Formation of Temporally Shaped Electron Bunches for Beam-Driven Collinear Wakefield Accelerators
FFormation of Temporally Shaped Electron Bunches forBeam-Driven Collinear Wakefield Accelerators
Wei Hou Tan, ∗ Philippe Piot,
1, 2 and Alexander Zholents Northern Illinois Center for Accelerator & Detector Development and Department of Physics,Northern Illinois University, DeKalb, IL 60115, USA Argonne National Laboratory, Lemont, IL 60439, USA (Dated: January 20, 2021)Beam-driven collinear wakefield accelerators (CWAs) that operate by using slow-wave structuresor plasmas hold great promise toward reducing the size of contemporary accelerators. Sustainableacceleration of charged particles to high energies in the CWA relies on using field-generating rel-ativistic electron bunches with a highly asymmetric peak current profile and a large energy chirp.A new approach to obtaining such bunches has been proposed and illustrated with the acceleratordesign supported by particle tracking simulations. It has been shown that the required particledistribution in the longitudinal phase space can be obtained without collimators, giving CWAs anopportunity for employment in applications requiring a high repetition rate of operation.
PACS numbers: 29.27.-a, 41.85.-p, 41.75.Fr
I. INTRODUCTION
In a beam-based collinear wakefield accelerator(CWA), a high-charge drive bunch generates an electro-magnetic field passing through a slow-wave structure (adielectric-lined or corrugated waveguide) or plasma. Thisfield, called the wakefield, is used to accelerate a witnessbunch propagating the structure in the same directionbehind the drive bunch [1–8]. An important figure ofmerit for a CWA is the transformer ratio,
R ≡ |E + / E − | ,where E + is the maximum accelerating field behind thedrive bunch, and E − is the maximum decelerating fieldwithin the drive bunch. For symmetric drive-bunch cur-rent distribution in time I ( t ), the transformer ratio islimited to R < I ( t ) can sig-nificantly enhance the transformer ratio [9], albeit at theexpense of reduced E + and E − [10].Bunch-shaping techniques investigated hitherto arephotocathode-laser intensity shaping [11–13], transverse-to-longitudinal phase-space exchange [14–16], and use ofmulti-frequency linacs [17]. Despite significant progress,they suffer either from their inability to deliver highlyasymmetric bunches or from prohibitively large beamlosses on collimators. Consequently, producing drivebunches with an asymmetric peak current profile whilepreserving most of the bunch electrons has been an activeresearch topic.Another important consideration for a drive buncharises from its proneness to the transverse beam-break-up(BBU) instability caused by the strong transverse forcesdue to the transverse wakefield [18–20]. A possible BBU-mitigation technique consists of imparting a large energychirp along the drive bunch [21–23] and creating a cur-rent profile I ( t ) that stimulates a dynamic adjustment ofthis chirp concurrently with the wakefield-induced bunch ∗ [email protected] deceleration in the CWA [24].The work reported in this paper was motivated by adesign of a high repetition rate CWA for use in a free-electron laser (FEL) facility described in Refs. [25, 26].This facility plans to employ up to ten FELs individu-ally driven by a dedicated CWA. A single conventionalaccelerator delivers ∼ I ( t ) and a large energy chirp tothe ten CWAs. Since the drive-bunch charge consid-ered in [25, 26] is up to 10 nC and the bunch repetitionrate up to 500 kHz, the electron beam carries significantpower. Therefore, using collimators to assist with thebunch shaping is prohibitive, and, consequently, prepar-ing the drive bunches doing otherwise becomes a primechallenge.To solve the problem, we undertook a new approachand distributed the task of obtaining the highly asym-metric I ( t ) over the entire drive bunch accelerator begin-ning from the photocathode electron gun and ending bythe final bunch compressor. To the best of our knowl-edge, our work demonstrates for the first time a pathwayto obtaining electron bunches with a highly asymmetric I ( t ), avoiding prohibitively large electron losses on col-limators. The employed technique is rather generic andcan be used for preparing the electron bunch peak currentdistribution with profiles different than those consideredin this paper.Although the main focus of the work was to obtain adrive bunch with the required distribution in the longitu-dinal phase space (LPS), an equally important additionalobjective, was to ensure the associated transverse emit-tances commensurate with the small CWA aperture. II. THE DRIVE BUNCH AND THEWAKEFIELD
We define the longitudinal charge distribution in theelectron bunch as q ( z ) and consider bunches localized on a r X i v : . [ phy s i c s . acc - ph ] J a n . . . . . . z/λ , - − − w a k e fi e l d ( M V / m ) E + = 94 .
