Electron beam transverse phase space tomography using nanofabricated wire scanners with submicrometer resolution
Benedikt Hermann, Vitaliy A. Guzenko, Orell R. Hürzeler, Adrian Kirchner, Gian Luca Orlandi, Eduard Prat, Rasmus Ischebeck
EElectron beam transverse phase space tomography using nanofabricated wire scannerswith submicrometer resolution
Benedikt Hermann , , ∗ Vitaliy A. Guzenko , Orell R. H¨urzeler ,Adrian Kirchner , Gian Luca Orlandi , Eduard Prat , and Rasmus Ischebeck Paul Scherrer Institut,5232 Villigen PSI, Switzerland Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg,91054 Erlangen, Germany Institute of Applied Physics, University of Bern,3012 Bern, Switzerland (Dated: February 17, 2021)Characterization and control of the transverse phase space of high-brightness electron beams isrequired at free-electron lasers or electron diffraction experiments for emittance measurement andbeam optimization as well as at advanced acceleration experiments. Dielectric laser accelerators orplasma accelerators with external injection indeed require beam sizes at the micron level and below.We present a method using nano-fabricated metallic wires oriented at different angles to obtainprojections of the transverse phase space by scanning the wires through the beam and detectingthe amount of scattered particles. Performing this measurement at several locations along thewaist allows assessing the transverse distribution at different phase advances. By applying a noveltomographic algorithm the transverse phase space density can be reconstructed. Measurementsat the ACHIP chamber at SwissFEL confirm that the transverse phase space of micrometer-sizedelectron beams can be reliably characterized using this method.
I. INTRODUCTION
High-gradient advanced accelerator concepts includingplasma and dielectric structure based schemes are devel-oped at various laboratories for future compact acceler-ators.The wavelength of the accelerating field in a plasma ac-celerator is given by the plasma wavelength which is typ-ically on the order of tens of micrometers [1]. A Dielec-tric laser accelerator (DLA) is operating in the optical tonear-infrared spectrum leading to structure apertures onthe order of a single micrometer [2]. Hence, suitable testbeams for external injection have to be generated andcharacterized down to the sub-micrometer level.Future compact free-electron laser facilities operating atsmall normalized emittances on the order of 50 nm rad [3]require profile monitors with micrometer resolution.Electron diffraction requires an even smaller emittanceto achieve the required coherence [4].Conventional beam profile monitors for ultra-relativisticelectron beams are scintillating screens, optical transi-tion radiation (OTR) screens and wire scanners. Screen-based methods provide single-shot two-dimensional in-formation, whereas conventional wire scanners providemulti-shot one-dimensional information.The thickness of the scintillating screen, the imaging lensand the camera pixel size limit the resolution of thismethod to around 5 µ m to 10 µ m [5, 6]. OTR screenswith sub-micrometer resolution have been demonstrated, ∗ [email protected] but their application is limited to uncompressed elec-tron bunches [7]. Typical wire scanners at free-electronlaser facilities consist of cylindrical metallic (aluminum ortungsten) wires with diameters down to 5 µ m [8]. Pro-jections of the transverse beam distribution can be mea-sured by moving the stretched wire through the beamand correlating the wire position to the signal of a down-stream beam loss monitor, which detects the scatteredparticle shower. Recent developments at PSI and FERMIled to single (one-dimensional) wire scanners fabricatedwith electron beam lithography reaching sub-micrometerresolution [9, 10]. Based on this technology we designeda wire scanner consisting of nine wires arranged radiallyat different angles, as a tool for precise beam profile to-mography at the ACHIP (Accelerator on a Chip Interna-tional Program) interaction chamber, which is installedin the Athos branch of SwissFEL at PSI (see Fig. 1). Thischamber is planned to support DLA research and devel-opment [11, 12]. A possible application of DLA tech-nology for FELs is the generation of a micro-bunchedpulse train using laser-based energy modulation followedby magnetic compression [13].The electrons at the ACHIP interaction point at Swiss-FEL possess a mean energy of 3 . a r X i v : . [ phy s i c s . acc - ph ] F e b User ExperimentsBC1 BC2Injector Linac 1 Linac 2 Linac 3 Undulatorssoft X-rayshard X-rays0.3 GeV 2.3 GeV 3.2 GeV AthosAramisup to 6 GeVGun
