Szilágyi, M

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Particle Accelerators1981 Vol.ll pp.213-2170031-2460/81/1104/0213$06.50/0© Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaA NOVEL APPROACH TO BEAM OPTICS DESIGN FOR PARTICLEACCELERATORSM. SZILAGYI*Deutsches Elektronen-Synchrotron DESY, HalnhurR, Ger/nany(Received February 3, /98/)A new computational method is proposed for optimal design of beam lines. The method is based on a dynanlic-programming recursive algorithm that minimizes an additively or multiplicatively expressed function of the desiredparameters.A formidable amount of effort has been made tocreate computer programs capable of optimizingthe number, length, position and strength of beamoptics elements (multipole lenses, bending mag-nets, etc.) used in particle accelerators, storagerings, beam-transport systems, etc., when var-ious constraints are imposed on the beam de-sign. I - 3 The available software is very powerfulin beam calculations with fixed parameters andis also capable of finding local optima if the num-ber of variable parameters and the number ofconstraints are not too high. The initial values ofthe parameters must be chosen from some ap-proximate solutions or rough physical intuition.If the starting values are too far from a localoptimum or the problem is over-constrained, nosolution can be found. An additional problemarises in connection with the weighting factorsthat can only be determined by experience andstrongly depend on the nature of the particularproblem. Totally different results may appear ifthese factors are changed ~ they can therefore beconsidered as additional unknown parameters.The results usually also fluctuate strongly if theorder of variation of the parameters is changed.The most important problem, however, is thateven if a local optimum is found, we have noguarantee that there are not other much bettersolutions.This situtation is quite similar to that in thefield of low-energy beam optics, where there isan additional complication that we are usuallyalso required to find electrode or pole-piece con-* New address: School of Applied and Engineering Phys-ics, Clark Hall, Cornell University, Ithaca, New York 14853U.S.A.213figurations that can be expected to have certainpredefined electron-optical properties. Electron-optical synthesis was of a trial-and-error naturein the past. The usual procedure was to changethe dimensions of electrodes or pole pieces untilthe desired optical properties were obtained.The author has proposed. an entirely differentapproach to electron optical synthesis. 4 Thenovel computational method involves restrictingthe possible electrostatic or magnetic field dis-tributions to those connecting the intersectionpoints of a rectangular grid by straight lines andgives a recursive algorithm for choosing distri-butions that minimize a given combination of ab-erration coefficients. This lnethod was success-fully applied to both magneticS and electrostatic6lens synthesis.The aim of this paper is to show that the abovemethod can also be applied to beam-optics designfor particle accelerators, at least for cases wherethe requirements can be expressed in an additiveor multiplicative way. This is certainly the casewhen some aberration must be minimized be-cause the aberration coefficients can always beexpressed in the form of a definite integral. Forthe sake of simplicity, however, we choose anillustrative example.Let us try to solve the following problem.Given a long straight channel with definite di-mensions and some spaces occupied by rf cavi-ties, the task is to find a suitably situated systemof quadrupole lenses that minimizes the overallsum of vertical q3 y ) and horizontal (~h) betatronfunctions measured at the medium planes of eachquadrupole along the whole channelG = min L (~Yk .+ ~h/J, (1)k214 M. SZILAGYIwhere k is the ordinal number of a given quad-rupole in the system.The values of the quadrupole excitations arenaturally constrained by saturation and energyconsiderations. Another constraint is imposed bythe fact that the cavities and quadrupoles cannotoverlap each other. Therefore, for the first tryone may suppose that the lengths and positionsof the quadrupoles are fixed. (Of course, this isnot a general requirement for the application ofthe method.) If we accept these constraints, theproblem is reduced to that of finding a set ofexcitation values hk in the given limits for thequadrupoles placed along the channel at arbitrary(not constant) distances from each other (Fig. 1).As a further simplification, for illustrative pur-poses one can use the thin-lens approximation,so it is assumed in Fig. 1 that the lenses are con-centrated in their mid-planes and therefore thath2 3 k-l k.. "t__---+---+--l"~_+-___Ii--___'I1,r,I---t----~- SFIGURE 1 The arrangement ofN quadrupoles along a chan-nel (s is the axial coordinate, h is the excitation). The arrowsrepresent that we do not know anything a priori about thevalues and signs of the excitations.the betatron functions do not change inside tnelenses. The lens action is taken into account bythe changes in the values of the derivatives ofthe betatron functions.We do not know anything a priori about thesolution. Even the '"signs" of the lenses are notgiven. (We consider a lens '"positive" if it is fo-cusing in the horizontal plane and defocusing inthe vertical one.)The solution of this simple problem is quitedifficult. Indeed, let us suppose that we have Nlenses and every lens can have M different ex-citation values. This is already a simplificationbecause the excitation is changed in a continuousway by changing the currents in the coils, but ifM is large enough it is quite a practical assump-tion. Then there are MN different possibilitiesavailable. If we want to try all these, we mustcalculate the vertical and horizontal betatronfunctions and their derivatives (M + M2 + M3+ ... + MN ) times for the lenses and do the sameamount of calculation for the drift spaces be-tween them in order to get all the informationnecessary to compare the results and choose thebest among them. If we have only 10 lenses andsuppose that M == 100 (which is not a very largenumber to approximate a continuous variation ofthe excitation between the negative and positivesaturation limits) we arrive at 1020 different pos-sibilities and an even larger number of calcula-tions and comparisons. It is.easy to agree thatthis' 'brute-force" approach is absolutely out ofthe question. If we try to use some existing op-timization software, we will certainly arrive at alocally '"optimal" solution but only if the numberof lenses is strongly limited. In case of N == 100,for example, there is no hope at all.But the hope is here! We shall try now to applyour method to the solution of this problem. First,we define a '"domain of existence" of the prob-lem. This means that the distances along the axisare limited by the actual length of the channeland the quadrupole excitations are limited by theconstraints. Then we design a rectangular' 'com-putational grid" which will restrict the excita-tions to M + 1 different values along the ordinate(zero excitations must not be excluded) while fix-ing the situations of the quadrupoles along theabscissa. The computational grid is shown in Fig.2. Our problem is now reduced to that of findingout the intersection points of this grid that de-termine the optimum distribution of quadrupoleexcitations hik , where i is the value of the exci-BEAM OPTICS DESIGN 215hAli(3)Gik = min [Fuk + Gilk - oJ,jvery deep mathematical principle is widely usedin economics and control engineering but. un-fortunately, almost unknown among physicists.We shall express the optimality principle bythe recursion relationwhere Gik is the result of the optimal procedureleading from the initial state to the k-th state,G;lk _ I) is the result of the optimal procedure lead-ing from the initial state to the (k - I)-th state andFUk is the contribution of the k-th lens. Of course.) is a variable and the aim of the procedure is tofind out that particular, optimal value of j forwhich the value of the sum of FUk + Gilk _ I) isminimal. The initial conditions GiO must be de-termined from the given initial values of the be-tatron functions and their derivatives. In our caseGiO = I3Ho + I3vo for every possible value of i.It is evident that Fuo = 0 for any values of i and}. Because in our case FUI does not depend on}, we can simply calculate the values of Gil forevery i and then begin the optimization procedureby calculating FU2 for a fixed value of i, but forevery possible value of} (see Fig. 2). Varying thevalue of} we finds\\ hik,, \ I'';' hi ~~ //1/ It' 1/It',I,/ II,~ hjlk-l1II I"'ll/1, 111,II/lit 2 3 k-I k N11/111//,1//1",,','111llilfI'l/111,"III1/II1///oM"25-I-3-2-4-5.M2FIGURE 2 The computational grid,Ihjl)Opt 3where. in general, the values of Gil are the sameas Gil at the first lens because the beginning ofthe k-th region of the computational grid ob-viously coincides with the end of the (k - I)-th(egion (what was "i" before becomes "r now).In this simplified model we assume that the be-tatron functions are calculated at the inputs ofthe lenses. thus combining the drift space andlens calculations into one step. We store thevalue of Gi2 together with hi2 and the correspond-ing optimal value of hil opt, as well as the betatronfunctions and their derivatives. All the other pos-sibilities have no interest for us anymore (dottedlines in Fig. 2). After evaluating M + I possi-bilities. we choose the best of them and abandonthe others.Now we change the value of i and find thecorresponding values ofGn and hil opt. Doing thatfor every possible value of i. we shall have a setof M + I data for each of the following param-eters: hi2 , Gn , hil opt, I3vi2. I3Hi2. aVi2 and aHi2where a = -13'/2 (the prime represents differ-entiation with respect to the axial coordinate s).tation expressed in the elementary units tih2hmax/M and k is the ordinal number of the lensunder consideration. For any given value of h ikthere are exactly M + I possible values in theprevious column of the grid. We shall denote theactual value of the excitation of the previous lensby h;lk _ I) (see Fig. 2). Then we can assign a value(2)to the linear segment connecting these two pointsin the computational grid. It is quite clear thatthe values of the betatron functions at the outputof the k-th lens must depend on the excitation iof the given lens and also on