Hsu, I C

CERN Document Server

Particle Accelerators, 1990, Vol. 34, pp. 43-52Reprints available directly from the publisherPhotocopying permitted by license only© 1990 Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaTHE DECOHERENCE AND RECOHERENCE OF THEBETATRON OSCILLATION SIGNAL AND ANAPPLICATIONIAN C. HSUtDepartment of Physics and Synchrotron Radiation CenterUniversity of Wisconsin-Madison, Madison, WI 53706 U.S.A. *(Received June 15, 1989)The understanding of the strongly synchrotron frequency modulated collective betatron oscillationsignal is achieved by both an analytical result and computer simulations. Those theoretical resultsagree well with the measurement data. One of the applications is the measurement of the value of thechromaticity times energy spread.1. INTRODUCTIONIt is observed in many accelerators and storage rings that after a horizontal kickeris fired to excite a single-bunch beam, the envelope of the betatron oscillationfrom a horizontal pickup is strongly modulated by the synchrotron frequency; see[Fig. l(a), (b)]. This can be understood qualitatively as being due to thecombined effects of the synchrotron motion and the machine chromaticity. 1The phenomenon of decoherence and later recoherence of collective oscilla-tions also occurs in plasma waves,2 where it is called an echo. If we apply anexternal disturbance to the plasma, the macroscopic field produced will decaythrough Landau damping (decoherence), but the disturbance in the microscopicmotions remains. If we apply a second disturbance, the microscopic motions maylater recohere, and a macroscopic field (the echo) may reappear in t4e plasmamany Landau-damping periods after the application· of the second disturbance. Inthe case of synchrobetatron oscillations in an accelerator, the phase adjustmentsthat cause recoherence are due not to a second external disturbance, but to theeffects of synchrotron oscillations on the betatron frequencies.We study this phenomenon by pulsing an injection kicker magnet to a 400-MeVsingle-bunch stored beam in Aladdin, which is a I-GeV electron storage ringserving as a light source. After the kicker pulse dies out, the beam will execute acoherent free betatron oscillation. A beam position monitor electrode which cansense the average position of the bunch is used to pick up the beam signal at eachturn. After the detailed study of the phenomenon, we will find that the value ofthe product of the chromaticity and the energy spread can be measured fromthese signals. This provides a new method for the measurement of thet Work supported by the National Science Foundation.* Present address: SLAC, P.O. Box 4349, Stanford, CA 94305.4344 IAN C. HSU(3)chromaticity or the energy spread, as soon as we get one of the two values bytraditional methods.3,42. THEORYTo study the theory of this phenomenon, we assume that every electron executesa free betatron oscillation with tune Vi. Due to the original synchrotronoscillation, each electron has an energy difference ~Ei' with respect to thesynchronous electron. If the horizontal chromaticity; is not zero, then electronswith different energies will execute free betatron oscillations with different Vi:(~P)i (~P)iVi = Vo +;-p = Vo +;-p cos (Vswt + 1/Ji), (1)where P is the momentum of the synchronous electron, (~P)i is the momentumoscillation amplitude of the electron i, Vs is the synchrotron oscillation tune, w isthe angular revolution frequency, and 1/Ji is the phase of the momentumoscillation of the electron i.After some number of betatron oscillation periods, the signal from the pickupbecomes smaller because electrons are no longer in phase and the displacement ofthe center of charge, D, becomes smaller:D(t) =~~ A cos ([ v;(t')w dt' + 4>), (2)where N is the total number of electrons. Notice here that the initial phase 4> isthe same for all electrons since they are given the same transverse impulse at thesame time.After one-half of a synchrotron oscillation period, the betatron oscillationsappear very incoherent. But, if nonlinear effects are negligible, after onesynchrotron period all the electrons will be in phase again and the displacementof the center of charge will be almost the same as just after the kicker was fired(except for a very small damping).To show this, we substitute Eq. (1) into Eq. (2) and expand it. We get:- A ~{ [;(~P)i ]D(t) =N ~ cos (vowt + 4» cos --(sin (vswt + 1/Ji) - sin 1/Ji)1=1 vsP. ( ~) . [;(~P)i(. ( ). )]}- sIn vowt + 'Y sIn ---;;p- sIn vswt + 1/Ji - sIn 1/Ji ·In an electron storage ring, due to the quantum excitation effect, (~P)i isGaussian-distributed with mean value 0, and 1/Ji is independently uniformlydistributed between -1r and 1r. Because the sine function is an odd function, wecan drop the last term in the above equation. It then becomes:- A ~{ [;(~P)i ]}D(t) =- L.J cos (vowt + 4» cos --(sin (vswt + 1/Ji) - sin 1/Ji)N i=1 vsP=A cos (vowt + 4»~(t), (4)DECOHERENCE AND RECOHERENCEFLTRAlTURNSFig. l(a)MAGCAl!a.aaAt~ I! a.aaat~I0.0001-------I~_ _ - .