Chen, C.

Wurtele, J.S.

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PFC/JA-89-17THEORY OF EFFICIENCY ENHANCEMENT IN CARM AMPLIFIERSBY MAGNETIC FIELD TAPERINGC. Chen and J.S. WurteleMarch, 1989Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, MA 02139USAThis work was supported by the U.S.Division of Nuclear and High EnergyDepartment of Energy,Physics.Submitted for publication in: Physical Review A.Theory of Efficiency Enhancement in CARM Amplifiersby Magnetic Field TaperingC. Chen and J.S. WurteleDepartment of Physics and Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139ABSTRACTThe efficiency of cyclotron autoresonance maser (CARM) amplifiers with piecewiselinear tapering of the magnetic field is analyzed. In the low current limit, we findthat increasing the magnetic field substantially enhances the efficiency if an effectivedetuning parameter is positive, while decreasing the magnetic field is advantageouswhen the detuning parameter is negative. For high current, high gain, CARM opera-tion the efficacy of tapering is found to be reduced in a parameter regime where thesaturation wave amplitude becomes of the order of an effective detuning parameter.PACS numbers: 42.52.+x, 41.70.+t2The cyclotron autoresonance maser (CARM), potentially a tunable high power co-herent radiation source from submillimeter to millimeter wavelengths, has attractedconsiderable attention. Extensive theoretical and computational effort has been madein the studies of the CARM interaction, including the theory of radiation amplificationin the cyclotron resonance maser (1-3], the kinetic theory of CARM with both planar[4] and circular [5] electromagnetic waves, as well as waveguide mode configurations[6], the nonlinear efficiency studies [7], the stabilization of the CARM instability byintense electron beams [8] and momentum spread [9], absolute instabilities [10], andsimulation studies of CARM amplifiers [11-15]. Experimental results on the CARMoscillator have been reported recently [16]. The CARM interaction takes place whenthe electrons undergoing cyclotron motion in a uniform axial magnetic field Boa, in-teract with an electromagnetic wave (w, k) propagating along the z-direction. Thecyclotron resonance condition isw - kilo' = , (1)where v, and -y are respectively the axial velocity and relativistic mass factor of theelectron beam, s is the harmonic number, weo = eBo/mc is the nonrelativistic cy-clotron frequency, m and e are respectively the electron mass and charge, and c isthe speed of light in vacuum. Making use of the phase velocity vo = W/kIl and takings = 1 (the fundamental cyclotron frequency), Eq. (1) becomesoco= = WD, (2)7(1 -3gg//34)-where 311 = v./c, 30 vb/c = w/cklj, and WD is known as the Doppler shiftedcyclotron frequency.In this rapid communication we analyze, for an arbitrary choice of parameters, the effi-ciency of single mode CARM amplifiers with piecewise linear tapering of the magnetic3field. The efficiency, denoted by 7, is defined as 1 = (< YO > - < -y >)/(< -yo > -1),where < y0 > mc2 and < -y > mc2 are respectively the average initial and final ener-gies of the electrons. In the low current limit, we find that increasing (or decreasing)the magnetic field substantially enhances the efficiency when an effective detuningparameter Def f is positive (or negative). (The definition of Deff will be given later.)For high current, high gain, CARM operation, this result remains applicable, exceptwhen the saturation wave amplitude is large compared with an effective detuningparameter.Consider a beam of relativistic electrons of density n undergoing cyclotron motion ina magnetic field fo(z,r) interact with a right-hand polarized electromagnetic wave(w, k6,) described by the vector potentialA(z, t) = A(z)[d. cos 4(z, t) - 6, sin 4(z, t)], (3)where q(z, t) = kz - wt + 8 (z) is the wave phase, and both the wave amplitude A(z)and the phase shift 8(z) vary slowly within one wavelength. In polar coordinates, themagnetic field is assumed to have an azimuthally symmetric form [13]Bo(z,r) = Bo.