Dynamic focusing has been proposed as a way to eliminate a conventional collimation and final focus system in linear colliders, and is a scheme that is more readily extended to colliders at several TeV center-of-mass energy. In this paper we examine several outstanding issues, in particular, the optimization of the lens and main beam parameters. Simulations of the lens-lens, lens-main, and main­main beam collisions using a modified version of the GUINEAPIG beam­beam code are in progress

Helm, D

Irwin, J

Schulte, Daniel

Thompson, K

CERN Document Server

OPEN-99-01717/06/99FURTHER DEVELOPMENTS IN DYNAMIC FOCUSINGJ.Irwin, D.Helm, K.Thompson, SLAC, Stanford, CA, D.Schulte, CERN, Geneva, SwitzerlandAbstractDynamic focusing has been proposed[1] as a way to elim-inate a conventional collimation and final focus system inlinear colliders, and is a scheme that is more readily ex-tended to colliders at several TeV center-of-mass energy.In this paper we examine several outstanding issues, inparticular, the optimization of the lens and main beamparameters. Simulations of the lens-lens, lens-main, andmain-main beam collisions using a modified version of theGUINEAPIG beam-beam code are in progress.1 INTRODUCTIONDynamic focusing has been discussed in earlier conferencepapers[1, 2, 3]. We continue with what we hope is a moreinsightful discussion of the equations controlling the lens-beam parameters, accompanied by the first particle track-ing simulations. The GUINEAPIG code[4] has been mod-ified to support very unequal bunch lengths (necessary forthe lens-main collison). A full simulation begins at theentry to the lens-lens collision after the particle distribu-tion has been shaped by the octupole modules[3], followedby simulation of the lens-lens collision, the lens-main col-lision and the main-main collisions. The dependence ofthe system on both the main-beam and lens-beam bunchlength are of special interest. The main-beam bunch lengthcan be varied to achieve maximum luminosity in the main-main collision, while the lens-beam bunch length shouldbe made as long as possible in order to minimize the syn-chrotron radiation[5] of the main beam in the lens-maincollision.2 THE PINCH EQUATIONThe central equation in the determination of lens-beam pa-rameters is the pinch equation[6]. For a diverging lensbeam, assuming for simplicity that the main beam hasa uniform longitudinal distribution, this equation has theform160(z,MfQ‘)2=< s2 >2=(2f − 2020): (1)where s is the displacment of the focal point,  is themain-beam beta function at the IP, ‘ is the distance be-tween the lens-main collision and the IP, z,M is the main-beam bunch length, 0 =pM, f is the final rmssize of the main beam at the IP taking into account beamenlargement from the pinch effect, and fQ is the focal∗Work supported by Department of Energy Contract DE-AC03-76SF00515.length experienced by the lens beam as it passes throughthe main beam. The ratio ‘= equals the demagnifica-tion  = M=0, where M is the size of the main beamat the lens-main collision. Introducing the formula for thefocal length fQ, and noting the presence of factors whichoccur in the definition of the main-beam disruption at theIP, we may write Eq. 1 asD0γMγQ=√60(2f20− 1)  (2)where D0 is the disruption parameter based on a beam size0 at the IP. Since the actual beam size is f , it is the dis-ruption parameter based on this beam size which is relevantto determining the luminosity enhancement. Therefore it isbetter to write this equation asDfγMγQ= cp where cp √60404f(2f20− 1)(3)As the main-beam bunch length is increased, the pinchaberration will cause the ratio f=0 to grow. cp has amaximum of cp =p15 = 3:87 at f=0 =p2, and avalue cp =√40=3 = 3:65 at f=0 =p1:5. Since itis a major advantage to have a large γ-ratio and a largedisruption, it will be advantageous to operate with a main-beam bunch length that gives an apparent 50% luminosityloss due to the lens pinch aberration, since this loss is morethan compensated by the enhancement achievable from thelarger disruption parameter. Simulations are underway tosee how this plays out when there is no assumption on thelongitudinal uniformity of the