34 MV/m (red) E − = 16 .
82 MV/m (red)modified doorstepdoorstep − − c u rr e n t( k A ) . . . − − Figure 1. Nominal (green trace) and modified doorstep dis-tributions with associated wakefields calculated using L = λ , χ = 0 and χ = − λ , respectively. The wakefields are com-puted for a bunch charge of 10 nC and use a single-modeGreen’s function, where f = 180 GHz and κ (cid:107) =14 . / pC / mcalculated using ECHO [30]. The transformer ratio for themodified doorstep distribution shown in the plot is R = 5 . the interval 0 ≤ z ≤ L , where z is the distance behindthe bunch head. Therefore, we have (cid:90) L q ( z )d z = Q, (1)where Q is the total bunch charge. Following [10], we usethe Green’s function G ( z ) consisting only of a fundamen-tal mode G ( z ) = 2 κ (cid:107) cos ( kz ) H ( z ) , where κ (cid:107) is the lossfactor of a point particle per unit length, k = 2 π/λ is thewave vector, λ is the wavelength, H ( z ) is the Heaviside step function. The longitudinal electric field within theelectron bunch can be written as [27, 28] E − ( z ) = 2 κ (cid:107) (cid:90) z cos [ k ( z − z (cid:48) )] q ( z (cid:48) )d z (cid:48) , z ≤ L, (2)which is a Volterra equation of the first kind for the func-tion q ( z ) with the trigonometric kernel cos [ k ( z (cid:48) − z )]. Ifwe assume that E − (0) = 0 at the bunch head, then thesolution of Eq. (2) is given by [29], q ( z ) = 12 κ (cid:107) (cid:20) E (cid:48)− ( z ) + k (cid:90) z E − ( x )d x (cid:21) , (3)where E − ( z ) is a known function, and its derivative istaken over z . Hence, q ( z ) is defined.In order to maintain the stability of the drive bunchin the CWA throughout its deceleration, we require thebunch’s relative chirp to be constant while being decel-erated by the wakefield E − ( z ), based on studies in [24].This requirement is achieved by having a small linearvariation in energy loss within the bunch, where headparticles lose more energies than tail particles such that χ ( s ) = 1 E (s) ∂E∂z ∝ E (cid:48)− ( z ) ≡ const , (4)where E (s) is the energy of the reference particle, and s is the distance propagated by the bunch in the CWA.This is accomplished by using the electron bunch produc-ing E − with a linear variation in z . Similar to Ref. [9],we solve Eq. (3) considering q ( z ) to be constant in therange 0 ≤ z < ξ with ξ = k arccos( χ/k ), in which casethe continuities of E − ( z ) and E (cid:48)− ( z ) are preserved overthe entire bunch length q ( z ) = (cid:40) q , ≤ z < ξ ,q (cid:104) − kξ sin( kξ ) + k ξ cos( kξ ) + (cid:0) k sin( kξ ) − k ξ cos( kξ ) (cid:1) z + k cos( kξ ) z (cid:105) , ξ ≤ z ≤ L , (5) q = 6 Q L + k cos( kξ )( L − ξ ) + 3 k sin( kξ )( L − ξ ) . Setting χ = 0, simplifies q ( z ) to one used in [9]. Figure 1shows an example of a modified doorstep distributionwith an associated wakefield calculated using L = λ and χ = − λ . In this example we considered a corrugatedwaveguide with radius a =1 mm and fundamental modefrequency f = 180 GHz, as discussed in Ref. [31]. Thecurrent profile has sharp features that are challenging torealize. Consequently, the distribution defined by Eq. (5) It has been shown in [10] that a multi-mode Green’s function isless effective in producing a high transformer ratio. is used only as a starting point to construct a practicallyrealizable distribution shown in Fig. 2 with similar finalproperties listed in Table I.
III. A PRELIMINARY DESIGN OF THE DRIVEBUNCH ACCELERATORA. Basic considerations
A block diagram of the drive bunch accelerator isshown in Fig. 3. It utilizes a commonly used con- − . − . . . z − h z i (mm)0123 c u rr e n t( k A ) (a) − . − . . . z − h z i (mm)975100010251050 E ( M e V ) (b) Figure 2. A target drive bunch peak current (a) and longi-tudinal phase space (b) distributions at the end of the drivebunch accelerator.Table I. Main parameters associated with the drive bunchdistribution shown in Fig. 2.
Bunch parameter Value Unit
Charge 10 nCReference energy 1 GeVRMS length 419 µ mPeak current 3.5 kARMS fractional energy spread 2 .
51 %RMS fractional slice energy spread 0 . figuration (see, for example, [32, 33]) and includes aphotocathode-gun-based injector, three linac sections,and two bunch compressors. Linac sections L1 and L2 arebased on 650 MHz superconducting (SRF) linac struc-tures, and linac section L39 is based on 3 . L1 L39 L2BC1 BC2injector
Figure 3. Block diagram of the drive bunch accelerator.
Using the known LPS distribution Φ f ( z f , E f ) at theend of the accelerator, we performed the one-dimensional(1D) backward tracking proposed in [11] to find the LPSdistribution Φ i ( z i , E i ) at the entrance of L1. We stoppedat L1 where the beam energy is approximately 50 MeVconsidering that 1D tracking may not be reliable at lowerenergies where transverse and longitudinal space chargeeffects are stronger. The assumption is that at this pointthe backward tracking will produce a plausible Φ i ( z i , E i )that can be matched by the injector. Specifically, weconstrained the peak current to I ≤
300 A and soughtΦ i ( z i , E i ) with minimal high-order correlations. A tracking program, twice [34], was developed forrapid prototyping of the longitudinal dynamics in thelinac without accounting for a transverse motion. Theprogram adopts an approach similar to that used in LiTrack [35]. An important feature of twice is its abil-ity to perform backward tracking including time-reversalof the collective effect, see Appendix A.The physics model implemented in twice includes thegeometric wakefields in the accelerating sections, longi-tudinal space charge effects (LSCs), and coherent syn-chrotron radiation (CSR). The Green’s functions neededfor modeling of the geometric wakefield effects in the650 MHz and 3 . echo software and the empirical formula documentedin Ref. [36].The backward tracking was performed to defineΦ i ( z i , E i ) using Φ f ( z f , E f ), shown in Fig. 2. The follow-ing constraints for the accelerator components were ob-served. First, the BBU-mitigation scheme implementedin the CWA requires a drive bunch with the negativechirp ∂E∂z <
0, which implies that the longitudinal disper-sions of BC1 and BC2 should be R (1)56 > R (2)56 > ∼
950 MeV in the linac part after the injector is needed.Third, an overall compression factor of ∼
10 is requiredfrom two bunch compressors.In order to enforce all these constraints, twice wascombined with the multi-objective optimization frame-work deap [37]. The optimization was performed by an-alyzing the LPS distributions upstream of BC1 and L1to extract the central energy of the beam slices at every z -coordinate and to fit the slice-energy dependence on z with the polynomial E ( z ) = c + c z + c z + c z , (6)where c i are constants derived from the fit. The opti-mizer was requested to minimize the ratio c /c in bothlocations. B. Discussion of 1D simulation results
A list of optimized accelerator settings found with twice backward tracking is given in Table II and theresulting Φ i ( z i , E i ) is shown in Fig. 4(a,b). The forwardtracking using this distribution recovers Φ f ( z f , E f ), asseen in Fig. 4(c,d). The excellent agreement betweenFig. 2(a,b) and Fig. 4(c,d) demonstrates the ability of twice to properly handle collective effects in both for-ward and backward tracking.Each accelerator component serves a special role in ob-taining the above-shown result. Linac section L1 pro-vides energy gain and operates far from the crest accel-eration to produce the required negative chirp. Linacsection L39 corrects a second-order correlation between E and z imprinted on the bunch by the injector and L1 Table II. Optimized parameters from the one-dimensionalmodel.
