Lebedev, A N

Martynchuk, N N

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Particle Accelerators, 1993, Vol. 41, pp. 163-172Reprints available directly from the publisherPhotocopying pennitted by licence only© 1993 Gordon & Breach Science Publishers, S.A.Printed in the United States of AmericaBEAM LOADING OF AN ACTIVE RF CAVITYA.N. LEBEDEV AND N.N. MARTYNCHUKP.N.Lebedev Institute ofPhysics, Russian Academy ofSciences117924 Moscow, Russia(Received 1 September 1992; infinalform 26 April 1993)Electron beam loading of a single-pass active cavity coupled with an external generator is considered in a fixedcurrent approximation. The loading effects can be controlled by proper choice of a generalized scattering matrixdescribing the feeder-cavity coupler. By varying phase shifts at the coupler and its transparency, one can minimizethe backward wave, which is a coherent sum of a reflected wave and a beam radiation field. Thus, the acceleratingfield and/or the limiting current can be increased. Moreover, the phase shift optimization can sustain efficientacceleration over a wide range of loading current variations. The same approach can be useful for lasertron systemsand for autoacceleration regimes.KEY WORDS: RF cavity, coupler, beam loading1 INTRODUCTIONOptimization of various resonant linear accelerators loaded with intense electron beamshas been considered in many publications (see, for example,1,2). Here we shall investigatea particular case of a standing wave structure formed by a chain of electrodynamicallyindependent single-gap cavities with external phasing. The system provides additionalcontrol flexibility for variations of an accelerated current and is of special interest for freeelectron laser drivers, as has been described in3 .A general approach to single-cavity matching can be formulated as minimization ofpower reflected from the coupler back to the generator. For ideal coupling of an unloadedcavity, i.e. for a vanishing backward wave amplitude, the load impedance from the feederside must be real. Besides, an optimal coupling coefficient must be provided to transmitthe power being dissipated inside the cavity. Then, the load impedance will be equal to thewave impedance of the feeder.If the cavity is loaded with an electron beam current of the same frequency, there appearsan additional field source, which provides coherent radiation that changes the phases andamplitudes of the backward wave and the total accelerating field. The total acceleratingvoltage across the gap decreases (beam-loading effect) of course, and ideal matching,meaning a zero amplitude backward wave, is generally impossible. However, the radiativefield remains coherent with the reflected wave, so the coupler parameters can be chosento minimize the total backward wave amplitude and the beam loading effect. In other163164 A.N. LEBEDEV and N.N. MARTYNCHUKwords, proper variation of independent coupler parameters as functions of the current cansustain the total accelerating field and/or increase the limiting current for which accelerationvanishes. Note that the load impedance of the cavity is not necessarily real in this case,and corresponding phase shifts which can be provided by proper dielectric inserts or byother distributed elements are of importance. A similar approach can be used for radiatingsystems in lasertrons or choppertrons to maximize the power irradiated by the beam.2 CAVITY DESCRIPTIONFor present purposes, the resonator can be treated as a device that transforms the electricfield wave coming from the generator into standing oscillations of the field in the accelerat-ing gap. Without specifying its geometry and assuming only that the dimension of the gap isconsiderably less than the distance from the gap to the coupler, one can consider the systemas a succession of two linear transformations described by the scattering matrixes SandC. The former refers to the coupler and the latter is related to the finite distance betweenthe coupler and accelerating gap, as schematically shown in Fig.l. Denoting the complexamplitudes of the field on the external and internal sides of the coupler by the indexes 0 and1, respectively, and the directions of propagation from and to the generator by the upperindexes ±, one can write from the definition of scattering matrix:iyIe£1 •I El.'.FIGURE 1: A model of an active cavity.BEAM LOADING OF AN ACTIVE RF CAVITY(~~) = s (~f).165If losses in the coupler are neglected and the transformation is considered reciprocal,the matrix S should be unitary and symmetrical, which yields four relations between itscomplex elements:SllSr1 + S21S~1 = 1,S12Sr2 + S22S~2 == 1,Sr1S12 + S22S~1 = 0,S12 - S21 = 0and leaves free only three real parameters:S = (-x eXkP(ia) k )X exp(-ia) exp(ij.L),(1)(2)o< k < 1.Here, k = VI - X2 is the coupling coefficient, JL the phase shift of the transmitted waveand JL - a the phase shift of the reflected wave. The transformation matrix of the resonatoritself should take into account the phase shift 8 during propagation of the wave from thecoupler to the gap (dependence of the field on time is chosen rv e-iwt ), irreversible ohmicdamping of amplitude described by the coefficient q < 1, and reversible coefficient ofamplitude transformation from coupler to gap G. In general formwhere(E1) = c (Et)Et E2 ' (3)c= (. 