FIG. 1. Schematic of the free-electron laser SwissFEL at PSI. The ACHIP chamber is located in the switch-yard to the Athosbeamline at a beam energy of 3 . space ( x − x (cid:48) and y − y (cid:48) ) is reconstructed with a novelparticle-based tomographic algorithm. This techniquegoes beyond conventional one-dimensional wire scannerssince it allows us to asses the four-dimensional transversephase space. We apply this algorithm to a set of wirescanner measurements performed with nano-fabricatedwires at the ACHIP chamber at SwissFEL and recon-struct the dynamics of the transverse phase space of thefocused electron beam along the waist. II. EXPERIMENTAL SETUPA. Accelerator Setup
The generation and characterization of a micro-metersized electron beam in the ACHIP chamber at SwissFELrequires a low-emittance electron beam. The beam sizealong the accelerator is given by: σ ( z ) = (cid:112) β ( z ) ε n ( z ) /γ ( z ) , (1)where β denotes the Twiss (or Courant-Snyder) param-eter of the magnetic lattice, γ is the relativistic Lorentzfactor of the electrons and ε n is the normalized emit-tance of the beam. With an optimized lattice a minimal β -function of around 1 cm in the horizontal and 1 . . × − . Forthis uncompressed and low-energy-spread beam we ex-pect chromatic enlargement of the focused beam size onthe order of 0 . . . B. ACHIP Chamber
The ACHIP chamber at SwissFEL is a multi-purposetest chamber, designed and built for DLA research. It islocated in the switch-yard of SwissFEL, where the elec-tron beam has an energy of around 3 . C. Nano-Fabricated Wire Scanner
Nano-fabricated wires are installed on the hexapod forthe characterization of the focused beam profile. The
FIG. 2. Inside view of the ACHIP chamber. Movablequadrupoles for focusing and re-matching are seen in the frontand back. The hexapod for sample positioning is located atthe center. Image adapted from [17] under Creative CommonsAttribution 3.0 licence. wire scan device consists of nine free-standing 1 µ m widemetallic (Au) stripes. The nine radial wires are sup-ported by a spiderweb-shaped structure attached to asilicon frame. A scanning electron microscope image ofthe wire scanner sample is shown in Fig. 3. We chose ninehomogeneously spaced wires for our design, since thisconfiguration allows us to access any wire angle withinthe tilt limits of the hexapod. The sample was fabricatedat the Laboratory for Micro and Nanotechnology at PSIby means of electron beam lithography. The 1 µ m widestripes of gold are electroplated on a 250 nm thick Si N membrane, which is removed afterwards with a KOHbath. The fabrication process and performance for thistype of wire scanner are described in detail in [9]. Thehexapod moves the wire scan device on a polygon pathto scan each of the nine wires orthogonally through theelectron beam. Hereby, projections along different angles( θ ) of the transverse electron density can be measured.The two-dimensional transverse beam profile can be ob-tained using tomographic reconstruction techniques. Thehexapod can position the wire scanner within a range of20 cm along the beam direction ( z ). By repeating thewire scan measurement at different locations around thewaist, the transverse phase space and emittance of thebeam can be inferred. D. Beam Loss Monitor
Electrons scatter off the atomic nuclei of the metallicwire and a particle shower containing mainly X-rays, elec-trons and positrons is generated. The intensity of the sec-ondary particle shower depends on the electron densityintegrated along the wire and is measured with a down-
200 μm 20 μm
