Solving the Vlasov-Fokker-Planck equation is a well-tested approach to simulate dynamics of electron bunches self-interacting with their own wake-field. Typical implementations model the dynamics of a charge density in a damped harmonic oscillator, with a small perturbation due to collective effects. This description imposes some limits to the applicability: Because after a certain simulation time coherent synchrotron motion will be damped down, effectively only the incoherent motion is described. Furthermore – even though computed - the tune spread is typically masked by the use of a charge density instead of individual particles. As a consequence, some effects are not reproduced. In this contribution, we present methods that allow to consider single-particle motion, coherent synchrotron oscillations, non-linearities of the accelerating voltage, higher orders of the momentum compaction factor, as well as modulations of the accelerating voltage. We also provide exemplary studies – based on the KIT storage ring KARA (KArlsruhe Research Accelerator) - to show the potentiality of the methods

Boltz, T.

Leonard Steinmann, J.

Mochihashi, A.

Müller, A.-S.

Schönfeldt, P.

English

Steinmann, J. L.

KITopen

ELABORATED MODELING OF SYNCHROTRON MOTION INVLASOV-FOKKER-PLANCK SOLVERSP. Schönfeldt∗, T. Boltz, A. Mochihashi, J. L. Steinmann and A.-S. Müller,Karlsruhe Institute of Technology , Karlsruhe, GermanyAbstractSolving the Vlasov-Fokker-Planck equation is a well-tested approach to simulate dynamics of electron bunchesself-interacting with their own wake-field. Typical imple-mentations model the dynamics of a charge density in adamped harmonic oscillator, with a small perturbation dueto collective effects. This description imposes some limitsto the applicability: Because after a certain simulation timecoherent synchrotron motion will be damped down, effec-tively only the incoherent motion is described. Furthermore –even though computed – the tune spread is typically maskedby the use of a charge density instead of individual parti-cles. As a consequence, some effects are not reproduced. Inthis contribution, we present methods that allow to considersingle-particle motion, coherent synchrotron oscillations,non-linearities of the accelerating voltage, higher orders ofthe momentum compaction factor, as well as modulations ofthe accelerating voltage. We also provide exemplary studies- based on the KIT storage ring KARA (KArlsruhe ResearchAccelerator) - to show the potentiality of the methods.INTRODUCTIONWhen it comes to the simulation of beam dynamics inthe longitudinal phase space of electron storage rings, theVlasov-Fokker-Planck equation∂ψ∂t+∂H∂p∂ψ∂q−∂H∂q∂ψ∂p=2τd∂∂p(pψ +∂ψ∂p)(1)allows an elegant formulation of the self-interaction of thebunch with its own wakefield. It describes the evolutionof the particle density ψ(q, p), here using the generalizedcoordinates q = z/σz,0, and p = (E − E0)/σδ,0, and theHamiltonian H. The quantities z, E , E0, τd, σδ,0, and σz,0describe the longitudinal position, the energy, the referenceparticle’s energy, the longitudinal damping time, the energyspread, and bunch length, respectively. The latter two ex-ist in the equilibrium state at small bunch charges. Theperturbation due to the collective effects is described as aperturbation to the Hamiltonian, which can be expressed interms of an impedance Z(ω). To follow this approach, anultra-relativistic (β = 1, but γ < ∞) line charge is assumed.In general, the evolution of the phase space can be ex-pressed using a mapM : ψ(q, p, t) → ψ(q, p, t + Δt). (2)For numerical stability, the map M can be split into a seriesof symplectic maps that are solutions to different parts of∗ patrik.schoenfeldt@kit.eduEq. (1): A rotation (R) solves the unperturbed, conservativeproblem and the additional potential due to collective effectsis modeled as an energy kick K . Damping and diffusion(RHS) are added in the spirit of a particular solution, and ex-pressed by the non-symplectic map D. They are