(z)6&. + Bo,(z,r)6,, (4)where Bo,(z,r) = -(r/2)(dBo.(z)/dz) is the radial component of the magnetic field.Using a system of units in which e = m = c = Bo0 (0) = 1, i.e. introducing the di-mensionless variables and parameters Bo2 (z) -- Bo,(z)/Boz(O), wCo(z) -+ wCO)/n,W -+ wP/, z --+ cz/fl, v, -+ v,/c, /v -+ vO/c, Pz -+ pZ/mc, Pt -+ pt/me, andA - eA/mc2 , with fl = eBo.,(0)/me, we can write the magnetically tapered CARMequations asdV wy Wo(z) wA 1 y d(z)d k- + + - )sinO + , (5a)dz P P Z Pt VO PZ dz4d-y wApt- - cos O, (5b)dz p,dp. k Apt Pt dBo(- O to (5c)dz P 2pzBoz dzdA 2 V_ O Pt-= - P (-- Cos ),(5d)dz 2k pd- w Pt s (5e)dz 2kA psm),(where w = 47ree2n/m is the electron plasma frequency squared, vZo is the axial velocityof the electron beam while entering the z > 0 interaction region. Detailed derivationsof the self-consistent equations describing the CARM amplifier with magnetic fieldtapering have been presented in Ref [13] and [14]. The first three equations describethe motion of the electrons in terms of the axial momentum pz, energy y, and phasekz - wt - tan1 (p/py) + 8(z). The last two describe the wave evolution, where(.. N-1  ' i ... denotes the average over all the electrons. In this N-particle modelthere are total of 3N + 2 equations since the transverse momentum pt is solved from-y (1+ p2  p2). From Eqs. (5b) and (5d), the total energy flux of the electronbeam and electromagnetic wave field, nvzo-y + (1/47r)wkA 2 , is a constant of motion.Moreover, the magnetic moments of the electrons are adiabatically conserved in theabsence of the wave field.In the linear instability regime, the electrons undergo a transition from random tobunched phase distributions such that more electrons give up their kinetic energy tothe electromagnetic wave. In the low current limit, since the wave amplitude andspatial growth rate are small, such phase bunching can be described as follows. Letus consider an untapered CARM amplifier operating at the frequencywcoW WD -+ Aw = ol- ~/# + Aw, (6)5where Aw and yo are respectively the frequency detuning and initial beam energy. Bydifferentiating Eq. (5a) and making use of Eqs. (5b), (5c), and (6), and the fact thatPz(z) - pzo = O(A) and pt(z) - pto = O(A), it is straight forward to show that thedynamics of the electron phase b is approximately govern by the pendulum equationd2- Deff ( k0 )A(z) cos b + O(A2 ), (7)where Opo = pto/pzo is the initial pitch angle of the electron beam. Here we haveintroduced an effective detuning parameterDff = 1 - v + [(_!I'-- 1) + (_ _ 1)2 . (8)Thus, the electrons bunch at the synchronous phase 0, = 7r/2 for Deff > 0 since thestable fixed point is located at (,0, db/dz) = (7r/2, 0) and the unstable fixed point at(V), db/dz) = (37r/2, 0). For Def f < 0, the synchronous phase is /, = 37r/2. Def f van-ishes when vo = 1 and Aw = 0, revealing the well-known cyclotron autoresonance phe-nomenon [7], where the electrons remain in synchronization with the electromagneticwave in the course of evolution. Typically, the growth rate is small when Def f = 0.To get an intuitive picture of efficiency enhancement using magnetic field tapering, itsuffices to analyze the motion of the electrons with phases close to the synchronousphase 0,. Since in the (b, p,) plane the phase dependence of the axial momenta ofthese electrons can be approximated as p,() ~ p(4,) + (dp.