Byrd, J M

Cheng Wen Hao

Zimmermann, Frank

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NONLINEAR LONGITUDINAL DYNAMICS STUDIES AT THE ALSJ. M. Byrdy , W-H Cheng, S. DeSantis, D. Li, LBNL, G. Stupakov, F. Zimmermann, SLACAbstractWe present a summary of results for a variety of studiesof nonlinear longitudinal dynamics in the Advanced LightSource, an electron storage ring. These include obser-vation of decoherence at injection, decay of an injectedbeam, forced synchrotron oscillations and diffusion fromone bunch to the next. All of the measurements were madeusing a dual–scan streak camera which allowed the real-time observation of the longitudinal distribution of the elec-tron beam.Introduction Synchrotron oscillations are inherently non-linear, mostly because of the sinusoidal radiofrequency(RF) voltage used to longitudinally focus the beam. Whenthe bunch length is small compared to the RF wavelength,the RF voltage over the length of the bunch can be con-sidered as linear, minimizing nonlinear effects. In elec-tron storage rings, this is almost always the case. How-ever, as the performance demands on electron rings grow,it becomes increasingly important to understand several ofthe subtle effects caused by the nonlinearities. These ef-fects are also fascinating unto themselves and their studyconstitutes a contribution to the physics of beams. This pa-per presents a summary of results of experimental studiesof several nonlinear effects in longitudinal beam dynamicsin the Advanced Light Source (ALS), an electron storagering optimized for producing high brightness synchrotronradiation. Machine parameters relevant to our experimentsare listed in Table 1. Due to space considerations, onlya qualitative description of the results are presented here.More theoretical and experimental details can be foundelsewhere[2, 5, 6].Section II describes the streak camera used for all of theresults in this paper. Section III presents a study of injec-tion transients with and without beam capture. Section IVpresents measurements of diffusion from one bunch to an-other. Section V describes measurements of forced non-linear synchrotron oscillations. Section VI summarizes thepaper.Dual scan streak camera To make a detailed study of thelongitudinal beam dynamics, we used a streak camera (SC)to observe the evolution of the longitudinal bunch distribu-tion. A schematic diagram of the Hamamatsu C5680 SC isshown in Figure 1. The SC converts the time structure of apulse of synchrotron radiation at optical wavelengths froma bend magnet into vertical deflection at the CCD camera.In our experiments, the vertical deflection plates are drivenThis work was supported by the U.S. Dept. of Energy under ContractNos. DE-AC03-76SF00098 and DE-AC03-76SF00515.y JMByrd@lbl.govParameter Description ValueE Beam energy 1.5 GeVC Circumference 196.8 mfrf RF frequency 499.664 MHzVrf RF voltage 1.1 MVh Harmonic number 328 Momentum compaction 1:6 10−3Qs Synchrotron tune 0.0075rad Long. rad. damping rate 5 10−5‘ RMS natural bunch length 4.5 mm RMS E=E 7:1 10−4Table 1: Nominal ALS longitudinal parameters."Fast" 125 MHzvertical sweep slow horz sweepA B C DE FGtime time"slow" timeFigure 1: Schematic diagram of the streak camera in syn-chroscan mode with dual sweep. A: photocathode, B: ac-cel. mesh, C: vert. deflection electrode, D: horz. defl. elec-trode, E: microchannel plate F: phosphor screen, G: CCDcamera.by a 125 MHz sinusoidal voltage synchronized to the RFfrequency. In addition, there is an optional slow horizon-tal deflection which allows observation of the longitudinalprofile as a function of time. The time scale of the hor-izontal sweep can be adjusted to observe several turns orthousands of turns and can be triggered to coincide with anevent such as injection. For sweep times longer than severalhundred turns, individual turns can no longer be resolvedand so the longitudinal profile appears as a continuous lineacross the image.The synchroscan principle is illustrated in Fig. 2 whichshows the 125 MHz vertical deflecting voltage with respectto the arrival times of 4 storage ring bunches. Only deflec-tions within the marked bands appear on the microchannel,allowing an effective means of gating a particular bunch.The phase of the deflecting voltage can be adjusted to placethe arrival of a single bunch on the center of the screen.Injection transients Optimizing the beam injection intoelectron rings is important for increasing the filling rate andreducing radiation backgrounds. Understanding the injec-0-7803-5573-3/99/$10.00@1999 IEEE. 382Proceedings of the 1999 Particle Accelerator Conference, New York, 1999Synchroscan voltage (a.u.)121086420Time (nsec)bunch 1bunch 2bunch 3bunch 4Figure 2: Vertical deflecting voltage shown with respect tothe arrival times of 4 storage ring bunches. Only