sequentiallyevaluated, so in totalM = D ◦ K ◦ R. (3)The application of the method for studying electron beamdynamics was first suggested by Warnock and Ellison [1]in 2000, and has since proven to show good convergence(e.g. [2, 3]). By splitting the rotation into an energy depen-dent drift followed by a location dependent RF kick [4]R = RK ◦ RD, (4)numerical stability can be further improved. An implemen-tation of the features discussed in this paper can be foundat [5], scripts to reproduce the results at [6]. The parametersused for the simulations are listed in Table 1. For the wake-field, we consider coherent synchrotron radiation shieldedby parallel plates [7].PASSIVE PARTICLE TRACKINGAs a charge density is considered, Vlasov-Fokker-Plancksolvers do not need to reproduce the trajectories of individualparticles. To still retrieve this information without the needto track a number of macro-particles that can represent thecharge density with sufficient accuracy, we suggest passiveparticle tracking. Equation 2 can be expressed as [1]ψ(r, t + Δt) ≈ ψ(M(t |t + Δt)ψ(t)(r), t), (5)where M(t |t + Δt)ψ(t) is a map that varies as ψ evolves intime. This notation mirrors the fact that it was evolved fromr ′ = M(t + Δt |t)(r), (6)Table 1: Physical Parameters Used for the SimulationsQuantity Symbol Value UnitAccelerating Voltage VRF 799 kVBeam energy E0 1.285 GeVEnergy spread σE 6040 keVBending radius R 5.559 mDamping time τd 10.4 msHarmonic number h 184Revolution frequency frev 2.716 MHzSynchrotron frequency fs,0 8.27 kHzFull beam pipe height g 32 mmThisisapreprint—thefinalversionispublishedwithIOP9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW PublishingISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK03205 Beam Dynamics and EM FieldsD11 Code Developments and Simulation TechniquesTHPAK0323283ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.−2 0 2Position (σz, 0)−3−2−10123Energy Deviation (σE,0) t=1.0 Ts0246810Charge Density (pC/σz,0/σE,0)Figure 1: Example charge density of a bunch in the poten-tial well distortion. Trajectories of five particles (light bluearrows) for a time corresponding to one unperturbed syn-chrotron period are displayed as an overlay. Also, there aretwo dark gray lines added that connect the positions at thefirst and the last time step, respectively. The dynamics havebeen calculated using the maps obtained based on the chargedensity, neglecting stochastic effects.which implies that a trajectory passing through r at time tpasses through r ′ at time t +Δt. Calculating the maps basedon the charge density provides good numerical stability, sowe derive them using the charge density and afterwards ex-press them in terms of the change of a particle’s position andenergy. To get a realistic picture, special care is needed forD: This map contains averaged quantum effects, which donot make sense on a single particle level. A first order prob-abilistic approach including both, damping and radiationexcitation, is [8]D : p(t + Δt) = p(t) ×(1 −2Δtτd)+ 2√ΔtτdN(σp), (7)whereN(σ) is a random number drawn from a zero-centeredGaussian distribution with the width σ. However, as thetracking is only performed for visualization purposes, it isalso valid to think of an “average particle”. An exampleusing this flavour of the passive tracking method is shown inFig. 1. It diplays the trajectories of five particles as an overlayto the charge density of an electron bunch in a distortedpotential well. Using this method, tune spread and tune shiftdue to collective effects are clearly visible.DYNAMIC ACCELERATING VOLTAGETypically, the unperturbed LHS of Eq. (1) is evaluatedto the first order, yielding a harmonic oscillator which isthen modeled using the rotation map R. A time-dependentacceleration could in principle be modeled by adding a driv-ing term to the RHS. We, instead, use the fact that we splitR into two contributions [see Eq. (4)] parallel to the twodimensions that span the phase space. This way, only theRF kick map RK has to be modified. For