Tomography is now a very broad topic with a wealth of algorithms for the reconstruction of both qualitative and quantitative images. One of the simplest algorithms has been modified to take into accou nt the non-linearity of large-amplitude synchrotron motion. This permits the accurate reconstruction of longitudinal phase space density from one-dimensional bunch profile data. The method is a hybrid one which incorporates particle tracking, and considerable effort has been invested to optimize the computer code so that it may also be compiled to exploit parallel architectures efficiently. A selec tion of the results obtained at different CERN accelerators is presented. The starting-point in each case is simply a "mountain range" of digitized bunch profiles

Hancock, S

Knaus, P

Lindroos, M

CERN Document Server

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN - PS DIVISION(RF)TOMOGRAPHIC MEASUREMENTS OF LONGITUDINAL PHASE SPACEDENSITYS. Hancock, P. Knaus, M. LindroosTomography is now a very broad topic with a wealth of algorithms for the reconstruction ofboth qualitative and quantitative images.  One of the simplest algorithms has been modified totake into account the non-linearity of large-amplitude synchrotron motion.  This permits theaccurate reconstruction of longitudinal phase space density from one-dimensional bunch profiledata.  The method is a hybrid one which incorporates particle tracking, and considerable efforthas been invested to optimize the computer code so that it may also be compiled to exploitparallel architectures efficiently.  A selection of the results obtained at different CERNaccelerators is presented.  The starting-point in each case is simply a "mountain range" ofdigitized bunch profiles.CERN/PS 98-0306th European Particle Accelerator Conference (EPAC98), June 22-26, 1998,  Stockholm,SwedenGeneva, Switzerland 14 August 1998AbstractTOMOGRAPHIC MEASUREMENTS OF LONGITUDINAL PHASE SPACEDENSITYS. Hancock, P. Knaus, M. LindroosCERN, Geneva, SwitzerlandAbstractTomography is now a very broad topic with a wealthof algorithms for the reconstruction of both qualitativeand quantitative images.  One of the simplest algorithmshas been modified to take into account the non-linearityof large-amplitude synchrotron motion.  This permits theaccurate reconstruction of longitudinal phase spacedensity from one-dimensional bunch profile data.  Themethod is a hybrid one which incorporates particletracking, and considerable effort has been invested tooptimize the computer code so that it may also becompiled to exploit parallel architectures efficiently.  Aselection of the results obtained at different CERNaccelerators is presented.  The starting-point in each caseis simply a “mountain range” of digitized bunch profiles.1  INTRODUCTIONThe underlying principle of tomography is to combinethe information in a sufficiently large number of profilesto be able to reconstruct unambiguously the fuller picturewith the extra dimension reinstated.  Thus, for example,many one-dimensional profiles of x-ray transparencytaken from different angles can give doctors an image ofa two-dimensional slice through a patient.The application of tomography to longitudinal phasespace becomes obvious once it is realised that a bunchperforming synchrotron motion is analogous to a patientrotating in a stationary body scanner.  On each turnaround the machine, a longitudinal pick-up provides asnapshot of the bunch projected at a slightly differentangle.  It only remains to combine such profiles tomo-graphically to obtain a two-dimensional picture of phasespace density[1].2  RECONSTRUCTIONBack projection is a key process by which the contentsof the bins of a one-dimensional histogram areredistributed over the two-dimensional array of cellswhich comprise the reconstructed image.  Given no apriori knowledge of the original two-dimensionaldistribution, the contents of one bin is shared over all thecells that could have contributed to that bin.  The backprojection of all bins of all profiles yields a firstapproximation to the original distribution.Algebraic Reconstruction Techniques[2] (ARTalgorithms) exploit the fact that the coefficients forsharing bins in back projection can also be used to projectthe contents of cells into those bins.  Hence a set ofprojections can be obtained from the first approximation.Back projection of the bin-by-bin difference between theoriginal set of profiles and this new one yields animproved approximation.  Further iterations convergemore rapidly if any cell whose contents has becomenegative is reset to zero.The problem with conventional ART is that itsstrategies for estimating the redistribution coefficients arebased on straight-line back projection geometry.  Thisimplies either rigid, circular motion of the two-dimen-sional distribution or that its projections be measuredsimultaneously.  An alternative approach is to considerhow the contents of one cell gets projected into the binsof a particular profile.  By launching a small number oftest particles which, initially, are uniformly distributedwithin the cell, the calculation of coefficients becomes asimple matter of counting how many particles end up ineach bin at the particular instant when the profile wasmeasured.  Thus, a hybrid algorithm which combinesparticle tracking with ART allows large-amplitudesynchrotron motion to be taken into account since thetrajectories of the test particles need not be assumedcircular.  Although iteration proceeds as before, there is aprice to be paid: a large map of (projection) coefficientsmust first be built and its inverse (for back projection)derived for every profile in the set of measured data.  Onthe other hand, since most of the computational effort isinvested in building the maps, it becomes trivial to repeata calculation with fresh data taken under the sameconditions. 3  SOME RESULTSThe first part of Fig. 1 shows a simulated distributioncomprising an unmatched bunch.  These particles weretracked over an entire synchrotron period to obtain72 profiles as the distribution filamented.  Taking theseprofiles as the input dataset, the new algorithm yields theaccurate reconstruction of the second part of Fig. 1.The method is very robust.  Since the iterationsconverge to the consensus of the information in theprofiles, trigger jitter and noise in the data have only aminor effect.  