Longitudinal phase space synthesis with tailored 3D-printable dielectric-lined waveguides
LLongitudinal phase space synthesis with tailored 3D-printable dielectric-linedwaveguides
F. Mayet, ∗ R. Assmann, and F. Lemery † DESY, Notkestrasse 85, 22607 Hamburg, Germany (Dated: September 28, 2020)Longitudinal phase space manipulation is a critical and necessary component for advanced accel-eration concepts, radiation sources and improving performances of X-ray free electron lasers. Herewe present a simple and versatile method to semi-arbitrarily shape the longitudinal phase space ofa charged bunch by using wakefields generated in tailored dielectric-lined waveguides. We apply theconcept in simulation and provide examples for radiation generation and bunch compression. Wefinally discuss the manufacturing capabilities of a modern 3D printer and investigate how printinglimitations, as well as the shape of the input LPS affect the performance of the device.
I. INTRODUCTION
Emerging advanced accelerator concepts require pre-cise control over the longitudinal phase space (LPS) ofcharged particle beams. Efficient beam-driven accelera-tion, for example, relies on longitudinally-tailored elec-tron bunch profiles which can be produced with an ap-propriate energy modulation and dispersive section [1–4].Phase-space linearization for bunch compression is espe-cially important to optimize the performance of multi-stage linacs and X-ray free electron lasers (XFELs) [5–7]. There are several ways to control the LPS. Energymodulation approaches via self-wakes in e.g. dielectricor corrugated structures provide attractive and simplemethods to produce microbunch trains and large peakcurrents [3, 8, 9]. Laser-based energy modulation tech-niques using magnetic chicanes are particularly useful forFEL seeding [10–14] and for beam acceleration [15]. Ar-bitrary laser-based phase space control was discussed in[16], illustrating the potential for producing different cur-rent profiles for various applications. Unfortunately how-ever, although the scheme works well in simulation, theapproach is complex to implement, requiring several un-dulators and magnetic chicanes in addition to the mod-ulating laser.In this paper we explore arbitrary waveform synthesisusing self-wakes produced in dielectric-lined waveguides(DLW). By using segmented waveguides with varyingcross sections, the excited wakefields carry different spec-tral contents throughout the structure, enabling controlover the energy modulation across the bunch. The depen-dence of the modal content on the DLW geometry allowsfor enough degrees of freedom to optimize such a seg-mented structure according to the desired output LPS.Due to the nature of the physical process, the scheme iscompletely passive, removing the need for synchroniza-tion with e.g. a modulating laser beam. In the following,the device is referred to as a longitudinal phase spaceshaper (LPSS). ∗ [email protected] † [email protected] The paper is structured as follows: Section II providesan overview on 1D wakefield theory, Section III discussesFourier synthesis for single-mode structures, Section IVprovides examples for multimode structure optimizationsusing computational optimization, Section V discussesthe impact of manufacturing limitations of modern 3Dprinters by investigating the effect of printing resolutionand segment transitions on the excited wakefields. Fi-nally, Section VI discusses the effect of slight variationsin the shape of the input LPS on a figure of merit of anoutput LPS, based on an example optimization case.
II. WAKEFIELD GENERATION IN A DLW
The theory of Cherenkov wakefield generation incylindrically-symmetric DLWs is well described in [17–19]. Here we follow [18], for a structure with inner radius r = a , outer radius r = b and dielectric permittivity (cid:15) r .The outer surface is assumed to be coated with a perfectconductor. See Fig. 1 for a schematic. A more rigoroustheoretical investigation could include conductive and di-electric losses in DLWs [20, 21]. ab FIG. 1. Schematic of a cylindrical dielectric-lined waveguide.The lining with dielectric constant (cid:15) r has an inner radius a and an outer radius b . It is coated with a thin metallic layeron the outside. In the ultrarelativistic limit, a point charge travellingon-axis ( r = 0) will excite a wakefield with a correspond-ing Green’s function with M modes [22, 23], G ( t ) = M (cid:88) m =1 κ m · cos(2 πf m t ) , (1) a r X i v : . [ phy s i c s . acc - ph ] S e p where κ m and k m are the loss factor and wave numberof the m th mode respectively and are calculated numeri-cally [18, 24]. This Green’s function is often also referredto as the single particle wake potential W z ( τ ) [ V /C ],where τ denotes the time difference between the pointcharge and a trailing witness charge. Note that it is de-fined by the boundary conditions and hence - in our case- the geometry of the DLW. By varying e.g. the innerradius a of a DLW, it can be seen that both wavelengthand amplitude depend strongly on the geometry of thestructure (see Fig 2). Considering that the amplitude ofthe longitudinal wakefield scales as 1 /a [18], it becomesapparent that potentially very high field strengths can bereached in small aperture DLWs. W a v e l e n g t h ( mm )
50 um100 um300 um A m p li t u d e ( G V / m / C )
50 um100 um300 um
FIG. 2. Plot of the numerically calculated wavelength andamplitude of a wakefield excited by an on-axis electron bunchin a single-mode DLW. The different colors correspond to dif-ferent thicknesses of the dielectric lining.