small variations,0 100 200 300Time (Ts, 0)−2.50.02.5Position (ps)Figure 2: Evolution of the synchrotron oscillation amplitudeover time. First, the simulated RF is stable. At T = 0, whiteGaussian phase noise with σq = 0.1◦= 0.56 ps is turnedon. The natural rms bunch length is σz,0 = 4.61 ps.the phase modulation results in an additive term δ+, the am-plitude modulation in a multiplicative term δ× [9]. So themap can be expressed asRK : p(q, t + Δt) = p(t) + δ+(t) + q × tan−1(1/(Δt fs,0))× [1 + δ×(t)], (8)with the synchrotron frequency fs,0. For the unperturbedcase (δ = 0), the map is constant in t. So, the formulation isequivalent to a simple rotation for this unperturbed case.As a first test, we consider Gaussian white noise to theRF phase. To do so, it has to be considered that the lengthof the time step Δt for the numerical solution of Eq. (1) canbe arbitrarily chosen. So, the random distribution is sam-pled more often for small Δt. This can be compensated byintroducing a correction factor [10] that scales the dynamicsto be equal to the case where acceleration happens once perrevolution period. In totalδ+ =√Δt frev × N(σq), (9)where σq is the RF phase spread σφ expressed in multiplesof the natural bunch length. An example of simulation us-ing σφ = 0.1◦ is displayed in Fig. 2. The RF phase noisedrives a coherent synchrotron oscillation with stochasticallyvarying amplitude. Its long-term mean amplitude is stronglydependent on the amplitude of the phase noise and can becalculated by [9]〈φ2〉 = π2 f 2s,0 σ2φ τd/ frev. (10)In a second step, we add RF amplitude noise withδ× =√Δt frev × N (σV/VRF) , (11)where σV is the spread of the peak accelerating voltage.Bursting spectrograms of exemplary results are displayedin Fig. 3. In general, the simple model only consider-ing the CSR impedance shielded by parallel plates repro-duces the main features: There are strong frequency linesat f ≈ 30 kHz and its multiples as well as the occurrence oflow frequency fluctuations at higher currents. The thresholdcurrents, however, differ by ≈ 25 μA. Introducing noisedoes not change this. Instead, it influences the backgroundThisisapreprint—thefinalversionispublishedwithIOP9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW PublishingISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK032THPAK0323284ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.05 Beam Dynamics and EM FieldsD11 Code Developments and Simulation TechniquesFigure 3: Spectrograms of emitted bursting coherent synchrotron radiation (see e.g. [11]). Left: Simulation data withoutRF noise. Center: Simulation considering RF phase noise with σq = 0.01◦ and RF amplitude noise of σV/V = 1 %. Right:Measurement data obtained using a broad-band Schottky diode from ACST [12]. Both simulations reproduce the mainfrequency at f ≈ 30 kHz and its harmonics. The instability threshold is 211 μA for the measurement and 237 μA for bothsimulations. Adding noise improves the agreement between simulation and measurement in another aspect: The unphysicaldrop of the background at the lowest current disappears and the minimum intensity between the peak frequencies is lowered,where an additional structure becomes visible. (See text for interpretation of current-independent fs,0 harmonics.)level in two ways: Without noise, the CSR ceases fluctu-ating below the threshold current. Considering noise re-moves this artifact. Furthermore, it actually lowers the noisefloor between the two main frequency components. In thegiven example, this allows to see an additional structure atf ≈ 50 kHz, which seems comparable to similar featuresin the measurement data. The simulated lines at the 2ndand 4th fs,0 harmonic are due to asymmetries driven by theRF noise. In the projection to the profile, they appear withmultiples fs,0. The fs,0 line in the measurement might beexplained by the different transmission from the emittedlight to the detector for different dispersive paths. In con-trast, only emission is considered for the simulation. Thedifference