The micro-bunching instability (MBI) is seeded by small charge-density fluctuations in the electron bunches, and is successively amplified by the combined effect of space charge and coherent synchrotron radiation, as the beam travels through magnetic compressors. The quantitative understanding of this effect demands for accurate numerical simulations. Here we report on the progress of an upgrading of a 2D Vlasov solver code toward a 4D grid-based Vlasov solver, including also the transversedynamics. The goal is to provide an accurate characterization of the MBI seeded by random noise present in the bunch distribution. We also comment on the advantages of our procedure with respect to other approaches, e.g. macroparticle simulations

Dattoli, Giuseppe

Migliorati, Mauro

Schiavi, Angelo

Venturini, Marco

MIGLIORATI, Mauro

SCHIAVI, ANGELO

G. DATTOLI

M. VENTURINI

Archivio della ricerca- Università di Roma La Sapienza

A four-dimensional Vlasov solver for microbunching instability in the injection system for X-ray FELs

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A FOUR-DIMENSIONAL VLASOV SOLVER FOR MICROBUNCHINGINSTABILITY IN THE INJECTION SYSTEM FOR X-RAY FELSM. Migliorati, A. Schiavi, Rome University ‘LA SAPIENZA’, Rome, Italy,G. Dattoli, ENEA Centro Ricerche Frascati, Rome, Italy,M. Venturini, LBNL, Berkeley, CA 94720-8211, USA.AbstractThe micro-bunching instability (MBI) is seeded by smallcharge-density fluctuations in the electron bunches, andis successively amplified by the combined effect of spacecharge and coherent synchrotron radiation, as the beamtravels through magnetic compressors. The quantitativeunderstanding of this effect demands for accurate numer-ical simulations. Here we report on the progress of anupgrading of a 2D Vlasov solver code toward a 4D grid-based Vlasov solver, including also the transverse dynam-ics. The goal is to provide an accurate characterization ofthe MBI seeded by random noise present in the bunch dis-tribution. We also comment on the advantages of our pro-cedure with respect to other approaches, e. g. macroparticlesimulations.INTRODUCTIONThe demand for high-brightness electron beams is cru-cial for the proper operation of the FEL-based next genera-tion of synchrotron light sources. Because existing linacsources can provide electron beams with required emit-tance and energy spread, but not with the necessary peakcurrent, some manipulations of the beam phase space dy-namics are required in order to increase the peak current,which would ensure the necessary gain to bring the laserinto saturation in a reasonable undulator length. Such amanipulation may occur through magnetic chicanes whichbecame a common tool to all 4th generation light sources,proposed or under construction, to realize the electronbunch compression enhancing the peak current. Althoughthey provide a very effective means to compress the bunch,their use can affect other qualities of the beam like trans-verse emittance and the uncorrelated energy spread. Inparticular, the finite dispersion generated by the bends incombination with collective effects may give rise to the so-called ’microbunching’ instability (MBI) [1], which mayresult into large charge-density fluctuations downstreamthe compressor and eventually into an unacceptably largeenergy spread at the end of the linac. The instability isseeded by the unavoidable small perturbations of the beamdensity present at injection due, for example, to noise inthe photo-gun laser or charge fluctuations caused by shotnoise. Much effort has been devoted over the last few yearsto study and to characterize the collective effects in the in-jection systems for X-ray FELs and the basic physics isnowadays well understood [2]. Accurate numerical mod-elling of the MBI still poses some challenges, partly due tothe high phase-space resolution necessary for an accuratesimulation. Within such a context we note that the use ofmacroparticle simulation methods, some of the most wide-spread tools for beam-dynamics studies, may be of limiteduse for a quantitative description of the MBI, unless thenumber of macroparticles used in the simulations is closeto that of the electrons in the physical bunch. This factoccurs because of the concurrence of the sampling noiseintroduced by the use of macroparticles, and of the highsensitivity of