Cohn, D.R.

Kreischer, K.

Lax, B.

Temkin, R.J.

Wolfe, S.M.

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HIGH FREQUENCY GYROTRONS AND THEIRAPPLICATION TO TOKAMAK PLASMA HEATINGR.J. Temkin, K. Kreischer, S.M. Wolfe,D.R. Cohn. and B. LaxSeptember 1978Plasma Fusion Center #RR-78-13HIGH FREQUENCY GYROTRONS AND THEIRAPPLICATION TO TOKAMAK PLASMA HEATINGR.J. Temkin, K. Kreischer, S.M. Wolfe,D.R. Cohn and B. LaxFrancis Bitter National Magnet Laboratory*and Plasma Fusion CentertMassachusetts Institute of TechnologyCambridge, Massachusetts 02139 USARUNNING TITLE: High Frequency Gyrotrons and Plasma HeatingKEY WORDS: gyrotron, cyclotron resonance maser, tokamak plasma heating*Supported by National Science FoundationtSupported by Department of Energy.W.NmwABSTRACTHigh frequency (> 200 GHz) gyrotrons are potentially useful forseveral important applications, including plasma heating and radar.For electron cyclotron resonance heating of a moderate-size, highpower density tokamak power reactor to ignition temperatures, agyrotron frequency around 200 GHz appears to be necessary. Thedesign of high frequency gyrotrons is discussed. Analysis of overallgyrotron efficiency indicates that high efficiency may be obtained infundamental electron cyclotron frequency (wc) emission at highfrequencies. The linear theory of a gyrotron operating at the funda-mental frequency is derived for the TE modes of a right circularcylinder cavity. An analytic expression is given for the oscillatorthreshold or starting current vs. magnetic field.-2-I. INTRODUCTIONGyrotrons are a specific version of the electron cyclotron reso-nance maser (CRM) developed in the Soviet Union by A.V. Gaponov andcoworkers [1-3] in the 1960's. A CRM is a fast wave interaction elec-tron beam tube which operates in a strong magnetic field and emitsradiation at the electron cyclotron frequency, wc, or its harmonics.Special features of the gyrotron include a cylindrically symmetric,temperature limited, magnetron injection type electron gun and a"quasi-optical" cavity which is relatively large compared to a wave-length (of order 2A - 5X in diameter by 5X - 20X long). Because ofthe relatively large cavity, gyrotrons are useful in generating highpower at wavelengths in the millimeter and submillimeter range. Powersof 22 kW, CW at 150 GHz, 1.5 kW, CW at 326 GHz and 1.1 MW, pulsed at100 GHz have been achieved [3,4]. Gyrotrons should prove useful inseveral applications, including heating of magnetically confined plasmasat electron cyclotron resonance and high frequency radar.-3-II. PLASMA HEATING WITH GYROTRONSThe high frequency, high power radiation generated by gyrotronsis useful for electron cyclotron resonance heating (ECRH) of tokamakplasmas. Successful ECRH experiments using gyrotrons have been car-ried out in the Soviet Union on the TM-3 tokamak [5]. Those experi-ments were performed at magnetic fields of up to 25 kG, .at powerlevels of up to 60 kW in 1 msec. pulses and at both fundamental (wC)and second harmonic (2wc) frequencies. Bulk heating of the plasma,with a central temperature rise of up to 200 eV, was observed. Simi-lar experiments are now planned for tokamak plasmas in several othercountries, including the U.S.We have investigated theoretically the characteristics of tokamakpower reactors which would be bulk heated by ECRH [6]. A set of reactoroperating characteristics was derived based onwave propagation andreactor engineering requirements. Of the heating modes considered,ordinary wave heating at w z wc appears to be the most promising. Forthis mode, w z wc should exceed the plasma frequency, wp, for goodpenetration into the center of the plasma. Operation of a moderate-size tokamak reactor at a fusion power density of 5 - 10 MW/m3 leadsto a requirement of a relatively high plasma density, n0 , of about4 - 5 x 1014cm-3. Since n0 determines w , setting w w c Z Wp resultsin frequencies (w/27) in the 200 GHz range for ECRH of reactor plasmas.For this case, the average plasma a, which is defined as the thermal/magnetic pressure ratio, is limited to less than 3.9% for a central-4-temperature below 15 keV. For extraordinary wave heating at 2wC'higher a operation (up to 7.8%) is allowed relative to ordinary waveheating at wc, and the tokamak toroidal magnetic field requirement islowered (by about /I2) but higher gyrotron frequencies (by about '2)are then needed.