FIG. 3. Scanning electron microscope images of the free-standing wire scanner device. Nine radial wires which areused for the wire scans are supported by a spiderweb-shapedstructure attached to a silicon frame. At the center of the ge-ometry a square simplifies the alignment of the wire scannerwith respect to the electron beam. Scanning the square hor-izontally and vertically across the beam provides 4 distinctpeaks (for beam sizes smaller than 50 µ m). The center of thegeometry can be referenced to the hexapod coordinate systemfrom the location of these peaks. stream beam loss monitor (BLM). The BLM consists ofa scintillating fiber wrapped around the beam pipe. Thefiber is connected to a photo-multiplier tube (PMT). Thesignal of the PMT is read-out beam synchronously in ashot-by-shot manner. To avoid saturation of the PMT,the gain voltage needs to be set appropriately. SwissFELis equipped with a series of BLMs, which are normallyused to detect unwanted beam losses and are connectedto an interlock system. For the purpose of wire scanmeasurements, individual BLMs can be excluded fromthe machine protection system. Details about the BLMsat SwissFEL can be found in [18]. For the wire scan mea-surement reported here, a BLM located 10 m downstreamof the interaction with the wire was used. III. TRANSVERSE PHASE SPACERECONSTRUCTION ALGORITHM
Inferring a density distribution from a series of pro-jection measurements is a problem arising in many sci-entific and medical imaging applications. Standard to-mographic reconstruction techniques, e.g., filtered backprojection or algebraic reconstruction technique [19] usean intensity on a grid to represent the density to be re-constructed. The complexity of these algorithms scalesas O ( n d ), where n is the number of pixels per dimensionand d is the number of dimensions of the reconstructeddensity. Typically, for real space density reconstruction, d is 2 (slice reconstruction) or 3 (volume reconstruction).In the case of transverse phase space tomography d equals4 ( x, x (cid:48) , y, y (cid:48) ), leading to very long reconstruction times.We developed a reconstruction algorithm based on amacro-particle distribution (instead of the intensity ongrid), where each macro-particle, from now on called par-ticle, represents a point in the four-dimensional phasespace. The complexity of this algorithm is proportionalto n p (number of particles) and is independent on thedimension of the reconstruction domain. The particledensity is then given by applying a Gaussian kernel toeach coordinate of the particle ensemble: G κ = 1 √ πρ κ exp (cid:18) − κ ρ κ (cid:19) , κ ∈ { x, x (cid:48) , y, y (cid:48) } (2)where we choose ρ x (cid:48) ,y (cid:48) = ρ x,y /z max , with z max the rangeof the measurement along z . Choosing the right kernelsize is important for an appropriate reconstruction of thebeam. It is dimensioned such that ρ x,x (cid:48) ,y,y (cid:48) represents thelength scale below which we expect only random fluctu-ations in the particle distribution, which are not repro-ducible from shot to shot. Note that despite the Gaussiankernel, this reconstruction does not assume a Gaussiandistribution of the beam, but is able to reconstruct arbi-trary distributions that vary on a length scale given by ρ x,x (cid:48) ,y,y (cid:48) .The ensemble of particles is iteratively optimized so thattheir projections match with the set of measured projec-tions. The algorithm starts from a homogeneous particledistribution. One iteration consists of the following op-erations. • Transport T ( z ) • Rotation R ( θ ) • Histogram of the transported and rotated coordi-nates • Convolution with wire profile • Interpolation to measured wire positions • Comparison of reconstruction and measurement • Redistribution of particlesIn the case of ultra-relativistic electrons transverse spacecharge effects can be neglected since they scale as O ( γ − )and hence T ( z ) becomes the ballistic transport matrix: T ( z ) = (cid:18) z (cid:19) (3)for ( x, x (cid:48) ) and ( y, y (cid:48) ). The rotation matrix is then appliedto ( x, y ): R ( θ ) = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) . (4)Afterwards, the histogram of the particles’ transportedand rotated x coordinates is calculated. Note that thebin width needs to be smaller than the width of the wire,to ensure an accurate convolution with the wire profile.This becomes important when the beam size or beamfeatures are smaller than the wire width. Next, the con-volution of the histogram and the wire profile is interpo-lated linearly to the measured wire positions ξ . Now, thereconstruction can be directly compared to the measure-ment: ∆ z,θ ( ξ ) = P mz,θ ( ξ ) − P rz,θ ( ξ )max ξ P rz,θ ( ξ ) , (5) where P mz,θ and P rz,θ are the measured and reconstructedprojections for the current iteration at position z andangle θ . The difference between both profiles quantifiesover- and under-dense regions in the projection. Then,∆ z,θ ( ξ ) is interpolated back to the particle coordinatesalong the wire scan direction, yielding ∆ iz,θ for the i -thparticle. Afterwards, we calculate the average over allmeasured z and θ :∆ i = 1 n θ n z (cid:88) θ,z ∆ iz,θ . (6)The sign of ∆ i indicates if a particle is located in an over-or under-dense region represented by the current particledistribution. According to the magnitude of ∆ i the newparticle ensemble is generated. A particle is copied or re-moved from the previous distribution with a probabilitybased on | ∆ i | . This process is implemented by drawinga pseudo-random number χ i ∈ [0 ,
1[ for each particle. Incase χ i < | ∆ i | /s max , particle i is copied or removed fromthe distribution (depending on the sign of ∆ i ). Other-wise, the particle remains in the ensemble. Here, s max isthe maximum of all measured BLM signals and is usedto normalize ∆ i for the comparison with χ i ∈ [0 , ρ . For the reconstruction of the measure-ment presented in Sec. IV, ρ x,y was set to 80 nm.The iterative algorithm is terminated by a criterion basedon the relative change of the average of the difference∆ iz,θ (further details in Appendix B). The measurementrange along z ideally covers the waist and the spacing be-tween measurements is reduced close to the waist, sincethe phase advance is the largest here. Since the algo-rithm does not assume a specific shape (e.g., Gaussian)of the distribution, asymmetries, double-peaks, or ha-los of the distribution can be reconstructed (an exam-ple is shown in Appendix C). Properties of the trans-verse phase space including, transverse emittance in bothplanes, astigmatism and Twiss parameters can be calcu-lated from the reconstructed distribution. To obtain thefull 4D emittance, cross-plane information, such as cor-relations in x − y (cid:48) or x (cid:48) − y need to be assessed. For thispurpose, the phase advance has to be scanned indepen-dently in both planes. This can be achieved with a mul-tiple quadrupole scan as explained for instance in [20, 21]but is not achieved by measuring beam projections alonga waist, as the phase advance in both planes is correlated.The presented phase space reconstruction algorithmcould also be adapted to use two-dimensional profile mea-surements from a screen at different phase advances tocharacterize the four-dimensional transverse phase space.The python-code related to the described tomographicreconstruction technique is made available on github [22]. FIG. 4. Reconstruction from simulated measurement. Theoriginal distribution in the transverse phase space is shownin the upper row. An astigmatism of − x − x (cid:48) ). The algorithm reconstructsthe transverse phase space based on a set of simulated wirescan projections. The result of the reconstruction is shown inthe lower row. The 1 σ -ellipse of a 2D Gaussian fit is drawnin blue for each histogram. A. Reconstruction of a Simulated Measurement
To verify the reconstruction algorithm, we generate atest distribution and calculate a set of wire scan pro-jections (nine projections along different angles at sevenlocations along the waist). The algorithm then recon-structs the distribution based on these simulated projec-tions. For this test, we choose a Gaussian beam distri-bution with Twiss parameters β ∗ x = 2 . β ∗ y = 3 . − ρ x,y (see Eq. 2) is 80 nm, which is aroundone order of magnitude smaller than the beam size in thistest. Figure 4 compares the original and reconstructedtransverse phase space at z = 0 cm. Good agreement( <
10 % error) is achieved for the emittances and astig-matism, which is manifested as a tilt in the x − x (cid:48) plane.For this numerical experiment, the algorithm terminatesaccording to the criterion described in Appendix B afteraround 100 iterations. The run-time on a single-core of astandard personal computer is around two minutes. Par-allelising the computation on several cores would reducethe computation time by few orders of magnitude. ε n (nm rad) β ∗ (cm) σ ∗ ( µ m)x 186 ±
15 3 . ± . . ± .