- + _.._ ~._-_ _._ _.-+-.._._ _._ -~ .._ _..__..-+- - -_ --'---o. ,. o. ,. o. '7Fractional part of tune45Fig. l(b)FIGURE 1 The measurement result for the case of non-zero chromaticity: (a) turn-by-turn signal;(b) the FFf of (a).46whereIAN C. HSU1 ~ [1;(~P)i. . ]lC(t) = - L.J cos p (sin (vswt + VJi) - sin VJi)N i=1 Vs(5)is the "envelope function", which is a slow modulation of the betatron oscillationA cos (vowt + 4J).Now we look at Eq. (5); whenever vswt = 2nn, 'iE reaches its maximum value,and whenever vswt = (2n + 1)n, lC reaches its minimum value. This is exactly thereason for the synchrotron frequency modulation we see in Fig. 1. The values of'iEmax and 'iEmin are1 ~ [1;(~P)i. ]'iEmax = -N~ cos --p- (sIn (2nn + VJi) - sin VJi) = 11=1 Vs1 ~ [1;(~P)i. . ]~min = N L,., cos --(sin «2n + 1)n + VJi) - sin VJi)i=1 vsP1 N [1;(~P)i . ]=N L cos --(-2 sin VJi) ·i=1 vsPBy replacing summation with integration and (~P)i/P with (~E)i/E, we get:1 fn foo (1; a€ ) 1 2/2'iEmin =-2 dVJ dxcos -2--xsinVJ ... f;Ce- xn -n -00 Vs E v2n(6)(7)(8)(9)=F(1 a€).Vs EFrom the above discussion we can also see the disappearance of thedecoherence phenomenon when the chromaticity is set to zero. That is the case inFig. 2. The universal function F is only dependent on the value of (1;/vs ) (a€ /E)and the distribution of the charged particles. Therefore if we are dealing with aproton beam, we should replace the Gaussian distribution in Eq. (8) by theparabolic distribution usually assumed for a proton beam. Fig. 5 is a plot of theuniversal function F for the electron beam case.The ratio of Dmax and Dmio is:whereandDmin A cos (vowtmin + 4J) 'iEminDmax = A cos (vowtmax + 4J) lCmax= cos (",owtmin + 4J) F(1 a€)cos (vowtmax + ep) Vs E '(2n + 1)ntmin=----VsW(10)If we look at only the envelope of the oscillation signal, we can drop the fastDECOHERENCE AND RECOHERENCEBUF.Al47IIL--->------'--~------~,00 ~oTURNSFig.2(a)MAGCAlI0.02+~o. 00t.. -----.___+_O.OB~--+­1000--+-------+--------+-._-o. 14 O. US o. 1 BFra.ctional part of tuneFig.2(b)FIGURE 2 The measurement result for the case of zero chromaticity: (a) turn-by-turn signal; (b)the FFT of (a).48 IAN C. HSUbetatron oscillation part. Therefore we getD mio (E a€)-;:r-- = F -- .Dmax Vs E(11)In the above equation Dmeans we only look at the envelope of the oscillationsignal. In this case, we can measure the energy spread, a€, by another method3first, and by measuring the value of D mio/ D max we can get the value ofchromaticity or vice versa. One remark before ending this section is that if thereexists some non-negligible tune spread due to nonlinear fields, then after thedecoherence the echo amplitude will be reduced and will affect the accuracy ofD mio/ D max . This can be easily checked by observing that the second maximumoscillation amplitude is as large as the first.3. SIMULATION RESULTFigure 3 and Fig. 4 are the results of computer simulations using Eq. (1) and Eq.(2) and the fast Fourier analysis spectrum of the results with both the cases of; = 0 and; =1= O. In those simulations (~P)i/P is Gaussian distributed with meanvalue 0 and standard deviation a€/E, and 1/Ji is uniformly distributed between -Jrand Jr. As we described in the previous section, the Gaussian distribution is onlyvalid for the high-energy electron case and is due to the quantum excitationeffect.In Table I, we compare {;(a€/E)}theory, which we got by measuring Dmio/Dmaxfrom the simulation result, with the product value of ; and aE / E (which we putinto the simulation) for several different values of chromaticity and energyspread. The last column is the percentage error which is calculated as{; a€} _ (;) x (a€)E theory E(;) X (~)In the above example, since D mio/ D max is measured from the simulation result,the accuracy can be very high. Therefore, the major error is due to the fact thatwe only use a finite number of particles to simulate the Gaussian distribution. Forthe results in Table I, we used 10,000 particles in the simulation. If we use, forexample, 5000 particles, the 1.0% error in the third row will go up to 3.9%. In areal measurement using the proposed method, the major error will be due to theinaccuracy of the measurement of D mio/ D max . This has been discussed in a paperreporting measurements of energy spread by the proposed method. 