the previous historyof the beam, i.e .. the excitations of the precedinglenses. which are taken into account by the singleindex}. We are able to use this single index byapplying R. E. Bellman' s optimality principle7 •which can be formulated in the following way:" The optimal procedure in a multistage decisionprocess has the property that. whatever the finaldecision is, the prior decisions must constitutean optimal procedure leading from the initial stateto the state preceding the final decision." ThisGi2 = min (FU2 + Gid.j(4)216 M. SZILAGYIA graph of M + I lines connecting the corre-sponding points hi2 and hi! opt for every possiblevalue of i will correspond to these sets of data(see the first region of Fig. 3).After completing the calculations at the secondlens, we move further to the third lens, repeatingthe whole procedure. We must only rememberthat the index "i" from the previous region al-ways becomes "j" in the new region. Thus theinitial conditions for the third lens are those givenas final values after the second drift space. Pro-ceeding in the same way, we shall have again aset of M + I data for each of the parameters thatmust be stored. Our graph can be continued (seesecond region in Fig. 3).Repeating this procedure N - I times, we ar-rive at the end of our calculations. We have Nsets of the values ofhib Gib hi<k- )Jopt, I3vib I3HibCJ..Vik and CJ..Hik (k = I, 2, "', N) and the wholegraph of all M + I possible optimal excitationdistributions can be completed. In Fig. 3 a simpleillustrative example is given for the case of M= 8, N = 5. We can choose that particular valueof i at the end which is best suited to our specialneeds or which corresponds to the smallest valuee G. Then we follow the optimum-value distri-bution in the graph starting from the given point(i, N) and proceeding in the backward directionstep by step. The thick line in Fig. 3 representsh 0Ah-1-2-3-41 2 3 4 5SFIGURE 3 The completed optimum-value graph.such an "optimal" distribution if i = - 2 is cho-sen as the final excitation value. For the com-pletion of the whole procedure, the betatron func-tions and their derivatives must be evaluated only(M + I) + (M + 1)2 (N - I) times, which makesa big difference in comparison with the brute-force approach. It means that the necessary com-puter time increases with the number of elementsonly linearly!The advantages of the method are obvious. Itis a global search with a linear dependence onthe number of elements as far as computationaltime is concerned. Any constraint that reducesthe number of possibilities at a particular stagesimplifies the computing procedure. (For exam-ple, any of the quadrupoles can be forced to zeroexcitation or a given excitation value). Aftercompleting the calculations we automaticallyalso have solutions for many sub-problems witha smaller number of elements. A system of lenseswith variable lengths can also be treated, as wellas many other combinations.Practical problems can be formulated in quitedifferent ways. For example, it might be desirableto obtain given values of the betatron functionsat some places and simultaneously fulfill somemore requirements at other steps, as is the casewhen a low-beta insert has to be designed. 8 Thenit is convenient to formulate the requirement sothat the sum of differences between the actualand desirable values must be minimized. We arewell aware of the fact that not all optimizationprocesses are like this, but our method certainlycan be applied to many practical problems ofbeam-line optimization. We can minimize emit-tance, momentum-compaction factor, phase ad-vance, chromaticity, etc. There is also a possi-bility to reduce the nonlinear effects of thecorrecting sextupoles. The minimization of thesum of betatron functions is important in rf-cavityregions. It is desirable to reduce the aberrationsin beam-transport systems, etc.Our aim in this short theoretical communica-tion has been to present only the method. Itspractical application to cases where previousmethods proved to be very difficult to use hasbegun now. The results will be reported sepa-rately.ACKNOWLEDGEMENTSThe author is indebted to Professor G.-A. Voss,Dr. D. Degele and Dr. J. Rossbach for very in-teresting discussions.BEAM OPTICS DESIGN 217REFERENCES1. K. L. Brown et aI., TRANSPORT-A Computer Programfor Designing Charged Particle Beam Transport Systems,Fermi National Accelerator Laboratory Report NAL-91,1974, and CERN 80-04, Geneva, 1980.2. E. Keil, Y. Marti, B. W. Montague, and A. Sudboe,AGS-The ISR Computer Program for Synchrotron De-sign, Orbit Analysis and Insertion Matching, CERN 75-13, Geneva, 1975.3. F. James and M. Roos, MINUIT-A System for FunctionMinimization and Analysis of the Parameter Errors andCorrelations, CERN Computer Center, D 506 (1977).4. M. Szilagyi, Optik 48,215 (1977).5. M. Szilagyi, Optik 49,223 (1977).6. M. Szilagyi, Optik 50, 35 (1978).7. R. E. Bellman and S. E. Dreyfus, Applied Dynamic Pro-gramming, Princeton University Press, Princeton, 1962.8. M. Szilagyi, Design Considerations for Low-Beta StraightSections at PETRA. DESY Internal Report, Hamburg,1981.

A Novel Approach To Beam Optics Design For Particle accelerators

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