Parameter Value Unit
Accelerating voltage L1 219 .
46 MVPhase L1 17 .
81 degFrequency L1 650 MHzAccelerating voltage L39 9 .
57 MVPhase L39 205 .
72 degFrequency L39 3 . R for bunch compressor 1 (BC1) 0 . T for bunch compressor 1 (BC1) − . .
69 MVPhase L2 28 degFrequency L2 650 GHz R for bunch compressor 2 (BC2) 0 . T for bunch compressor 2 (BC2) 0 .
22 m − −
10 0 10 z − h z i (mm)0100200300 c u rr e n t( A ) (a) − −
10 0 10 z − h z i (mm)535455 E ( M e V ) (b) − . − . . . z − h z i (mm)0123 c u rr e n t( k A ) (c) − . − . . . z − h z i (mm)975100010251050 E ( M e V ) (d) Figure 4. Current (a,c) and LPS (b,d) distributions obtainedfrom the backward-tracking optimization (a,b) and trackedup BC2 end (c,d) to confirm the agreement with the targeteddistribution shown in Fig. 2. before it enters BC1. Linac section L2 operates even fur-ther off-crest to impart the necessary large chirp requiredfor maintaining beam stability in the CWA. Both bunchcompressors shorten the bunch lengths and impact theLPS distributions. The values of T selected in bothbunch compressors ensure achieving Φ f ( z f , E f ) despitethe large energy chirp. The use of a negative T inBC1 and a positive T in BC2 enables the generationof a doorstep-like initial distribution without giving riseto a current spike, where T has the effect of shiftingthe peak of current [38, 39]. In this paper, we adopt theconvention that T with a negative (resp. positive) signshifts the peak of current distribution to the tail (resp. head).The result of the backward-tracking optimization pro-vides only a starting point for obtaining a more realis-tic solution. For instance, the zigzag feature observedin the tail of the LPS distribution in Fig. 4(b) is chal-lenging to create. In the following sections, we discusshow 1D backward tracking results guide the design ofa photocathode-gun-based injector and the downstreamaccelerator lattice. IV. INJECTOR DESIGN
Given the required initial LPS distribution obtainedfrom the backward tracking, the next step is to explorewhether such LPS distribution is achievable downstreamof the injector; our approach relies on temporally shapingthe photocathode laser pulse [40].The injector beamline was modeled using the particle-in-cell beam-dynamics program astra , which includesa quasi-static space-charge algorithm [41]. The programwas combined with the deap multivariate optimizationframework to find a possible injector configuration andthe laser pulse shape that realize the desired final bunchdistribution while minimizing the transverse-emittancedownstream of the photoinjector.The injector configuration consists of a 200 MHzquarter-wave SRF gun [42–44], coupled to a 650 MHz ac-celerator module composed of five 5-cell SRF cavities [45].The gun includes a high-T c superconducting solenoid [46]for emittance control.In the absence of collective effects, the photoemittedelectron-bunch distribution mirrors the laser pulse dis-tribution. In practice, image-charge and space-chargeeffects are substantial during the emission process anddistort the electron bunch distribution. Consequently,devising laser-pulse distributions that compensate for theintroduced deformities is critical to the generation ofbunches with tailored current profiles. The laser pulsedistribution is characterized by I ( t, r ) = Λ( t ) R ( r ), whereΛ( t ) and R ( r ) describe the laser temporal profile and thetransverse envelope, respectively. In our simulation, weassumed the transverse distribution to be radially uni-form R ( r ) = H ( r c − r ), where H ( r c − r ) is Heaviside stepfunction and r c is the maximum radius. The temporalprofile is parameterized asΛ( t ) = Af ( t ) S ( a ( t − f )) S ( − b ( t − g )), where (7) f ( t ) = h, ≤ t < ch + d ( t − c ) d − , c ≤ t ≤ , elsewhere , where A is the normalization constant; and a , b c , d , f , g , and h are the parameters controlling the bunch shape.The smooth edges at both ends are characterized by a , b , f , g via the logistic function S ( u ) = 1 / (1 + e − u ); c deter-mines the length of the constant part of the laser pulseanalogous to the length of the bunch head of the doorstepdistribution; and h determines the relative amplitude ofthe constant laser pulse; see Fig. 5. The overall shape . . . . . . n o r m a li ze d m a c r o p a rt i c l e d i s tr i bu t i o n idealw/ Cs Te responsew/ Cs Te response & laser BW
Figure 5. Programmed macroparticle distributions at thephotocathode surface: for an optimized laser pulse (bluetrace), taking into account the photocathode response (orangetrace), and both the cathode response and finite bandwidth(BW) of the laser pulse (green trace). The laser bandwidthis taken to be δf = 2 THz. resembles a smoothed version of the door-step distribu-tion. The laser-shape parameters introduced in Eq. (7),the laser spot size, the phase and accelerating voltage ofall RF cavities, and the HTS solenoid peak magnetic fieldwere taken as control parameters for the optimization al-gorithm. The beam kinetic energy was constrained notto exceed 60 MeV. In order to quantify the final distribu-tion, we used the Wasserstein’s distance [47] to quantifyhow close the shape of the simulated macroparticle dis-tribution at the injector exit I ( o ) ( z ) was to the shape ofthe target macroparticle density distributions I ( t ) ( z ) ob-tained from backward tracking results. Specifically, theWasserstein’s distance is evaluated as D = (cid:88) i =1 N b || I ( t ) i − I ( o ) i || N b , (8)where I ( t,o ) i are the corresponding histograms of themacroparticles’ longitudinal positions over the interval i defined as [ z i + δz, z i − δz ], with δz ≡ max ( z ) − min ( z ) N b being the longitudinal bin size and N b the number ofbins used to compute the histogram. Additionally, weneed to have a small beam transverse emittance. Hence,the Wasserstein’s distance and the beam transverse emit-tance were used as our objective functions to be mini-mized.An example of the optimized injector settings is sum-marized in Table III, and the evolution of the associated Table III. Optimized parameters for the injector and beamparameters at s = 11 .