- 5 -III. DESIGN OF 1IGH FREQUENCY GYROTRONSWe have found that a complete design of a high frequency gyrotronoscillator can be carried out using a combination of analytic expressionsand numerical results primarily derived from the Soviet literature [7].The design utilizes an adiabatic electron gun theory [8], numericalresonator mode analysis [9] and computer calculations of cavityefficiency [10, 11]. The present design approach is useful for pre-liminary design and for evaluation of technological feasibility. Acompletely optimized final design, however, would obviously requiremore extensive computational work and modeling.In the present discussion, we concentrate on one aspect ofgyrotron design, namely the evaluation of the overall gyrotron efficiency,Il. Using our model [7], we have calculated the efficiency vs. wavelengthfor moderate output power gyrotron oscillators operating at the fundamental(Wc) or second harmonic (Z c) in the wavelength range 0.5 to 5.0 mm.The calculation as sumes an electron beam with V= 30 kV, I = 1 A and aperpendicular to parallel velocity ratio of 1.5 in the cavity region. Thecavity is assumed to oscillate in a TE 0 3 1 mode with the electron beamcoinciding with the second radial maximum of the E field.The overall efficiencyis given by [10 ]7 =[ - 5 -- OO +TIQM D +QOM lwhere Q is the ohmic Q, Q is the diffractive Q and Q is the totalOM Dresonator Q, i.e.,~ = IM + QD()ei is given by [10]:- 6 -ef 2 n1 (3)1 + (v/vI)where TI is the efficiency for converting transverse energy of theelectron beam into electromagnetic energy inside the gyrotron cavity.The theory used for calculating il is that of Nusinovich and Erm [11]as modified in Gaponov et al. [10] . This theory assumes a Gaussian RFfield distribution along the gyrotron axis. The theory is presented as aset of plots of efficiency as a function of normalized beam current, I0,and normalized cavity length, [10]. In order to evaluate 10 and ,we use the assumed values of V, Ivi/v1 and TE 0 3 1 mode. The ohmic Q,QOM' is evaluated for a TE 0 3 1 mode copper cavity. The calculationsare performed for both an ideal cavity with 0OU and for a cavity with0.75 QOM, which corresponds to more realistic losses through fabricationerrors and wall heating.The cavity diffractive Q, Q D is a function of cavity length andshape. For the present analysis, we assume a cavity with untaperedwalls, such as that shown in Fig. 3b of Ref.[ 9] . For such a cavity,a Gaussian RF field distribution, exp -(2z/LC) 2, is a reasonableapproximation, where L c is roughly equal to the cavity length L andtherefore L c = L is assumed here. The diffractive Q, QD' is given byL)2=D K 4w ()where, for the cavity shape assumed, K is between 2 and 3 9].With the above assumptions and procedure, the overall efficiency, 11,may be calculated as a function of wavelength, X, and of cavity length, L.- 7 -For each X, L was varied to optimize n. These optimized values areplotted in Fig. 1. Both fundamental and second harmonic emissionwere treated, and both the ideal QOM and the value 0.75 QOM wereused, marked "minimum" and "realistic" losses, respectively.The existence of an optimum length L may be explzinedqualitatively as follows. For very small lengths, I and are smalland the interaction between the electron beam and cavity field is weak.A small value of iL is obtained and hence a small value of n. As L in-2creases, I increases. However, QD increases as L and eventuallyQD OM, so that Eq. (1) provides a decreasing value of 1 even ifis still increasing with L. Figure 1 also shows experimental resultsof Zaytsev et al. [4 ] for fundamental (upper point) and second harmonicemission. The experimental results were obtained for beam