06y 278 ±
18 3 . ± . . ± . ε n , Twiss β -function at thewaist β ∗ , and corresponding beam size σ ∗ of the reconstructedtransverse phase space distribution. IV. RESULTS
We have measured projections of the transverse elec-tron beam profile at the ACHIP chamber at SwissFELwith the accelerator setup, wire scanner and BLM detec-tor described in Sec. II. All nine wire orientations are usedat six different locations along the waist of the electronbeam. This results in a total of 54 projections of the elec-tron beam’s transverse phase space. Lowering the num-ber of projections limits the possibility to observe inho-mogeneities of the charge distribution. The distance be-tween measurement locations is increased along z , sincethe expected waist location was around z = 0 cm. All54 individual profiles are shown in Fig. 5. In each sub-plot, the orange dashed curve represents the projectionof the reconstructed phase space for the respective angle θ and longitudinal position z . The reconstruction repre-sents the average distribution over many shots and agreeswith most of the measured data points. Discrepanciesarise due to shot-to-shot position jitter, charge fluctua-tions, or density variations of the electron beam. Theeffect of these error sources is discussed further in Ap-pendix A. The evolution of the reconstructed transversephase space along the waist is depicted in Fig. 6. Theexpected rotation of the transverse phase space aroundthe waist is clearly observed. The position of the waistis found to be at around z = 6 . β -function of the distribution by fitting a2D Gauss function to the distribution in the ( x , x (cid:48) ) and( y , y (cid:48) ) phase space. The 1- σ ellipse of the fit is drawnin blue in all sub-plots of Fig. 6. We use the followingdefinition for the normalized emittance: ε n = γA σ /π, (7)where A σ is the area of the 1- σ ellipse in transversephase space. The values for the reconstructed emittance,minimal β -function ( β ∗ ) and beam size at the waist aresummarized in Table I. The measurement range (8 cm)along the waist with β ∗ = 3 . ° . V. DISCUSSION
The reconstructed phase space represents the averagedistribution of many shots, since shot-to-shot fluctua-tions in the density cannot be characterized with multi-shot measurements like wire scans. Errors induced by
FIG. 5. Measured (blue crosses) and reconstructed (orange dashed) profiles of the electron beam distribution. The verticalaxes are identical for all sub-plots and show the BLM signal or reconstruction in arbitrary units. Sub-plots in the same columncorrespond to the same z location of the wire scanner and sub-plots in the same row correspond to the same projection angle θ . The grey area depicts the uncertainty of the reconstruction. For the last column ( z = 8 cm) the scan range did not coverthe entire beam profile for all scans due to a misalignment of the electron propagation direction and the z -axis of the hexapod,which results in a transverse offset of the wire scanner device with respect to the electron beam. This effect is the largest forthe last scan ( z = 8 cm) since the wire scanner was aligned to the beam axis at z = 0 cm. total bunch charge fluctuations and position jitter of theelectron beam could be corrected for by evaluating beam-synchronous BPM data. Since the BPMs in the Athosbranch were still uncalibrated, their precision was insuf-ficient to correct orbit jitter in our measurement. Thisissue is considered further in Appendix A.The expected waist is located at the center of the cham-ber ( z = 0 cm), whereas the reconstructed waist is found6 . β -function at thewaist ( β ∗ ) was measured to be around 3 . β ∗ x = 1 cm, β ∗ y = 1 . . . FIG. 6. 2D histograms of the phase space reconstructed from wire scan measurements. Sub-plots in the same columncorrespond to the same z location. The first row shows the x − x (cid:48) and the second row shows the y − y (cid:48) phase space. The lastrow depicts the corresponding beam profile ( x − y ). The 1 σ -ellipse of a 2D Gaussian fit is drawn in blue for each histogram.FIG. 7. Evolution of the reconstructed beam size aroundthe waist. The dashed vertical lines indicate wire scan mea-surement locations along z . At each of the six locations wirescans are carried out along nine different angles. Since we ex-pected the waist to be around z = 0 cm, the distance betweenmeasurements is reduced here. the emittance [15]. In contrast, the tomographic wirescan technique presented here reconstructs the transversephase space averaged over many shots. Afterwards, aGaussian fit estimates the area of the distribution in the transverse phase space. Both large shot-to-shot jitter andnon-Gaussian beams can give rise to differences betweenthe results of the two techniques.The wire scan acquisition time could be reduced by us-ing fewer projection angles. This could be done, if lessdetailed information on the beam distribution is accept-able, e.g., if only projected beam sizes are of interest, twoprojection angles are sufficient. The optimal number ofangles depends on the internal beam structure and thebeam quantities of interest. A. Resolution Limit
The ultimate resolution limit of the presented tomo-graphic characterization of the transverse beam profiledepends on the roughness of the wire profile. With thecurrent fabrication process, this is on the order of 100 nmestimated from electron microscope images of the free-standing gold wires. This is one to two orders of magni-tude below the resolution of standard profile monitors forultra-relativistic electron beams (YAG:Ce screens) [5, 6].
