It is interesting to study the matching capability of a triplet for a large range of parameters and to cover all the possible solutions. Fully analytical treatments can be used for this purpose, when working with the thin lens approximation. Closed solutions and the conditions in which they exist are described for the special case of triplets with a symmetric geometrical arrangement. They define all possible matchings, which correspond to the requests or limits on the Twiss parameters that are specified, and then define the range of values that are obtainable for the betatron functions. Numerical extension to thick lenses gives the complete solutions for the retained cases. Applications are presented on insertion and lattice problems, with emphasis on the design for a possible isochronous ring that is part of the injection chain of the CLIC drive beam

D'Amico, T E

Guignard, Gilbert

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ANALYSIS OF A SYMMETRIC TRIPLET AND ITS APPLICATION TORING INSERTIONST. d’Amico, G. Guignard, CERN, 1211 Geneva 23, SwitzerlandAbstractIt is interesting to study the matching capability of a tripletfor a large range of parameters and to cover all the possi-ble solutions. Fully analytical treatments can be used forthis purpose, when working with the thin lens approxima-tion. Closed solutions and the conditions in which theyexist are described for the special case of triplets with asymmetric geometrical arrangement. They define all pos-sible matchings, which correspond to the requests or limitson the Twiss parameters that are specified, and then de-fine the range of values that are obtainable for the betatronfunctions. Numerical extension to thick lenses gives thecomplete solutions for the retained cases. Applications arepresented on insertion and lattice problems, with emphasison the design for a possible isochronous ring that is part ofthe injection chain of the CLIC drive beam.1 INTRODUCTIONThe design of a new type of isochronous ring for the in-jection chain of the CLIC drive beam [1] needs a matchingsection between two adjacent isochronous cells. The pur-pose of this section is twofold. First the betatron function atthe entrance and at the exit of it should have the same valueand its slope should be equal and opposite. Second the to-tal phase advance including the isochronous cell should ben=m where n and m are small integers, in order to mini-mize the geometrical aberrations. The symmetry of the be-tatron function leads quite naturally to choose a symmetriclattice for the matching section. We selected a symmetrictriplet for its compactness and because it provides the re-quired degrees of freedom. Several authors [2-7] have ob-tained analytical solutions in the thin-lens approximationfor different configurations of quadrupole multiplets whichare more general than a simple symmetric triplet. Theirnumber of free variables is different from the six degrees offreedom of the transfer matrix and implies extra constraintsor a mismatch factor [8]. In addition the adaptation of theanalytical development of the general triplet to the symmet-ric case is cumbersome and does not take advantage of thelattice symmetry in an easy way.l1G2Firstquadrupolel3G4G4Secondquadrupolel3G2Thirdquadrupolel1Figure 1: Schematic of a symmetric tripletA geometrically symmetric triplet as shown in Fig.1 hasfour free variables (the two drift lengths l1; l3and the mag-netic gradients G2; G4respectively of the first and half ofthe second quadrupole). It can easily be proved [9] thatthe diagonal elements of the transfer matrix for each planeare equal. Thus the number of independent values whichcan be fixed by selecting the Twiss parameters at the en-trance and at the exit of the triplet is also four. The numberof solutions is finite and does not depend upon additionalassumptions. This observation has convinced us that a spe-cific study would be useful not only for the design of anisochronous ring but also to explore, in an exaustive way,all the existing solutions for a more general matching prob-lem. A particularly interesting one is related to the experi-mental insertions of collider rings such as the LHC. Tripletsare indeed conveniently used to match the double waist atthe interaction point to the rest of the lattice on either side.2 COMPATIBLE BETATRON