5 In that paperthe authors compare the energy spread measured by the bunch length measure-ment method and the proposed method. Since the beam current in thatmeasurement was well below the microwave instability, the energy spread shouldbe a constant for different chromaticities and for different currents. The authorscompare the constancy of the energy spread for different current and differentDECOHERENCE AND RECOHERENCE 49CHROMATICITY = 0.001.0If.)...,.°2 0.5~.a~...,.s::Q)S 0.0Q)u'"]..0-0.5-1.0o 200 400 600TURNSENERGY SPREAD = 2.40 E-04Fig.3(a)800 1000500000If.)oS~ 400000.a~Q)"'0==';::; 300000Q.,S<:20000010000000.170.15 0.16Fractional part of tuneFig.3(b)FIGURE 3 The simulation result for the case of zero chromaticity: (a) turn-by-turn signal; (b) theFFf of (a).50 IAN C. HSUCHROMATICITY = 9.001.0(I)'2 0.5='..Q~-=~a 0.0~un5].is-0.5-1.0o 200 400 600 800 1000TURNSENERGY SPREAD = 2.40 E-04Fig.4(a)0.170.15o2500012500050000~"'tl~ 75000Q..e<fI2'a=' 100000..ci~0.16Fractional part of tuneFig.4(b)FIGURE 4 The simulation result for the case of non-zero chromaticity: (a) turn-by-turn signal; (b)the FFf of (a).DECOHERENCE AND RECOHERENCE 51TABLE IComparison with SimulationaE x 10-4 ~ m(i) x 10-4 Drnio {SO<'} X 10-4 % errorE Drnax E theory9.0 21.60 0.263 21.86 1.22.4 6.0 14.40 0.442 14.72 2.23.0 7.20 0.764 7.27 1.09.0 64.80 0.083 65.60 1.27.2 6.0 43.20 0.129 42.44 -1.83.0 21.60 0.274 21.0 -2.5chromaticities. The result from the proposed method is about a factor of threebetter than that from the bunch length measurement method.4. CONCLUSIONAfter analyzing the decoherence and the recoherence of the coherent transversebetatron oscillation, we have concluded with a method for the measurement of;(aE/E). The advantages of this method are:1. It is machine-independent. Even for a proton machine, we need only changethe particle distribution function in Eq. (8) to the parabolic distributionusually assumed.1.00.80.6F0.40.20.0o 2 3-.t~vljEFIGURE 5 The universal function F as function of ;aE/vsE.52 IAN C. HSU2. It is current-independent. The result depends on Dmio/ Dmax in which currentcancels out.3. It is independent of the kicker strength, provided we do not kick so hard as toget into the nonlinear region.4. It can be used even at very small currents, in which case the bunch lengthmeasurement method for measuring energy spread is difficult.However, the disadvantage of this method is that the measured value is theproduct of two unknowns. If we want to get one of the unknowns by this method,the result will depend on the accuracy of the other value which must be obtainedby another method. The linearity of the signal pickup system is important. It isalso clear from Fig. 5 that, for (;aE/vsE) > 2.0, a small measurement error inDmio/ Dmax will produce a large error in ;aE/ vsE.ACKNOWLEDGMENTThe author wants to thank W. Trzeciak, K. Kleman, and the operation group ofSRC for the experimental data presented in this paper and A. Chao (SSC) and D.Eisert (SRC) for a valuable comm~nt on the simulation part of this paper.Helpful discussions with K. Symon (UW-Madison) are appreciated.REFERENCES1. D. Edwards, R. P. Johnson, and F. Willeke, Part. Accel., 19, 145 (1986).2. J. H. Malmberg, C. B. Wharton, R. W. Gould, and T. M. O'Neil, Phys. Rev. Lett., 20, 95 (1968).3. S. Chattopadhyay, M. Cornacchia, A. Jackson, and M. S. Zisman, SRC Note, 45 (1985).(Available by request from the Synchrotron Radiation Center, University of Wisconsin-Madison.)4. W. Trzeciak, and I. Hsu, SRC Note, 75 (1987). (Available by request from the SynchrotronRadiation Center, University of Wisconsin-Madison.)5. I. Hsu, and W. Trzeciak, Proc. IEEE Particle Accelerator Conf., March 1989.

The Decoherence And Recoherence Of The Betatron Oscillation signal And An application

Abstract

Similar works

Full text

Available Versions

CERN Document Server