B. Comparison to other Profile Monitors
The scintillating screens (YAG:Ce) at SwissFELachieve an optical resolution of 8 µ m, and the smallestmeasured beam sizes are 15 µ m [6]. At the Pegasus Lab-oratory at UCLA beam sizes down to 5 µ m were measuredwith a 20 µ m thick YAG:Ce screen in combination withan in-vacuum microscope objective [5]. Optical transi-tion radiation (OTR) based profile monitors are only lim-ited by the optics and camera resolution [23]. At the Ac-celerator Test Facility 2 at KEK this technique was usedto measure a beam size of 750 nm [7]. However, OTRprofile monitors are not suitable for compressed electronbunches (e.g., at FELs) due to the emission of coherentOTR [24].At the SLAC Final Focus Test Beam experiment a laser-Compton monitor was used to characterize a 70 nm widebeam along one dimension [25]. The cost and complexityof this system, especially for multi-angle measurements,are its main draw-backs.Concerning radiation hardness of the nano-fabricatedwire scanner, tests with a single wire and a bunch chargeof 200 pC at a beam energy of 300 MeV at SwissFEL didnot show any sign of degradation after repeated measure-ments [9]. VI. CONCLUSION
In summary, we have presented and validated a noveltechnique for the reconstruction of the transverse phasespace of a strongly focused, ultra-relativistic electronbeam. The method is based on a series of wire scansat different angles and positions along the waist. Aniterative tomographic algorithm has been developed toreconstruct the transverse phase space. The technique isvalidated with experimental data obtained in the ACHIPchamber at SwissFEL. The method could be appliedto other facilities and experiments, where focused high-brightness electron beams need to be characterized, forinstance at plasma acceleration or DLA experiments formatching of an externally injected electron beam, emit-tance measurements at future compact low-emittanceFELs [3], or for the characterization of the final-focussystem at a high-energy collider test facility. For the lat-ter application, the damage threshold of the free-standingnano-fabricated gold wires needs to be identified and ra-diation protection for the intense shower of scattered par-ticles needs to be considered. Nevertheless, the focusingoptics could be characterized with the presented methodusing a reduced bunch charge.
ACKNOWLEDGMENTS
We would like to express our gratitude to the SwissFELoperations crew, the PSI expert groups, and the entire ACHIP collaboration for their support with these exper-iments. We would like to thank Thomas Schietinger forcareful proofreading of the manuscript. This research issupported by the Gordon and Betty Moore Foundationthrough Grant GBMF4744 (ACHIP) to Stanford Univer-sity.