FUNCTIONS FOR ASYMMETRIC TRIPLETThe equality of the diagonal elements of the transfer matrixfor each plane forces an important constraint on the beta-tron function in that plane at the entrance and at the exitof a symmetric triplet. Neglecting the special case whenthe phase advance is an odd multiple of , we may distin-guish two situations. The first one which can be called amirror symmetry triplet occurs when the betatron functionhas equal values and its derivative has opposite values, thephase advance being a free parameter. This is the case ofinterest for building up an isochronous module which maybe chained up to form a ring. An example is shown in Fig.2. The second configuration occurs when the betatron func-tion has different values at each end of the symmetric tripletwith its derivatives being free parameters. This generalityis limited by the fact that the phase advance can no longerbe freely chosen and is given by :cot =21+ 121  2where 1; 1and 2; 2are the Twiss functions at theentrance and exit of the triplet (see Fig. 1) respectively.Thus we have the choice only between two phase advanceswhich differ by . The more general case, which will becalled a matching triplet can be of interest in several typesof insertion. Its application to a collider final focus, wherea round beam at the interaction point is matched to the stan-dard periodic FODO cell, has been studied. An example isgiven in Figs 3 and 4.Figure 2: Horizontal and vertical -functions of a symmet-ric triplet which inverts the slopes of these functions.0 2 4 6 8 10 12s [m]050100150200250300β    Figure 3: Hor. and vert. -functions of the first solution ofa symmetric triplet which transforms a round beam with awaist into a round beam with opposite divergences.3 DETERMINATION OF THE FOURPARAMETERS OF A SYMMETRIC TRIPLETIn the thin lens approximation, the transfer matrices for thehorizontal and vertical planes can be expressed as functionsof the two drift lengths l1; l3and the quadrupole strengthsg2= K2lq; g4= K4lqwhere lqis the quadrupole lengthand K2;K4are the normalized gradients. To further sim-plify the problem the corresponding elements in each planecan be combined by taking half the sum and half the differ-ence of those in the first column (only two elements in eachtransfer matrix are actually independent). Thus we obtainthe following system of equations:8>><>>:c`1+ `23g2g4= ad`1+ `3(g2+ g4) = b`3g2(g2+ 2g4) = c`23g22g4+ g2+ g4= d ;(1)0 2 4 6 8s [m]0102030405060β    Figure 4: Hor. and vert. -functions of the second solutionof a symmetric triplet which transforms a round beam witha waist into a round beam with opposite divergences.where a; b; c; d are defined by:a = (th;11+ tv;11  2)=4 ; b = (th;11  tv;11)=4c = (th;21+ tv;21)=4 ; d = (th;21  tv;21)=4 :where th;nmand tv;nmare the elements of the horizontaland vertical transfer matrices respectively. The full analysisof (1) can be found in [9]. Neglecting the very special caseswhen either c = 0 or c = d and assuming bc   ad 6= 0,the general solution of (1) is given by :`1=1c2  d2(bc  ad)z   (bd  ac)g2=c2  d2(bc  ad)(1  z2)`3=(bc  ad)(1  z2)z(c2  d2)g4=z2(d  cz)1  z2where z is a solution of the cubic equationz3 2dcz2+ z +d2  c2c(bc  ad)= 0 :Of course only real values of z chosen such that the corre-sponding drift lengths are positive, should be retained.4 APPLICATIONSThe motivation of the study was to obtain the parametersof the matching triplet between two isochronous modulesproviding given phase advances in the two planes. The be-tatron function and its derivative were determined by theisochronous module while the phase advances were chosenin such a way that the total phase advances for the periodiccell were =3 in the horizontal plane and 3=2 in the ver-tical plane. Thus every three cells in the horizontal planeand two in the vertical plane, kicks due to field errors maycancel out. This is a typical application of mirror symmetryand the results are quite good (Fig. 2). A second interestingapplication is the matching from an interaction point to thestandard FODO lattice. This would be schematically thecase in circular colliders such as LHC when are neglectedthe constraints imposed on the value of the first drift length.At the interaction point, the values of the betatron functionsin the two planes are equal and relatively small (“low ”)while