parametersand cavity -modes that, although not identical to those used here, areroughly equivalent. However, the cavity profile is not specified inRef. [4] .Figure 1 indicates that, for the present analysis which assumesmoderate electron beam power and a TE 0 3 1 cavity mode, fundamentaloperation is significantly more efficient than second harmonic operation.This difference in efficiency is explained as follows. The transverseefficiency, lI, can achieve a large value, almost 0.80, in either w c or2Wc operation, according to the theory of Nusinovich and Erm. However,in second harmonic operation the gain is lower and relatively large valuesof cavity Q are required to achieve a large i . At short wavelengths, QOM'which goes as X/2, is small so that a large cavity Q may exceed Q0M andlead to a low overall efficiency (cf. Eq. (1)). The present results for-8-the relative efficiency of wc and 2wc are to some extent dependent on thebeam parameters and model assumed. Somewhat different resultsm ght be obtained if a different cavity shape and RF field profile wereassumed. The relative efficiency in 2wc operation will also increase ifthe beamvoltage and/or current are raised.- 9 -IV. LINEAR CRM THEORYWe have derived an expression for the starting current, Ist of agyrotron operating at the fundamental frequency (w w c) for the case ofa right circular cylinder cavity. The method used is similar to that usedin a calculation by Hirshfield et al. [13] for the TE011 mode. Thepresent calculation is an extension to the general TE mode. Basically,starting with the Vlasov and Maxwell's equations, the perturbation of thedistribution function of the electron beam due to the RF E and B fieldsin the cavity can be calculated using the method of characteristics. Thisin turn is used to calculate the power transfer that occurs between thebeam and the resonant field. Under certain conditions the field gainsenergy and emission is possible. Relativistic effects are included byretaining the velocity dependence of the electron mass and cyclotron frequency.Our expression for Ist was derived for a weakly relativistic electronbeam. For simplicity, we present here only the result for an electronbeam with no spread in parallel or perpendicular velocity, ie.,f(v) 6(v - w)6(v11 - u). It can be written as:2Tmo/u3W2 l u 1 x)Tso (=( FG~(x)[(k 2Je r wk(v 2  - 2)J' (where F - MP mm(V mpr I 2(m' (k r) + (m/k 1r)2(k r)]G(x) = (-x?)-2 sin2 ((x + 1)7wq/2)H~x) = 1 dG(x) ( Wc -2)(x) = x 2 (x) dx \k c /- 10 -(w - ox = ___WHere me is the relativistic electron mass, w is the frequency of thestanding wave, w c is the relativistic gyrofrequency of the electrons,Vmp is the pth zero of J'm(x) = 0, kH = qn/L-, k = vmp/R where R isthe cavity radius, and r is the radial distance of the beam from thecavity axis. This equation is valid in the limit of small electrongyroradius (ie, k rL - << 1). Note that the starting current ispositive and emission is possible only when H (x) < - (ou/k 11w2The above equation was compared to Hirshfield's result [2] andfound to reduce to his expression for the TE011 mode, except for a factorof two[14]. Agreement is found in the weakly relativistic limit withresults presented by Symons and Jory [15] for the TE01q mode. Chu [16]has recently solved this problem for the TE0pq modes (i.e. m = 0) usinga fully relativistic approach. By comparing his expression with ourequation, it is found that using the fully relativistic approach intro-duces rather small changes as long as the beam voltage is low and thedevice operates at the fundamental frequency. For example, if we assumew = 1.5 u, the present weakly relativistic expression agrees with thefully relativistic expression within 9% for a beam energy of 30 keV orless and to within 18% for a beam energy of 60 keV or less. This levelof accuracy should be satisfactory for most purposes since the presentcalculations are idealized in two major respects. First, the sinusoidaldistribution of RF field along the magnetic axis, which is assumed in- 11 -the present calculation