The overall wake potential V ( t ) produced by a bunchcan be calculated by convolving its current profile I ( t )with W z ( τ ). Therefore V ( t ) = − (cid:90) t −∞ I ( τ ) W z ( t − τ ) dτ. (2)The field excitation can also be described in terms of thefrequency dependent bunch form factor F . Then V ( t ) = q · M (cid:88) m =1 F m κ m · cos(2 πf m t ) , (3)where q is the total charge of the bunch. A strong modeexcitation therefore requires a bunch with an appropriatespectral content i.e. a relatively short bunch, or alsoby having a relatively short rise time in e.g. a flat-topdistribution [9, 25].In a cascaded arrangement of multiple DLWs, outsideof experimental constraints due to e.g. limitations inbeam transport, the energy modulations via wakefieldsfrom different structures can be concatenated. The fol-lowing section illustrates the broad potential for a set ofcascaded, or a single segmented structure to produce aversatile range of energy modulations. We note that theusage of segmented structures, and the produced effectsof transient wakes is discussed in Section V. III. LPS SHAPING IN SINGLE-MODESTRUCTURES
Fourier synthesis provides a simple way to producea large variety of waveforms which have various appli-cations in conventional electronics. Here we explorehow Fourier synthesis can be applied to charged parti-cle beams using self-wakes imparted in high-impedancemediums, e.g. DLWs. We are specifically interested inthe Fourier series for odd functions, since the wakefieldat the head of the bunch must be zero.In the simplest case, each of the individual segmentsof an LPSS is a single mode structure with a specific fun-damental mode frequency and amplitude. As discussedabove, the wake function W z ( τ ) for such a structure issimply given by W z,m ( τ ) = κ m · cos(2 πf m τ ) . (4)Using this and Eq. 2, the energy modulation imparted bya single DLW segment n can be estimated as (cf. [26])∆ E n ( t ) = − l n · κ m ( n ) · (cid:90) t −∞ I ( τ ) cos(2 πf m ( n ) ( t − τ )) dτ, (5)where l n is the length of the n th DLW segment. The totalenergy modulation imparted by an N -segment structurecan hence be estimated as∆ E tot ( t ) = N (cid:88) n =0 ∆ E n ( t ) (6)(see Section V for a discussion on the effects of sharp seg-ment transitions on the resulting wakefields). Assumingan idealized flat-top current profile I ( τ ), the total energymodulation reduces to∆ E tot ( t ) = N (cid:88) n =0 A n · sin(2 πf m ( n ) t ) , (7)where A n is the amplitude factor of the n th segment.Considering the scaling laws shown in Fig. 2, arbitraryLPS shapes can be constructed via Fourier composition.The amplitude A n of each frequency component can beadjusted by choosing an appropriate l n . It should benoted that the harmonic content of the input currentprofile must be sufficient to excite the desired modes.Eq. 7 essentially corresponds to an ordinary Fouriersine series. A saw-tooth wave, for example, can be con-structed by summing up only even harmonics with propernormalization. Hence, the Fourier series for a given fun-damental frequency f is given by F saw ( t ) = A · ∞ (cid:88) n =0 n + 2 sin( π (2 n + 2) f · t ) , (8)where A is an amplitude scaling factor. Another simpleexample is a square wave. Its Fourier series only containsodd harmonics. Thus F squ ( t ) = A · ∞ (cid:88) n =0 n + 1 sin(2 π (2 n + 1) f · t ) . (9)Fig. 3 visualizes the two modulation types for differentvalues of N . A m p li t u d e ( a r b . un i t ) Saw Tooth
N = 1N = 3N = 100.0 0.2 0.4 0.6 0.8 1.0Longitudinal Coordinate (arb. unit)1.51.00.50.00.51.01.5 A m p li t u d e ( a r b . un i t ) Square Wave
N = 1N = 3N = 10
FIG. 3. Plot of amplitude vs. long. coordinate for an arbi-trary saw-tooth modulation (Eq. 8) and an arbitrary squarewave modulation (Eq. 9) for N = 1, N = 3 and N = 10. In order to explore possible use cases of such energymodulations we investigated the effect of applying linearlongitudinal dispersion ( R ) to the the phase space. Inthis work we adopt the convention that the head of thebunch is at z < R <