(O,)/d)( 0 - 0,), thechange in d/dz due to the magnetic field change SBOz (or Swco) is then given by theTaylor expansion(_) _ [ 1 dp (0,)( - ,)] (9)dz Pz() pz(4,) pz(0,,) db6where pz,(,) is the axial momentum of the electron with ) = k,. We shall argue inthe following that increasing (or decreasing) the magnetic field enhances the efficiencywhen D, 1 1 is positive (or negative). For Df f > 0, both d-y(4,)/d/ and dp.(.,)/ddbare negative before nonlinear saturation. This occurs because the synchronous elec-trons with b > 0, = 7r/2 lose energy and axial momenta while those with '0 < 0,gain energy and axial momenta, as seen from Eqs. (5b) and (5c). Eq. (9) impliesthat as the magnetic field increases the synchronous electrons are forced to the rightin the (0,-y) plane, so that the number of the electrons with phases situated in theinterval (7r/2, 37r/2) increases or the electrons continue losing energy on the average.In short, increasing (or up tapering) the magnetic field yield efficiency enhancementwhen Def f > 0. Similarly, decreasing (or down tapering) the magnetic field enhancesthe efficiency when Dej < 0.Fig. la shows the untapered and corresponding optimally tapered efficiencies as func-tions of the relative frequency detuning Aw/w, obtained from self-consistent simula-tions with Eq. (5), cold electron beams, and piecewise linear tapering. For the resultsin this figure the simulations had v4 = 1, corresponding to the CARM operating invacuum, -yo = 2.37, and Opo = 0.53. The dimensionless electron plasma frequencyU, = 0.05, which is representative of the low current limit. In order to obtain the ta-pered efficiencies, upward tapering was used for the high frequency branch (O > WD),while downward tapering for the low frequency branch (w < WD). To achieve optimalefficiency we start tapering somewhat before nonlinear saturation occurs. Usually theabsolute value of the slope for optimal tapering ranges from 0.04 to 0.08'Bo.(0). Asimilar plot is shown in Fig. lb for v4 = 1.03 and w, = 0.005, where only the negativeDe11 branch is plotted since the growth rate is small for the positive Df f branch andlow current. This figure corresponds to the CARM operating with a waveguide mode.(Since the transverse variation of the rf field and the forces due to the longitudinalrf field can be neglected under the conditions that 1 - v.-2 << 1 - V'O/vo, the elec-tron Larmor radius rL << 1/k±, and the electron beam radius r, << 1/ki [13], theone-dimensional model given by Eq. (5) provides a good description for the CARM7operating in waveguide.)For high current CARM operation, the validity of the pendulum equation (7) breaksdown because the term of the order of A2 can no longer be ignored in the expansion.Indeed, when deriving Eq. (7), one has to differentiate for instance the sine term inEq. (5a), which contributes a term of the order of A 2 sin(2O) to the right hand sideof Eq. (7). For example, if the term proportional to A 2 sin(24) dominates, then theelectrons will equally bunch at two phases differing by 7r. To the lowest order, taperingdoes not yield net gain in energy extraction because the electrons bunched at 0, giveup energy, while those at 0, +7r gain energy, or vice versa. Typically, multiple phasebunching occurs and the efficacy of tapering is reduced when the saturation waveamplitude A,.t is of the order of DCff k 2Go/pO. A qualitative criterion for taperingto result in efficiency