67 m from the photocathode surface.The RF-cavity phases are referenced with respect to themaximum-energy phases.
Parameter Value Unit
Laser spot radius 2 .
810 mmLaser duration 91 psRF gun peak electric field 40 MV/mRF gun phase 1 .
71 degCavity C1 peak electric field 13 .
25 MV/mCavity C1 phase 11.28 degCavity C2 phase -15.05 degCavities C2 to C5 peak electric field 20 MV/mCavities C3 to C4 phase 0 degCavity C5 phase 20 degCavity C1 distance from the photocathode 2.67 mSolenoid B-field 0.2068 TShape parameter a b c d f g h . .
06 mmFinal beam transverse emittance 8 . µ mFinal beam rms radius 1 .
64 mm beam parameters along the beamline are presented inFigs. 6 and 7. The final bunch distributions 11.5 m down-stream of the photocathode appears in Fig. 6. The beamtransverse phase space indicates some halo population.Ultimately, an alternative laser-shaping approach imple-menting a spatiotemporal-tailoring scheme could providebetter control over the transverse emittance while pro-ducing the required shaped electron beams [40]. We alsofind, as depicted in Fig. 6, that the current distributiontends to have a peak current lower than that desired fromthe backward tracking result shown in Fig. 4. Althoughhigher currents are possible, they come at the expenseof transverse emittance. Consequently, the distributiongenerated from the injector was considered as an inputto the one-dimensional forward tracking simulations. It-erations of one-dimensional forward tracking simulationstudies were done to further cross-check accelerator pa-rameters needed for the beam-shaping process. We espe-cially found that the desired final bunch shape at 1 GeVcan be recovered by altering the L39 phase and ampli-tude. Furthermore, the small slice rms energy spread σ E <
10 keV simulated from the injector [see Fig. 4(b)]renders the bunch prone to microbunching instability.Consequently, a laser heater is required to increase theuncorrelated energy spread.The correspondingly revised diagram of the acceleratorbeamline shown in Fig. 8 was used as a starting pointto investigate the performance of the proposed bunch-shaping process with elegant tracking simulations tak-ing into account the transverse beam dynamics.Another challenge associated with the bunch formationpertains to the temporal resolution of the bunch shapingprocess. Ultimately, the laser pulse shape can only becontrolled on a time scale δt ≥ / (2 πδf L ) limited by thebandwidth of the photocathode laser δf L . Contemporarylaser systems are capable to δt ≤
150 fs (RMS) [48]. Ad-ditionally, the electron bunch shape is also affected by thetime response of the photoemission process. Given therequired charge of ∼
10 nC, we consider a Cs Te photo-cathode with temporal response numerically investigatedin Ref. [49, 50]. Recent measurements confirm that Cs Tehas a photoemission response time below 370 fs [51]. Fig-ure 5 compares the optimized ideal laser pulse shape de-scribed by Eq. (7) with the cases when the photocathoderesponse time and the laser finite bandwidth are takeninto account. The added effects have an insignificant im-pact on the final distribution due to relatively slow tem-poral variations in the required peak current distribution.
V. FINAL ACCELERATOR DESIGN
The strawman accelerator design developed with thehelp of 1D simulations provides guidance for the finaldesign of the accelerator.
A. Accelerator components a. Linacs:
For the 650 MHz L1 and L2 SRFlinacs we adopted cryomodules proposed for the PIP-II − −
10 0 10 z − h z i (mm)0 . . . c u rr e n t( k A ) (a) −
10 0 10 z − h z i (mm)57 . . . E ( M e V ) (b) − . . . x (mm) − . . . x ( m r a d ) (c) − . . . y (mm) − . . . y ( m r a d ) (d) σ E ( k e V ) Figure 6. Current profile (a) with associated LPS (b), andhorizontal (c) and vertical (d) phase-space distributions sim-ulated with
Astra at the end of the photoinjector (11 .
