. Reyolds ad G. J. Morales Dowloaded 8 Sep 000 to Redistributio subjet to AIP opyright, see

6 FIG. 3. Frequey spetrum of the saled mageti field B z for three differet values of the pith agle p 87 dashed lie, 88 solid lie, ad 89 dotted lie. The spatial loatio is /r 0.7, z0, the beam veloity is 0.01, ad This is due solely to the extraordiary mode, for the polarizatio of the ordiary mode is B z E 0. This expressio otais both the derease of peak amplitude with harmoi umber for a give radial positio, show as the dotted lie i Fig., as well as the depedee of peak amplitude o /r for a give harmoi umber. The radial depedee of B z for e ad for three differet values of the azimuthal agle, 0, 90, ad 180, is show i Fig. 4. Far from r, the fields are well-approximated by a power law i, as predited by Eq. 6. Of ourse, as beomes large, the small-argumet expasio is o loger valid, ad the fields beome evaeset. Near the ylotro radius, the width of the harmoi resoaes ireases, implyig that may terms otribute sigifiatly to the sum, ad the radial depedee departs from the simple power law. This is the mathematial maifestatio of the fat that the idued plasma urret ear the beam is strog ad has a omplex spatial struture. The followig piture emerges: FIG. 4. Radial depedee of the saled mageti field B z for three differet values of azimuthal agle 0 solid lie, 90 dashed lie, ad 180 dotted lie. The parameters are / e, 0.01, 10 4, ad p Eah term i the sum has axial ad azimuthal depedee give by Eq. 15, ad a radial depedee give by Eq. 6 at the harmoi, ad by Eq. 3 far from the harmoi. For large pith agles ad far from the radial loatio of the beam, the frequey spetrum osists of sharp resoaes at the ylotro harmois. I this parameter regime, these resoaes are well-approximated by oe term i the ifiite sum the other terms are expoetially small. Hee, the field has the same spatial depedee as the sigle, domiat, term. 3 Close to the radial loatio of the beam, r, ad for small pith agles, may terms otribute sigifiatly to the magitude of the field. I this ase, the field magitude at a give frequey is a superpositio of those terms that are losest, i the sese of Eq. 17. These properties are due to the Bessel futios whih appear i the expressio for the field, ad hee are a osequee of the helial geometry. Beause all ompoets of E ad B deped o this same produt of Bessel futios, they all exhibit similar struture. B. Upper-hybrid regime: pe << uh This frequey regime is haraterized by eletrostati wave propagatio. As log as pe ad uh are far from the harmois of e, the value of N is large for all terms i the sum, the approximatio i Eq. 19 a be used, ad beause the eletrostati mode has a real ad positive N, the fields propagate. Although the dispersio relatio is well approximated by the eletrostati dispersio relatio, k k 0, the fields have a sigifiat mageti ompoet, hee we otiue to use B z as harateristi of the field struture. The eletromageti mode may be igored beause it is evaeset with a short deay legth Lk 1 r, where the iequality is due to the fat that k N. To illustrate the behavior i the upper-hybrid regime, we hoose the value of 0.5, whih implies pe 4.5 e ad uh 4.61 e. This upper-hybrid regime orrespods to the frequey iterval betwee the 4 ad 5 harmois so that the large N approximatio is appropriate for all terms i the ifiite sum. Figure 5 shows the mageti field due to the 5 ad 6 terms at a radial positio of r, outside of the beam. The osillatios i the field are due to the Bessel futios J 5 (k r ) ad J 6 (k r ), where k k 0 N /.Asapproahes uh, approahes zero, ad the osillatios are more losely spaed. The field i this limit approahes the value b z N U r os k r Beause these waves propagate away from the beam, it is useful to disuss the power radiated by the beam ad its radiatio resistae. The total power radiated per-uit-legth P may be foud by itegratig the Poytig vetor S over a ylider of legth L ad radial positio r that is etered o the beam Phys. Plasmas, Vol. 3, No. 1, Deember 1996 M. A. Reyolds ad G. J. Morales 471 Dowloaded 8 Sep 000 to Redistributio subjet to AIP opyright, see