0. Fig. 4 shows contourplots of both the beam current within a single funda-mental modulation wavelength λ = 1 mm, as well as theharmonic content of the bunch vs. the longitudinal dis-persion R for different values of N . The idealized inputcurrent is assumed to be flat-top. We also assume a coldbeam in order to be able to explore the full mathematicallimits of the scheme. The investigation is carried out forboth a saw-tooth modulation (cf. Eq. 8), as well as fora square wave modulation (cf. Eq. 9). It can be seen,as longitudinal dispersion is applied, interesting featuresemerge.In the case of the saw-tooth modulation first the higherfrequency modulation on the rising part of the saw-tooth(see Fig. 3) is compressed. Then, as R increases,the minimum and maximum of the saw-tooth converge,which results in a current spike. The current spike is moredefined as N increases, which can be attributed to a lesspronounced Gibbs ringing at the sharp edges of the saw-tooth, as well as an overall flattening for higher values of N . This behaviour is also represented by the ellipsoidalshape visible in the contour plots of the beam current vs. d z and R , which becomes narrower as N increases (seeFig 4). It is interesting to note that as the amplitudeof the high frequency modulation along the rising partof the saw-tooth varies, different parts of the rising edgerequire different values of R for optimal compression.This is clearly visible in the contour plots. For symme-try reasons, always two sub-microbunches emerge. Byadjusting R , a specific pair of microbunches with a de-fined relative distance can be selected. It has to be noted,however, that - depending on the modulation depth -these sub-structures require very low slice energy spreadto be significant vs. the background. If the respectivebunching factor b n should not be reduced by more thanroughly a factor of 2, then δ mod /δ sl ≤ n has to be sat-isfied, where n is the harmonic number of f and δ mod and δ sl are the relative modulation depth and slice energyspread respectively; cf. [27].In case of the square wave modulation the plots showa different behaviour. As R increases, first a singlecurrent spike is formed, which corresponds to the sharpedge of the energy modulation. As the edge becomessharper (higher N ), optimal bunching occurs for smallervalues of R . Increasing R beyond optimal bunchingreveals a very particular rhombus pattern in the contourplot, which is explained by the fact that the negativeand positive plateaus of the square wave are shifted ontop of each other. The higher the value of N , the moreintricate the rhombus pattern becomes. It is interestingto note that - by applying appropriately high R - thetwo plateaus of the square wave modulation will formtwo sub-microbunch trains at their own distinct energylevels ( E ± ∆ E ).The saw-tooth and square wave modulation are onlytwo examples of possible Fourier series based LPS mod-ulations. Many other interesting waveforms might exist,which are not discussed here. In order to show how dras-tic even small changes to a particular Fourier series defini-tion can be, one can consider squaring the normalizationfactor in Eq. 9. This yields, instead of a sharp squarewave, a smooth rounded wave. The definition now reads F rnd ( t ) = A · ∞ (cid:88) n =0 n + 1) sin(2 π (2 n + 1) f · t ) . (10)Fig. 5 shows both the shape of an N = 11 round wavemodulation, as well as contour plots analogous to Fig. 4.It can be seen that applying R to this kind of modu-lation at first glance leads to a dependence similar to asimple sine wave modulation. The main difference, how-ever, is that the beam current of the sub-microbunches,which occur after over-bunching, shows multiple addi-tional maxima of similar magnitude compared to theinitial single microbunch. In the case of a simple sinemodulation the peak current would decrease rapidly.As the number of additional maxima increases with N ,this means that using a high- N round wave modulation,one can obtain high-quality sub-microbunches with semi-continuously adjustable relative spacing. FIG. 4. Contour plots of both the beam current within a single fundamental modulation wavelength λ = 1 mm, as well as theharmonic content of the bunch vs. the longitudinal dispersion R . The scan was performed for N ∈ [1 , , , , , E = 100 MeV is constant along the bunch. It has atotal bunch length of 1 mm and Q = 42 pC. The assumed maximum modulation depth of the lowest frequency component is500 keV. Note that a high slice energy spread would lead to blurring out the small features in the respective phase spaces. Herewe assume a cold beam in order to explore the full mathematical potential of the scheme. FIG. 5.
