SLAC - PUB June 1987 P/E) A Conceptual Design of Final Focus Systems for Linear Colliders*

Transcription

1 SLAC - PUB June 1987 P/E) A Conceptua Design of Fina Focus Systems for Linear Coiders* KARL L. BROWN Stanford Linear Acceerator Center Stanford, CA Introduction Linear coiders are a reativey recent deveopment in the evoution of partice acceerators. This report discusses some of the approaches that have been considered for the design of Fina Focus Systems to demagnify the beam exiting from a inac to the sma size suitabe for coisions at the interaction point. The system receiving the most attention is the one adopted for the SLAC Linear Coider. However, the theory and optica techniques discussed shoud be appicabe to the design efforts for future machines. 2. The Probem ---e The principa probem in designing Fina Focus Systems (FFS) for inear coiders is the eimination or minimization of the chromatic distortions introduced by the fina ens system nearest to the interaction point (I.P.). If the beam exiting from the inac is monoenergetic and has a sma enough emittance, then the FFS design for coiders is reativey simpe. It coud consist of a simpe first-order optica system demagnifying the monoenergetic inac beam to the sma size needed for coisions at the I.P. The utimate imitation woud then be the residua geometric (monoenergetic) aberrations of the optica system. The magnitude of these. aberrations woud be a function of the phase space emittance from the inac and the natura third-order geometric aberrations of quadrupoe focusing systems. Unfortunatey the beams from microwave inear acceerators are not monoenergetic. They have a finite momentum spread, and one is forced to cope with the chromatic distortions caused by this momentum spectrum. There are severa possibe approaches to soving this probem: 1. A reduction in the momentum spread or the emittance of the beam exiting from the inac ceary eases the severity of the probem. 2. The magnitude of the chromatic distortion is a function of the distance,, of the first ens from the I.P. The coser the ens system is to the I.P. the smaer the chromatic distortion. Hence strong, compact enses are an asset to soving the probem. 3. Sextupoes can be introduced into the optica design to cance the principa second-order chromatic aberrations, eaving the residua third-order chromatic and geometric aberrations as the fundamenta imitation to achieving a sma spot at the I.P. 4. Aternativey one might choose to ive with the sma residua second-rder chromatic aberrations by choosing sma aperture but very strong enses as the fina enses in the FFS. Such systems were studied and considered for the initia SLC design but were utimatey rejected in favor of a %extupoe corrected soution. * This work WM supported by the U. S. Department of Energy under contract number DGAC03-76SF Invited tak presented at the Joint US-CERN Schoo on Partice Acceerators: Topica Course on Frontiers on Partice Beams South Padre, Texas, October 23-29, 1986

2 A Unfortunatey sextupoe corrected soutions create other probems that must be minimized in order to find a satisfactory soution. Sextupoes require dipoes to be inserted into the chromatic correction system in order for the sextupoes to seectivey coupe to the chromatic aberrations that one is trying to eiminate. These dipoes, in turn, introduce emittance growth to the beam via the fuctuations in the synchrotron radiation energy osses. This compication demands a carefu seection of the pacement and strength of the dipoes so as to minimize this effect. Sextupoes create an additiona probem which utimatey imits the smaest spot size that can be achieved at the I.P. Whie sextupoe famiies can be designed such that both the secondorder geometric and chromatic aberrations vanish! the cross-couping of the eextupoe famiies generates residua third- and higher-order aberrations. For the SLC case these residua aberrations imit the effective p* at the I.P. to 8 mm. Without the sextupoe corrections the SLC woud have a p* of 7 ems for the same I. Thus a factor of nine has been gained in thii particuar appication. 3. Optics Definitions and Notation For the theoretica discussion in this report, we use the six dimensiona phase space parameters defined by the TRANSPORT 21 notation; namey -rc- - x= where 6 = (p - po)/po. The first- and second-order optics is represented by the R and T matrix eements as foows: z 2 Y Y 1 6 (1) zi = 2 RijZj + 2 TijkXjZk + higher-order terms j= j,k= (2) where the R matrix is commony expressed in the form: The objective in the design of the fina focus system for the SLC was to find a way of eiminating a of the second-order aberrations, Tijk, and be eft with ony the appropriate inear terms pus the residua third- and higher-order aberrations. These residua terms then determined the that coud be achieved at the interaction point of the coider. 2

3 4. Chromatic Aberrations Assuming an upright eipse, i.e., cyo = 0, at the beginning of the fina focus system, then p at the I.P. as a function of momentum may be expressed by the equations or p(6) = R;1(6)Po + y -&)&I = R;(S)p; + R&(6) (4 where po is the vaue at the beginning of the FFS, and 6 = Ap/po. R1 and R12 are matrix eements of the R transformation matrix measured from the beginning of the FFS to the I.P. &(6) and &2(b) are a function of the chromatic aberrations in the system as foows: h(6) = h(0) + T11e.6 + U1~66~ +... R12(6) = R12(0) i- T12s6 i- u12s~6~ +... (5) : where R11(0) and R12(0) are the vaues of the matrix eements for the centra momentum PO. Substituting Eqs. (5) into (4) and coecting terms, we obtain the resut c NW0 = [R:,P,2 + R:,] + 2 [RIIT~P; + RI&~] [(T:6 + 2%u66)P: + Tf26 + 2R2U266)] b2 + higher order geometric and chromatic distortions. (6) The idea objective woud be to devise an aberration Eq. (6) vanish except for the first term free optica system where a terms in P(6) = &PO = P*(o) where R1 = M is the monoenergetic demagnification of the beam enveope at the I.P. As a step in this direction, we now wish to derive some chromatic properties of teescopic _ systems which have been usefu in soving our probem. We restrict the discussion to the x pane for simpicity, but the resuts and concusions are appicabe to both transverse panes, x and y. PROPERTIES OF TELESCOPIC SYSTEMS Shown beow in Fig. 1 is a representative one-dimensiona, teescopic system. For monoenergetic trajectories, teescopic systems have point-t-point [R12(0) = 0] and parae-t-parae [R21(0) = 0] imaging A3 Ft F2 Fig. 1. A one-dimensiona symmetric teescopic system. 3

4 I In addition, teescopic systems with the symmetry shown in Fig. 1 have some interesting chromatic properties. To iustrate this, we define the foowing quantities.ap 1 1-C 1 1-C c=-; -=-; -= -; 1 = ; P F 11 F2 12 andm=- where F and F2 are the foca engths as a function of momentum of the two enses comprising the system and M is the demagnification of the beam enveope for the centra momentum po. Note the reationship between c and (1 - c) = & or 6 c=1+6 c has been used in this derivation so as to avoid a power series expansion in the matrix eements and thus simpify the resut. The transformation matrix R(e) for the teescopic system shown in Fig. 1 is then _- or finay -A4 + (A4 + 1)G c R(c) = -& (L - c2) -&+(1+$# * * Comparing Eq. (7) with Eq. (5), we observe that, in addition to Rs(O) = 0, TIIS = 0 for this teescopic system. To appreciate the importance of this, we substitute the resut into Eq. (6) and find W)Po = R:& + 2 [O] 6 (7) + [2RU166@: + Tf26] b2 + higher-order terms. (8) So for teescopic systems having the symmetry shown in Fig. 1, we find that the first derivative of p with respect to 6 vanishes and we are eft with the 62 term as the first term of importance. If now we equate 2%u166~: = -T&6 (9) then the 62 term aso vanishes and the teescope becomes achromatic to second order in (g). Note that no sextupoes have been used to obtain this resut! Whie this is an interesting theoretica resut one finds that for most physicay reaizabe situations -.~ %6 >> 2Rdh66P:. At this point one has to decide whether to toerate the second+rder chromatic distortion generated by the quadrupoes via the Tr2s aberration or, aternativey, eiminate Trss by introducing sextupoes and defer the optica distortions to higher-order terms. 4