7 FIG. 5. Frequey depedee of b z5 thi lie ad b z6 thik lie for the parameter values of 0.5, 0.01, p 88, ad a spatial loatio /r, z0. FIG. 6. Frequey spetrum of the saled mageti field B z i the vauum regime. The spatial loatio is /r, z0, ad the parameters are 0.01, p 88, ad 1. For ompariso, the dashed lie is the spetrum for 10 4, with all other parameters the same. P 1 L 0 L dz 0 d S. 8 The otributios from the edaps of the ylider ael. A radiatio resistae per-uit-legth, whih is a measure of the effiiey of the beam, may be defied as R I P, 9 where I q U ñ. I this upper-hybrid frequey regime, the resistae is approximately R N 1 os k r r For the parameters i Fig. 5, the magitude of eah term is approximately N ohm r e m, 31 if e is measured i s 1. Eletrostati waves are also radiated above the lower-hybrid frequey. C. Vauum regime: uh < For frequeies larger tha the upper-hybrid frequey, N a be real ad positive for small values of N, i.e., the ordiary mode propagates for pe, while the extraordiary mode propagates for uh. This implies that for frequeies very ear the th harmoi, the th term desribes a propagatig eletromageti field. The otributio of other terms is expoetially small beause for them N is large. To illustrate this behavior, we hoose 1 whih plaes all harmois above the upper-hybrid frequey, so that both modes will propagate for small N. A typial ear-field spetrum is show i Fig. 6, for /r, 0.01, ad p 88. Also show is the spetrum for 10 4 evaeset regime for ompariso. As i the evaeset regime, sharp resoaes our beause the parameter / e is small, ad the stregth of eah term deays expoetially. However, the beam radiates ear the ylotro harmois, ad the radiatio resistae per-uit-legth is agai a useful quatity. Ulike the upper-hybrid regime, the regios i frequey spae orrespodig to propagatio do ot overlap for differet, so that the radiatio resistae per-uit-legth for e o the harmois beomes where R U U J k r e / H H J k r k r J k r, k r U, k r U H 3 33a. 33b The two terms i the square brakets represet the power radiated ito the ordiary mode ad extraordiary mode, respetively. As beomes large ompared with pe, the radiated fields losely approximate those that would be radiated by a beam i vauum. That is, approahes l i this highfrequey limit, ad the argumets of the Bessel futios approah U /.IfU /1 as well, i.e., the beam is orelativisti, the radiatio resistae takes a simple form 1 U R e U! whih for large beomes R e U e U 4 U 4, 34 U, 35 so that the resistae dereases with harmoi umber as, as expeted. 47 Phys. Plasmas, Vol. 3, No. 1, Deember 1996 M. A. Reyolds ad G. J. Morales Dowloaded 8 Sep 000 to Redistributio subjet to AIP opyright, see

8 IV. COMPARISON WITH EXPERIMENT I the experimet performed at UCLA by Stezel ad Golubyatikov, 4 the parallel ompoet of the flutuatig mageti field is measured usig a Ḃ probe, whih osists of a irular wire oil i whih the idued emf drive by the hagig mageti flux through the loop is deteted. The Ḃ probe is.3 m i radius, with a thikess of.6 mm. This radius is approximately equal to the ylotro radius of the beam eletros, the exat ratio depedig o the beam s perpediular veloity. The bakgroud plasma i whih the beam is immersed is weakly magetized, e pe, ad its temperature is low eough to warrat the approximatio of the dieletri tesor by that of a old plasma. There are three experimetal fats whih, whe otrasted with the theoretial idealizatio preseted here, make the quatitative ompariso betwee experimet ad theory diffiult. First, ad most importat, is the fiite size of the Ḃ probe. The experimetal tehique aot detet variatios i the mageti field o a sale smaller tha the size of the probe. For typial beam eergies of 30 V 100 V, the orrespodig ylotro radii are r 1.7 m 3 m. This meas that the probe is ot muh smaller tha the beam, ad is