In addition, since the test particles are onlytracked for a relatively brief period to build the maps,moderate errors in the machine parameters used in thetracking model are not serious.-30 -20 -10 0 10 20 30-60-40-200204060nsMeV-30 -20 -10 0 10 20 30-60-40-200204060nsMeV0.20.40.60.81Figure 1:  Simulated proton distribution and its reconstruction (on the same density scale).-200 -100 0 100 200-6-4-20246nsMeV0.20.40.60.81Figure 2:  Measured profiles and reconstructed contour plot of a phase-modulated bunch at 1 GeV.-100 -50 0 50 100ns-100 -50 0 50 100-15-10-5051015nsMeV0.20.40.60.81Figure 3:  Quasi-adiabatic bunch splitting at 3.5 GeV/c.The 72 profiles of Fig. 2 were measured every 48 turnsat the CERN Booster during attempts to move emptyphase space into the centre of a normal bunch by phasemodulation of the rf.  Without tomography, it would bedifficult to deduce anything from the bunch shapeinformation.Arbitrarily complex rf systems can be treated.  The“mountain range” plot of Fig. 3 shows a series ofmeasurements every 50 turns during bunch splitting[3] inthe CERN PS.  Splitting is achieved by increasing thesecond-harmonic voltage component of a dual-harmonicrf system while reducing the fundamental.  The data span10 ms and the details of the two voltage programmesduring this period were incorporated in the trackingmodel of the algorithm to produce the correspondingreconstructed image.  Since the origin in time at whichthe test particles are launched can be chosen freely withinthe measurement interval, the reconstruction can be atany instant during that period.  In Fig. 3, the reconstruct-ion is at the time of the 72nd profile (out of 92).  Ananimation of the entire process can be made by repeatingthe calculation for all time slices.Discrepancy[2] expresses in a single figure of merit theresidual differences between the projections of areconstructed distribution and the original profiles.  Assuch it may be used to monitor convergence.  It is anobvious conjecture that, in the limit of many iterations,the lowest discrepancy should be obtained when theparameters used in the tracking model match most closelythe reality of the machine.  Reconstruction of thesimulated data of Fig. 1 assuming different rf voltagessupports this idea.  The discrepancy after 100 iterationsexhibits a sharp minimum at precisely the value of the rfvoltage that was used to generate the data in the firstplace (see Fig. 4).  Arguably, this constitutes a voltagemeasurement.0.9 0.95 1.05 1.1Vnorm0.00010.00020.0003DiscrepancyFigure 4:  Discrepancy versus normalized rf voltage. 4  COMPUTER CODEThe algorithm was originally developed inMathematica¥, a choice mainly motivated by the rich setof built-in functions for graphics and for the manipulationof arrays.  However, the objective was merely to establisha proof of principle.  Consequently, with the aim ofreducing the execution time by a factor of at least 100,the code was translated into Fortran 90.  This was donewith the view ultimately to use parallel architectures.In the original Mathematica¥ code there is extensiveuse of floating-point operations and many very large andvery sparse matrices.  In the Fortran 90 version, thenumber of floating-point operations is reduced by the useof integer representation until the final step in eachnumber manipulation, while a pointer-like re-allocatablevector representation was created to deal with the sparsematrices.  The built-in pointer facility was avoidedbecause it would make the use of parallel architectureswith shared memory impossible.The execution time of the present version of the code issome 400 times faster than that of the Mathematica¥ one.This enables the production of well-resolved imageswithin minutes, which is entirely adequate for machinedevelopment work.  It also makes possible the productionof animated sequences of dynamic processes withinreasonable execution times.  Extensive work is now inprogress to further optimize the code for faster processorsand parallel architectures.  The long-term goal is tointegrate the measurement as an operational tool withinthe controls system of the PS Complex. 5  CONCLUSIONSAn algorithm has been developed for longitudinalphase space tomography in which the contents of thereconstructed array is effectively rotated instead ofinclining profile bins in order to make a projection.  Thisallows a different mapping to be applied to each cell inthe array so that rigid, circular motion of the phase spacedistribution need not be assumed.Leptons are excluded due to synchrotron radiation, butsimulated proton data have shown the method to be bothaccurate and robust.  Valuable tomographic measure-ments have been made during machine experiments.The algorithm is a hybrid one and, consequently,arbitrarily complex rf systems can be catered for bymodifying a small part of the code.6  ACKNOWLEDGEMENTSThe authors wish to thank Mike Metcalf for helpfuldiscussions and advice on programming in Fortran 90. REFERENCES[1] S. Hancock, “A simple algorithm for longitudinalphase space tomography”, CERN/PS/RF/Note 97-06,1997.[2] R. Gordon, “A tutorial on ART”, IEEE Trans. Nucl.Sci., NS-21, pp78-93, 1974.[3] A. Blas, R. Cappi, R. Garoby, S. Hancock,K. Schindl, J-L. Vallet, “Beams in the CERN PSComplex after the rf upgrades for LHC”, theseProceedings.

Tomographic Measurements of Longitudinal Phase Space Density

Tomography is now a very broad topic with a wealth of algorithms for the reconstruction of both qualitative and quantitative images. One of the simplest algorithms has been modified to take into account the non-linearity of large-amplitude synchrotron motion. This permits the accurate reconstruction of longitudinal phase space density from one-dimensional bunch profile data. The method is a hybrid one which incorporates particle tracking, and considerable effort has been invested to optimize the computer code so that it may also be compiled to exploit parallel architectures efficiently. A selection of the results obtained at different CERN accelerators is presented. The starting-point in each case is simply a “mountain range ” of digitized bunch profiles. 

Tomographic Measurements of Longitudinal Phase Space Density

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