in the instability threshold not being explained bynoise suggests to take additional impedances into account.In principle, Eq. (8) also holds for intentional RF modu-lations. However, they are typically driven with an ampli-tude [13–16] where the applicability of the linear approxima-tion might be questioned. So, for these cases, the sinusoidalshape of the accelerating voltage should be considered. Itcan be expressed asRK : p(t + Δt) = p(t)− frevΔt e [VRF sin(ωV q(t) + q0) + V0]/σδ,0, (12)where q0 is the synchronous phase, ωV is the RF frequencyin units of q, and eV0 is radiation loss of an electron per turn.When (ωV q(t) + q0) ≈ 0, this formulation is equivalent tothe linear approximation.NONLINEAR MOMENTUMCOMPACTIONTo complete the picture, we also include higher orders ofthe momentum compaction factorαc =∞∑n=0αn(EE0)n=ΔL/L0ΔE/E0, (13)where L0 marks the length of the design orbit and ΔL thedifference in orbit length due to an energy offset ΔE . Thisnonlinearity introduces a tune-spread that can e.g. Landaudamp instabilities [2]. The modified drift map RD now readsRD : q(t + Δt) = q(t) + η′c × p(t), (14)where η′c = α′c−1/γ2 is the slip factor in the q, p coordinates.If αn = 0 ∀n > 0, the map is identical to the one originallydeveloped for the rotation.SUMMARY AND OUTLOOKWe presented additions to Vlasov-Fokker-Planck solversthat allow to solve problems that were previously not ad-dressed by this simulation method. As a premiere example,we considered RF noise. In the simulation of bursting CSR,the addition allows to see frequency components that wherenot reproduced before. Systematic studies modeling inten-tional RF phase modulation and using nonlinear momentumcompaction are planned and first tests already show promis-ing results.ACKNOWLEDGMENTSFunded by the German Federal Ministry of Education andResearch (Grant No. 05K16VKA) and Initiative and Net-working Fund of the Helmholtz Association (contract num-ber: VH-NG-320). T. Boltz, P. Schönfeldt and J. L. Stein-mann want to acknowledge the support by the HelmholtzInternational Research School for Teratronics (HIRST).REFERENCES[1] R. L. Warnock and J. A. Ellison, “A general method for prop-agation of the phase space distribution, with application toThisisapreprint—thefinalversionispublishedwithIOP9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW PublishingISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK03205 Beam Dynamics and EM FieldsD11 Code Developments and Simulation TechniquesTHPAK0323285ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.the sawtooth instability”, in Proc. 2nd ICFA Advanced Accel-erator Workshop, Los Angeles, California, USA, Nov. 1999,pp. 322-348. doi:10.1142/9789812792181_0020[2] K. L. F. Bane, Y. Cai, and G. Stupakov, “Threshold studies ofthe microwave instability in electron storage rings”, in Phys.Rev. ST Accel. Beams, vol. 13, p. 104402, 2010.[3] E. Roussel, C. Evain, C. Szwaj, and S. Bielawski, “Mi-crobunching instability in storage rings: Link between phase-space structure and terahertz coherent synchrotron radiationradio-frequency spectra”, in Phys. Rev. ST Accel. Beams, vol17, p. 010701, 2014.[4] P. Schönfeldt, M. Brosi, M. Schwarz, J. L. Steinmann, andA.-S. Müller, “Parallelized Vlasov-Fokker-Planck solver fordesktop personal computers”, in Phys. Rev. Accel. Beams,vol. 20, p. 030704, 2017.[5] “Inovesa source code”, doi:10.5281/zenodo.597356[6] “Inovesa examples”, github.com/Inovesa/examples[7] J. Murphy, S. Krinsky, and R. Gluckstern, “Longitudinalwakefield for an electron moving on a circular orbit”, in Part.Accel., vol. 57, pp. 9-64, 1997.[8] K. L. F. Bane and K. Oide, “Simulations of the longitudinalinstability in the SLC damping rings”, in Proc. 1993 ParticleAccelerator Conference, Washington D.C., USA, May 1993,pp. 3339-3341. doi:10.1109/PAC.1993.309645[9] K. W. Ormond and J. T. Rogers, “Synchrotron oscillationdriven by RF phase noise”, in Proc. 1997 Particle AcceleratorConference, Vancouver, Canada, May 1997, p. 1822-1824.doi:10.1109/PAC.1997.751029[10] J. Schestag, “Study of noise in the simulation of collective ef-fects of relativistic electron bunches”, Bachelor thesis, Karls-ruhe Institute of Technology, Karlsruhe, Germany, 2018.