67 mfrom the photocathode). In plot (b), the red trace representsthe slice RMS energy spread σ E . . . . B z ( T ) (a) e n e r g y ( M e V ) (b) ε n x , n y ( µ m ) (c) . . . . . s (m)0246 σ x , y ( µ m ) (d) E z ( M V / m ) ε n z ( µ m ) σ z ( µ m ) Figure 7. Axial electric E z (red trace) and magnetic B z (bluetrace) fields experienced by the reference particle as it propa-gates along the optimized photoinjector (a) with correspond-ing kinetic energy (b), transverse (blue) and longitudinal (red)beam emittances (c), and sizes (d) evolving along the injector. L1 L2 SRF gun
C1 to C5 L39Laser heater
BC1
BC2
Figure 8. Updated accelerator design, with the addition ofthe injector beamline and a laser heater section. project [52]. The linac L1 consists of two cryomodules,and L2 has eight cryomodules. Each cryomodule includessix cavities containing five cells. We assume that in CWoperation each cavity provides up to 20 MV / m averageaccelerating gradients. The quadrupole magnet doubletsare located between cryomodules and produce a pseudo-periodic oscillation of the betatron functions. The twocavities used in the 3 . b. Bunch compressors: We use an arc-shaped bunchcompressor consisting of a series of FODO cells, whereeach cell contains two quadrupoles and two dipole mag-nets. The latter configuration nominally provides a pos-itive R [53–56] R (cid:39) θ L total N sin ( ψ x / , (9)where θ total is the total bending angle, L total is the totalpath length, N cell is the total number of FODO cells, and ψ x is the horizontal phase advance per cell. The dipolemagnet bending angles can be used to tune the R . Thebending angle or dipole polarity from cell to cell does notneed to be identical, but the number of cells should beselected to realize a phase advance ψ x, total = 2 nπ (with n integer) over the compressor to achieve the first-orderachromat.The second-order longitudinal dispersion produced bythe bunch compressor is given by [57, 58] T = (cid:90) L (cid:20) η ,x ( s (cid:48) ) ρ ( s (cid:48) ) + η (cid:48) x ( s (cid:48) )2 (cid:21) d s (cid:48) , (10)where L is the length of the beamline, ρ is the bendingradius, η ,x ( s ) ≡ ( E / ∂ x ( s ) /∂E is the second-orderhorizontal dispersion function, and η (cid:48) x ( s ) is the derivativeof the dispersion function. We incorporate 12 sextupolemagnets to control the T and 12 octupole magnets tocancel the third-order longitudinal transfer-map element U computed over BC1. If needed, a non-vanishingvalue of U can enable higher-order control over theLPS correlation [38].The sextupole and octupole magnets are also used tozero the chromatic transfer-map elements T , T , and U , resulting in the bunch compressors being achro-matic up to the third order.Figure 9 displays the BC1 configuration along withthe evolution of the betatron functions and relevant hor-izontal chromatic ( η x , η ,x , and η ,x ) and longitudinalaccumulated transfer-map elements ( R → s , T → s , and U → s ) up to third order as a function of the beamlinecoordinate s . It has two arcs, one bending the beam tra-jectory by 22 . ◦ and another one bending it back. Eachbending magnet has the bending angle θ = 2 . ◦ . Thisdesign eases the requirement on the sextupole-magnetstrength required to provide a T <
0; see Table II.The strengths of the sextupole magnets were optimizedusing elegant to achieve the required T across BC1while obtaining a second-order achromat by constraining T = T = 0. The three pairs of sextupole magnetsin the second arc are mirror-symmetric to the first threepairs, with opposite-polarity magnet strengths. Duringthe design process, the first pair of sextupole magnetswas inserted close to the region of the first arc with thehighest dispersion for tuning the desired T ; its mir-ror symmetry pair was placed in the second arc andseparated by 2 π phase advance. Another two pairs ofsextupole magnets were subsequently inserted for tuning T . Similarly, their mirror symmetry pairs were sepa-rated by 2 π phase advance. Finally, six pairs of octupolemagnets were inserted to zero the overall U i i = 1 , , R and T to be positive, which is naturally provided bythe arc bunch compressor introduced earlier. It has a to-tal bending angle of 32 . ° , and each dipole has a bendingangle of 4 . ° . Similar to BC1, we used sextupole- and β x ( m ) (a) 246 β y ( m ) − . . . η x ( m ) (b) 0 . . R → s ( m ) − η , x ( m ) (c) − . . T → s ( m ) s (m) − η , x ( m ) (d) − . . . U → s ( m ) Figure 9. Layout of bunch compressor BC1 (top diagram)with evolution of associated betatron function (a) and perti-nent linear (b), second-order (c), and third-order (d) transfer-map elements along the beamline (with s = 0 correspondingto the beginning of BC1). In plots (b-d) the left and right axesrefer to the horizontal chromatic functions η i,x and accumu-lated longitudinal transfer-map elements from 0 to location s along BC1. In the top diagram the red, blue, green, and pur-ple rectangles correspond, respectively, to quadrupole, dipole,sextupole, and octupole magnets. octupole-magnet families to adjust both T and U and produce the third-order achromat. The BC2 latticeappears in Fig. 10 along with the evolution of the beta-tron functions and relevant chromatic elements. Finally,the layout of the two bunch compressors is presented inFig. 11. c. Matching sections: All accelerator componentsare connected using matching sections composed ofquadrupole magnets and drift spaces.The evolution of the betatron functions from the in-jector exit up to the end of BC2 appears in Fig. 12.Throughout the entire accelerator, the betatron functionsare maintained to values β x,y <
30 m.
B. Tracking and optimization
The beam distribution obtained at the exit of the in-jector was used as input to elegant for tracking andoptimization. We found that we need to increase theslice energy spread to ∼
75 keV using the laser heater β x ( m ) (a) 246 β y ( m ) . . η x ( m ) (b) 0 . . R → s ( m ) − . . . η , x ( m ) (c) 0 . . T → s ( m ) s (m)0 . . η , x ( m ) (d) − . . . U → s ( m ) Figure 10. Layout of bunch compressor BC2 (top diagram)with evolution of associated betatron function (a) and perti-nent linear (b), second-order (c), and third-order (d) transfer-map elements along the beamline (with s = 0 correspondingto the beginning of BC2). In plots (b-d) the left and right axesrefer to the horizontal chromatic functions η i,x and accumu-lated longitudinal transfer-map elements from 0 to location s along BC2. The top diagram follows the same conventions asin Fig. 9. (a) ∼ .
68 m ∼ .
03 m (b) ∼ .
99 m ∼ .
63 m
Figure 11. The geometry of the bunch compressors BC1 (a)and BC2 (b), where red, blue, green, and purple rectanglesare quadrupoles, dipoles, sextupoles, and octupoles magnets,respectively. to suppress the microbunching instability [59, 60]. How-ever, in this study, we numerically added random noisewith Gaussian distribution to the macroparticles’ energyusing the scatter element available in elegant . Thus,Fig. 13 shows the actual LPS distribution used at thebeginning of the accelerator in tracking studies.The accelerator settings obtained with twice wereused as a starting point in the accelerator optimizationincluding transverse effects. The fine-tuning of the above-described accelerator components was accomplished us-ing elegant . A multi-objective optimization was ap-plied to determine the twelve accelerator parameterscontrolling the longitudinal dynamics, i.e., voltages andphases of L1, L2, L39, and values of R , T in twobunch compressors. The resulting beam distribution ob-tained downstream of BC2 was then used to compute thewakefield generated in a 180 GHz corrugated waveguideconsidered for the role of the wakefield accelerator in [31].The resulting peak accelerating field and transformer ra-tio were then adopted as objective functions to be max-imized with the accelerator parameters as control vari-ables. The trade-off between peak accelerating field andtransformer ratio was quantified in Eq. (30) of Ref. [10],hence providing a good measure to verify whether ouroptimization reaches the optimal Pareto front. The op- Table IV. Main accelerator parameters and beam parametersat the end of BC2.
