Chan, K C D

Rangarajan, G

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Particle Accelerators, 1990, Vol. 33, pp. 135-140Reprints available directly from the publisherPhotocopying permitted by license only© 1990 Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaTRANSIENT RESISTIVE-"WALL EFFECTS OF A BUNCHEDELECTRON BEAM IN A WIGGLERGOVINDAN RANGARAJANDepartment of Physics and Astronomy, University of MarylandCollege Park, MD 20742, U.S.A.K.C.D.CHANLos Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.Abstract The effects of transverse resistive-wall instability on the operationof a free-electron laser (FEL) are analyzed. The equation of motion for abunched beam was previously solved analytically and a steady-state solu-tion shown to exist. The possibility that the transient state could dominatethis steady state is now considered. The maximum transient amplitude isobtained for the case where the focusing force dominates the resistive-wallforce. An analytic expression for this amplitude when the bunches come inwith random initial displacements is also derived. Results are compared withnumerical simulations and are applied to various FEL's to determine if thetransverse resistive-wall instability could pose a problem in their operation.INTRODUCTIONWhen an electron bunch travels through a pipe of small radius in the wiggler ofa free-electron laser (FEL), it generates a wakefield behind it if the pipe is notperfectly conducting. This wakefield affects the transverse motion of the bunchesthat follow and can lead to beam loss. This possibility was considered in ourearlier papers!,2. Transverse displacement of the bunches was shown to attaina steady-state value. Criteria were established to determine when this steady-state displacement could grow exponentially with the length of the wiggler leadingto instability. However, only a heuristic treatment of the transient state (thatprecedes the steady state) was given. Since the transient state could have a largedisplacement leading to beam loss even if the steady state is well behaved, it isimportant to analyze the transient state in a more careful fashion. This is the goalof this paper.The analysis is restricted to the case where the dominant effect is from thefirst mode in an expansion of the wakefield. This single-mode solution is a goodapproximation to the complete solution when the beam-pipe thickness is small.First, an integral representation of the solution is obtained. Second, this integral[1825]/135136/[1826] G. RANGARAJAN AND K. C. D. CHANrepresentation is used to derive analytic formulae characterizing the transient statefor two different cases - the case where only the first bunch has an initial displace-ment (called the "single pulse" case henceforth) and the case where bunches comein with random initial displacements. Results are compared with numerical simu-lations and are also applied to various proposed FEL's to determine whether thetransient state could lead to beam breakup. Because of space limitations, proofsof equations are not given. These details can be found in Ref. 3.INTEGRAL REPRESENTATION OF SINGLE MODE SOLUTIONThe beam is considered to be a series of bunches traveling with speed v (in thez direction) in a pipe of inner radius b, outer radius d, thickness T( == d - b), andconductivitya. Each bunch carries a charge q and there is a fixed time interval ~between any two bunches. The transverse displacement from the axis of the ]{thbunch is denoted by ~(z == ct, ]{). Thus t == 0 denotes the time when the bunchenters pipe (z == 0).The equation of motion for ~(t, ]{) has been derived earlier2 . If the firstmode dominates, it is