Bekefi, G.

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PFC/JA-91-15Double Stream Cyclotron MaserBekefi, G.June, 1991Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, MA 02139 USAThis work was supported by the National Science Foundation. Reproduction, translation,publication, use and disposal, in whole of part, by or for the United States governmentis permitted.Submitted for publication in Applied Physics Letters.DOUBLE STREAM CYCLOTRON MASERG. BekefiDepartment of PhysicsandResearch Laboratory of ElectronicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139ABSTRACTThe double stream cyclotron maser is a novel source of millimeter wavelengthradiation in which two copropagating electron beams are caused to gyrate in a uniformaxial magnetic field. The interaction of the slow cyclotron space charge wave on onebeam with the fast cyclotron space charge wave on the other beam leads to highfrequency bunching. The desired operating frequency is proportional to the electroncyclotron frequency (or a harmonic thereof) and inversely proportional to the differencein beam velocities, and can be achieved at low beam energies and axial magneticfields. The linear instability growth rate is calculated from the fully relativistic Vlasovequation for the case of cold electron beams.PACS numbers: 42.55 Tb; 42.60 Fc; 52.75 Ms1Electrons with relativistic velocities gyrating and drifting in a uniform axial mag-netic field Bi, are known' to generate intense, coherent electromagnetic radiation atcentimeter and millimeter wavelengths. This radiation is a manifestation of the "neg-ative mass" instability2 of the copropagating right circularly polarized wave travelingin the magnetic field direction. The instability arises from a relativistic mass shift ofthe electron gyrofrequency 1 = (eBj1/m'-y) with - = [1 - (V - (v/c)2] -21. Theresulting emission occurs at a radiation frequencyW (1+ 311)_Yj(mf2), (m = 1, 2,3...) (1)where 01 = vil/c and yl = [1 - 12 ~. The gyrotron3 (with -yl ~ 1) and the cyclotronautoresonance maser 4 (with -yl > 1) are special cases of the above instability.The double-stream cyclotron maser is a novel source that does not rely on rela-tivistic effects for instability growth, but on the mutual interaction of charge-densityperturbations of two copropagating beams traveling at somewhat different speeds. Theadvantage of the double stream system over the conventional one-beam cyclotron maseris the much higher frequency that can be achieved for the same experimental parame-ters. As will be shown below, the interaction frequency is now given by,W2 ~- 2 (0jjgg )2(_Y /AYIj)(Mfj) (2)where A-y1 is the energy difference between the two electron beams. Comparing Eqs. (1)and (2), it follows that W2/wI ~ (-yj 1/Ayjj) so that, for example, a 20% energy differ-ence can increase the interaction frequency fivefold. Conversely, a desired operating2frequency can be now obtained at relatively low beam energies and axial magneticfields.The double stream cyclotron maser is illustrated schematically in Fig. 1. Twowell intermingled electron beams of different axial velocities v11,1 and V1,2 are spun upand allowed to gyrate in a uniform axial magnetic field B11. The appropriate velocitydifference can be obtained, for example, by means of a split-cathode' technique in whichhalf of the cathode is connected to the accelerator via a resistive load. The spinningup of the beams can be performed using a magnetron injection' gun configuration;however, in order to minimize axial velocity spread, a short section of bifilar helicalwiggler 7 is preferable (see Fig. 1). As a result, the electron beams acquire transversevelocities vJ,1 and VJ.,2.The spun-up beams interact in a cylindrical drift tube which also acts as a wave-guide. In this three-dimensional configuration, the wave vector k has both axial (kii)and transverse (k1 ) components, with k1 being determined by the beam radius andthe transverse guide dimensions. Excitation of a space-charge wave (E,f 11 k ), atthe cyclotron frequency and its harmonics comes about solely because the transversecomponent of the rf electric field Erf exerts a tangential force on the electron in itshelical motion. This orbit perturbation leads to harmonic generation. In order toachieve significant wave amplitude at a given harmonic, the perpendicular wavelengthmust be comparable in magnitude to the electron Larmor radius, or, kivw/fL ~ 1. Itshould be noted that the type of two-stream instability discussed here differs from theconventional8 one characterized by frequency oscillations tied to the electron plasma3frequency w2 = Ne2 /miE0 ,-y rather than the electron cyclotron frequency [I. Indeed,in such cases, calculations9 and the resulting microwave amplifier tubes' 0 are based onthe assumption that v1 ~ 0, so that coupling to the electron cyclotron motion is of noconcern.Our calculations of the wave frequency and growth rate are based on a solution"of the relativistic Vlasov equation and Maxwell's equations in free space. In place ofa fully self-consistent solution of the unneutralized two-beam system, a neutralizingbackground of infinitely massive ions is assumed. The wave with Ef 1k and an exp[j(wt - k 'r )] dependence has a linear (small signal) dispersion equation of the form,KL(k,w) = (k - K - k)/k2 = 0, where K is the tensor dielectric coefficient and KLthe effective longitudinal component given by12KL=1- 0 dv_ dvW, 2rfo xbeams JOJ-oo00 [ ( [J.2(p+ [k 2 - kI I ~Jn (P(3E v _ dp I C M,1= 0. (3)M--00 .( P -k 1 vi -m)2Here fo = f0 (v±, v11) is the unperturbed velocity distribution function for each beam,and Jm (p) is the mth order Bessel function with argument given by p = kjv 1 /.We now assume for simplicity that the two beams are completely superimposedand that they fill entirely a cylindrical waveguide of radius a. In that case8 k1 = ul,,/awhere ul,, is the sth root of J1(u) = 0. Then,p = uisvw/f2a (4)= ulsrL/a4where rTL is the Larmor radius of the electron orbit (rL < a). The values of I and sselect the waveguide mode. For the lowest transverse magnetic TMO,1 mode of interesthere, us = 2.405. For beams with cold electrons, the distribution function has theform,fo(v., v11) = 6(vi - voi)6(vj - v011). (5)27rvOISubstituting Eq. (5) in Eq. (3) yields the sought-after dispersion equation:KL = 0 =1- - k 2  +beams m=--o (o - kg og - ml)k- J .(6)20(w - k -m v ) - m()In the case of one beam alone, the above dispersion equation shows that aroundeach harmonic number mll = w - k11vlj, there are two narrow propagation bands,one above and one below each integral multiple of Q. The propagation bands extendapproximately ±WpJm(p) on either side of mfl. The lower, slower branch around eachharmonic is a negative energy wave1 3 , while the upper, faster wave is a positive energywave. At high plasma densities (w, > m11) the negative energy wave of the mthharmonic can come into phase velocity synchronism with the positive energy waveassociated with the (m - 1)th harmonic, and a high frequency micro-instability candevelop14. However, this phenomenon is of little interest for unneutralized electronbeams considered here since such beams will become macroscopically unstable unless8W, < n/vZ.5The situation is different when two beams with different axial velocities interactwith one another as is illustrated in Fig. 2. Now, phase velocity synchronism can beachieved at low plasma densities p < 0. For the case of almost paraxial propagation,k l > k-, the second term of Eq. (6) can be neglected; it is identically equal to zerowhen p is so chosen as to maximize Jm (p). Then, the dispersion characteristics foreach beam-cyclotron mode can be approximated byw ~ kjjvi + m01(7)w ~ kjjv 2 + nf 2 .with n, m equal ±1, ±2.... For, say, n = -m, and beams whose velocities are not toodifferent, Eqs. (7) yieldW - MO V2 + VIV2 - V 1  (8)= 2(,|3 ||yj)2(-y||/A-y|)(mQ)where now 0, 3 1 and 3|| represent averages over the two beams. We note that in thevicinity of the above frequency, w ~ kil(vi + v2 )/2, the wave is dispersionless, and hasa phase velocity equal to the average beam velocity.The accompanying spatial growth rate, Im(k), of the convective instability (thesystem is an amplifier rather than an oscillator) is derived from Eq. (6). This ismost readily accomplished by noting that for sufficiently small growth rates, Im(k) ~Im(w)/vg, where vg is the group velocity of the unstable wave. Thus, solving forcomplex w and real k one then obtains,Im(w) -- wp,(p)/2j||.()In deriving the growth rate, it is assumed that the two beams have equal plasmafrequencies, or more precisely that w,,1Jm(P1)111j,1 = W,,2Jm(P2)/h1j.2 = W)Jm(P)/ 11 -6For purposes of illustration we consider two cases: the interaction of two slowbeams where relativistic effects are very small, and the case of two beams with mildlyrelativistic energies. In both cases the energy differences A-yl are taken to be consider-ably smaller that the mean beam energies, but much larger than the parallel thermalenergy spreads expected in each beam. The results of the calculations are summarizedin Table I. The conversion efficiency 77 (at saturation) quoted in the Table is estimatedfrom 8 ?7 ; AgI/(b - 1).We see that high frequency operation can be achieved at low beam energies andlow magnetic field strengths. The instability growth rate is appreciable, but is limited(see Eq. (9)) by the need to maintain the ratio wp/f2 < iv/ (Brillouin condition8 ),and thereby prevent beam break-up in the unneutralized system. Introduction' ofa cold background plasma to provide space-charge neutralization offers an interestingoption. It allows an increase in w, which increases the growth rate, the beam currentdensity and the rf power. In addition, by properly tuning the plasma density of thebackground plasma, the associated space charge wave can be made synchronous 17 withthe unstable beam cyclotron wave discussed above. This is readily seen by deriving thewave phase velocity (as found by setting v1 =vil = 0 in Eq. (6)):( 2 (w2 _ w2)( 2 - 2)(--) ma = (U2 2 -W 2 +W2 (10)Setting (W/kli) p s (V1 + v2 )/2 assures phase velocity synchronism with the beamcyclotron wave of Eq. (8). This three wave interaction may enhance' 6 ", 7 the bunchingprocess.7In conclusion, we have described the characteristics of a novel double-stream cy-clotron maser that can achieve high oscillation frequencies using two mildly relativis-tic beams propagating in weak magnetic fields. To interact with the beam-cyclotronmodes, the electrons must first acquire perpendicular energy just as in gyrotrons andcyclotron autoresonance masers. The proposed interaction with n = -m results ina slow wave (w/k 11 < c), and in this respect suffers from the same disadvantagesas other erenkov type systems. Thus, generation of electromagnetic radiation re-quires the presence of a slow-wave structure such as a dielectric loaded, or corrugatedwaveguide. Alternately, our two-beam system could be incorporated in a klystronl 8geometry, and the rf power in the growing space-charge bunches extracted in a reso-nant cavity. However, a different choice of mode numbers m and n can lead to a fastwave interaction. In the general case, the interaction frequency and phase velocity arefound to be w ~ E(m,32 - n,1)/(02- ,1) and (w/k 11c) ~ (m32 - n,31)/(m - n). For,m > 0, n > 0, (m 0 n) and for sufficiently large beam energies, w/kIIc can be equal to,or greater than, unity, and interaction with an electromagnetic wave can take place.Now, however, the validity of the quasi-static approximation (Ehf f k), on whichEq. 