Parameter Value Unit
Accelerating voltage L1 193 .
22 MVPhase L1 21.64 degFrequency L1 650 MHzAccelerating voltage L39 9 .
73 MVPhase L39 202.52 degFrequency L39 3.9 GHz R for bunch compressor 1 (BC1) 0 . T for bunch compressor 1 (BC1) -0.1294 m U for bunch compressor 1 (BC1) 0 mAccelerating voltage L2 857 .
92 MVPhase L2 26.05 degFrequency L2 650 MHz R for bunch compressor 2 (BC2) 0 . T for bunch compressor 2 (BC2) 0 . U for bunch compressor 2 (BC1) 0 mFinal beam energy 998 MeVFinal beam bunch length 414 µ mFinal beam normalized emittance, ε nx µ mFinal beam normalized emittance, ε ny µ mPeak accelerating wakefield |E + | . / mPeak decelerating wakefield |E − | . / mTransformer ratio R . timal accelerator settings and final beam parameters aresummarized in Table IV. The LPS distribution at theend of the accelerator is shown in Fig. 14. We also calcu-lated that the ∼ .
26 MV / mwith a transformer ratio of 5 propagating in a corrugatedwaveguide. Figure 15 demonstrates that our optimiza- ( m ) β y β x − . . . η x ( m ) Figure 12. Evolution of the betatron (left axis) and horizontal dispersion (right axis) functions along the proposed linac. Thevertical dispersion is zero throughout the linac. The magnetic-lattice color coding for the element follows Fig. 9 with theaccelerating cavities shown as gold rectangles. −
20 0 z − h z i (mm)0 . . . c u rr e n t( k A ) (a) −
10 0 10 ζ (mm)575859 E ( M e V ) (b) σ E ( k e V ) Figure 13. Current profile (a) and associated LPS (b) dis-tributions simulated with
Astra at the end of the photoin-jector (see Fig. 6) with added uncorrelated fraction energyspread following a Gaussian distribution with RMS spread σ E /E = 1 . × − . In plot (b) the red trace represents theslice RMS energy spread σ E . tion has reached the optimal set of solutions, where thePareto front closely follows the analytically calculatedtradeoff curve [10]. The obtained current profile pro-duces a wakefield amplitude ∼
15% lower than the oneexpected from the ideal distribution for a transformer ra-tio
R (cid:39)
5. Such an agreement gives confidence in our op-timization approach based on the trade-off between peakaccelerating field and transformer ratio. The simulationsalso indicate that the horizontal transverse emittance in-creases to ε nx = 31 µ m due to the CSR and chromaticaberrations in the electron bunch having large correlatedenergy variations. Although significant, this emittancedilution is still acceptable.Our main result is shown in Fig. 16. It compares thefinal distribution and wakefield with that of the targetdistribution and wakefield from Fig. 2. A good agreementmanifests that, indeed, the drive electron bunch with ahighly asymmetric peak current profile can be obtainedwithout employing the collimators.A comparison of Tables III and IV indicates that the z − h z i (mm)024 c u rr e n t( k A ) (a) − z − h z i (mm)10001050 E ( M e V ) (b) − . . . x (mm) − . . . x ( m r a d ) (c) − . . . y (mm) − . . . y ( m r a d ) (d) σ E ( M e V ) Figure 14. Current (a) with associated LPS (b), and trans-verse horizontal (c) and vertical (d) phase-space distributionssimulated with elegant at the end of BC2 using the op-timized linac and bunch-compressor settings summarized inTable IV and the injector distributions from Fig. 13. In plot(b) the red trace represents the slice RMS energy spread σ E . final accelerator settings optimized by elegant deviateless than 10% from those obtained with twice . It jus-tifies the strategy taken in this study to solve the diffi-cult problem of formation of temporally shaped electronbunches for a beam-driven collinear wakefield acceleratorin two steps.The nonlinear correlation observed in the tail of theLPS distribution downstream of BC2 [see Fig. 14(b, bluetrace)] originates from the CSR. As the beam is com-pressed inside the bunch compressors, its tail experiencesa stronger CSR force due to its peak current being higher0than the rest of the bunch. It is worth noting that ele-gant uses a 1D projected model to treat the CSR effect.The applicability of such a 1D treatment is conditionedby the Derbenev’s criterion [61], which suggests thatprojecting the bunch distribution onto a line-charge dis-tribution may overestimate the CSR force, particularlywhen the bunch has a large transverse-to-longitudinalaspect ratio A ( s ) ≡ ( σ x ( s ) /σ z ( s )) (cid:112) ( σ x ( s ) /ρ ( s )). Inour design, the condition A (cid:28) A < z − x ) correlations dueto CSR effects; see Fig. 17. Although the associatedprojected-emittance dilution is tolerable, the electrons inthe longitudinal slices with the horizontal offsets seen inFig. 17(c) will excite transverse wakefields in the CWAand ultimately seed the BBU instability. These offsetscome from CSR-induced energy loss occurring in the BC2 transformer ratio p e a k fi e l d ( M V / m ) ← R R κ k | Q | pareto front Figure 15. Comparison of the Pareto front with the analyticaltrade-off curve between the peak field and transformer ratiodescribed by Eq. (30) of Ref. [10]. Each blue dot represents anumerically simulated configuration with the red star repre-senting the configuration with parameters listed in Table IV. -1.0 0.0 1.0 2.0 z − h z i (mm) − − w a k e fi e l d ( M V / m ) E + = 94 .
