The operation of synchrotron light sources with short electron bunches increases the emitted CSR power in the THz frequency range. However, the spatial compression leads to complex longitudinal dynamics, causing the formation of micro-structures in the longitudinal bunch profiles. The fast temporal variation and small scale of these micro-structures put challenging demands on their observation. At the KIT storage ring KARA (KArlsruhe Research Accelerator), diagnostics have been developed allowing direct observation of the dynamics by an electro-optical setup, and indirect observation by measuring the fluctuation of the emitted CSR. In this contribution, we present studies of the micro-structure dynamics on simulated data, obtained using the numerical Vlasov-Fokker-Planck solver Inovesa, and first applications on measured data. To deal with generated data sets in the order of terabytes in size, we apply the machine learning technique k-means to identify the dominant micro-structures in the longitudinal bunch profiles. Following this approach, new insights on the correlation of the CSR power fluctuation to the underlying longitudinal dynamics can be gained

Boltz, T.

Brosi, M.

Bründermann, E.

Müller, A.-S.

Schwarz, M.

Schönfeldt, P.

Yan, M.

Boltz, Tobias

Brosi, Miriam

Bründermann, Erik

Müller, Anke-Susanne

Schwarz, Markus

Schönfeldt, Patrik

Yan, Minjie

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Studies of Longitudinal Dynamics in the Micro-Bunching Instability Using Machine Learning