the instability to small initial perturbations. Asimulation employing Nmp randomly deposed macroparti-cles overestimates indeed the amplitude of the shot noiseby a factor√N/Nmp, where N is the number of physicalelectrons per bunch. An effective alternative to macroparti-cles simulations is represented by codes that solve directlythe Vlasov equation ruling the beam dynamics. The VlasovSolvers (V-S) yield the evolution of the phase space beamdensity as a function defined on a grid. V-S are thereforeimmune from sampling noise problems and may thereforebe used as a more effective tool to characterize the insta-bility. A first successful demonstration of the use of V-Smethods for single-pass systems was reported in [3, 4]. Sofar, however, the developed solvers treat two-dimensionalproblems and are therefore suited to model the longitudinalphase space only of beams with negligible transverse emit-tance. These methods have proved useful for both beam dy-namics studies and lattice optimization when some heuris-tic and approximate account of the effect of the transversedynamics on the longitudinal motion is included [4]. How-ever, a more accurate characterization of the instability re-quires a complete description of the coupling between thelongitudinal and horizontal motion. In this paper we reporton the progress we have recently made toward the goal ofdeveloping a fully operational 4D V-S for application topeak current enhancement in Linac dedicated to FEL oper-ation. The goal is to develop a solver with the capabilityof following the beam through the various elements of anactual lattice and to account for the collective effects asso-ciated with the on set of MBI, namely longitudinal space-charge, coherent synchrotron radiation, and possibly RFwakes. We report an outline of the core algorithm dedi-cated to the solution of the Vlasov equation modelling thebeam propagation, a discussion of some of the technicalproblems and the numerical challenges posed by such a V-S, and preliminary results from some numerical tests.VLASOV EQUATIONThe evolution equation that describes the beam dynam-ics of a charged beam density distribution Ψ is the Vlasovequation∂Ψ∂s= HΨ with Ψ|s=0 = Ψ0 (1)where s refers to the propagation coordinate.The formal solution of eq. (1) can be written, for a smallpropagation interval ∆s and if H is not explicitly a timedependent operator, asΨ = exp (H∆s)Ψ0 (2)representing the Vlasov evolution operator. Actually, inwriting eq. (2), we are neglecting any contribution due totime ordering corrections, that arises whenever the operatorH is explicitly time dependent and does not commute withitself at different times. With these assumptions we neglectthird order terms in the integration step ∆s, the symplectic-ity is however automatically preserved by the exponentialform of the evolution operator.The operator H encloses the physical properties of thepropagation problem and contains the beam dynamics. Inabsence of any collective effects generated by wake fields,it is a differential operator, describing the beam propaga-tion through magnetic lens systems [5]. If we include thewake fields effects, H becomes an integral operator andeq. (2) becomes non linear [3]. For example, in case ofbeam transport through a bending magnet of radius R andlength L, in a 4D beam dynamics, we can writeH =xR∂∂z− θ ∂∂x+(k2βx−δR)∂∂θ+∂∂δe2NE0L∫ ∞−∞W (z′ − z)ρ(z′)dz′ (3)with x and θ the transverse coordinates, z and δ the longitu-dinal ones, kβ the magnet focusing strength, N the numberof particles in the bunch, ρ(z) the projection of the densitydistribution Ψ on the longitudinal axis, and W (z) the lon-gitudinal wake function, providing, for the problem we areconsidering, the CSR wake field that we write as [6]W (z) =14piε02(3R2)1/3∂∂zz−1/3 z > 0 (4)Accordingly, we neglect the screening effect of conductingwalls and we consider only the steady state radiation froman ultra-relativistic particle in a long magnet [7].Equations analogous to eq. (3) can be written for thequadrupoles, drifts and RF cavities, which are the beamtransport devices used for our study of the MBI. Further-more, the longitudinal space charge wake field, of thekind [8]14piε0γ2(1 + 2 logba)δ′ (5)with δ′ the derivative of the Dirac delta function, has beenused in the drift sections by considering a transversally uni-form bunch