26 MV/m (red) E − = 18 .
81 MV/m (red)resulttarget − − c u rr e n t( k A ) -1.0 0.0 − − Figure 16. Target (from Fig. 2) and optimized final currentdistributions (respectively shown as green- and red-shadedcurves) with associated wakefields (respectively displayed asgreen and red traces). The transformer ratio for the simulateddistribution is R = 5 . that breaks the achromatic property of this beamline.Understanding the impact of this distribution feature inthe CWA linac along with finding mitigation techniquesis a current research focus. − z − h z i (mm) − x ( mm ) (a) − z − h z i (mm) − s li ce o ff s e t s ( mm ) (b) h x i h y i − z − h z i (mm) − y ( mm ) (c) − z (mm) s li ce r m ss i ze s ( mm ) (d) σ x σ y ε nx ε ny s li cee m i tt a n ce ( µ m ) Figure 17. Final ( z, x ) (a) and ( z, y ) (c) beam distributionscorresponding to the data shown in Fig. 14, and slice anal-ysis for positions (cid:104) x (cid:105) and (cid:104) y (cid:105) (b) and RMS beam size andemittances (d). C. Impact of errors
In order to validate the robustness of the proposed de-sign, it is instructive to investigate the sensitivity of theproposed shaping technique to shot-to-shot jitters of theamplitude and phase of the accelerating field in the linac’sstructures. Consistent with LCLS-II specifications [62],we considered the relative RMS amplitude jitter of 0.01%and the phase jitter of 0.01 degree. For simplicity, weassume that the injector produced identical bunches, asshown in Fig. 6, and performed 100 simulations of the ac-celerator beamline (from the injector exit to the exit ofBC2) for different random realizations of the phase andamplitude for linacs L1, L2, and L39. The errors in linacsettings were randomly generated using Gaussian prob-ability function with standard deviations of 0.01% and0.01 ◦ . Figure 18 presents the wakefield averaged over the100 simulations and indicates that a stable transformerratio 5 . ± .
05 can be maintained owing to the stablebeam produced in the superconducting linac.Likewise, we observe the impact of charge fluctuationon the shaping to be tolerable. Cathode-to-end simula-tions combining astra and elegant indicate that a rel-ative charge variation of +2% (resp. -2%) yields a relativechange in the transformer ratio of -2% (resp. +1%) and1 -1.0 0.0 1.0 2.0 z − h z i (mm) − − w a k e fi e l d ( M V / m ) E + = 94 .
26 MV/m E − = 18 .
84 MV/m wakefield avg.reconstructed current profile − c u rr e n t( k A ) -0.5 0.0 0.5 − − Figure 18. Wakefields obtained from 100 simulations withjitter in linacs L1, L2, and L39. All cavities are taken to haverelative jitter in accelerating voltage of 0.01% and phase jitter0 . ◦ . The red line shows the average wakefield while the blueshaded region represents the fluctuation of wakefields due tojitter over 100 random realizations of the linac settings. Theaverage transformer ratio is 5 . ± .
05. The reconstructedcurrent profile (green-shaded curve) is obtained numericallyusing Eq. (3). a relative variation in peak field of -1.7% (resp. +1.7%);see Fig. 19. − z − h z i (mm) − − w a k e fi e l d ( M V / m ) − c u rr e n t( k A ) Q = 10 nC Q = 10 . Q = 9 . − . . . − − Figure 19. Current distribution (shaded curves, left axis) andassociated wakefields (traces) for the nominal charge and ± . VI. SUMMARY
We have presented the design of an accelerator capa-ble of generating 1 GeV electron bunches with a highlyasymmetric peak current profile and a large energy chirprequired for a collinear wakefield accelerator. It has beenachieved without the use of collimators. Our approachis based on ab-initio temporal shaping of the photocath-ode laser pulse followed by nonlinear manipulations ofthe electron distribution in the longitudinal phase spacethroughout the accelerator using collective effects andprecision control of the longitudinal dispersion in twobunch compressors up to the third order. Finding theoptimal design consisted of first implementing a simpli- fied accelerator model and using it for backward trackingof the longitudinal phase space distribution of electronsthrough the main accelerator to provide the longitudinalphase space distribution required from the injector. Theprogram twice was developed to support such a capabil-ity and used to optimize the global linac parameters andtime-of-flight properties of bunch compressors. Second,the simulation of the photo-injector using astra was per-formed to generate the required distribution. Third, thelinac design was refined using elegant to account for thetransverse beam dynamics. Finally, formation of longi-tudinally shaped drive bunches capable of producing inthe collinear wakefield accelerator a transformer ratio of ∼ / mhas been numerically demonstrated.Although the proposed accelerator design is promis-ing, we note that further work is required to investi-gate whether the same accelerator can accelerate the low-charge, low-emittance “witness bunches” that would beaccelerated to multi-GeV energies in the collinear wake-field accelerator and used for the generation of x-rays inthe downstream free-electron laser. Discussion of thisresearch is the subject of a forthcoming publication. ACKNOWLEDGMENTS
The authors are grateful to Dr. Stanislav Baturin(NIU) for useful discussions. WHT thanks Y. Park(UCLA) for several discussions on simulation studies.This work is supported by the U.S. Department of En-ergy, Office of Science, under award No. DE-SC0018656with Northern Illinois University and contract No. DE-AC02-06CH11357 with Argonne National Laboratory.