Top:
Plot of an N = 11 round wave modulation ac-cording to Eq. 10. Bottom:
Contour plots of both the beamcurrent within a single fundamental modulation wavelength λ = 1 mm, as well as the harmonic content of the bunch vs.the longitudinal dispersion R . The scan was performed for N = 11. The idealized input current and modulation depthis assumed to be the same as described in Fig. 4. IV. ARBITRARY MULTIMODEOPTIMIZATION
So far we have investigated Fourier shaping of an ide-alized input LPS. In order to work with arbitrary inputdistributions, a more sophisticated optimization routinemust be used. This is especially true if multi-mode DLWsegments are to be included, as the number of degrees offreedom gets too large for manual optimization. Hencea routine based for example on the particle swarm al-gorithm (PSO) must be employed [28]. The algorithmvaries all geometric parameters of the individual seg-ments at the same time in order to find a global minimumof a given merit function. This merit function is given inthe LPSS case by the similarity of the resulting LPS tothe desired LPS shape. Since segment radius, length andwall thickness can be varied, the resulting number of in-dependent variables is 3 N , where N is the number of seg-ments. For the LPSS study presented here, the PSO wasimplemented using PyOpt [29]. At each iteration stepeither a simulation using a specifically generated inputfile for a numerical simulation code, or a semi-analyticalcalculation based on Eq. 2 is carried out. If space-chargeeffects are neglected, the difference between the numeri-cal simulation using ASTRA [30] and the semi-analyticalapproach was found to be negligible. Hence, the muchfaster semi-analytical calculation was used for the simu-lations shown in the following discussion.As an example optimization goal the linearization of anincoming LPS obtained from close to on-crest accelera-tion was chosen. This scenario is interesting, because the resulting LPS shows a clear signature of the sinusoidalRF field of the linac structures, which would limit theachievable bunch length in subsequent compression. Inorder to keep the number of free parameters manageable,the number of LPSS segments was limited to 10. The op-timizer was configured to bring the Pearson’s R value ofcentered ˜ nσ z regions within the final distribution as closeto 1 as possible. Here ˜ n ∈ N . Table I summarizes thepossible ranges of values for the geometry parameters ofthe 10 individual segments. TABLE I. LPSS optimization variable ranges for each of the10 segments.
Parameter Value
Inner radius [0 . , .
5] mmDielectric thickness [50 , µ mSegment length [1 , For the input we consider three different electron bunchdistributions with 10 pC, 100 pC and 200 pC of totalcharge and a mean energy of ∼
100 MeV, based on nu-merical simulations of the ARES linac at DESY [31].This is done in order to provide a realistic example, whichcould be used as the basis for future experimental veri-fication of the scheme. Fig. 6 shows a schematic of theARES lattice. If both linac structures are driven at theirrespective maximum gradients of ∼
25 MV / m, a finalmean energy up to ∼
150 MeV is possible. The decisionto limit the example working points to ∼
100 MeV is apractical one, as the overall LPSS structure length gener-ally increases with the required LPS modulation strengthand the experimental chamber at ARES imposes strictspace limitations. The three working points were opti-mized to minimize transverse emittance at the interac-tion point ( z = 16 . TABLE II. Beam parameters of the three ARES linac workingpoints (WP) at the interaction point ( z = 16 . Parameter WP1 WP2 WP3
Charge 10 pC 100 pC 200 pCTWS injection phase − ◦ − ◦ − ◦ E
108 MeV 109 MeV 108 MeV σ E /E . · − . · − . · − σ t
673 fs 1 .
95 ps 2 .
65 ps ε n ,x,y
146 nm 370 nm 465 nm
S-Band RF GunS-Band Travelling Wave Structure 1 S-Band Travelling Wave Structure 2SolenoidsGun Solenoids Quadrupole Dipole Collimator X-Band PolariX TDSExperimental Area 1 TDS + Diagnostics BeamlineMagnetic CompressorExperimentalChamber Spectrometer
FIG. 6. Schematic of the layout of the ARES linac at DESY (not to scale). The LPSS interaction is simulated to take placein the experimental chamber of
Experimental Area 1 at z = 16 . In order to first investigate the effect of limiting theoptimization goal to specific ˜ nσ z regions within the LPSon the resulting LPSS geometry, four different optimiza-tion runs were performed. As input, WP3 was chosen(see Table II). Fig. 7 shows the results. Starting from anoverall linearity of the input LPS of R = 0 . R = 0 . n = 1. It is apparent thatif the whole LPS is taken into account (i.e. a 6 σ ROI),the results are noticeably worse than for a restricted ROI.This can be attributed to the fact that, due to the Gaus-sian time profile of the input LPS, the beam currentin the head region of the bunch is low and hence thestrength of the excited wakefield is weak. Thus, it is dif-ficult for the optimizer to find configurations where thisregion is linearized sufficiently well, subsequently spoil-ing the overall linearity of the LPS. Excluding this headregion of the LPS from the optimization, on the otherhand, improves the performance significantly. In an ex-periment at ARES for example, the region outside of theROI could be cut using the slit collimator implementedin the magnetic compressor (see below). d E ( M e V ) Longitudinal Phase Space6 sigma4 sigma2 sigma1 sigma
Structure Length (cm)
Minimum Aperture Radius (mm)
Linearity In Sigma Region
FIG. 7.
Left : Comparison of LPSS linearization results, de-pending on the size of a defined region of interest (ROI) withinthe bunch. The solid part of the lines corresponds to the re-spective ROI.