enhancement isAsat < ID.f |(k po). (10)PZOIn Fig. 2 we plot the untapered efficiency (dashed curve), corresponding taperedefficiency (solid curve), and dimensionless ratio Aatpzo/1De1 1k 20PO (dotted curve)as functions of Deff, where downward tapering is used and the parameters are thesame as in Fig. 1b, except now w, = 0.3.We conclude that in the low current limit the efficiency of CARM amplifiers can besubstantially enhanced by increasing (or decreasing) the magnetic field if an effec-tive detuning parameter is positive (or negative). Moreover, for high current, highgain, CARM operation this result remains valid, as long as condition (10) holds ap-proximately. When condition (10) is violated, the effect of tapering on the couplingbetween the electrons and electromagetic wave field becomes delicate and requiresfurther investigations.8ACKNOWLEDGEMENTThis work was supported by the U.S. Department of Energy, Division of Nuclear andHigh Energy Physics.9REFERENCES1. R.W. Twiss, Aust. J. Phys. 11, 564 (1958).2. J. Schneider, Phys. Rev. Lett. 2, 504 (1959).3. A.V. Gapunov, Addendum. Izverstia VUZ. Radiafizika 2, 837 (1959).4. E. Ott and W.M. Manheimer, IEEE Trans. Plasma Sci. 25, 1 (1975).5. R.K. Chu and J.L. Hirshfield, Phys. Fluids 21, 461 (1978).6. P. Sprangle, and W.M. Manheimer, Phys. Fluids 18, 224 (1975), and P. Sprangle,and A.T. Drobot, IEEE Trans. Microwave Theory Tech. 25, 528 (1977).7. V.L. Bratman, G.G. Ginzburg, G.S. Nusinovich, M.I. Petelin, and P.S. Strelkov,Int. J. Electron 51, 541 (1988).8. R.C. Davidson, and H.S. Uhm, Phys. Fluids 29, 2713 (1986).9. R.C. Davidson, and P.H. Yoon, preprint10. J.A. Davies, Phys. Fluids B, (1989).11. A.T Lin, Int. J. Electron 57, 1097 (1984).12. A.T. Lin, and C.C. Lin, Int. J. Infrared Mill. Waves 6, 41 (1985).13. A.W. Fliflet, Int. J. Electron. 61, 1049 (1986).14. K.D. Pendergast, B.G. Danly, R.J. Temkim, and J.S. Wurtele, IEEE Trans.Plasma Sci. 16, 122 (1988).15. B.G. Danly, K.D. Pendergast, R.J. Temkin, and J.A. Davies, In Proc. SPIE, Vol.873,143 (1988).16. I.E. Botvinnik, V.L. Bratman, A.B. Volkov, N.S. Ginzburg, G.G. Denisov, B.D.Kol'chugin, M.M. Ofitserov, and M.I. Petelin, Pis'ma Zh. Eksp. Teor. Fis. 35, 418(1982).10FIGURE CAPTIONSFigure 1The untapered (dashed line) and corresponding optimal tapered (solid line) efficienciesare plotted as functions of detuning parameters in the low current limit. a) The choiceof vo = 1 corresponds to the CARM operating in vacuum. Other parameters usedin the simulations are w, = 0.05, yo = 2.37, and 6 po = 0.53. Upward tapering wasused for the high frequency branch with Aw > 0, and downward tapering for the lowfrequency branch with Aw < 0. b) A similar plot for w, = 0.005 and vo = 1.03,corresponding to the CARM operating in a waveguide mode for the negative Doffbranch.Figure 2The untapered efficiency (dashed line), corresponding optimal tapered efficiency (solidline) and ratio A,.tpzO/|DeffIk2 po (dotted line) as functions of Deff for vo = 1.03,WP = 0.3, yo = 2.37, and Opo = 0.53.110.00Figure 1(a)100.08 0.0 -60.0[-0. I 1 , I I ,I I - I~r -r*-*--*~~ ~ 0-0-@~*= = -.40.01-20.0-0.0- 0.10 -0.05 0.05 0.10I ,.,1280.060.0-o 40.0-40.0-00.0 1 I I I I . . ,-0.065 -0.060 -0.055 -0.050DeffFigure 1(b)1350.040.030.0020.0110.00.0-0.15 -0.10- 0.05Def f00000000000 00000- 0.000 w=.. *--.-t w --0Figure 22.52.01.5 0N01.0a.:4-00.5O000.00

Theory of efficiency enhancement in CARM amplifiers by magnetic field tapering

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Theory of efficiency enhancement in CARM amplifiers by magnetic field tapering

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