5 For the moment we consider the first aternative and evauate the magnitude of the Tr2s aberration for an a-quadrupoe teescopic FFS. Equation 8 may then be truncated and rewritten in the foowing form: A or /3(S) H R;r(0)po + T1;612 P(6) = P*(O) + (RIIT~,]~~~ P (O) where /3*(O) = RI,(O)/3 o is the monoenergetic p at the interaction point. We now wish to cacuate the average vaue p(6) and find its minimum. The resut ceary depends upon the distribution function representing the momentum spectrum of the inac. We sha consider two possibe cases: a gaussian and a rectanguar distribution. For the gaussian distribution, the average vaue of p(6) is B(6) = p*(o) + ir1~;;;2u61 (10) where ug is the sigma of the gaussian momentum distribution For a rectanguar momen&.n distribution the resut is - function. [R;Gd2 B(6) = P*(O) + t.6%zz - 3 (12) where 6,,, is the maximum vaue of 6 in a rectanguar (constant vaue) distribution function representing the momentum spectrum of the inac. If now we wish to find the minimum vaue of p(6), using /3*(O) as the variabe, the resut is obtained by equating the two terms in each equation. The resut is &&nin = 2/3 (O) = 2[Rd m]m (13) for the gaussian distribution and for the rectanguar distribution.?(s)min = 2p*(0) = 2[&&26] - % (14) The rectanguar distribution is more representative of the SLC inac in its present mode of operation, so for the remainder of this report we sha assume Eq. (14) to be vaid. 6. The Reationship between Path Length and Chromatic Aberrations Using the optics formaism deveoped in SLAC Report-75: it can be shown that L RrTma = / (s,)2ds (15) where Rrr? rss is measured from the beginning of the FFS to the interaction point(*), but s: is the R22 matrix eement measured from the I.P.(*) back through to the beginning of the fina focus system.

6 I Evauating the integra in Eq. (15), we find A (16) where z (s) is the ange an arbitrary trajectory makes with respect to the centra trajectory at a position s. z (O) is its ange at the interaction point (*) and z(o)=o. A is the path-ength difference between this arbitrary trajectory and the centra trajectory. using TRANSPORT notation, We define Ts22 as foows: then & = T622 -A- - - L (s:)2ds / * = 2T622 (17) Ts22 is the normaized second-order path ength difference measured from the interaction point (*) back through the system. Simiar resuts may be derived for the y pane optics. Note that the integra is proportiona to the square of the ange a trajectory makes at any position s in the system. Therefore the main contribution to the integra comes from the first ens system nearest to the I.P. where the anges are argest. This reationship between the transverse chromatic aberrations and the path ength of the monoenergetic trajectories through a system can be shown to be a genera resut of Hamitonian Mechanics. It is reated to the fact that E and t are conjugate variabes in the Hamitonian description of the system. EXAMPLES : Let s consider this integra for some representative ens systems. Consider first a simpe thin ens modue of ength L, = if as shown in Fig. 2. Fig. 2. A thin ens modue A8 6

7 The integra for this modue may be cacuated by observing that 8: = 1, by definition, from the interaction point (*) to the input face of the ens. The distance to the ens is F. 8: = 0 after passing through the ens. Thus, A L (s;)2ds = 2T622 = F = $. / : Thus we earn that short foca ength enses are desirabe if one wishes to minimize the chromatic distortion in a fina focus system. Consider now a symmetric tripet ens system as shown in Fig A7 Fig-3. A-symmetric tripet ensmodue.. _- If we impose the condition that the chromatic distortion be equa in both transverse panes (x and y), i.e., that T622 = T644, then in a we designed tripet appy: modue the foowing empirica scaing aw has been found to., Bo b [ 1 >4 abp - 3 (19) where I* is the distance to the input face of the first ens in the tripet; EQ a = is the % fied gradient in each ens (assumed to be the same for a eements of the tripet) and Bp is the magnetic rigidity of the beam. For an optimized symmetric tripet system, it has aso been empiricay determined that and RT26 = L(a:)2da = 2Tf,22 fi: LT / * LT fi! 41. (20) Evauation of the path ength integra in Eq. (20) s h ows that approximatey one-fourth of the tota chromatic distortion is generated by 1 and the remaining three-fourths is generated by the path-ength excursions within the tripet ens structure itsef. 7