Right:
Total LPSS structure length, minimalsegment aperture radius within the structure and linearity ofthe output LPS within the respective ROI, depending on theROI size.
In addition to the degree of linearity in the respective˜ nσ z region, Fig. 7 also shows that two important geom-etry parameters depend on the ROI as well. First, theoverall structure length decreases with the ROI. This isof practical importance, not only in terms of beam trans-port through the structure, but also in terms of manu-facturing. Second, the minimum aperture radius of thestructure increases with a decreasing ROI, which is im-portant from a beam transport point of view and in ac-cordance with the time profile of the bunch and the de-pendence of the wake field strength per charge as shownin Fig. 2. Taking these results into account, it is clearlyworth considering trading - in case of a Gaussian timeprofile - less than 5 % of the total bunch charge for themuch better linearization performance of a 4 σ ROI.Based on the results discussed above, optimizationruns were performed for all of the three ARES work-ing points, considering both a full 6 σ and a smaller 2 σ linearization ROI. Fig. 8 shows the detailed results. Itcan be seen that in all cases a significant improvementof R can be achieved within the ROI. Better results areobtained in the case of the limited ROI, as expected.Furthermore, the geometries of the resulting LPSS struc-tures are shown. The shorter in time the input LPS,the shorter the resulting LPSS. This is partly due to thesmaller required modulation depth, but also due to howthe wakefield amplitude scales with the required innerradii of the segments ( E z ∝ /a ; see Fig. 2). In orderto accommodate a typical focused beam envelope, theindividual segments of the LPSS structures are sortedsuch that the tightest segment is placed at the center ofthe structure, which then has increasing inner radii to-wards both entrance and exit. The results show that asimilar degree of linearization can be achieved, regardlessof the different bunch lengths across the different work-ing points. The shape of the respective resulting struc-ture does vary significantly however, due to the requiredmodal content. A. Other Optimization Goals
As already discussed above, not only the linearizationwithin a defined ROI can be set as an optimization goal.Another interesting case could be the removal of any cor-related energy spread, aiming for a completely flat LPS.
FIG. 8. LPSS optimization results for input LPSs based on the ARES working points shown in Table II. The optimizationgoal was to achieve R = 1 across the full 6 σ ROI ( top row ), as well as a centered 2 σ ROI ( bottom row ). From left toright: WP1, WP2, WP3. Each plot shows the LPS before and after the LPSS interaction. The color and thickness visualizethe current profile. The gray shaded areas correspond to the 2,4 and 6 σ regions respectively. The head of the bunch is onthe left (negative z values). Below the main plot, the geometry of the final segmented DLW is visualized, with the orange linecorresponding to the inner radius and the blue line to the outer radius. Fig. 9 shows the result of such an optimization, basedon the 10 pC WP1 as shown in Table II. It can be seenthat the phase space is significantly flattened within the4 σ ROI. Note that this kind of structure could be usedto prepare an LPS for further modulation as shown, forexample, in section III.
B. Example Case: Bunch Compression
It was shown in simulation that at ARES, based onmagnetic compression and a slit-collimator, sub-fs bunchlengths can be achieved [32, 33]. Starting from an ini-tial bunch charge of 20 pC a final rms bunch length of0 .
51 fs was achieved, 1 .
75 m downstream of the chicaneexit (cf. Fig. 6). The remaining charge after the slit is0 .
79 pC, which corresponds to a ∼ TABLE III. Beam parameters of different ARES workingpoints (WP) 1 .
75 m downstream of the chicane exit ( z =30 . nσ subscript refers to the LPSS opti-mization ROI. TWS: Travelling Wave Structure. Parameter WP,Zhu WP4, σ WP4, σ Initial charge 20 pC 10 pC 10 pCFinal charge 0 .
79 pC 2 . .
18 pCTWS injection phase − ◦ − ◦ − ◦ Chicane R − . − . − . . . . E . . . σ E /E . · − . · − . · − σ t .
51 fs 0 .
84 fs 0 .
73 fs ε n ,x . µ m 0 . µ m 0 . µ m ε n ,y . µ m 0 . µ m 0 . µ m I p .
62 kA 1 .
32 kA 2 .
18 kA
FIG. 9. LPSS optimization results based on the ARES work-ing point WP1 shown in Table II. The optimization goal wasto completely remove any correlated energy spread within a4 σ ROI. The color and thickness visualize the current profile.The gray shaded areas correspond to the 2,4 and 6 σ regionsrespectively. The head of the bunch is on the left (negative z values). Below the main plot, the geometry of the final seg-mented DLW is visualized, with the orange line correspondingto the inner radius and the blue line to the outer radius. we can achieve similar beam parameters, but at highermean energy and higher final peak current. To this endWP4, which is a modified version of WP1 (cf. Table II),where the TWS structures are driven at − ◦ is used ina start-to-end simulation using ASTRA, the LPSS opti-mization routine and IMPACT-T [34]. Up to the LPSSstructure the simulation includes space charge forces viaASTRA and after that both space charge and CSR viaIMPACT-T. Full linearization in a 4 σ optimization ROIwas considered as the LPSS optimization goal. The re-sulting beam parameters 1 .