sometimes larger. The preset theory, however, determies the field loally. Beause of the mathematial form i whih we have expressed the fields a ifiite sum of Bessel futios, it is diffiult to itegrate the field over the area of the probe, exept whe the probe is loated either ompletely iside the beam (r ) or ompletely outside (r ). Seod, while the theory sigles out a speifi pith agle ad beam veloity, the experimetal beam has a fiite spread i these quatities produed by the broadbad oise iheret i the ijetio proess. Thus, there is o well-defied beam axis. Third, the beam is ijeted from a speifi loatio ad does ot exted ifiitely i z. This fat leads to the experimetal observatio that whe the probe is ear the ijetio poit, the field spetrum is broadbad, orrespodig to the oise of the ijetio. However, whe the probe is far from the ijetio poit, those modes that are ot oheret are filtered out, ad spetra more losely resemblig those i Fig. are obtaied. This implies that the theory a oly be used to explai measuremets take far from the disturbig ifluee of the ijetor. The simple axial ad azimuthal depedee predited by Eq. 15 is observed usig iterferometri tehiques. For large pith agles ( p 88 ) ad frequeies ear e ad its first harmoi (/ e 1,), the observed depedee is learly e i ad the measured z depedee is uiform, i.e., k 0. The phase hage i z is also measured for itermediate frequeies ( e e ) ad k is foud to be osistet with Eq. 16, where is the losest harmoi, i.e., due to that term whih has the largest magitude. The preset theory agrees well with these observatios. The observed radial depedee at the first harmoi (/ e ), depited i Fig. 1 of Ref. 4, also is i agreemet with the preset theory as show i Fig. 7. This theoretial plot is a ut alog x for y0. That is, egative values of x orrespod to 180 while positive values sigify 0. The miimum at x0 ad the deay for xr FIG. 7. Radial depedee of the saled mageti field B z theoretial preditio for the first two harmois / e 1.01 solid lie ad / e dashed lie. The parameters are p 88, 0.01, z0, ad 0, 180. Compare with Fig. 1 of Ref. 4. show i Fig. 7 is see i the experimet. However, the measured field dereases more slowly, a feature expeted from the fiite size of the probe. At the ylotro frequey ( e ), however, there is a sigifiat disrepay. While the experimetal measuremet does show a sharp drop i magitude for r, it remais large for r ad does ot show a miimum at 0. The preset theory does ot predit this type of behavior, most likely beause it does ot ilude ay dissipatio mehaisms whih might be preset i the experimet. The frequey depedee of the mageti flutuatios is perhaps the most strikig result of the experimet, ad is explaied by the preset theory. The theoretial spetrum i Fig. 8, where the parameters were hose to approximate the experimetal oditios, agrees qualitatively with the experimetal spetrum see i Fig. 6 of Ref. 4. A quatitative ompariso is diffiult to make, for the reasos metioed earlier. Due to the depedee of the resoae width o pith agle ad radial loatio, ad the experimetal spread i these quatities, the resoaes of the observed spetra FIG. 8. Frequey spetrum of the saled mageti field B z. The spatial loatio is /r 0.7, z0, ad the parameters are p 88, , ad Compare with Fig. 6 of Ref. 4. Phys. Plasmas, Vol. 3, No. 1, Deember 1996 M. A. Reyolds ad G. J. Morales 473 Dowloaded 8 Sep 000 to Redistributio subjet to AIP opyright, see