Appendix A: One-dimensional tracking model
A simple one-dimensional tracking program twice [34]was developed for rapid assessment of the longitudinaldynamics of electrons in linear accelerators. The programadopts an approach similar to the one used in
LiTrack [35], where only the accelerator components affecting thelongitudinal beam dynamics are considered and modeledanalytically. A detailed description of twice is publishedin [34]. In brief, the beam is represented by a set of N macroparticles with identical charges Q/N and given aset of initial LPS coordinates ( z i , E i ). A transformation( z f , E f ) = f ( z i , E i ) is applied to obtain final coordinatesin the LPS. a. Single particle dynamics In twice the transformation for a macroparticle withcoordinates ( z i , E i ) passing through a radiofrequency2(RF) linac is given by (cid:18) z f E f (cid:19) = (cid:18) z i E i ( z i ) ± eV cos( kz i + ϕ ) (cid:19) , (A1)where V , k , and ϕ are, respectively, the acceleratingvoltage, wave-vector amplitude, and off-crest phase as-sociated with the accelerating section, and e is the elec-tronic charge. In the latter and following equations the ± sign indicates the forward (+) and backward (-) trackingprocess detailed in Sec. A 0 c. Similarly, the transforma-tion through a longitudinally dispersive section, such asa bunch compressor, is given by (cid:18) z f E f (cid:19) = z i ± (cid:20) R E i − E E + T (cid:16) E i − E E (cid:17) (cid:21) E i , (A2)where E is the reference-particle energy assumed to re-main constant during the transformation, and R ≡ E ∂z f ∂E i and T ≡ E ∂ z f ∂E i are the first- and second-order longitudinal-dispersion functions introduced by thebeamline. It should be noted that, given our LPS coor-dinate conventions, a conventional four-bend “chicane”magnetic bunch compressor has a longitudinal dispersion R >
0. The latter equation ignores energy loss, e.g.,due to incoherent synchrotron radiation, occurring in thebeamline magnets. b. Collective effects In twice , we implemented collective effects as an en-ergy kick approximation using the transformation (cid:18) z f E f (cid:19) = (cid:18) z i E i ( z i ) ± ∆ E ( z i ) (cid:19) , (A3)where ∆ E ( z ) represents the energy change associatedwith the considered collective effect. The treatment ofcollective effects is modeled as a z -dependent energykick ∆ E ( z ) taken downstream of beamline elements asspecified for the forward and backward tracking withthe diagram shown in Fig. 20. The implemented col-lective effects include wakefields modeled after a user-supplied Green’s function, one-dimensional steady-statecoherent synchrotron radiation (CSR), and longitudinalspace charge (LSC) described via an impedance. The col-lective effects require the estimation of the beam’s chargedensity, which is done in twice either using a standardhistogram binning method with noise filtering or via thekernel-density estimation technique [63].In order to model the impact of a wakefield, the chargedistribution q ( z ) is directly used to compute the wakepotential given a tabulated Green’s function W ( z ) = (cid:90) z q ( z (cid:48) ) G ( z − z (cid:48) )d z (cid:48) . (A4) Drift spaces
LSC
Forward tracking
Linac
Backward tracking wakefieldLSCR56
T566 CSR
R56
T566
CSR
Figure 20. Treatment of collective effects as energy kicksdownstream of beamline elements. In forward (resp. back-ward) tracking, transformations of beamline elements (resp.energy kicks) were applied, followed by energy kicks (resp.beamline elements).
The change in energy is computed as ∆ E ( z ) = LW ( z ),where L is the effective length where the beam experi-ences the wakefield.The LSC is implemented using a one-dimensionalmodel detailed in [64], where the impedance per unitlength is Z ( k ) = i Z πγr b − I ( ξ b ) K ( ξ b ) ξ b , (A5)where ξ b ≡ kr b /γ ; I and K are modified Bessel func-tions of the first and second kind, respectively; and k , Z and r b are, respectively, the wave-vector amplitude,impedance of free space and a user-defined transversebeam radius, and γ is the Lorentz factor. Given thecharge density, the Fourier-transformed current density˜ I ( k ) is derived from ˜ I ( k ) = F [ cq ( z )] , (A6)with F representing the Fourier transform. The changein energy is computed as∆ E = −F − [ eZ ( k ) ˜ I ( k ) L ] , (A7)where F − is the inverse Fourier transform, and L is theeffective distance along which the LSC interaction occurs.In order to account for LSC during acceleration, γ isreplaced by the geometry average √ γ i γ f of the Lorentzfactors computed at the entrance γ i and exit γ f of thelinac section.Finally, CSR energy kicks are applied downstream ofthe dispersive beamline elements. For instance, a CSRenergy kick can be applied after a dispersive element withuser-defined length and angle described by R and T .The effect of CSR is described using a one-dimensionalmodel commonly implemented in other beam-dynamicsprogram [65]. To simplify the calculation, only the3steady-state CSR is considered in twice . The energyloss associated with CSR is obtained from [66] ,∆ E ( z ) = ρθ d E d ct = − θ γm e c r e e (cid:90) z −∞ ∂q ( z (cid:48) ) ∂z (cid:48) I csr ( z, z (cid:48) )d z (cid:48) , (A8)with the integral kernel defined as I csr ( z, z (cid:48) ) = 4 u ( u + 8)( u + 4)( u + 12) , (A9)where θ is the angle, m e c is the electron rest mass en-ergy, r e is the classical electron radius and the variable u is the solution of γ ( z − z (cid:48) ) ρ = u + u . CSR introducesan energy loss strongly dependent on the bunch length,which varies within the dispersive sections used to com-press the bunch. Consequently, the longitudinally disper-sive beamlines are segmented into several elements withindividual ( R , T ) parameters. A CSR kick is ap- plied after each of the elements. A conventional chicane-type bunch compressor is usually broken into two sections(two mirror-symmetric doglegs) but can in principle bedivided into an arbitrary number of segments to improvethe resolution at the expense of computational time. c. Backward tracking An important feature of twice is its capability to trackthe beam in the forward or backward directions (indi-cated by the ± sign in Eqs. (A2) and (A3)) in the pres-ence of collective effects [so far LSC, CSR, and wake-field effects are included]. The effects of LSC and wake-field are straightforward to implement as they only in-volve a change in energy, while handling of the CSR re-quires extra care since the particles’ positions also changethroughout the dispersive section. Therefore, an en-ergy kick is applied after the beamline element in theforward-tracking mode and before the beamline elementin backward-tracking mode, as shown in Fig. 20. Al-though the treatment of CSR is not exact, it nonethelessprovides a good starting point to account for the effect. [1] G. Voss and T. Weiland, The wake field accelerationmechanism , Tech. Rep. DESY-82-074 (DESY, 1982).[2] R. J. Briggs, T. J. Fessenden, and V. K. Neil, Electronautoacceleration, in
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