75 m downstream of the chi-cane exit are summarized in Table III, where WP4,0 σ refers to our working point without LPSS linearizationand WP4,4 σ to the case employing the optimized LPSSstructure. The final longitudinal phase spaces are shownin Fig. 10. It can be seen that using a passive LPSSstructure upstream of the magnetic bunch compressor in˜ nσ linearization mode yields bunches with similar beamquality, but at 26 % higher mean energy. At the sametime, even though the initial charge is 50 % less, the finalcharge is higher, due to the larger slit width. This is pos-sible due to the high degree of linearization in the LPSSROI. The peak current is noticeably higher in both WP4cases ( ∼ × w.o. the LPSS and ∼ . × using the LPSS).We note that the transverse phase space of WP4 wasnot fully optimized as part of this study, which meansthat the transverse properties of the beam could be im-proved in future iterations of this particular workingpoint. FIG. 10. Numerical simulation of the longitudinal phase spaceand current profile of the ARES working point WP4 shown inTable III, 1 .
75 m downstream of the chicane exit ( z = 30 . Top:
Bunch compression without applying the LPSS opti-mization, i.e. no structure.
Bottom:
Bunch compressionafter applying a 4 σ linearization with an optimized LPSSstructure. Finally it should be noted, that at higher overallcharges significant energy modulation due to CSR canspoil the linearity of the LPS during bunch compression.This, however, could be included into future versions ofthe LPSS optimization routine as the virtual last elementof the LPSS structure.
V. REALISTIC STRUCTURESA. Segment Transitions
Our previous discussion has treated the LPSS as a se-ries of individual successive DLW segments. In orderto calculate the resulting energy modulation, the indi-vidual wakefields of the segments were summed up andapplied to the input LPS. Although this is a good firstapproximation, in reality there are two issues with thisapproach. First, the sharp transitions between the seg-ments will disturb the wakefield slightly. Second andmost importantly, this kind of segmented structure can-not be produced, because in some cases it turns out that a i +1 > b i , which would mean that the ( i + 1)th seg-ment could not actually be attached to the i th segment.It is hence necessary to include transition elements be-tween the individual segments. These elements could forexample be short linearly tapered sections. Althoughadding such a transition would enable production of thestructure, it also alters the resulting wakefield. In or-der to investigate this effect, ECHO2D [35] simulationswere performed. The longitudinal monopole wakefield,excited by a Gaussian current with an arbitrarily chosen σ t = 500 fs, was compared for three different cases:1. The sum of the resulting wakefield of two individ-ually simulated DLW segments of length l and l ,2. The two segments directly behind one another.(sharp, unrealistic transition),3. The two segments connected with a linearly ta-pered transition region of length l t .Note that the overall length L of the structure is thesame for both case 2 and 3. This means that for case3 the individual segments are shortened by 0 . · l t each.Hence, case 2 is essentially case 3 with l t = 0. See Fig. 11for an illustration of the three different cases.Fig. 12 shows the integrated residual difference be-tween the wakefield obtained from the case 1 and case 3geometries using a drive bunch with σ t = 500 fs vs. differ-ent values of l t . The exemplary dimensions of the DLWsegments are a = 0 . b = 0 . l = 10 mm, a = 0 . b = 0 . l = 10 mm. The dielectric isdefined by (cid:15) r = 4 . , µ r = 1 and the metal coating, whichis assumed to be a perfect conductor, has a thickness of0 . l t can be found depending on the area of interest aroundthe peak of the drive current. It has to be noted thatalthough this minimum does not depend strongly on thelongitudinal dimensions of the segments, it does dependon the transverse dimensions a i and b i (and on σ t , as thewhole composition of the structure depends on it). Itis hence implied that each transition has to be uniquelyoptimized. This, however, can be directly factored intothe optimization routine discussed above (extending thenumber of degrees of freedom from 3 N to 4 N − r ( mm ) r ( mm ) FIG. 11. Illustration of the DLW geometry used in theECHO2D simulations. All cases include a (lossless) metalcoating of 100 µ m thickness. The blue lines correspond to theoutline of the metal coating and the orange lines to the outlineof the dielectric. : Single segment of length l = 10 mm. : Single segment of length l = 10 mm. : Segments rightnext to