[11] M. Brosi et. al., “Fast mapping of terahertz bursting thresholdsand characteristics at synchrotron light sources”, Phys. Rev.Accel. Beams, vol. 19, p. 110701, 2016.[12] ACST GmbH, http://www.acst.de/?view=thz_detector[13] N. P. Abreu, R. H. A. Farias, and P. F. Tavares “Longitudinaldynamics with RF phase modulation in the Brazilian electronstorage ring”, Phys. Rev. ST Accel. Beams, vol. 9, p. 124401,2006.[14] H. Huang et al., “Experimental determination of the Hamil-tonian for synchrotron motion with RF phase modulation” inPhys. Rev. E, vol. 48, pp. 4678-4688, 1993.[15] F. Orsini and A. Mosnier, "Effectiveness of rf phase modula-tion for increasing bunch length in electron storage rings", inPhys. Rev. E, vol. 61, pp. 4431-4440, 2000.[16] M.A. Jebramcik et al., “Coherent harmonic generation in thepresence of synchronized RF phase modulation at DELTA”in Proc. IPAC ’16, Busan, Korea, May 2016, pp. 2847-2850.doi:10.18429/JACoW-IPAC2016-WEPOW013Thisisapreprint—thefinalversionispublishedwithIOP9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW PublishingISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK032THPAK0323286ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.05 Beam Dynamics and EM FieldsD11 Code Developments and Simulation Techniques

Elaborated Modeling of Synchrotron Motion in Vlasov-Fokker-Planck Solvers

Journal of Physics: Conference SeriesPAPER • OPEN ACCESSElaborated Modeling of Synchrotron Motion in Vlasov-Fokker-PlanckSolversTo cite this article: Patrik Schönfeldt et al 2018 J. Phys.: Conf. Ser. 1067 062025 View the article online for updates and enhancements.This content was downloaded from IP address 129.13.72.197 on 22/11/2018 at 09:031Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd1234567890 ‘’“”9th International Particle Accelerator Conference, IPAC18 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1067 (2018) 062025  doi :10.1088/1742-6596/1067/6/062025Elaborated Modeling of Synchrotron Motion inVlasov-Fokker-Planck SolversPatrik Schönfeldt1, Tobias Boltz1, Akira Mochihashi1, JohannesLeonard Steinmann1 and Anke-Susanne Müller11 Karlsruhe Institute of Technology , Karlsruhe, GermanyE-mail: patrik.schoenfeldt@kit.eduAbstract. Solving the Vlasov-Fokker-Planck equation is a well-tested approach tosimulate dynamics of electron bunches self-interacting with their own wake-field. Typicalimplementations model the dynamics of a charge density in a damped harmonic oscillator,with a small perturbation due to collective effects. This description imposes some limits tothe applicability: Because after a certain simulation time coherent synchrotron motion willbe damped down, effectively only the incoherent motion is described. Furthermore – eventhough computed – the tune spread is typically masked by the use of a charge density instead ofindividual particles. As a consequence, some effects are not reproduced. In this contribution, wepresent methods that allow to consider single-particle motion, coherent synchrotron oscillations,non-linearities of the accelerating voltage, higher orders of the momentum compaction factor,as well as modulations of the accelerating voltage. We also provide exemplary studies - basedon the KIT storage ring KARA (KArlsruhe Research Accelerator) - to show the potentiality ofthe methods.1. IntroductionWhen it comes to the simulation of beam dynamics in the longitudinal phase space of electronstorage rings, the Vlasov-Fokker-Planck equation∂ψ∂t+∂H∂p∂ψ∂q− ∂H∂q∂ψ∂p=2τd∂∂p(pψ +∂ψ∂p)(1)allows an elegant formulation of the self-interaction of the bunch with its own wakefield. Itdescribes the evolution of the particle density ψ(q, p), here using the generalized coordinatesq = z/σz,0, and p = (E − E0)/σδ,0, and the Hamiltonian H. The quantities z, E, E0, τd,σδ,0, and σz,0 describe the longitudinal position, the