3 is based, is open to question.Studies are in progress concerning effects of electron beam temperature, lack ofcomplete spatial overlapping of the two beams, and nonlinear saturation. In the non-linear regime, loss of synchronism will occur as a result of changes in Y1, -Y2 and A&y .This limits the degree of bunching that can be achieved. Equations (8) suggests thattapering the magnetic field as a means of "efficiency enhancement" may be possible.8The calculations in this paper are applicable to beams propagating in an assumedneutralizing background of stationary plasma ions. In the absence of the neutralizingplasma, the results are a good approximation provided that the beam plasma densitiesare sufficiently low, such that (w,/11) 2 < 1. As the density increases and the Brillouincondition (w,/11) 2 = is approached, a self-consistent calculation is mandatory.9REFERENCES[1] V. Granatstein and I. Alexeff (editors), "High Power Microwave Sources", (ArtechHouse, Boston 1987).[2] Y.Y. Lau and D. Chermin, Phys. Rev. Lett., 52, 1425 (1984); also D. Chermin andY.Y. Lau, Phys. Fluids, 27, 2319 (1984).[3] K. Felch, H. Huey and H. Jory, J. of Fusion Energy, 9, 59 (1990).[4] W.A. Fliflet, Int. J. Electron., 61, 1049 (1986); also A.C. DiRienzo, G. Bekefi, C.Chen and J.S. Wurtele, Phys. Fluids, (in press).[5] J. Fink, H.B. Schilling and V. Schumacher, J. Appl. Phys., 51, 2995 (1980); It waspointed out to me by D.L. Goodman and D.L. Birx that in the case of an inductionlinac, the beam voltage difference (generated by a separate power supply) can bevaried independently of the accelerator voltage, thereby achieving a great deal offlexibility.[6] J.M. Baird and W. Lawson, Int. J. Electron., 61, 953, (1986).[7] H.B. Mitchel and J.M. Wachtel, IEE Proceedings, 131, 178 (1984); also, V.L. Brat-man, G.G. Denisov, N.S. Ginzburg and M.I. Petelin, IEEE J. Quant. Electron.,QE-19, 282, (1983).[8] R.C.Davidson, " Theory of Nonneutral Plasmas", W.A. Benjamin, 1974.[9] J.R. Pierce, Proc. L.R.E., 37, 980 (1949); V.N. Shevchik, G.N. Shvedov and A.V.Soboleva, "Wave and Oscillatory Phenomena in Electron Beams at MicrowaveFrequencies", (Pergamon Press, 1966); also G. Bekefi and K.D. Jacobs, J. Appl.Phys., 53, 4113 (1982).[10] A.V. Haeff, Proc. I.R.E., 37, 4 (1949).[11] A. Bers, (private communication).[12] A. Bers and C.E. Speck, Massachusetts Institute of Technology, Research Labora-tory of Electronics, Quarterly Progress Report, 78, 110 (1965).[13] A. Bers and S. Gruber, Appl. Phys. Letters, 6, 27 (1965).10[14] S. Gruber, M.W. Klein and P.L. Auer, Phys. Fluids, 8, 1504 (1965).[15] P.A. Sprangle, R.A. Smith and V.L. Granatstein, Naval Research Laboratory Re-port, No. 3911, 1978.[16] Y. Carmel, K. Minami, R.A. Kehs, W.W. Destler, V.L. Granatstein, D. Abe andW.L. Lou, Phys. Rev. Lett., 62, 2389 (1989).[17] A.T. Lin, Phys. Rev. Lett., 65, 717, (1990).[18] Y.Y. Lau, M. Friedman, J. Krall and V. Serlin, IEEE Trans. Plasma Science,18, 553 (1990); also D.L. Goodman, D.L. Birx and B.G. Danly, MassachusettsInstitute of Technology, Plasma Fusion Report No. PFC/JA-90-43 (1990).11TABLE I. Summary of Operating Parameters of a Double Stream Cyclotron Maserfor Ca,/D)2 (1/9) and (kvL/)m1.841.12PARAMETER I IIVOW(kv) 87.5 275A V(kV) 25.0 50J(A/cm) 23 100Bz(kG) 1.0 2.1W/2R(GHz) 50 150Gain (dB/m) 80 68efficiency q (%) 29 18harmonic: m,n 1,-1 1,-iwaveguide mode TM, TM,a(cm) 0.5 0.5v.Ic 0.2 0.3FIGURE CAPTIONSFigure 1: Schematic drawing of a double-stream cyclotron maser.Figure 2: Sketch of the dispension characteristics for cyclotron modes, n = -m (see Eqs. (7)).The heavy lines denote the slow, negative energy waves on each beam. The dashedbell-shaped curve shows the growth rate Im(w).13zowLJ.iwaDw.c o0&w-.w0'0OwwC.,LC.)14.9.4r74- 44C03C c C ~E E'D4II15

Double stream cyclotron maser

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Double stream cyclotron maser

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