given aswhered2~(t ]{) K-ldt; + w6e(t, K) = G L exp(-(K -l)tJ./T)e(t, I)1=0G _ eqv- m/1raTb3 'T == 21rabT.c2(1)(2)(3)Here, the factor G incorporates resistive-wall effects and T is the characteristictime of diffusion of the magnetic field through the wall of the beam pipe. Thequantity Wo in the above expressions is the frequency of the slow betatron motionin the wiggler and / is the usual relativistic factor. The sum over I in Eq. (1)represents a sum of the interactions between the ]{th bunch and the wakefields ofall bunches ahead of it.Equation (1) has been solved exactly3. For the sake of simplicity, we givebelow only the solution for the case when all the bunches are assumed to havezero initial velocity:e(t,K) = ~ exp(-ktJ./T)~ (1rwot)1/2 (Gt)n ~ (k -1)L...J L...J 2 2wo n! k-nk=O n=OX In_l(WOt)~(O,]{ - k). (4)2A BUNCHED ELECTRON BEAM [1827]/137(5)(6)To obtain a transient state solution, it is convenient to transform the aboveresult into an integral representation (see Ref. 3 for the derivation):K~(t, !{) == L (_I)k exp(-k~/T)R1(t, k) ~(O,!{ - k)k=Owhere1 f 1 [( 2 Gu2 )1/2 ]R 1(t, k) == -. du 2k+1 cos Wo + --2 t.21rl U 1 + uThe contour in Eq. (6) encloses the origin and Iu 1< 1 everywhere on the contour.Equation (5) is now in a form that can be used to derive the transient statesolution.TRANSIENT STATE FOR SINGLE PULSEIn this Section, the transient state is derived for a beam where the first bunch isdisplaced off-axis by an amount d and the subsequent bunches follow on-axis. Thiscase will be referred to as the "single pulse" case. It is assumed that w5 ~ G, anassumption valid in most practical situations. The results obtained are comparedwith numerical simulation.Setting ~(O, 0) == d, ~(O, k) == 0 for all k > 1 in Eq. (5), the integral represen-tation of ~(t, !{) for a single pulse is given as follows~(t,K)=d(-l)K eXP(-Kb./T)2~i f dUu2~+1 cos[(w~+1~U~2r/\l (7)A saddle point evaluation of the integral gives3 :d (2!{Gt) 1/4 [!{~ (!{Gt) 1/2]~(t,!{) ~ -- exp -- + --!{ vI8i Wo T Wo{ [ Gt (}'?"Gt) 1/2] }x Re exp iwot - ii + i 4wo - i ~o . (8)At a given location z == vt along the wiggler, ~(t, !{) reaches a maximum when1/2 [ ] 1/2!{1/2 ~ (Gt) ~ + (Gt) _3~ ~m 4wo 2~ 4wo T 2~ (9)The normalized (setting d == 1) maximum transient displacement for a single pulseis found to bec ( zr ) ~ 1 (2!{mGt) 1/4 [_ !{m~ (!<mGt) 1/2]':,m t, .1'\m - r ro= exp T + .!1.. m y81r Wo Wo(10)138/[1828] G. RANGARAJAN AND K. C. D. CHAN5 I I I I4 . • numerical simulation- . -3x theoretical prediction-. I--.:I2 •- •• I",.......•e •»J' 1 - I--"'-'".5 ..0 - •• ... I--......-1 . -. ~..I •.....-2 I • ..-3 I I I I0.5 0.6 0.7 0.8 0.9000 (1098-1)FIGURE 1 Variation of the logarithm of normalized maximum transversedisplacement ~m in the hypothetical FEL as a function of the betatron fre-q~ency Wo for the case of a single pulse. The results obtained using theanalytical formula (Eq. (10)) are compared with those obtained using nu-merical simulations.Five proposed FEL's were studied in Ref. 3. In all these cases, the steady statewas found to dominate the transient state. And since the steady state is wellbehaved in all these cases, resistive wall instability should not be a problem.To verify the validity of Eq. (10) when the transient state is dominant, ahypothetical FEL with extreme values of parameters is considered. This FEL hasI == 4.6A, pipelength == 600m, b == 0.18cm, d == 0.198cm, T == 0.46JLs, and G ==1.4 X 1013s-2 . Figure 1 gives the comparison between numerical simulations andEq. (10) for different values of the focusing strength (i.e. wo). The two results areseen to be in good agreement for large wo's. However they start to diverge