Appendix A: Error Estimation1. Position Errors
The uncertainty of the position of the wire scannerwith respect to the electron beam is affected by the read-out precision of the hexapod ( < <
10 nm) and position jitter of theelectron beam, which at SwissFEL is typically a few-percent of the beam size. The orbit of the electron beamis measured with BPMs along the accelerator. Unfortu-nately, the BPMs along the Athos branch of SwissFELhave not been calibrated (the measurement took placeduring the commissioning phase of Athos). Neverthe-less, we tried correcting the orbit shot-by-shot based onfive BPMs and the magnetic lattice around the interac-tion point. However, it does not reduce the measuredbeam emittance, as their position reading is not preciseenough to correct orbit jitter at the wire scanner locationcorrectly. Therefore, we do not include corrections to thewire positions based on BPMs. The reconstructed beamphase space represents the average distribution for manyshots including orbit fluctuations. After the calibrationof the BPMs in Athos we plan to characterize the effectof orbit jitter to wire scan measurements in detail.
2. Amplitude Errors
Jitter to the BLM signal is introduced by read-outnoise of the PMT ( < . s max /σ noise , where s max isthe maximum of the signal and σ noise refers to the stan-dard deviation of the background.
3. Uncertainty of the Reconstruction
Due to the error sources mentioned above themeasured projections are not fully compatible witheach other, i.e., the reconstructed distribution cannotmatch to all measured data points. The error of thereconstructed phase space density and the derivedquantities is estimated by a procedure similar to themain reconstruction algorithm. The reconstructeddistribution is now taken as input. Instead of averagingover all projections, the iteration is performed foreach projection individually. Hence, a set of n z × n θ distributions is generated, in which each distributionmatches best to one measured projection. All derivedquantities, such as the emittance or β -function, arecomputed for each distribution and the error is taken asthe standard deviation of this set. Appendix B: Termination Criterion forReconstruction Algorithm
The algorithm to reconstruct the phase space from wirescan measurements iteratively approximates the distribu-tion that fits best to all measurements (see Sec. III). Theiteration is stopped when a criterion based on the rel-ative change from the current to the previous iterationis reached. We define p k as the average probability fora particle to be added or removed to the ensemble initeration k . p k = 1 n p n θ n z (cid:88) i,θ,z | ∆ iz,θ | (B1)The iteration terminates when the relative change of p k reaches a tolerance limit τ : | p k − p k − || p k | < τ (B2) For the case of the presented data set τ = 0.005 is foundto provide stable convergence and a consistent solution.Around 110 iterations are required to reach the termina-tion criterion. Appendix C: Reconstruction of non-Gaussian Beams
Our particle based tomographic reconstruction algo-rithm does not assume any specific shape for the densityprofile. Therefore, asymmetric density variations, such astails of a localized core can be reconstructed. To demon-strate this capability of our tomographic technique, weshow here a measurement of a non-Gaussian beam shapeand compare the result to a 2D Gaussian fit. This mea-surement was performed with different machine settingsthan the measurement presented in Sec. IV. The elec-tron bunch carried a charge of around 10 pC. The trans-verse beam profile was characterized with nine wire scansat different angles at one z position. Therefore we canonly reconstruct the two-dimensional ( x , y ) beam profile.The measurement and the tomographic reconstructionare shown in Fig. 8. For comparison, we add the result ofa single two-dimensional Gaussian fit to all nine measuredprojections (Fig. 9). 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Tomographic reconstruction of a beam with non-Gaussian tails. The nine measured projections are indicated bycrosses in the small nine sub-plots. The reconstruction result is shown in the larger sub-plot on the right ( x , y profile). Theprojections of the reconstruction are shown as solid lines in the corresponding sub-plots. The colors correspond to the differentprojection angles as indicated by dashed lines in the 2D profile plot on the right. The tomographic reconstruction is able torepresent the core and tails of the beam.FIG. 9. The result of a single Two-dimensional Gaussian fit to approximate nine measured projections. The measurementand the beam profile are shown analogously to the tomographic result shown in Fig. 8. In contrast to the tomographicreconstruction, the Gaussian fit is not able to represent the tails correctly.[13] B. Hermann, S. Bettoni, T. Egenolf, U. Niedermayer,E. Prat, and R. 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