0q G exp(i8)G- 1 eXP(i8))o .There still remains to relate the amplitudes of waves E~ leaving and coming to the gap(Fig. 1). In the absence of a beam, the assumptions of standing oscillations in the gap wouldmean automatically that Et = E2, which in the presence of losses could be realizedphysically for cophasal excitation of the resonator by two couplers symmetrically locatedrelative to the gap. To avoid complicated calculations, we limit ourselves to the case ofsymmetrical excitation.A modulated beam passing through the gap in the plane of symmetry creates an additionalfield of radiation E r coherent with the other fields and propagating from the gap to thecouplers. Therefore,E2 = Et +Erand E a, the total accelerating field in the gap will be(4)166 A.N. LEBEDEV and N.N. MARTYNCHUK(5)Using the algebraic linear relations (3) - (5), we now can express the amplitude of the waveleaving the resonator Eo and the accelerating field Ea in terms of the amplitude of theradiation field Er and the amplitude of the incident wave Et, which we shall consider realfor determining the beginning of phase reckoning. Thus, we obtainwherez+ == 2qk exp( iJ-l + i8)° 1 - Xq2 exp(i~) ,Z- = [q2 exp(i~) - X] exp(ijl + ia)° 1-xq2exp(i~)'z _ 1 + Xq2 exp(i~)r - 1 - Xq2 exp(i~)'~ == 28 - ex + J-l.(6)(7)Returning to what has been discussed above one can see that in the absence of the loadingcurrent the standing wave coefficient Z == (Et +Eo) / (Et - Eo) is equal to unity whenZo == O. The latter means that two requirements mentioned above:(8)must be satisfied. The first shows that the cavity is tuned, and the second indicates that thecoupling coefficient must be small for small ohmic losses.The coefficient of ohmic damping q introduced above is closely related, of course, to theinherent Q-factor of the resonator. To elucidate this relationship, we note that the squareof the modulus of the resonance denominator in the expressions for impedances (7) can bewritten in the form(9)and becomes small for Xq2 ---+ 1, ~ ---+ O. The total phase shift ~ is, of course, a functionof the frequency wand becomes zero for w == wo, where Wo is the fundamental frequencyof the operating mode. In the neighborhood of this resonance value,(10)whereBEAM LOADING OF AN ACTIVE RF CAVITY 167The quantity QoDQo = (11)2(X-1/ 2q-1 - X1/ 2q)can be identified as a quality factor of the resonator, determined by the width of theresonance curve. In the limit, k ~ 0; q ~ 1(12)where the two r.h.s. terms determine the Q-factors connected with extraction of power andwith ohmic losses, respectively. The coefficient of proportionality D- 1 , the value of whichis of the order of unity, depends, of course, on the concrete geometry of the resonator.3 OPTIMIZATION OF THE ACCELERATING FIELDTo determine the power transmitted to the beam, one has to know a relation between thecurrent I ei , and the radiation field Er . For a narrow accelerating gap, one can suppose thatE r = -pIei " i.e., the field is out of phase with the current, exciting a TEM radiation wavein the line with wave impedance p. Then, the power isW = Re(EaIe-i,) = I Re(GZtEte-i , - ZrpI). (13)One can see that an effective accelerating voltage Va = WIIGEt (in units of Et) can beoptimized with variation of the injection phase ry to match it with the phase of the effectiveaccelerating field. ThenVo = zt - Re(Zr)J = 2qk [(1- Xq2}2 +4xq2sin2(~/2)J-1/2 - (14)-J(l - X2q4) [(1 - Xq2)2 + 4Xq2sin2(~/2)J -1 .where J = pIIGEt (remember that Et is supposed to be real and positive, definingrelative phase shifts).Eq.(14) describes a two-parameter family of straight lines. Each shows a linear decreasein the accelerating voltage with current starting from a "cool" cavity value and vanishingat a certain limiting current. The value at J = 0 is maximized with ~ = 0, and with theoptimal coupling coefficient k = J1=Q4 = Q-1/2:(15)2qVmax = . ;:;---::;r.V 1- q4(Of course q ~ 1 means an infinitely large Q-value of the cavity and hence makes equation(15) not valid). For larger coupling, the field amplitude decreases because of radiation168 A.N. LEBEDEV and N.N. MARTYNCHUKthrough the coupler; for smaller ones - because of ohmic losses. Note that k == Q-l/2provides at the same time a maximal sensitivity to loading, i.e., this choice minimizes thelimiting current to the value J == 2q/ (1 + q4).According to the above considerations, one can optimize the loading characteristic inFig.