lens beam. A sensible set ofparameters satisfying Eq. 3 are Df = 1:2 (giving a round-beam enhancement H  4), a γ ratio of about 100, and ademagnification of 33.3 THE POWER-RATIO EQUATIONWe have previously shown[3] that‘fQ=PMP0=γMNMγQNQ0where NQ0 =γMre : (4)The charge in the uniform disk of the lens beam is(Q=M )2NQ0  3NQ0, where the ratio (Q=M )2 ischosen so that only about 10% of the main beam tail is notproperly focused. Using the γ-ratio as determined by Eq. 3,Eq. 4 becomes‘fQ= cp2f20FreγMDF H: (5)F , which has the dimension of a flux, is basically a ratioof luminosity to beam power, coming from the luminosityequation:NMH2f= 4LEBPB F : (6)Assuming beam power is allowed to increase as the squareroot of the energy, F=γM is increasing as energy to the 3/2,while all other parameters in Eq. 5 tend to remain constant.Hence it is difficult to hold ‘=fQ constant at energies of 5TeV c.m. and above. However the product DfH and can change some.Eq. 5 may also be regarded as an equation for fQ:fQ =1cp202fγMFreDfH : (7)To maintain a diverging lens-beam geometry, we requireQ  3fQ, so fQ cannot be allowed to get too small.4 THE LENS-LENS COLLISIONAs previously noted, the lens-lens collision can be used forself-alignment[3]. In that case‘fQ= 2√2γMγQ+ 2 : (8)For a γ ratio of 100, we find ‘=fQ = 30:2. Table 1gives parameters (not yet fully optimized) for c.m. energiesfrom 0.5 TeV to 3 TeV with ‘=fQ  30. In all cases,2Q = R2Q=2, demagnification M=0 = ‘=  35,=fQ = 0:67, (f=0)2 = 1:5, (Q=M )2 = 3,NQ=NQD = 1:2, fQ = z,Q, A  z,M= = 0:18,Df = 1:27, and an analytic estimate of luminosity en-hancement factor HD=3.67. The main beam power PB =80 MW for 1 TeV c.m. and scales aspE, but the luminos-ity L departs somewhat from E2 scaling.5 THE DEMAGNIFICATIONPARAMETERFor c.m. energies larger than 5 TeV, Eq. 7 indicates a largerdemagnification and a larger disruption are desirable. Fromthe chromatic condition(1)2=(‘‘1)2+‘‘1(‘‘12 − 2)¯2 + ¯4 (9)and for ‘=‘1 = 400, the demagnification  can be as largeas 30 and one can still have a 2% bandwidth. However, thetolerances on the lens beam quality become more stringentfor larger demagnification.6 LENS-BEAM CURRENTThe lens-beam parameters in Table 1 appear reasonableenough. However the round main-beam parameters havea small bunch charge and imply a train with many bunches.Table 1: Parametersc.m. energy [TeV] 1/2 1 1 1/2 3γM 5.E5 1.E6 1.5E6 3.0E6γQ 5.E3 9.9E3 1.5E4 3.E4z,M [m] 182 129 70.2 82.8z,Q [m] 1170 824 449 529(γ)M [m] .543 .221 .103 .0733(γ)Q [m] 2.88 1.17 .544 .389NM [1010] .204 .0829 .0385 .0276NQ [1010] 2.44 .989 .460 .329‘ [cm] 3.54 2.50 1.36 1.61B[%] 2.3 5.3 10.4 13.8Υavg 0.06 0.19 0.59 1.10nB (rep rate 120 Hz) 565 983 1727 1708L [1034cm−2sec−1] 0.5 1.0 2.25 2.70Lens-main collision:Q[mm] (waist in front) .781 .552 .301 .354M [nm] 1160 440 181 117Q [nm] 2010 762 313 203Main-main collision:f [nm] 40.6 15.4 6.32 4.100 [nm] 33.2 12.6 5.16 3.35 [mm] 1.01 .716 .389 .459To maintain the efficiency of the main linac it is necessaryto maintain a beam current (during the pulse) of about 0.5Amps. Remarkably this optimum does not depend much onthe fundamental frequency of the accelerating structures.For the main-beam bunch charge at 1 TeV the requiredspacing equals the C-band wavelength. Furthermore, sinceNQT =NQTNQD2Q2MNQ0  3:6NQ0 ; (10)we haveNQTNM=γMγQPQPM=γMγQNQTNQ0PQ0PM 12 : (11)This implies the lens-beam current would be about 6 Amps.This is a large current for a ring (the PEP LER is designedfor 2 Amps) and implies a major perturbation to the rf sys-tem as the 84-meter bunch train passes. We have not de-cided on the best way to overcome this problem. One so-lution is to have a lower current in the damping ring anduse a combiner ring to compress the bunch train much as inthe two-beam accelerator drive beam complex. And sincethe beam is only in the combiner ring for a few revolutions,that ring is simple and need not have an rf system.7 LENS-BEAM PROPERTIESWe have described two physical configurations of thelens beam beamlines: i) injection from a damping ringinto a jitter-correction and beam-shaping loop into the IPregion[2], and ii) injection from a damping ring into a linacfollowed by