each other (sharp, unrealistic transition). : Twosegments connected with a linearly tapered transition regionof length l t = 1 mm. It was shown that the integrated difference between acase 1 and 3 geometry can be minimized by adjusting l t . Fig. 13 shows the longitudinal wake for all three ge-ometry cases based on a simulation using the exemplaryparameters from above and an optimized l t of 953 µ m.In addition to the wakefields, the absolute and relativedifference compared to case 1 is plotted for both the case2 and 3 geometry respectively. It can be seen that, de-pending on the area of interest along the drive bunch,the error can be very small and is generally smaller than10 %. The error can be large, however, towards the tail ofthe bunch. The significance of this effect depends a lot on R e s i d u a l E rr o r ( a r b . un i t ) Total6 sigma4 sigma2 sigma
FIG. 12. Normalized integrated residual difference betweenthe wakefield obtained from the sum of two singular DLWsegments and a combined device with a linearly tapered tran-sition region of length l t , as show in Fig. 11. The differentcurves correspond to the 6 σ, σ and 2 σ parts of the drivebunch, as well as the complete simulation box (total). <
16 % of the charge is affected. Recalling Fig. 12,the goal should in general be to minimize the effect ofthe transition in the region of highest charge density. Insummary, it can be concluded that it is possible to findtransition regions, which minimize the difference of theproduced wakefield compared to the summed up wake-field of individual segments, as used in the optimizationroutine discussed above. W _ l ( k V / p C ) Longitudinal Wake p1p2p1+p2SmoothHard 0.0 0.5 1.0 1.5z (mm)0246810 D e l t a ( % ) Relative Difference (p1+p2)-Smooth(p1+p2)-Hard0.0 0.5 1.0 1.5z (mm)0.150.100.050.000.050.100.15 D e l t a ( k V / p C ) Absolute Difference (p1+p2)-Smooth(p1+p2)-Hard
FIG. 13. Comparison of the wakefield obtained using thegeometries illustrated in Fig. 11. l t = 953 µ m, which is thevalue determined by the optimization scan shown in Fig. 12.The shaded areas correspond to the 6 σ, σ and 2 σ parts ofthe drive bunch. B. Manufacturing
The optimization shown above does not include anyassumption about possible inaccuracies due to the man-ufacturing process. In reality, the exact shape of theindividual segments is determined by the tolerances dur-ing production. Assuming a 3D-printed structure, theparameters a i , b i and l i are determined by the trans-verse and longitudinal printing resolution and on howthe structure is printed (flat or standing). We considerthe ASIGA MAX X27 3D printer [36] and its printingresolution as an example. This particular printer hasa longitudinal resolution ρ z of 10 µ m (minimum layerthickness) and a lateral resolution ρ xy of 27 µ m (DLPpixel size). Fig. 14 shows the comparison between thelinearization using an ideal LPSS and an LPSS, whichwas optimized taking the aforementioned printing res-olution into account. Here we model the effect suchthat ˜ a i = (cid:98) a i /ρ xy (cid:99) · ρ xy /
2, ˜ b i = (cid:100) b i /ρ xy (cid:101) · ρ xy / l i = (cid:100) l i /ρ z (cid:101) · ρ z , where the tilde denotes the radii and length of the segments after applying the printer resolu-tion. The results show that the limited printing resolu-tion only has a small impact on the final linearization.It has to be noted, that the chirp across the ROI is dif-ferent, but only because it was not part of the particularoptimization goal. d E ( M e V ) BeforeAfter, 4 sigma ROI, ideal (R=-0.99998)After, 4 sigma ROI, ASIGA (R=-0.99997)
FIG. 14. Comparison of LPSS optimization results for aGaussian input current profile. The optimization goal wasto achieve R = 1 in a 4 σ region of interest. The input beamparameters correspond to WP3 (see Table II). Blue:
IdealLPSS, orange:
LPSS taking a lateral printing resolution of27 µ m and a longitudinal printing resolution of 10 µ m into ac-count (as can, for example, be achieved with an ASIGA MAXX27 3D printer). VI. ROBUSTNESS OF THE SCHEMEA. Input LPS
As discussed above, an LPSS must be specifically tai-lored to the incoming LPS. In reality the actual shape ofthe input LPS varies according to the stability of certainaccelerator machine parameters. The LPS in particularis influenced by the stability of both amplitude and phaseof the accelerating fields, but also by dispersive sectionsand collective effects, such as coherent synchrotron ra-diation (CSR). It is hence interesting to investigate theeffect of the actual shape of the input LPS on the outputLPS. To this end, the third ARES working point (WP3)with 200 pC of total charge and a Gaussian time profilewith σ t = 2 .