(2), changing Ll from 0 at J == q/ (1 +q4) to 1r at J == q/yI1=q4. At the left boundaryof this interval, the optimized voltage is Vo == qjyI1=q4, at the right side it is equal toq/(l + q4). (Note that all curves in Fig.(2) are symmetric with respect to the bisectrixVo == J). Within the optimization intervalq2VoJ == ------(1 + q4)yI1=q4' (16)so the transmitted power and acceleration efficiency are independent of the loading current.For larger currents, i.e., for q/yI1=q4 < J < 2q/yI1=q4, the voltage is low and againdecreases linearly with current when Ll == 1r. Note that for an arbitrary coupling coefficientthe phase optimization is available in the intervalqk J qk---. < <---1+ Xq2 1- Xq2Vo3.53Z.5Z1.510.500.5 1 1.5 2FIGURE 2: Accelerating voltage vs beam current (q=O.9):1.- ~=O,2.- ~=7r,3.- ~=~opt,4.-k=kopt 'J(17)BEAM LOADING OF AN ACTIVE RF CAVITY10.80.60.40.20 00.5 1 1.5 2 Z.5 a JFIGURE 3: Optimal coupling coeffcient k and detuning ~ vs beam current (q=O.9).169JFIGURE 4: Transmitted power vs beam current (q == 0.9): 1.- ~=~oPt, 2.-k=kopt .17010.80.60.2oA.N. LEBEDEV and N.N. MARTYNCHUK................................,...............~.......................................-..-................. JQfl20.2 0.4 0.6 0.8 1FIGURE 5: Reflected power vs beam current (q=O.9):1.- A=O, 2.- A=1r, 3.- A=Aopt , 4.- k=kopt •P10.80.6o.~0.20 0.5 .1 1.5 2 2.5 3FIGURE 6: Accelerating voltage vs beam current for Q»l:1.- A=O, 2.- A=Aopt, 3.- k=kopt 'and yields the voltageBEAM LOADING OF AN ACTIVE RF CAVITY 171(18)q2k2Va = J(1 _ X2q4) ,which has an upper limit Va == q for k == 1, when the interval shrinks to a point J == q.The existence of the upper limit indicates that one can optimize Va with k. Its value hasto drop from~ at J == 0 down to zero at J == q and then to increase with ~ == 1rup to~ when the current approaches the maximal limiting value J == 2q/~.The corresponding envelope equation can be written in a parametric form:(19)o< X < q2.Thus, optimization with phase shift and with coupling coefficient can increase the acceler-ating voltage and/or the limiting current. Moreover, the phase optimization allows constantefficiency while the current varies over wide limits. The corresponding dependencies areshown in Figs.(3), (4),(5). Note, however, that to use effectively the resonant properties ofthe system and to keep Va >> 1 one has to deal with Q >> 1 and J << 1.The optimization in this limit gives:txtfor ~ == 0, X == q2;Va == 1/4J (20)for ~==O, X==X(J).Ql/2Va = JQl/2 + (PQ + 1)1/2The corresponding universal curves are shown in Fig. (6).4 RADIATION REGIMESThe above considerations can also be applied to systems where the modulated beam-cavityinteraction is used to obtain RF power. In these cases, the backward wave power P and thebeam radiation losses are the objects of optimization. The optimal choice of the injectionphase "y gives:(21)172 A.N. LEBEDEV and N.N. MARTYNCHUKTaking into account that according to (7)Iz-12 == 1-IZ+121- q4o 0 4q2one can easily get the maximal value(22)(23)Pmax = 1 + q2J2~t2 (1 - q4)'which can be reached with a certain optimizing value (J, ~). In the limit ~t == 0 whenthe second term in (23) dominates Pmax == q212 /G2(1- q4). Thus introducing an externalwave for additional deceleration of the beam, one cannot get back more than the injectedpower plus the "natural" radiation power.In the case of an autoacceleration regime, or two-beam acceleration scheme, the internalfield amplitude ~a is to be maximized irregardless of the exciting beam phase. Accordingto (7) the maximum of Zr can be reached with ~ == 0 and X == 1 and(24)Note, however, that all calculations above are relevant mainly to relativistic electron beamsbecause in a fixed current approximation particle energy variations in the gap should besmall enough not to change essentially a particle velocity.5 CONCLUSIONSThe model considered above indicates that an accelerating voltage across a cavity gapcan be optimized by proper choice of coupler phase shift ~ and coupling coefficientk, depending on the accelerated current. Phase shift optimization can provide constantefficiency of acceleration over a wide range of beam current variations. The maximized RFpower which can be obtained from modulated beams in lasertrons or choppertrons cannotbe increased by external deceleration.ACKNOWLEDGEMENTSThe authors would like to thank O.A. Smith and L.A. Bankovskaya for the help in preparingthe paper.REFERENCES1. P. Lapostolle, A. Septier. Linear accelerators. North-Holland Publishing Company. Amsterdam. 1970.2. G. Dome. RF system: Wave guides and cavities, AlP Conference Proceedings 153 Physics of ParticleAccelerators, SLAC Summer School (1985).3. A free electron laser at P.N. Lebedev Institute. BEAMS-92 Proceedings. Washington, USA. 1992.

Beam Loading Of An Active Rf Cavity

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