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STUDIES OF LONGITUDINAL DYNAMICS IN THE MICRO-BUNCHINGINSTABILITY USING MACHINE LEARNINGT. Boltz∗, M. Brosi, E. Bründermann, P. Schönfeldt, M. Schwarz†, M. Yan and A.-S. MüllerKarlsruhe Institute for Technology, Karlsruhe, GermanyAbstractThe operation of synchrotron light sources with shortelectron bunches increases the emitted coherent synchrotronradiation (CSR) power in the THz frequency range. How-ever, the spatial compression leads to complex longitudinaldynamics, causing the formation of micro-structures in thelongitudinal bunch proﬁles. The fast temporal variation andsmall scale of these micro-structures put challenging de-mands on their observation. At the KIT storage ring KARA(Karlsruhe Research Accelerator), diagnostics have been de-veloped allowing direct observation of the dynamics by anelectro-optical setup, and indirect observation by measuringthe ﬂuctuation of the emitted CSR. In this contribution, wepresent studies of the micro-structure dynamics on simulateddata, obtained using the numerical Vlasov-Fokker-Plancksolver Inovesa. To deal with generated data sets in the orderof terabytes in size, we apply the machine learning tech-nique k-means to identify the dominant micro-structures inthe longitudinal bunch proﬁles. Following this approach,new insights on the correlation of the CSR power ﬂuctuationto the underlying longitudinal dynamics can be gained.INTRODUCTIONDue to the self-interaction with its own CSR, short elec-tron bunches in a storage ring are subject to complex lon-gitudinal dynamics. Above a given threshold current, thisresults in the formation of dynamically changing, but per-sistent, micro-structures within the electron bunch. As thelongitudinal charge distribution varies, this also leads to ma-jor ﬂuctuations in the emitted CSR power and is thus calledmicro-bunching instability. Experimental observations ofthis phenomenon have been made at various synchrotronlight sources, including e.g. [1–3]. Additionally, the underly-ing longitudinal dynamics can be simulated by numericallysolving the Vlasov-Fokker-Planck equation, where the CSRself-interaction can be added as a perturbation to the Hamil-tonian. Such theoretical studies have been primarily focusedon prediction of the threshold current, leading e.g. to theformulation of an empirical scaling law [4].This contribution focuses explicitly on the longitudinal dy-namics above the threshold current and places special empha-sis on the characteristics of the occurring micro-structures.Therefore, the dynamics are simulated for a given machinesetting across multiple bunch currents using the in-housedeveloped simulation code Inovesa [5]. Considering therequirements for reasonable temporal and spatial resolution,simulation of the longitudinal dynamics quickly accumu-∗ tobias.boltz@kit.edu† now at CERN, Geneva, Switzerlandlates to large sets of data. In order to explore the generateddata, we apply the machine learning method k-means to iden-tify the dominant micro-structures within the longitudinalproﬁles of a given bunch current. In this contribution, theapplication of the k-means method is aimed at describingthe dynamically changing bunch proﬁles by merely a fewdiscrete states. The cluster separation is therefore expectedto be rather low (e.g. silhouette scores of Sk ≈ 0.5). Never-theless, it can still yield very useful information in regard ofunderstanding the underlying dynamics.While the ﬁrst part deals with the identiﬁcation of thedominant micro-structures on the longitudinal bunch proﬁleswithin a ﬁxed current and their correlation to the ﬂuctuatingemission of CSR power, the second part extends the scope tostudies of the micro-structure characteristics across diﬀerentbunch currents and machine settings.ANALYSIS OF MICRO-STRUCTUREDYNAMICSIn the studies of the longitudinal dynamics in the micro-bunching instability, it was found that two major regimesexist above the threshold current [6]. In the regular burstingregime directly above the threshold, the CSR power signal02468ρ(z)(pC/ps)-10 -5 0 5 10−8−6−4−202468z/c (ps)Δρ(z)(10−1pC/ps)Figure 1: Cluster centers found on the longitudinal pro-ﬁles for an exemplary bunch current (Iex = 1.15mA) inthe sawtooth bursting regime. Diﬀerent colors indicate therespective cluster.9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW PublishingISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK03005 Beam Dynamics and EM FieldsD11 Code Developments and Simulation TechniquesTHPAK0303277ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.has an almost sinusoidal shape, whereas in the sawtoothbursting regime at higher currents, distinct burst of CSRemission can be observed. The results of the aforementionedapplication of the k-means method will be discussed foran exemplary bunch current (Iex = 1.15mA) within thesawtooth bursting regime.In the upper part of Fig. 1, the cluster centers found by thek-means method searching for four clusters in the longitudi-nal bunch proﬁles are shown. Here, two of four cluster cen-ters (line color blue and orange) display distinct modulationson the longitudinal proﬁles. An elegant way to emphasizethese modulations, is to subtract from each cluster center theglobal mean proﬁle (calculated over all longitudinal proﬁles)as shown in the lower part of the ﬁgure. Here, the clustercenters are clearly distinguishable and, in the case of theblue and orange cluster, display the phase and paraphaseof an almost sinusoidal modulation on the longitudinal pro-ﬁles. Additionally, the shaded areas denote the standarddeviations of the longitudinal proﬁles within the respectivecluster, showing to which degree the cluster centers concurwith the actual longitudinal proﬁles in the respective cluster.In contrast, the red and green cluster centers describe mainlylongitudinal proﬁles with diﬀerent bunch lengths. This isfor example indicated by the slightly higher values of thered cluster center (corresponding to a longer bunch) at theedges of the proﬁle.Furthermore, the correlation of these ﬁndings with theemitted CSR power can be studied. During the clusteringprocedure, each longitudinal proﬁle ρ(z, ti), correspondingto the speciﬁc time step ti , is assigned with a cluster labely(ti). This temporal sequence of integers y(ti) contains theentire information about the generated clustering results andcan be mapped to the corresponding sequence of CSR powervalues Pcsr(ti). This is illustrated in Fig. 2, where each valueof Pcsr(ti) is colored according to the corresponding clusterlabel y(ti) of this time step.Obviously, the blue and orange cluster contain the longi-tudinal proﬁles corresponding to the four distinct bursts ofCSR emission, while the red and green cluster describe thephases in between. The points of maximal CSR emissionare thus perfectly coinciding with the occurrence of micro-0 50 100 150 200 2503456t (Ts)Pcsr(arb.unit)Figure 