and those in refs. 13, 15 and 16, is only anapproximation to the distribution in a real gyrotron cavity. Gaponovet al [10] have calculated Ist for an RF electric field with a Gaussianrather than a sinusoidal dependence along the cavity axis. Comparinghis equation with our result shows that a Gaussian distribution canreduce the starting current by as much as a factor of two. Secondly,the velocity spread in the beam, which, for simplicity, was omitted fromthe expression for Ist presented here, can also significantly alter thevalue of Ist. The present calculation should, however, prove useful inpredicting the possibility of mode competition in a gyrotron since Istis given for all TE modes of the resonator.- 12 -V. CONCLUSIONSHigh frequency gyrotrons would be useful for several applications,including -bulk ECRH of tokamak plasmas to ignition temperatures. Fora tokamak reactor, about 75 MW of heating in a pulse of order 5 - 10 sec.is required, which could be achieved by grouping together a number of100 - 200 kW sources. For good wave penetration in a moderate size,high power density tokamak reactor, a frequency of about 200 GHz isrequired [6]. Theoretical considerations as well as recent Soviet results[3] indicate that the required efficient, high-power, high-frequency gyro-trons needed for tokamak reactor heating should be feasible.- 13 -REFERENCES[1] For a review, see V.A. Flyagin, A.V. Gaponov, M.I. Petelinand V.K. Yulpatov, IEEE Trans. Microwave Theory and Tech.MTT-25 (1977) 514.[2] J.L. Hirshfield and V.L. Granatstein, IEEE Trans. MicrowaveTheory and Tech. MTT-25 (1977) 522.[3] A.A. Andronov, V.A. Flyagin, A.V. Gaponov, A.L. Gol'denberg,M.I. Petelin, V.G. Usov and V.K. Yulpatov, Proceedings of Int'l.Conf. on Submillimeter Waves and Their Applications, Guildford,U.K. 1978 (to be published).[4] N.I. Zaytsev, T.B. Pankratova, M.I. Petelin and V.A. Flyagin,Radio Eng. Electron. Phys. 19, No. 5 (1974) 103.[5] V.V. Alikaev, G.A. Bobrovskii, V.I. Poznyak, K.A. Razusnova,V.V. Sanikov, Y.A. Slolov and A.A. Shmavin, Sov. J. PlasmaPhys. 2 (1976) 212.[6] S.M. Wolfe, D.R. Cohn, R.J. Temkin and K. Kreischer, MIT PlasmaFusion Center Research Report No. RR-78-3 (1978, unpublished).[7] R.J. Temkin and S.M. Wolfe, MIT Plasma Fusion Center ResearchReport No. RR-78-9 (1978, unpublished).[8] A.L. Gol'denberg and M.I. Petelin, Radiophysics and QuantumElectron. 16, No. 1 (1973) 106.[9] S.N. Vlasov, G.M. Zhislin, I.M. Orlova, M.I. Petelin and G.G.Rogacheva, Radiophysics and Quantum Electron. 12, No. 8 (1969)972.[10] A.V. Gaponov, A.L. Gol'denberg, D.P. Grigor'ev, T.B. Pankratova,M.I. Petelin and V.A. Flyagin, Radiophysics and Quantum Electron.18, No. 2 (1975) 204.[11] G.S. Nusinovich and R.E. Erm, Elekron Tekh., Ser. 1, Elektron.SVCh, No. 2 (1972) 55.[12] R.J. Temkin, S.M. Wolfe and B. Lax, Francis Bitter National MagnetLaboratory Memorandum Report Gl (1977, unpublished).- 14 -[13] J.L. Hirshfield, I.B. Bernstein, and J.M. Wachtel, IEEE JournalQuantum Electronics 1, No.6 (1965) 237.[14] The equation for Ist for the TE011 mode in Ref. 2 yields twicethe value obtained with the present equation. An extra factorof two in Ref. 2 appears to occur when Eq. 3 is divided by thesubsequent equation.[15] A.S. Symons and H.R. Jory in Proceedings of the Seventh Symposiumon Engineering Problems of Fusion Research (Knoxville, Tenn. 1977).[16] K.R. Chu, Naval Research Laboratory Report 3672 (1978).- 15 -FIGURE CAPTIONFig. 1. Theoretical overall gyrotron efficiency vs. wavelength isplotted for fundamental (Wc) and second harmonic (2wC) opera-tion. The model and assumed parameters are described in thetext.C,)wU)0-J20z Iz w0 c0w .00 cz I0()20 000 0 0to) ITj m(I)co 00Jwo0o0 0w w(I)n00e 0o 0C\j -I'.)~:J.wwNL0C4)ww(azwI-00w9-I-C-,zw-JwN>ADN3101J.J3 N08.i.O Jko -l-lV83AO

High frequency gyrotrons and their application to tokamak plasma heating

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High frequency gyrotrons and their application to tokamak plasma heating

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