density with circular cross section a and a cir-cular beam pipe of radius b. This wake field is supposed tobe the main cause leading to the MBI [9].Once the expression of the operator H is known for eachdevice of the bunch compressor, we can write the explicitsolution of eq. (2). In order to do that, we observe that anyexpression of H of the kind of eq. (3) can be always writtenas the sum of the beam transport matrix without collectiveeffects A0, and the non linear term due to the wake field,that is H = A0 + F (z)∂/∂δ.With this assumption, the operator H consists of twoparts that we decouple by using the operator splitting tech-niqueeH∆s = e12∆sA0e∆sF (z)∂∂δ e12∆sA0 (6)Such a decoupling realizes a kind of interaction picture inwhich the collective effects are separated from the ordinarytransport part whose action on the beam distribution func-tion is known exactly. We have indeede12∆sA0Ψ0 = Ψ(R(−∆s2)X0)(7)with R(s) the transport matrix of the element and X0 theinitial coordinates vector. The action of the non linear partcan be evaluated in an analogous way and it is readily un-derstood since the exponential operator accounting for itseffect is just a shift operator in the coordinate δ, provid-ing a translation of the same coordinate by ∆sF (z) in thedistribution function, i. e.e∆sF (z)∂∂δΨ(z, δ, x, θ) = Ψ (z, δ +∆sF (z), x, θ) (8)This is the way we transport the distribution functionthrough the compressor devices in our V-S.NUMERICAL RESULTSThe simulation code TEO [3] was upgraded to the 4D do-main. The initial e-beam distribution Ψ was sampled on auniform dense Cartesian grid (zi, δj , xk, θl). The beam wasthen advanced by a discrete step∆s along the beam line us-ing the operator method described in the previous section.For each grid point the advanced distribution Ψ′ was ob-tained by numerical interpolation of the distribution Ψ atthe beginning of the step using the method of local char-acteristics. Namely, Ψ′(zi, δj , xk, θl) = Ψ(zˆi, δˆj , xˆk, θˆl),where the origin of the characteristic zˆi, δˆj , xˆk, θˆl was ob-tained by applying the exponential operator as in (8). La-grange polynomials of the 5th order where used for the 4Doff-grid interpolation of the distribution.The first tests of the Vlasov solver have been performedin order to verify if the 4D dynamics of the simple trans-port without wake fields is well represented. In Fig. (1) wehave used a magnetic bunch compressor with parametersclose to those of SPARX BC2 [10] and have compared thetransverse emittance with that obtained with the transportmatrix method. The agreement is excellent. Also with thelongitudinal dynamics there is total agreement. One prob-lem arising in the use of a Vlasov solver is related to thehigh resolution of the grid necessary to evidence the MBI.Figure 1: Emittance comparison between the Vlasov solverand the beam transport matrix.If we consider, for example, the longitudinal phase space atthe entrance of the compressor, as shown in Fig. (2), we cansee that there is a strong correlation between the variablesz - δ. The same happens in the transverse plane. Using anuniform orthogonal grid, a large fraction of the mesh points(> 90%) corresponds to empty regions in the phase space,and therefore represents a waste of memory and compu-tational resources. Studying the MBI in 2D (longitudinalphase space only) we have found that grids of the orderof 1000x500 nodes were needed in order to follow the in-stability at short wavelenghts, say 10 µm and below. In4D a very coarse grid with a hundred points per directioncorrespond to 1004 mesh nodes, and requires typically 2GB of memory space at runtime. Preliminary investiga-tions show that high resolution in the longitudinal phasespace z − δ calls for a comparable resolution in the trans-verse space x−θ, because the two spaces are strongly cou-pled by the dynamics. It is clear that increasing by a factor10 the grid resolution is not feasible, since it would cor-respond to a factor of 104 increase in the computationalresources. Presently we are considering two options foraddressing this problem: the distribution support could beremapped to a uniform orthogonal space using a coordi-nate transformation similar to that presented for a 2D solverin [4], or the phase space could be divided into parallelslices which closely follow the distribution support. Thefirst