their derivatives are null. At the exit to the triplet it isassumed that the betatron functions in the two planes crosseach other with opposite slopes. Thus,h;1= v;1= 1; h;1= h;2= 0h;2= v;2= 2; h;2=  h;1= 2The horizontal and vertical transfer matrices areTh= 1=021  21  22Tv= 1=0 21  21  2 2where 0=p12+ 21  2 and 1= 1=1; 2=(1 + 22)=2. It is possible to show [11] that two and onlytwo sets of triplet parameters satisfy these transfer matricesif we neglect a difference of  in the corresponding phaseadvances. The only condition is that 2should not be zeroor equal top(2  1)=1. These two sets of parametersare given by:a) l1=2zr;1+0j1 2jg2=1 2211 z2r;1l3= 2l1=z2r;1g4=2z3r;1g220where zr;1is the (only) real solution of the cubic equationz3+ z + 20=2= 0b) f = 1=g2=21 2l1= jf=2jp0(0+ j2j)l3= jf jp1 + j2=0jg4=  g2=2 :This treatment has been included in the packageBeamOptics developed using the Mathematica language[10]. When l1is imposed the approach described here canbe extended by adding a drift of free length at the exit ofsuch a symmetric triplet. It is then possible to examinethe behaviour of important quantities such as the lengthof this drift and the maximum value reached by the beta-tron functions inside the triplet; and this according to thevariation of the triplet parameters in a predefined range,and to specific constraints on the betatron functions at theend of the added drift. Two cases have been investigated: waist in one plane and equal values of the betatron func-tion with opposite slopes. All the possible solutions can bedetermined for canonical triplets (g4=  g2), general sym-metric triplets and even triplets where the third quadrupolestrength is slightly modified. This study was applied to in-sertions of LHC type and provided interesting results onthe ranges of acceptable triplet parameters [11].5 CONCLUSIONSIt has been shown that in the thin lens approximation a geo-metrically symmetric triplet is a structure simple enough tobe handled analytically and yet powerful enough to be ap-plied to several interesting and quite different lattices. Themethod provides in a direct way all the existing solutionspermitting a complete study of the required lattice. Exis-tence conditions can only be made explicit in special casesbut readily constructed as an abacus whenever needed. Itcan be shown [9] that the extension to thick lenses is eas-ily obtained by solving a system of two nonlinear equa-tions functions of the normalized gradients K2;K4. Ofcourse such results are obtainable also by standard numeri-cal matching programs although they may have difficultiesin converging if not conveniently guided. Moreover theiruse is cumbersome and time consuming when included ina chain of specific codes written in other languages. Thethin lens approximation obtained by the method describedhere is an excellent starting point for such algorithms.6 REFERENCES[1] T.E. d’Amico, G. Guignard, ’First-order design of a newtype of isochronous arc’, CERN SL/95–120, 1995.[2] E. Regenstreif, ’Phase-space transformations by means ofquadrupole multiplets’, CERN 67–6, 1967.[3] G. Wu¨stefeld, R. Maier and B. Simon, ’The analytical lat-tice approach for the ring design of BESSY II’, Proc. FirstEuropean Part. Accel. Conf. (EPAC), Rome, 1988.[4] B. Zotter, ’Calculation of the parameters of a quad tripletfrom the transfer matrices’, CERN/ISR–TH/73–43, 1973.[5] B. Autin, E. Wildner, ’Applications of symbolic techniquesto structural beam optics’, Proc. Fifth European Part. Accel.Conf. (EPAC), Sitges (Spain), 1996.[6] O. Napoly, ’Thin lens telescopes for final focus systems’,CERN/LEP–TH/89–69, CLIC Note No. 102, 1989.[7] P.J. Bryant, K. Johnsen,’The principles of circular accelera-tors and storage rings’, (Cambridge University Press, 1993).[8] P.J. Bryant, private communication.[9] T. E. d’Amico, ‘General treatment of a matching quadrupoletriplet which is symmetric around its median plane’, CLICNote 322, January 1997.[10] B. Autin, T. d’Amico, V. Ducas, M. Martini, E. Wildner,‘Analytic lattice design with BeamOptics’ Proceedings ofthis Conference.[11] T. E. d’Amico, G. Guignard, ‘Constraints in the use of asymmetric triplet for the LHC low  insertion ’, to be pub-lished.

Analysis of a Symmetric Triplet and its Application to Ring Insertions

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