8 If the tripet and singet ens systems occupy the same physica ength, i.e., LT = Ls, then we concude that the tota chromatic distortion in the tripet system is approximatey twice that found in the singet system! Substituting the resuts of Eq. (20) for the symmetric tripet into Eq. (14) for a rectanguar momentum distribution, we find for an a quadrupoe FFS (no sextupoes) the resut A (21) For a gaussian momentum distribution of the inac, the resut is The inequaity signs in Eqs. (21) and (22) are incuded because the path ength integra has been evauated ony for the tripet ens nearest to the interaction region. Additiona contributions to. the integra wi come from the remaining portions of the teescopic system. The tota p wi be of the order (+M) times the above vaues, where M is the demagnification of the teescopic system. Since M is normay_yry sm_a, Eqs. (21) and (22) are usuay good approximations to -., the fina answer. (22) BEAM-BEAM DISRUPTION EFFECTS If the disrupted beams must pass back through the apertures of the FFS optica eements, then the maximum disruption ange, 6D, becomes an important factor in the design. The maximum radius of the beam enveope within the quadrupoe tripet depends upon 80. For an optimized tripet, the resut is found empiricay from TRANSPORT to be abeam g 31 ed. (23) For a gaussian beam profie, in the transverse coordinates x and y, 8~ has the form@ e2 where N is the number of partices in a bunch, rc = 3 is the cassica eectron radius, Y = --& and u = m, where p = J(S)mi,. c is the beam emittance and Kp is a design safety factor to aow for the beamstrahung pinch effect. Kp shoud be determined from computer simuations. In the SLC design a vaue of Kp = 3 has been used. For two intersecting beams having gaussian cross sections, the uminosity is defined as t N2f =q7tq2. o- cmw2sec- where f is the repetition rate and Q is the gaussian sigma beam size at the interaction point given in meters.

9 A Equations (19), (23), (24) and (25) may be combined for the Uoptimized tripet to yied the inequaity Bet* o- K PT 0 t 4 kgauss m. (26) This equation appies for a round beam spot at the interaction point. It tes us that for a given uminosity there is a minimum product of Bc and 1 required. This resut is independent of whether sextupoes are used for the chromatic correction process. As wi be seen ater for a given uminosity, sextupoes aow a arger 1 than woud otherwise be possibe with an uncorrected system. Combining Eqs. (21) and (25), the attainabe uminosity for the tripet without sextupoe corrections is f! 5 (g) (&,,J cm-2sec- (27) ; for a beam with a rectanguar momentum spectrum. If satisfactory uminosity cannot be achieved with just a simpe symmetric tripet, then the introduction of sextupoes to eiminate the quadrupoe chromxi?dist&tions, manifested by the T26 and 2 34s aberrations, is a ogica recourse. This is what was done for the present SLC design. 6. The Fina Focus System for the SLAC Linear Coider The basic parameters for the initia SLC design were the foowing: Beam Energy 2 60 GeV. (Nomina Design Vaue = 50 GeV.) Linac Emittance, c = 3 *1O-1o meters. o Linac Momentum Spread 6,,, = &. (The distribution is assumed to be rectanguar.) Required 1 to accommodate the partice detectors For the Mark11 For the SLD I*= 2.8 meters 1*= 2.2 meters t& = fi 5 2 pm round spot (incuding optica distortions). Assumed maximum disruption ange mrad (for magnet aperture purposes). The maximum number of partices/beam N = 5. Oo. Beam repetition rate = 180 pps (ater reduced to 120 pps). The task was to design a fina focus system meeting the above specifications which was aso compatibe with physica constraints dictated by the SLAC site. To achieve these specifications, it was decided to adopt a sextupoe corrected soution. Severa possibe FFS systems were studied during the preiminary design studies! Eventuay the design converged to the basic optica system shown in the bock diagram beow. Detais 1st of the competed design have been reported at the 1987 Washington Acceerator Conference. 9

10 I In the remainder of this report, we discuss the conceptua design considerations eading up to the adopted SLC design. i 7. Bock Diagram of SLC Fina Focus System SW 5BosAo Fig. 4. A bock diagram of the SLC fina focus system. A segments of the bock diagram are teescopic transformers. The fina ens system is a symmetric quadrupoe-tripet. The chromatic correction section is a modified second-order achromat! ] Sextupoes and dipoes are needed -in this region to eiminate the second-rder chromatic aberrations in the x and y transverse panes which originate in the fina ens system nearest to the interaction region. The matching section- need&d to match the p and n functions at the exit of the SLC arcs. FFS BEND INITIAL TRANSFORMER It---- [:-;I -1 M,=& CHROMATIC CORRECTION SECTION /----I~ -:+b -+--j A, J * A v SF SD SF SD SF SD FINAL TRANSFORMER SF SD *II Soft Bend IP Fig. 5. Detaied schematic diagram of the SLC fina focus system. 10