option would require a more complex redefinition ofthe Hamiltonian operator, and could introduce numericalnoise due to the remapping of the longitudinal coordinate.Therefore we are currently implementing the second optionwhich requires just a careful book keeping of grid, withoutany changes to the computational kernel.On the basis of the considerations just presented, it isworthwhile to note that Vlasov solvers for particle acceler-ator problems require large computational resource and, atleast with present computers, it seems very hard to addressthe full 6D problem at a resolution high enough to resolvethe CSR dynamics at very short wavelengths.CONCLUSIONSWe have discussed an accurate analysis of the micro-bunching instability. Macroparticle codes suffer by intrin-sic numerical noise problems, while the possible limita-Figure 2: Longitudinal phase space at the entrance of thecompressor. Linear color scheme.tions of the V-S method is due to the grid number of pointswhich may become prohibitively large with many uselesspoints, since the phase space is strongly correlated. Theessence of the instability itself is such a correlation, whichgives rise to bunching and hence to coherent emission. TheMBI dynamics share strong analogies with the FEL dynam-ics itself, whose final effect on the beam is that of creating alarge (uncorrelated) energy spread. The interplay betweenFEL and saw-tooth type instabilities (a different flavour ofMBI) has shown that they are competing effects [11], there-fore MBI dampers based on FEL heater devices have beenproposed [9]. A quantitative understanding of the MBI willtherefore provide elements to design such a tool to inhibitthe growth of the instability.REFERENCES[1] E.L. Saldin, E.A. Schneidmiller, and M.V. Yurkov, Nucl. In-strum. Methods Phys. Res., Sect. A 483, 516 (2002).[2] S. Heifets, G. Stupakov, and S. Krinsky, Phys. Rev. ST Ac-cel. Beams 5, 064401 (2002); Z. Huang and K.-J. Kim,Phys. Rev. ST Accel. Beams 5, 074401 (2002).[3] G. Dattoli, et al., NIM A 574 (2007) pp. 244-250.[4] M. Venturini, R. Warnock, and A. Zholents, Phys. Rev. STAccel. Beams 10, 054403 (2007); M. Venturini, Phys. Rev.ST Accel. Beams 10, 104401 (2007); M. Venturini and A.Zholents, Modeling Microbunching from Shot Noise UsingVlasov Solvers, to appear in NIM-A (2008).[5] A.J. Dragt, “Lectures in non-linear orbit dynamics”, Proc.AIP Conf., vol. 87, AIP, Washington, 1982.[6] S. Heifets and G. Stupakov, PRST AB 5, 054402 (2002).[7] J. B. Murphy, S. Krinsky, and R. L. Gluckstern, Part. Ac-cel. 57, 9 (1997); Y. S. Derbenev et al., DESY Report No.TESLA-FEL 95-05, 1995.[8] A. Chao, Physics of Collective Beam Instabilities in HighEnergy Acceleratos (Wiley, New York, 1993).[9] E.L. Saldin, et al., Nucl. Instrum. Methods Phys. Res., Sect.A 528, pp. 355-359 (2004).[10] C. Vaccarezza, private communication.[11] R. Bartolini, et al., PRL, vol. 87, N. 13, 134801 (2001).This document was prepared as an account of worksponsored by the United States Government. Whilethis document is believed to contain correctinformation, neither the United States Governmentnor any agency thereof, nor The Regents of theUniversity of California, nor any of theiremployees, makes any warranty, express or implied,or assumes any legal responsibility for theaccuracy, completeness,or usefulness of anyinformation, apparatus, product, or processdisclosed, or represents that its use wouldnot infringe privately owned rights. Referenceherein to any specific commercial product,process, or service by its trade name, trademark,manufacturer, or otherwise, does not necessarilyconstitute or imply its endorsement,recommendation, or favoring by the United StatesGovernment or any agency thereof, or The Regentsof the University of California. The views andopinions of authors expressed herein do notnecessarily state or reflect those of theUnited States Government or any agency thereof,or The Regents of the University of California.Ernest Orlando Lawrence Berkeley NationalLaboratory is an equal opportunity employer.

A FOUR-DIMENSIONAL VLASOV SOLVER FOR MICROBUNCHING INSTABILITY INTHE INJECTION SYSTEM FOR X-RAY FELS

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A four-dimensional Vlasov solver for microbunching instability in the injection system for X-ray FELs

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