a jitter-correction and beam-shaping loop[3].If one desires a lens-beam system that extends to 3 TeVc.m., the pure damping ring solution becomes very diffi-cult. For this reason, and because of a need for bunch com-pression, we presently favor the linac solution.The lens beam must have the following systems andproperties: an energy about 1/100th of the main beam energy, an emittance given by(γ)Q  14Q2fQ2Q2MNMre ; (12)which is about 1 m-r at 1 TeV c.m. and does not gettoo much smaller, a final bunch length of about z,Q = 0:5 mm, number of electrons per bunch  1010, a current entering the IP of about 6 Amps, bunch-to-bunch jitter at the IP of less than 1%, a Q that would be about 2 mm in the absence of alens-lens collision and that is about 0.5 mm with thelens-lens collision, a transverse distribution approximating a uniform diskat the lens-main collision, recapture of positrons and reinjection into a dampingring with a transverse emittance that can be redampedto the design emittance in a few damping times.To meet these requirements this system probably con-tains two damping rings at about 2 GeV, bunch compression systems upon extraction from thedamping ring, combiner rings following the bunch compression,probably containing jitter damping systems, an efficient S-band or C-band linac for acceleration tofinal lens-beam energy, loops after the linac each containing octupole mod-ules for beam shaping, and perhaps feed-forward jittercontrols, chromatically-corrected incoming final-focus systemsfor an incoming Q of about 2 mm, crab cavities at the entrance to the detector region, recapture loops after the IP, a linear wiggler for reducing at least positrons to thedamping ring energy, a reinjection loop into the damping rings.This is not a simple beam system, and perhaps it willnot represent a cost savings compared to the conventionalsystem. However, a centralized campus for main damp-ing rings and low-energy linacs would allow the lens-beamlinac to double as a main-beam pre-linac (to 10 GeV), andthe returning lens-beam electrons could be used to createpositrons. Major cost reductions and improved operabil-ity should follow from the round beam parameters withtheir lower charge, the wider focusing bandwidth (2%) thatreduces energy compensation tolerances, the absence of acollimation system, and absence of final focusing elementsin the detector. The system scales to energies of 3 TeV c.m.and perhaps higher, and is self-aligning at 3 TeV c.m. andbelow.8 DISCUSSION AND CONCLUSIONSSimulations of the entire series of beam manipulations,from shaping of the lens beam by octupoles, through thelens-lens collision, lens-main collisions, and main-maincollisions are in progress. In Figure 1 we show an ex-ample of a main beam distribution, after GUINEAPIGsimulation of the lens-main collision and then transportby ‘ = 25 mm to the main-main interaction point. Thefull simulations have yet to be completed; of particular im-portance will be to demonstrate in detail that it is possibleto get sufficiently uniform lenses at both the lens-lens andlens-main collisions. With the caveat that such simulationsbear out our expectations, dynamic focusing appears to be apromising alternative to conventional final-focus and colli-mation systems. We note also that a first experiment[7] hasshown that the crab cavity phases can be measured with therequired precision of 0.01 degrees at X-band.Figure 1: Main beam distribution after lens-main collisionand transport by ‘ = 25 mm to main-main collision.9 REFERENCES[1] J.Irwin, Proceedings IEEE Particle Accelerator Conference(PAC97), Vancouver, B.C., 12-16 May 1997.[2] J.Irwin, 8th Workshop on Advanced Accelerator Concepts(AAC98), Baltimore, Maryland, 5-11 July 1998.[3] J.Irwin, 19th International Linear Accelerator Conference(LINAC98), Chicago, 23-28 Aug 1998.[4] D.Schulte, Ph.D. thesis (1996), TESLA-97-08.[5] K. Oide, Phys.Rev.Lett., 61, 1713 (10 October 1988).[6] Ref 2. Eq. 1 for pinch. There was a factor of 2 error in theoriginal equation which is corrected here.[7] S.Tantawi, private communication.

Further Developments in Dynamic Focusing

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