11 A schematic diagram of the optics is given in Fig. 5 showing the individua optica enses required as taken from SLAC-PUB The functions of these individua subsystems are as foows: THE FINAL FOCUS BEND (FFB) Removes dispersion which at the ends of the Arcs is given by t]% = m andq,=q;=$,=o. Transfers the Arc beams (with pz = 8.70 m, pv = 1.14 m and o=,s = 0) to the next subsystem with a transfer matrix equa to the negative identity matrix. Provides a arge, achromatic bend needed to satisfy goba geometry constraints. Provides quad/skew-quad pairs separated in phase by - r/2 to remove the accumuated anomaous dispersion caused by perturbations in the Arcs. THE INITIAL TRANSFORMER (IT) Initiates demagnification and provides a round beam, with &, = 0.12 m and (Y%,~ = 0, to the next subsystem. Provides trims on the two ast quadrupoes to adjust /3& at the I.P. Provides a region where&he e $-;c- path-ength difference was adjusted. THE CHROMATIC CORRECTION SYSTEM (CCS) Corrects the dominant second-rder chromatic aberrations, T2e and T346, at the I.P. without introducing other second-order geometric or chromatic aberrations. Provides s bend geometry to reduce synchrotron radiation (SR) background in the partice detector. Provides space for a ow fied soft bend magnet to further reduce SR background. THE FINAL TRANSFORMER (FT) Competes the demagnification process. The magnification in this transformer is the same in the z and y panes, so that at the I.P. the idea beam is round, with an optimum p:,y = m and (Y%,~ = 0. Aows independent adjustment of the axia position of the beam waists to match the position of coincidence at the I.P. Provides a skew quad for suppression of cross-pane couping between z, y and z, y at the I.P. 8. Characteristics of Second-Order Aberrations The foowing optics principes were used in the SLC design so as to avoid or minimize the accumuation of second-order aberrations. Separated-function quadrupoes do not introduce second-order geometric aberrations. Identica sextupoe pairs separated by a minus unity (-1) transformation matrix can seectivey coupe to chromatic aberrations without introducing second-order geometric aberrations. 11

12 Identica dipoe pairs separated by a minus unity (-1) transformation matrix do not introduce second-rder geometric aberrations. The above principes are the basis of the second+nder achromat I1 which can more generay be stated by the theorems isted beow. : 9. The Second-Order Achromat THEOREM A If one combines four or more identica ces consisting of dipoe, quadrupoe, and sextupoe components with the parameters chosen so that the tota first-order transfer matrix is unity (+I) in both transverse panes, then a geometric (on momentum) second-order aberrations wi vanish. THEOREM B If the two sextupoe famiies are adjusted so as to make one second-order chromatic aberration vanish in each transverse pane, then a second-order aberrations (geometric, chromatic, and path ength) wi vanish except for those path-ength terms which depend ony upon the momentum of the partices. The non-vanishing terms are Rss and T566. If both theorem A and %&em 6 are satisfied, then we-ca the resufig system a Second- Order Achromat. An optica attice having these properties is shown in Fig V - A VW- 11 OF QD QF OD QF QD QF QD A12 Fig. 6. Optica attice of a typica four-ce second-order achromat. The optica mode adopted for the chromatic correction section of the SLC, as shown in Fig. 5, is a modified second+rder achromat. The essentia difference is that in the chromatic correction section the dipoes are positioned ony in paces of ow q so as to minimize emittance growth caused by fuctuations in the synchrotron radiation energy oss. In this modified achromat, the quadrupoes and sextupoes sti satisfy repetitive symmetry such that a second-order geometric aberrations cance. The dipoes are positioned so as to minimize emittance growth from synchrotron radiation quantum fuctuations and the two sextupoe famiies are adjusted so as to cance the Trzs and Ts4e second order chromatic aberrations for the entire &a focus system. 10. Residua Third-Order Aberrations Having introduced the above symmetries into the design, a of the second-order aberrations wi vanish or at east become sufficienty sma that they are of no practica consequence. We are then eft with the residua third-order geometric and chromatic aberrations as the imiting factors in the achievabe uminosity. 12