energy, the reference particle’s energy, thelongitudinal damping time, the energy spread, and bunch length, respectively. The latter twoexist in the equilibrium state at small bunch charges. The perturbation due to the collectiveeffects is described as a perturbation to the Hamiltonian, which can be expressed in terms of animpedance Z(ω). To follow this approach, an ultra-relativistic (β = 1, but γ < ∞) line chargeis assumed.In general, the evolution of the phase space can be expressed using a mapM : ψ(q, p, t)→ ψ(q, p, t+ ∆t). (2)21234567890 ‘’“”9th International Particle Accelerator Conference, IPAC18 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1067 (2018) 062025  doi :10.1088/1742-6596/1067/6/062025Table 1. Physical parameters used for the simulationsQuantity Symbol Value UnitAccelerating Voltage VRF 799 kVBeam energy E0 1.285 GeVEnergy spread σE 6040 keVBending radius R 5.559 mDamping time τd 10.4 msHarmonic number h 184Revolution frequency frev 2.716 MHzSynchrotron frequency fs,0 8.27 kHzFull beam pipe height g 32 mmFor numerical stability, the map M can be split into a series of symplectic maps that aresolutions to different parts of Eq. (1): A rotation (R) solves the unperturbed, conservativeproblem and the additional potential due to collective effects is modeled as an energy kick K.Damping and diffusion (RHS) are added in the spirit of a particular solution, and expressed bythe non-symplectic map D. They are sequentially evaluated, so in totalM = D ◦K ◦R. (3)The application of the method for studying electron beam dynamics was first suggested byWarnock and Ellison [1] in 2000, and has since proven to show good convergence (e.g. [2, 3]).By splitting the rotation into an energy dependent drift followed by a location dependent RFkick [4]R = RK ◦RD, (4)numerical stability can be further improved. An implementation of the features discussed in thispaper can be found at [5], scripts to reproduce the results at [6]. The parameters used for thesimulations are listed in Table 1. For the wake-field, we consider coherent synchrotron radiationshielded by parallel plates [7].2. Passive particle trackingAs a charge density is considered, Vlasov-Fokker-Planck solvers do not need to reproduce thetrajectories of individual particles. To still retrieve this information without the need to tracka number of macro-particles that can represent the charge density with sufficient accuracy, wesuggest passive particle tracking. Equation 2 can be expressed as [1]ψ(r, t+ ∆t) ≈ ψ(M(t|t+ ∆t)ψ(t)(r), t), (5)where M(t|t+ ∆t)ψ(t) is a map that varies as ψ evolves in time. This notation mirrors the factthat it was evolved fromr′ = M(t+ ∆t|t)(r), (6)which implies that a trajectory passing through r at time t passes through r′ at time t + ∆t.Calculating the maps based on the charge density provides good numerical stability, so we derivethem using the charge density and afterwards express them in terms of the change of a particle’sposition and energy. To get a realistic picture, special care is needed for D: This map contains31234567890 ‘’“”9th International Particle Accelerator Conference, IPAC18 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1067 (2018) 062025  doi :10.1088/1742-6596/1067/6/062025−3 −2 −1 0 1 2 3Position (σz, 0)−3−2−10123Energy Deviation (σE,0)0246810Charge Density (pC/σ z,0/σE,0)Figure 1. Example charge density of a bunch in the potential well distortion. Trajectoriesof five particles (light blue arrows) for a time corresponding to one unperturbed synchrotronperiod are displayed as an overlay. Also, there are two dark gray lines added that connect thepositions at the first and the last time step, respectively. The dynamics have been calculatedusing the maps obtained based on the charge density, neglecting stochastic effects.averaged quantum effects, which do not make sense on a single particle level. A first orderprobabilistic approach including both, damping and radiation excitation, is [8]D : p(t+ ∆t) = p(t)×(1− 2∆tτd)+ 2√∆tτdN (σp) , (7)where