65 ps (see Section IV) is used. The sensi-tivity of the linearity parameter R within a 4 σ ROI isdetermined for four different parameters, with the firsttwo parameters being the amplitude and phase of theaccelerating field, which define the curvature of the in-coming LPS. The third parameter is σ t , which in real-ity, of course, non-trivially depends on multiple factors,but is here varied independently, while keeping the to-tal bunch charge constant. The fourth parameter is thebunch charge Q , keeping σ t constant. Fig. 15 summa-1rizes the results of the four scans. The results show thatthe relative change in R is very little ( (cid:28) . D e l t a R ( % )
10 0 10Delta E0 (MeV)0.030.020.010.0050 0 50Delta Sigma_t (fs)0.030.02 D e l t a R ( % )
10 0 10 Q (pC)0.060.040.02
FIG. 15. Relative change of the linearity factor R vs. fourdifferent parameters, which influence the input LPS. B. Systematic Manufacturing Errors
In addition to the uncertainty in the shape of the inputLPS, there can also be systematic errors in the geome-try of the LPSS itself. In order to investigate this, twoscenarios were studied. The first one is a constant error∆ r of both the inner and outer radii, i.e. ˜ a i = a i + ∆ r and ˜ b i = b i + ∆ r . The second scenario is a constant dif-ference in wall thickness, meaning ˜ a i = a i − ∆ r/ b i = b i +∆ r/
2. The range is chosen to be according to thelateral resolution of the ASIGA printer discussed above.Hence ∆ r ∈ [ − , µ m. Fig. 16 summarizes the re-sults of the scan. The LPSS optimization scenario is thesame as before. It can be seen that the change in globalaperture has a very small effect on R ( < .
01 %). Thewall thickness, on the other hand, has a ∼ × strongereffect, with a slight asymmetry. It is still a small effectwith | ∆ R | < . r . Theslightly asymmetric behaviour might be explained by thenon-linear dependence of the amplitude and frequencyof the wake towards smaller inner radii (cf. Fig. 2) inconjunction with an increase in the modal content as thethickness of the dielectric lining increases. A more thor-ough study of this behaviour would be interesting, butexceeds the scope of this work. VII. CONCLUSION AND OUTLOOK
A completely passive LPSS solution, based on seg-mented DLWs was presented and studied both analyti-cally and numerically. The results based on the idealized
30 20 10 0 10 20 30Delta (um)0.080.060.040.020.00 D e l t a R ( % ) a+delta, b+delta (Global aperture changed)a-delta/2, b+delta/2 (Wall thickness changed) FIG. 16. Relative change of the linearity factor R vs. aconstant error ∆ r for two different systematic error scenarios.The same ∆ r is applied to all segments. single-mode Fourier synthesis, coupled with a longitu-dinally dispersive section, reveal phase space configura-tions, which could be interesting for applications, espe-cially in the context of radiation generation (multi-colormicrobunch trains, sub-microbunches with tunable rela-tive spacing, etc.).Arbitrary multimode optimization was investigated,which enables application of the method to arbitrary in-put phase spaces. The results shown here are promis-ing, as the exemplary goal of full linearization of the in-put LPS within a given ˜ nσ z ROI of an input LPS wasachieved in a semi-analytical simulation to a very highdegree. The input LPS used for the study were chosen tobe realistic and are based on numerical simulation of anexisting accelerator, the ARES linac at DESY. Motivatedby these results, a start-to-end simulation of a possibleexperiment at the ARES linac was performed yieldingsub-fs bunches comparable to reference working points,but at ∼
26 % higher mean energy and ∼ . × largerpeak current, starting from 50 % less initial charge.It was furthermore shown, based on ECHO2D simula-tions, that it is possible to integrate short transition re-gions between the segments, which enables realistic struc-ture shapes that can be produced with a 3D-printer. Theoptimization routine used in this work can export its re-sult as 3D models suitable for direct import into a 3Dprinting software. Fig. 17 shows a rendering of such a file.The structures can be made from metallized 3D-printedplastic, or even 3D-printed quartz [37]. Depending onthe specific printing process, longer structures might beconstructed of two or more cascaded macro segments.The robustness of the scheme was investigated for theLPS linearization example and found to be satisfactorybased on accelerator stability, as well as manufacturingtolerance considerations. This together with the low costof the devices alleviates the fact that each LPSS deviceis specific to a given accelerator working point; multiplestructures could be installed and swapped in as needed.Further studies could focus on transverse effects inLPSS structures, as potentially triggered dipole modesmight lead to deflection. Also material-dependent charg-2 FIG. 17. Section view of a 3D rendering of a potential printedand metallized LPSS structure. The 3D model was obtaineddirectly from the optimization routine. ing of the dielectric could be studied. Finally, the LPSSoptimization routine could be updated to take expecteddownstream LPS modulation, due to e.g. collective ef-fects, into account.
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