deflectionswithin the marked bands are observed.injected beam after 1/4 periodstored beam∆E/EφFigure 3: Illustration of quadrupole bunch oscillations atinjection for bunch mismatched to the storage ring RFbucket.tion process and subsequent transient is particularly impor-tant in damping rings for linear colliders and also for con-tinuous injection (i.e. top–up.) This section describes mea-surements of the longitudinal beam distribution followinginjection in the ALS.When an electron bunch is injected into the storage ring,it’s length and energy spread will either grow or damp tothe natural bunch length and energy spread of the storagering over several radiation damping times[1]. In the ALS,the injected bunch length is about five times the naturalbunch length and the injected energy spread is close to thenatural energy spread. If this mismatched bunch is injectedinto the center of an RF bucket, it rotates in the bucket atthe synchrotron frequency, modulating the bunch length asshown schematically in Fig. 3. However, the nonlinearityof the RF voltage creates a dependence of synchrotron fre-quency on amplitude, causing the bunch head and tail toadvance slower in phase than the bunch center, causing afilamentation of the bunch distribution as shown in the re-sults of a computer simulation in Fig. 4.Shown in Fig. 5a is a SC image of beam 0.3 msec follow-ing injection into an empty storage ring. The vertical axisrepresents the longitudinal bunch length in units of timeand the horizontal axis is time. The bunch length is mod-ulated at half the synchrotron period of 87 sec due to therotation of the mismatched bunch shape as shown in Fig. 3.-6x104 -4-20246∆E/E-1.0 -0.5 0.0 0.5 1.0φNturn =4560Figure 4: Phase space distribution 4560 turns following in-jection for ALS injected beam conditions illustrating phasespace filamentation.-150-100-50050100150Time (psec)0.200.150.100.050.00Time (msec)c) simulation of b)-150-100-50050100150Time (psec)b) 2.7 msec after injection-150-100-50050100150Time (psec)a) 0.3 msec after injectionFigure 5: a) Measured SC image 0.3 msec following injec-tion showing modulation of the bunch length. b) SC image2.4 msec later showing filamentation of the large ampli-tude particles. c) A computer simulated image found fromtracking.Fig. 5b shows the SC image 2.4 msec later. The bunch fil-amentation is evidenced by the tails forming on the bunchdistribution. This compares very well with a simulated SCimage as shown in Fig. 5c, where each vertical slice is theprojection of the phase space distribution as shown in Fig. 4onto the phase axis.It is also interesting to examine the decay of an injectedbeam without RF capture (i.e. no RF voltage). As the beamloses energy, its orbit spirals inwards and it eventually islost. The ring orbit period also decreases according to therelationTT0= tot (1)where  is the momentum compaction, and tot is the totalfrequency energy lost. Assuming the energy loss per turn isconstant and  is independent of energy, the path difference383Proceedings of the 1999 Particle Accelerator Conference, New York, 19996004002000Time (psec)120100806040200Time (µsec)Injection transient for uncaptured beam (no RF) Figure 6: Transient of a decaying, uncaptured electronbeam at injection. The calculated drift for parameters givenin Table 1 is shown as a white line.-40x10-3 -2002040Fractional Energy Offset-4 -2 0 2 4Phase Offset (rad)Figure 7: Phase space orbits including radiation damping.The RF buckets are no longer closed, with attractors forfollowing buckets lying above each bucket.is given bytot =f20 turn2t2 (2)If the energy loss/turn is known, this may be another tech-nique for measuring the momentum compaction.Fig. 6 shows a SC image at injection with the storagering RF turned off. As the uncaptured beam decays, itarrives earlier on each turn w.r.t the synchroscan voltageshown in Fig. 2, creating an increasing vertical deflectionon the SC. Also shown is the calculated decrease in orbitperiod using parameters given in Table 1. We are currentlyanalysing the evolution of the bunch shape.Bunch-bunch diffusion Ignoring radiation damping,which acts as a frictional damping to the synchrotronoscillations, synchrotron oscillations within the RF bucketare stable. Electrons with synchrotron orbits outside ofthe RF bucket slowly lose energy and eventually are lost.Generally, this approximation is good because radiationdamping times in electron storage rings are typically smallcompared with synchrotron periods. However, radiationdamping provides a mechanism by which electrons withsynchrotron orbits outside the RF bucket have a smallprobability of being recaptured into a subsequent RFbucket. Given an electron bunch stored in a ring, electronscan be ejected from the RF bucket by processes such aslarge–angle intrabeam (Touschek) scattering or