N (σ) is a random number drawn from a zero-centered Gaussian distribution with thewidth σ. However, as the tracking is only performed for visualization purposes, it is also validto think of an “average particle”. An example using this flavour of the passive tracking methodis shown in Fig. 1. It diplays the trajectories of five particles as an overlay to the charge densityof an electron bunch in a distorted potential well. Using this method, tune spread and tuneshift due to collective effects are clearly visible.3. Dynamic accelerating voltageTypically, the unperturbed LHS of Eq. (1) is evaluated to the first order, yielding a harmonicoscillator which is then modeled using the rotation map R. A time-dependent acceleration couldin principle be modeled by adding a driving term to the RHS. We, instead, use the fact that wesplit R into two contributions [see Eq. (4)] parallel to the two dimensions that span the phasespace. This way, only the RF kick map RK has to be modified. For small variations, the phasemodulation results in an additive term δ+, the amplitude modulation in a multiplicative termδ× [9]. So the map can be expressed asRK : p(q, t+ ∆t) = p(t) + δ+(t) + q × tan−1(1/(∆tfs,0))× [1 + δ×(t)], (8)with the synchrotron frequency fs,0. For the unperturbed case (δ = 0), the map is constant int. So, the formulation is equivalent to a simple rotation for this unperturbed case.As a first test, we consider Gaussian white noise to the RF phase. To do so, it has to beconsidered that the length of the time step ∆t for the numerical solution of Eq. (1) can bearbitrarily chosen. So, the random distribution is sampled more often for small ∆t. This canbe compensated by introducing a correction factor [10] that scales the dynamics to be equal to41234567890 ‘’“”9th International Particle Accelerator Conference, IPAC18 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1067 (2018) 062025  doi :10.1088/1742-6596/1067/6/0620250 50 100 150 200 250 300 350Time (Ts, 0)−2−1012Position (ps)Figure 2. Evolution of the synchrotron oscillation amplitude over time. First, the simulatedRF is stable. At T = 0, white Gaussian phase noise with σq = 0.1◦ = 0.56 ps is turned on. Thenatural rms bunch length is σz,0 = 4.61 ps.the case where acceleration happens once per revolution period. In totalδ+ =√∆tfrev ×N (σq) , (9)where σq is the RF phase spread σφ expressed in multiples of the natural bunch length. Anexample of simulation using σφ = 0.1◦ is displayed in Fig. 2. The RF phase noise drivesa coherent synchrotron oscillation with stochastically varying amplitude. Its long-term meanamplitude is strongly dependent on the amplitude of the phase noise and can be calculated by [9]〈φ2〉 = π2f2s,0 σ2φ τd/frev. (10)In a second step, we add RF amplitude noise withδ× =√∆tfrev ×N (σV/VRF) , (11)where σV is the spread of the peak accelerating voltage. Bursting spectrograms of exemplaryresults are displayed in Fig. 3. In general, the simple model only considering the CSR impedanceshielded by parallel plates reproduces the main features: There are strong frequency lines atf ≈ 30 kHz and its multiples as well as the occurrence of low frequency fluctuations at highercurrents. The threshold currents, however, differ by ≈ 25µA. Introducing noise does not changethis. Instead, it influences the background level in two ways: Without noise, the CSR ceasesfluctuating below the threshold current. Considering noise removes this artifact. Furthermore,it actually lowers the noise floor between the two main frequency components. In the givenexample, this allows to see an additional structure at f ≈ 50 kHz, which seems comparable tosimilar features in the measurement data. The simulated lines at the 2nd and 4th fs,0 harmonicare due to asymmetries driven by the RF noise. In the projection to the profile, they appear withmultiples fs,0. The fs,0 line in the measurement might be explained by the different transmissionfrom the emitted light to the detector for different dispersive paths. In contrast, only