9 FIG. 9. Depedee of the frequey spetrum of the saled mageti field B z o the pith agle p of the beam. The parameters are , 10 4, ad the spatial loatio is /r 0.7, z0. Compare with Fig. 8 of Ref. 4. have a miimum width, regardless of the average parameters of the beam. A lear depedee of resoae width o pith agle is ideed observed i the experimets see Fig. 8 of Ref. 4. The sesitivity of the experimetal resoae width o p, however, is ot as strog as predited by the preset model. Figure 9 shows that the resoaes disappear whe p 80, while experimetally they are see for pith agles as small as 70. Beause experimetally, the resoae width depeds strogly o the distae betwee the probe ad the ijetio poit, ad, sie this is ot addressed i the preset theory, it is diffiult to offer a explaatio for the disrepay. However, it is worth otig that while a geeral tedey is for the o-ideal experimetal realities to destroy theoretially sharp features, i this ase the observed behavior is otrary to this tred. The liear beam modes exited by spiralig eletros, as alulated by the preset theory, agree qualitatively with the experimetal results. Some of the disrepaies a be uderstood i light of the method of measuremet, i.e., a spatial itegratio of B z rather tha a loalized measuremet, ad by takig ito aout the experimetal limitatios ad the theoretial idealizatios. However, the two poits of disagreemet are puzzlig: the radial struture for e, ad the resoae width for small pith agles. V. CONCLUSION The eletromageti flutuatio spetrum geerated by a modulated, free-streamig, helial eletro beam has bee alulated ad ompared to a reet laboratory experimet. 4 For the frequey regime of the experimet ( e pe ), the spetrum osists of loalized, evaeset fields whih shows a series of sharp resoaes at the eletro-ylotro harmois. This behavior is well desribed by the exitatio of beam modes whih satisfy the oditio e k U. The width ad magitude of these resoat peaks a be explaied by the form of the fields both ear the harmois ad far from the harmois. It is foud that the peaks are futios of the pith agle ad the spatial loatio. The axial ad azimuthal depedee of the fields is also desribed by the model developed i this study. A disrepay arises betwee the experimetal results ad the theoretial preditio for the radial depedee of the field at e. The fiite size of the measurig istrumet, a mageti loop whih itegrates the flux, as well as the idealizatio of the theory, whih iludes either a veloity modulatio or a spread i pith agle, are the probable auses for the disrepay. I additio, the preset theoretial formulatio is geeral eough to desribe flutuatios geerated i other frequey regimes as well as those geerated by io beams. This geerality allows the alulatio of quatities suh as the radiated power i a straightforward maer, without resortig to approximatios. Two of these other frequey regimes, ear ad above the upper-hybrid frequey uh, have bee osidered. Near uh the beam radiates a broad spetrum of eletrostati waves whih propagate away from the beam, ad propagatig eletromageti modes are emitted ear those harmois of e that are above uh. ACKNOWLEDGMENTS We thak Professor R. Stezel for illumiatig disussios oerig his experimetal results. This work is sposored by the Offie of Naval Researh. 1 N. J. Lopes Cardozo, F. C. Shuller, C. J. Barth, C. C. Chu, F. J. Pijper, J. Lok, ad A. A. M. Oomes, Phys. Rev. Lett. 73, J. E. Maggs, i Artifiial Partile Beams i Spae Plasma Studies, edited by B. Gradal Pleum, New York, 198, p T. Neubert ad P. M. Baks, Plaet. Spae Si. 40, R. L. Stezel ad G. Golubyatikov, Phys. Fluids B 5, G. Ladauer, J. Nul. Eergy C Plasma Phys. 4, J. Lavergat ad R. Pellat, J. Geophys. Res. 84, J. Lavergat ad R. Pellat, i Ref., p J. Lavergat ad T. Leher, IEEE Tras. Ateas Propag. AP-3, J. Lavergat, T. Leher, ad G. Matthieusset, Phys. Fluids 7, K. J. Harker ad P. M. Baks, Plaet. Spae Si. 33, K. J. Harker ad P. M. Baks, Plaet. Spae Si. 35, E. Caobbio ad R. Croi, Phys. Fluids 9, ; R. Croi ad E. Caobbio, i Proeedigs of the Seveth Iteratioal Coferee o Pheomea i Ioized Gases, Beograd, 1965, edited by B. Perovi ad D. Tosi Gradeviska Kjiga, Beograd, 1966, Vol. II, p E. Caobbio, Nul. Fusio 1, J. D. Jakso, Classial Eletrodyamis, d ed. Wiley, New York, 1975, Se W. P. Allis, S. J. Buhsbaum, ad A. Bers, Waves i Aisotropi Plasmas MIT Press, Cambridge, MA, 1963, Chap Phys. Plasmas, Vol. 3, No. 1, Deember 1996 M. A. Reyolds ad G. J. Morales Dowloaded 8 Sep 000 to Redistributio subjet to AIP opyright, see