quantumexcitation. We refer to the recapture of a fraction of theejected electrons as a diffusion process. This sectionpresents a brief description and measurements of thisdiffusion process at the ALS. More details are givenelsewhere[2]. Similar studies have been done at UVSORand the Photon Factory[3].Although the diffusion process is interesting from thepoint of view of accelerator physics, it can also be impor-tant for operation of a synchrotron light source. For ex-ample, one operational mode at several facilities requireslarge single light pulses separated by at least several hun-dred nanoseconds to accommodate the time–of–flight tech-niques used by experimenters. In this mode at the ALS, thering is filled with two electron bunches separated by 328nsec. The storage ring injection process typically fills sev-eral unwanted parasite bunches which are removed priorto an experiment using a common RF knockout technique.However, the diffusion process can create additional para-site bunches over time, possibly contaminating the experi-ment.The effect of the radiation damping on longitudinalphase space is illustrated in Fig. 7. The RF bucket is nolonger a closed area but rather a basin of attraction for elec-trons within the boundaries shown. For a given bucket,the attractors of subsequent buckets lie above the bucket.Electrons ejected from the bucket have a small probabil-ity of landing on an attractor and damping to a subsequentbucket. The diffusion rate can be calculated as the productof the the probablity of exciting an electron to an ampli-tude and the width and position of the attractor. Note thethe process is not symmetric in that only particles ejectedwith too much energy can be recaptured. Also note thatother particle loss mechanisms are ignored.Our experiment consisted of filling a single bunch to alevel of several milliamps and recording its current and thatof the following parasite bunch as a function of time. Therelative current in the parasite bunch was measured usingthe SC simply as a means of gating the synchrotron lightsignal such that it could be measured independently fromthe main bunch.Measurement of the diffusion into the parasite bunch rel-ative to the initial charge in main as a function of time isshown in Figure 8. The two cases are for RF voltages of1.1 and 0.78 MV with corresponding synchrotron tunes of7.7e-3 and 6.4e-3. Diffusion is faster for lower RF voltagessince the attractors are closer to the bunch.One technique for eliminating the diffusion is to adjustthe momentum acceptance of the lattice to be approxi-mately equal to the RF acceptance such that any particleejected from the bucket is lost. This already occurs in theALS in some conditions, albeit unintentionally. Studies[4]indicate that the ring momentum acceptance decreases be-low the RF acceptance when the wiggler is in its fullyclosed position. We no longer observed the diffusion in384Proceedings of the 1999 Particle Accelerator Conference, New York, 19992.52.01.51.00.50.0Fraction of initial main bunch (x103 ) 6050403020100Time (min)  Qs=7.7e-3, tau=97.7 min-mA Qs=6.4e-3, tau=47.3 min-mAFigure 8: Measured charge in the parasite bunch relativeto the main bunch for 2 different RF voltages. The solidlines for each data set are the calculated diffusion using themeasured Touschek lifetime for each RF voltage.6005004003002001000Amplitude (psec)upward sweepνs=0.00732, φm=1.2 mrad6005004003002001000Amplitude (psec)8.5x10-3 8.07.57.06.56.0Frequency/f 0downward sweepνs=0.0073, φm=7.5 mradFigure 9: Measured response curve for a modulation fre-quency swept upwards and downwards through the reso-nance.this condition.Forced oscillations The subject of forced, or driven, syn-chrotron oscillations has many applications in the physicsof beams[5]. Forced synchrotron oscillations occur whenthe beam is driven by modulations of the bending fields orvoltage or phase modulations in the RF system. Both ofthese effects can have significant impact on operation, par-ticularly in efforts to achieve very short electron bunches.This section presents highlights from our studies of phasemodulation of synchrotron oscillations. More details aregiven elsewhere[6].Synchrotron oscillations are very similar to those of apendulum, which has a response curve as shown in Fig. 9.When the frequency of the driving force is below the bifur-cation frequency, the response has two stable points as alsoshown in a Poincare` map of the phase space in a frame ro-tating at the driving frequency. The beam can be distributedbetween the two islands. If the excitation frequency isswept through the bifurcation point, the actual distributiondepends on the initial beam distribution as well as the beamcurrent. Our experiments consisted of mapping the fixedpoints of the system as a function of modulation amplitudeand frequency. We