13 In TRANSPORT notation we find that the important residua third-order aberrations in the x pane are the foowing:, I and u266 = (+ 62), U446 = (+t26) u1222 = (+ 3), U244 = (+ d2) *. Simiar residua aberrations occur in the y pane optics. The design is optimized by observing that U 126s is the dominant residua aberration in the x pane. The other three aberrations can be shown to be functions of the magnitude of the dipoe strengths used in the CCS. The stronger the dipoe, the weaker these aberrations become. However this has to be baanced against the emittance growth caused by the dipoes. The design is adjusted so ss to make the contribution from these three aberrations approximatey equa to that from the UZes term. The combination is then foded with the monoenergetic beam size so as to achieve the minimum p(s). An approximate equation has been derived representing this minimization process; the resut is the foowing: (28) _- where B(S),;, = 2p (O) is the average beta function at the interaction point. (RrrUr266) is evauated for the entire FFS measured from the input to the interaction point when I = 1, i.e., U26~j is proportiona to 1. Using TRANSPORT to evauate u12es, the net resut for the fina SLC design is P(J)min 1 oo 1 6kaz (29).-- where as before S,,, is the maximum 4 for a rectanguar inac momentum distribution and 1 is the distance to the interaction point from the exit face of the ast quadrupoe in the symmetric tripet ens system nearest to the I.P. (measured in m). Combining this resut with Eq. (25) for the uminosity, we obtain L,,, 5 (z)(k) -10m6cmm2sec-. (30) These equations are approximate but seem to be vaid for S,,, > &. For 6,,, ess than &, the residua geometric aberrations begin to dominate and the assumptions break down, at east for the SLC design. It is interesting to compare this equation for uminosity with that for the tripet aone. We find that so for the SLC specification of 6,,, = &, the gain in uminosity is 6 as a resut of using the z sextupoe corrected soution. 13

14 11. Summary As a summary, a cookbook soution for designing SLC type systems is given beow ss a guide to the reader. Use teescopic transformers for the basic optics modues. Design the transformer nearest the interaction region so as to minimize the path engths between the monoenergetic trajectories. Use a modified form of the second-order achromat for the design of the chromatic correction section. Choose the ocation of dipoes in the chromatic correction section so as to minimize emittance growth. Choose the dipoe strength so as to minimize the overa effect of third-order aberrations and emittance growth. REFERENCES 1. K.L. Brown, A Second-Order Magnetic Optica Achromat, SLAC-PUB-2257, (February 1979), and IEEE Transactions on Nucear Science, Vo. NS-26, No. 3, June 1979, p For a more up-t&&ate discussion of the second-rder achromat see K.L. Brown and R.V. Servranckx, First- and Second-Order Charged Partice Optics, SLAGPUB-3381, Juy Presented at the 1983 BNL Summer Schoo for Physics of High Energy Partice Acceerators. 2. K.L. Brown, A First- and Second-Order Matrix Theory for the Design of Beam Transport Systems and Charged Partice Spectrometers, SLAC Report No. 75, or Advances in Partice Physics 1, (1967). and K.L. Brown, D.C. Carey, C. Isein, and F. Rothacker,. TRANSPORT, A Computer Program for Designing Charged Partice Beam Transport Systems, SLAC-91 (1973 rev.), NAL-91, and CERN R. Hoebeek, Disruption Limits for Linear Coiders, Nuc. Instr. and Meth. i84, (1981). 4. K.L. Brown, as given in the SLC Conceptua Design Report, SLAC-Report-229, June 1980; and in the Proceedings of the SLC Workshop, SLAC-Report-247, March J.J. Murray, K.L. Brown and T. Fieguth, The Competed Design of the SLC Fina Focus System, SLAC-Report-4219, February Presented at the Partice Acceerator Conference, Washington, DC, March

15 SLAC - PUB Juy 1987 (N) ERRATUM A CONCEPTUAL DESIGN OF FINAL FOCUS SYSTEMS FOR LINEAR COLLIDER* KARL L. BROWN Stanford Linear Acceerator Center Stanford &4&erstTy, Stanford, Caijorraia, This manuscript was submitted to two different Conferences: Invited tak presented at the Joint US-CERN Schoo on Partice Acceerators: Topica Course on Frontiers on Partice Beams, South Padre, Texas, October 23-29, 1986 and aso Tak presented at the BB Factory Workshop, Los Angees, Caifornia, January 26-30, 1987 * Work supported by the Department of Energy, contract D E - A C SF