emissionis considered for the simulation. The difference in the instability threshold not being explainedby noise suggests to take additional impedances into account.In principle, Eq. (8) also holds for intentional RF modulations. However, they are typicallydriven with an amplitude [13–16] where the applicability of the linear approximation mightbe questioned. So, for these cases, the sinusoidal shape of the accelerating voltage should beconsidered. It can be expressed asRK : p(t+ ∆t) = p(t)− frev∆t e [VRF sin(ωV q(t) + q0) + V0]/σδ,0, (12)where q0 is the synchronous phase, ωV is the RF frequency in units of q, and eV0 is radiationloss of an electron per turn. When (ωV q(t)+q0) ≈ 0, this formulation is equivalent to the linearapproximation.51234567890 ‘’“”9th International Particle Accelerator Conference, IPAC18 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1067 (2018) 062025  doi :10.1088/1742-6596/1067/6/062025Figure 3. Spectrograms of emitted bursting coherent synchrotron radiation (see e.g. [11]).Left: Simulation data without RF noise. Center: Simulation considering RF phase noise withσq = 0.01◦ and RF amplitude noise of σV/V = 1 %. Right: Measurement data obtained usinga broad-band Schottky diode from ACST [12]. Both simulations reproduce the main frequencyat f ≈ 30 kHz and its harmonics. The instability threshold is 211µA for the measurementand 237µA for both simulations. Adding noise improves the agreement between simulationand measurement in another aspect: The unphysical drop of the background at the lowestcurrent disappears and the minimum intensity between the peak frequencies is lowered, wherean additional structure becomes visible. (See text for interpretation of current-independent fs,0harmonics.)4. Nonlinear momentum compactionTo complete the picture, we also include higher orders of the momentum compaction factorαc =∞∑n=0αn(EE0)n=∆L/L0∆E/E0, (13)where L0 marks the length of the design orbit and ∆L the difference in orbit length due toan energy offset ∆E. This nonlinearity introduces a tune-spread that can e.g. Landau dampinstabilities [2]. The modified drift map RD now readsRD : q(t+ ∆t) = q(t) + η′c × p(t), (14)where η′c = α′c − 1/γ2 is the slip factor in the q, p coordinates. If αn = 0 ∀n > 0, the map isidentical to the one originally developed for the rotation.5. Summary and outlookWe presented additions to Vlasov-Fokker-Planck solvers that allow to solve problems that werepreviously not addressed by this simulation method. As a premiere example, we considered RFnoise. In the simulation of bursting CSR, the addition allows to see frequency components thatwhere not reproduced before. Systematic studies modeling intentional RF phase modulationand using nonlinear momentum compaction are planned and first tests already show promisingresults.6. AcknowledgmentsFunded by the German Federal Ministry of Education and Research (Grant No. 05K16VKA)and Initiative and Networking Fund of the Helmholtz Association (contract number: VH-NG-320). T. Boltz, P. Schönfeldt and J. L. Steinmann want to acknowledge the support by theHelmholtz International Research School for Teratronics (HIRST).61234567890 ‘’“”9th International Particle Accelerator Conference, IPAC18 IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1067 (2018) 062025  doi :10.1088/1742-6596/1067/6/062025References[1] Warnock R L and Ellison J A 2000 A general method for propagation of the phase spacedistribution, with application to the sawtooth instability Tech. Rep. SLAC-PUB-8404 SLAC[2] Bane K L F, Cai Y and Stupakov G 2010 Phys. Rev. ST Accel. Beams 13(10) 104402[3] Roussel E, Evain C, Szwaj C and Bielawski S 2014 Phys. Rev. ST Accel. Beams 17(1)010701 URL https://link.aps.org/doi/10.1103/PhysRevSTAB.17.010701[4] Schönfeldt P, Brosi M, Schwarz M, Steinmann J L and Müller A S 2017 Phys. Rev. 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Elaborated Modeling of Synchrotron Motion in Vlasov-Fokker-Planck Solvers

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