also studied the diffusion process fromone island to another and related it to the Touschek scatter-ing process.An example SC image at a condition where the phasemodulation frequency is just below the bifurcation fre-quency is shown in Fig. 11. As the modulation frequency isswept upwards above the bifurcation frequency, the smalleramplitude island disappears as can be seen from the re-sponse curve in Fig. 9. The measured fixed points of thesystem for a upward and downward sweep of the modula-tion frequency are shown in Fig. 9. The hysteresis typicalof such as system is also evident in the difference betweenthe direction of the swept frequency.To measure the interisland diffusion rates, we filled asingle bunch with current and used the SC to measure thelongitudinal profile as a function of time after initiating thephase modulation at a fixed frequency. The modulation fre-quency was always below the bifurcation frequency, ensur-ing the presence of two stable islands. For each beam cur-rent, we measured the diffusion as a function of modulationfrequency in order to vary the island size and separation.We extracted the initial diffusion rate by comparing therelative rate of increase of the peak signal in the island A(outer island) compared to the island B (central island.) Aplot of the diffusion rates as a function of modulation fre-quency for several beam currents is shown in Figure 12.Calculations of the diffusion rate with Touschek scatteringas the diffusion mechanism and the separatrix of island Bas the momentum acceptance show approximate agreementwith the measurements.We have also studied the structure of higher resonances-1.0-0.50.00.51.0(2I)1/2sin(ψ−νmθ)-1.0 -0.5 0.0 0.5 1.0(2I)1/2 cos(ψ−νmθ)SFP BSFP A UFP Ca) well below bifurcation frequencyFigure 10: Poincare` map of the phase space in a frame ro-tating with the modulation frequency when exciting belowthe below bifurcation frequency.385Proceedings of the 1999 Particle Accelerator Conference, New York, 1999-200-1000100200Time (psec)200150100500 fm<fcFigure 11: SC Image of the longitudinal profile when themodulation frequency is just below the bifurcation fre-quency. The beam populates both stable phase space is-lands.0.0010.010.1Diffusion rate (1/sec) 11.211.111.010.910.8Modulation Frequency (kHz) 0.55 mA 1.0 mA 2.0 mAFigure 12: Measured and predicted diffusion rates as afunction of modulation frequency for three bunch currents.at multiples of the synchrotron frequency. At twice the syn-chrotron frequency, two or three phase space islands areexpected to form, depending on the modulation frequency.An example is shown in Fig. 13, which shows the bunchsplit into two islands. The calculated phase space structureis also shown. At three times the synchrotron frequencythree or four islands can form. An example of this reso-nance is also shown in Fig. 13.700600500400300Time (psec)0.100.080.060.040.020.00Time (msec)2*νs resonance; 2 outer islands only 550500450400350Time (psec)0.100.080.060.040.020.00Time (msec)3*νs resonance; 3 outer islands only Figure 13: Measured SC image showing formation of twoand three islands for excitation near 2*fs and 3*fs. Thecalculated phase space structure is also shown.Conclusions The nonlinearity of synchrotron oscillationscreates several interesting and subtle effects in electronstorage rings. These experiments also demonstrate that thedual–scan SC is an invaluable tool for understanding lon-gitudinal beam dynamics in electron storage rings.The author would like to thank Dr. Albert Hofman andProf. S. Y. Lee for many useful discussions on longitudinaldynamics and Dr. Alex Lumpkin, Jim Hinkson, and MikeChin for helping set up and understand the streak camera.1 REFERENCES[1] H. Mosshammer, Phys. Rev. E 48, 1390, (1993).H. Mosshammer, Nucl. Inst. Methods A 323, 553, (1992).H. Mosshammer, Phys. Rev. E 48, 2140, (1993).[2] J. M. Byrd, G. Stupakov, Proc. of the 1998 Conf. on QuantumAspects of Beam Physics, ed. P. Chen (1998).[3] T. Kasuga, H. Yonehara, M. Hasumoto, and T. Kinoshita, Jpn.J. Appl. Phys. 28 (1989) 541. M. Tobiyama, A. Higuchi, T.Mitsuhashi, T. Kasuga, S. Sakanaka, Proc. of the 1995 Part.Accel. Conf. (1995) 3300.[4] W. Decking, D. Robin, Proc. of the 16th Advanced BeamDynamics Workshop, Arcidosso, (1998).[5] M. Ellison, et. al., Phys. Rev. Lett. 70, 591, (1993). H. Huang,et. al., Phys. Rev. E 48, 4678, 1993. G. Tsironis, S. Peggs,T. Chen, 1990 EPAC, p1753. J. Ellison, H.-J. Shih, M. Kum-mer, Phys. Rev. E 49, 2484, (1994). D. Li et al., Phys. Rev. E48, 1638, (1993). D. Li et al., Nucl. Instr. Methods A 361, p.205–223 (1995).[6] J. M. Byrd, W.H. Cheng, F. Zimmermann, Phys. Rev. E 57,4706, (1998).386Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

Nonlinear longitudinal dynamics studies at the ALS

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