2: Color-coded CSR power, displaying the correlationbetween the found modulations on the longitudinal proﬁlesand ﬂuctuations in the emitted CSR.Figure 3: Phase space cluster means (left) corresponding tothe blue and orange cluster centers displayed in Fig. 1. Toemphasize the occurring micro-structures, the global meanover all phase space densities is subtracted (right).structures with high amplitude. Between these bursts ofemitted CSR, there are transition periods of the longitudinaldynamics where the bunch length reduces gradually.As the entire longitudinal phase space density can beobtained from simulation, the found micro-structures canbe studied directly in phase space. One way to do so, isto calculate the mean phase space density within the foundclusters, as displayed in Fig. 3 for the blue and orange cluster.Similarly to the above, in order to emphasize the dynamicstructure, the global mean phase space density is subtracted.Here, several distinct micro-structures are visible. By com-parison to the results presented in Fig. 1, it can be inferredthat the highest values in the amplitude of the modulation onthe longitudinal proﬁles originate from multiple maxima inthe phase space density that get added up in the projectionon the z-axis. Due to synchrotron motion, the phase spacedensity is rotating and thus leading to an oscillation betweenphase and paraphase of the modulation on the longitudinalproﬁle.Although the observed CSR power signal changes signif-icantly with increasing bunch current, micro-structures ofsimilar appearances were found for all currents above thresh-old, including the regular bursting regime. While the generalappearance, i.e. the number of structures and their overallshape, is consistent throughout the diﬀerent regimes of CSRemission, the temporal changes of their amplitude are themain reason for the diﬀerences in emitted CSR power.MICRO-STRUCTURECHARACTERISTICSIn order to study the micro-structure characteristics acrossdiﬀerent bunch currents, the cluster centers found by the k-means method are further analyzed to determine the mainfrequency fmod and amplitude Amod of the modulation onthe longitudinal proﬁles, as shown in Fig. 4. For the simu-lated wide bunch current range from 0.5mA to 2.0mA, themodulation frequencies are distributed mainly around twolevels. While the modulation frequencies around 50GHz9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW PublishingISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK030THPAK0303278ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.05 Beam Dynamics and EM FieldsD11 Code Developments and Simulation Techniques0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.050100150200250Ith IexI (mA)f mod(GHz)(a) modulation frequencies0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.002468Ith IexI (mA)Amod(10−1pC/ps)(b) modulation amplitudesFigure 4: Modulation frequencies (a) and amplitudes (b) across diﬀerent currents. As higher frequencies correspond tomodulations caused by micro-structures in phase space, values higher than 75GHz are colored red. The same color schemeis then applied to the displayed modulation amplitudes.correspond to cluster centers describing mainly changes inbunch length, the higher frequencies correspond to distinctmicro-structures on the longitudinal proﬁles. The latter startoccurring directly above the threshold Ith and stay around aslightly decaying value. Despite the slight decay, which ismainly caused by bunch lengthening, the number of micro-structures is still the same for all these currents.The amplitude of the micro-structures (Fig. 4b) starts ris-ing directly above the threshold current until reaching a sat-uration value at I ≈ 1.1mA. Eventually for I > 1.8mA, theamplitudes of modulations due to changes in bunch lengthare in the same magnitude as those caused by the micro-structures.While the number of micro-structures in phase space isfound to be constant across diﬀerent currents for a given setof machine parameters, this no longer holds for modiﬁca-tions of speciﬁc parameters e.g. synchrotron frequency orthe height of the vacuum gap g = 2h. Figure 5 shows theresults of systematic simulations for diﬀerent vacuum gaps.Compared to the formulation found by K.F.L. Bane et al. [4]for the threshold current, the number of micro-structuresnstr and modulation frequency fmod show interestingly thesame dependence on the vacuum gap, i.e. linear to h−1.5.Additionally it should be noted, while the number of micro-16 18 20 22 24 26 28 30 3268101214g = 2h (mm)nstr150200250300350f mod(GHz)ﬁt function: f (h) = a · hb + cparameter: b = −1.53 ± 0.37ﬁt function: f (h) = a · hb + cparameter: b = −1.59 ± 0.35Figure 5: Number of micro-structures nstr and modulationfrequencies fmod for diﬀerent vacuum gaps g = 2h.structures is assumed to be a discrete quantity and counteddirectly by studying phase space, the modulation frequencyis not necessarily bound to the same constraints. Never-theless, the modulation frequency decreases stepwise withincreasing height of the vacuum gap, indicating an intrinsicdiscretization of the occurring micro-bunching.SUMMARY AND OUTLOOKThe longitudinal dynamics in the micro-bunching insta-bility can be studied on simulation data using the Vlasov-Fokker-Planck Solver Inovesa. Careful analysis of the ob-tained data led to new insights on the underlying dynam-ics, e.g. the number of micro-structures in phase space isobserved to be constant above the threshold for otherwiseﬁxed machine settings. Additionally, dependency of furthercharacteristics of the micro-structures on various machineparameters has been found, which coincides with the es-tablished formulation for the bursting threshold. This inter-esting ﬁnding might stimulate further understanding of themicro-bunching instability.ACKNOWLEDGEMENTT. Boltz, M. Brosi and P. Schönfeldt want to acknowledgethe support by the Helmholtz International Research Schoolfor Teratronics (HIRST).REFERENCES[1] M. Abo-Bakr et al., PAC’03 5, 3023-3025 (2003).[2] W. Shields et al., Journal of Physics: Conference Series 357,012037 (2012).[3] C. Evain et al., EPL (Europhysics Letters) 98, 40006 (2012).[4] K.L.F. Bane et al., Phys. Rev. ST Accel. Beams 13, 104402(2010).[5] P. Schönfeldt et al., Phys. Rev. Accel. Beams 20, 030704 (2017).[6] T. Boltz, Master’s Thesis, doi: 10.5445/IR/1000068253(2017).9th International Particle Accelerator Conference IPAC2018, Vancouver, BC, Canada JACoW PublishingISBN: 978-3-95450-184-7 doi:10.18429/JACoW-IPAC2018-THPAK03005 Beam Dynamics and EM FieldsD11 Code Developments and Simulation TechniquesTHPAK0303279ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2018).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.

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Studies of Longitudinal Dynamics in the Micro-Bunching Instability Using Machine Learning

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