slightlyfor small wo's. This is to be expected since Eq. (10) is valid only for focusingforces large compared to the resistive-wall forces. As an additional check, the aboveresults were compared with those obtained earlier. When the nonexponential termsin Eq. (8) are ignored, our results3 agree with the less general results derived byNeil and Whittum4 • We include the nonexponential terms in our analysis sincethey can contribute significant corrections to the maximum amplitude.A BUNCHED ELECTRON BEAMTRANSIENT STATE FOR A RANDOM INITIAL DISPLACEMENT[1829]/139In this Section, we consider the case where the bunches come in with randominitial displacements. An expression for the root-mean-squared displacement isderived and is compared with numerical simulations.First, we rewrite the solution for a single pulse in the following form:where~(t, ]{) ~ ~o [j(t, ]{) + j*(t, ]{)]1 (2]{Gt) 1/4 []{~ (]{Gt) 1/2]j(t, ]{) ~ -- exp -- + --]{ V327r Wo T Wo[..7r .Gt . (]{Gt) 1/2]x exp zwo t - z- + z- - z -- .8 4wo Wo(11)(12)and j*(t, ]{) is the complex conjugate of j(t, ]{). When bunches come in withrandom initial displacements, ~(t, ]{) can be written as a linear superposition ofsolutions of the above form5 • Thus,K~(t, !{) ~ L[j(t, k) + j*(t, k)] ~(O,]{ - k).k=O(13)(14)After considerable manipulations including use ?f the facts that initial dis-placements of two successive bunches is uncorrelated and that f(t, k) oscillatesonly very slowly with k, the final result is obtained as3 :< ~2(t, (0) > ~ 2 1/2 (wo ) 1/4 lor3 / 4 I ~ (t lor ) 12~2 - 7r G '\.m ~m, '\.m .< ~O > twhere < ~2(t, (0) > is the rms value of ~(t, !{) as !{ ---7 00, < ~5 > is the rmsinitial displacement, and !{m and ~m(t, !{m) are given by Eqs. (9) and (10).This result has been compared with numerical simulations for our hypothet-ical FEL. Figure 2 shows the results of numerical simulation for Wo == 9 X 108s-1with < ~5 >1/2== Imm. It gives a value of rv 1.2cm for < ~2(t, (0) >1/2. From Eq.(14) and Figure 1 we obtain a theoretical value of rv 1.3cm for < ~2(t, (0) >1/2.Thus we get a good agreement between theory and simulation.SUMMARYThe transient state of a bunched electron beam in a wiggler was analysed in detailusing the integral representation of the complete solution. Analytic expressions for1815i~I I. \1 \, IJ' "' II ~ I'V129Time (Jls)6G. RANGARAJAN AND K. C. D. CHAN3\ fA I\ \/VV140/[1830]4321~e() 0'-"u..J'-1-2-3-40FIGURE 2 Numerical simulation of the transverse displacement of thebeam in the hypothetical FEL when the bunches come in with random initialdisplacements. The rms value of the initial displacements was taken to beImm. The betatron frequency has the value 9 X 108s-1 .the maximum transient amplitude were derived for the case where only the firstbunch is displaced off-axis and for the case where bunches come in with randominitial displacements. This led to the conclusion that transient state could leadto beam loss if the normalized maximum amplitudes given in Eqs. (10) and (14)exceed unity. However, no such instability was found in the five proposed FEL'sthat were studied.REFERENCES1. G. Rangarajan and K. C. D. Chan, Proc. 1988 Linear Accelerator Confer-ence, Los Alamos National Laboratory Report LA-VR-88-3006 (1988).2. G. Rangarajan and K. C. D. Chan, Phys. Rev. A 39,4749 (1989).3. G. Rangarajan and K. C. D. Chan, submitted to Phys. Rev. A (1989).4. V. K. Neil and D. H. Whittum, Lawrence Livermore National LaboratoryReport VCRL 96712 (1988).5. R. L. Gluckstern and R. K. Cooper, IEEE Trans. Nucl. Sci. NS-32, 2398(1985).

Transient Resistive-wall Effects Of A Bunched Electron Beam in A Wiggler

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