An Adiabatic Phase-Matching Accelerator
Francois Lemery, Klaus Floettmann, Philippe Piot, Franz X. Kaertner, Ralph Assmann
AAn Adiabatic Phase-Matching Accelerator
F. Lemery , K. Floettmann , P. Piot , , F. X. K¨artner , , R. Aßmann DESY, Notkestrasse 85, 22607 Hamburg, Germany Department of Physics, University of Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany Department of Physics, and Northern Illinois Center for Accelerator & Detector Development,Northern Illinois University DeKalb, IL 60115, USA Fermi National Accelerator Laboratory, Batavia, IL 60510, USA (Dated: December 25, 2017)We present a general concept to accelerate non-relativistic charged particles. Our concept employsan adiabatically-tapered dielectric-lined waveguide which supports accelerating phase velocities forsynchronous acceleration. We propose an ansatz for the transient field equations, show it satis-fies Maxwell’s equations under an adiabatic approximation and find excellent agreement with afinite-difference time-domain computer simulation. The fields were implemented into the particle-tracking program astra and we present beam dynamics results for an accelerating field with a1-mm-wavelength and peak electric field of 100 MV/m. The numerical simulations indicate thata ∼ ∼
10 MeV over ∼
10 cm. Thenovel scheme is also found to form electron beams with parameters of interest to a wide range ofapplications including, e.g., future advanced accelerators, and ultra-fast electron diffraction.
PACS numbers: 29.27.-a, 41.85.-p, 41.75.Fr
High-energy charged-particle accelerators haveemerged as invaluable tools to conduct fundamanetalscientific research. Circular high energy colliders con-tinue exploring nuclear and high-energy landscapes,searching for hints beyond the standard model. Linearaccelerators capable of forming high-quality electronbunches have paved the way to bright, coherent X-raysources to probe ultrafast phenomena at the nanometer-scale with femtosecond resolutions in condensed matter,life science and chemistry. Accelerators have alsofound medical applications such as, e.g., high-resolutionimaging and oncology.Modern klystron-powered conventional accelerators in-corporate radio-frequency (RF) accelerating structuresoptimized to provide suitable accelerating fields typicallyin the frequency range f ∈ [0 . ,
10] GHz (i.e. wavelengthsrespectively in the range λ ∈ [3 , .
03] m). Unfortunately,power requirements and mechanical breakdowns in ac-celerating cavities have limited the permissible electricfields to E (cid:46)
50 MV/m, leading to km-scale infras-tructures for high-energy accelerators. These limitationshave motivated the development of advanced accelerationtechniques capable of supporting high accelerating fields.Accelerating structures based on dielectric waveguides orplasmas operating in a higher-frequency regime [ O (THz)]have been extensively explored in the relativistic regime.A key challenge in accelerating low-energy non-relativistic beams with higher frequencies stems from thedifference between the beam’s velocity and accelerating-mode’s phase velocity. This difference leads to “phaseslippage” between the beam and the accelerating fieldwhich ultimately limits the final beam energy and quality.Scaling to higher frequencies (i.e. shorter wavelengths)exacerbates the problem [1–4]. A figure of merit conven- tionally used to characterize the beam dynamics in thelongitudinal degree of freedom during acceleration of anon-relativistic beam is the normalized vector potential α = ( eE λ ) / (2 πmc ), where e and mc are respectivelythe electronic charge and rest mass, and E is the timeaveraged accelerating field. Conventional electron pho-toinjectors typically operate in a relativistic regime of α (cid:38)
1; retaining relativistic field strengths while scalingto smaller wavelengths (following E ∝ λ − ) is challeng-ing beyond RF frequencies but is now routinely attainedusing high-power infrared lasers in plasmas operating at f ∼ α acceleration in e.g.dielectric-laser accelerators (DLA) at optical wavelengthsis interesting nonetheless owing to the foreseen compactfootprints, relatively large gradients and high-repetitionrates. DLA has pursued side-coupled grating accelerators[5, 6] to address phase-slippage with tapered grating peri-ods. Likewise, proton accelerators have established radiofrequency quadrupoles (RFQs) [7] and drift-tube linacswith comparably long wavelengths to achieve low- α ac-celeration.In this Letter, we show analytically that alongitudinally-tapered dielectric-lined waveguide (DLW)can support electromagnetic fields with a longitudinally-dependent phase velocity. Therefore, by properlytailoring the spatial taper profile of the DLW, one canestablish an electromagnetic field with instantaneousphase velocity v p ( z ) matching the beam velocity β ( z ) c along the direction of motion ˆ z [8, 9]. The conceptis shown to be able to accelerate a non-relativisticelectron beam ( ∼
200 keV) generated out of a compactlow-power RF gun to relativistic energies ∼
10 MeVwithin a few centimeters. We hypothesize an ansatzfor the transient field equations supported in a tapered a r X i v : . [ phy s i c s . acc - ph ] D ec DLW, show they verify Maxwell’s equations and validatethem against a finite-difference time-domain (FDTD)electromagnetic simulation. We finally implement thetransient field equations in the beam dynamics program astra [10], present start-to-end simulations and vali-date the concept to form bright electron bunches suitedfor the production of attosecond X rays via inverseCompton scattering [11]. We especially find that asingle derived tapered waveguide can have a versatilerange of operation, yielding electron bunches with abroad set of properties of interest for various applications.A cylindrical-symmetric DLW consists of a hollow-coredielectric waveguide (with relative dielectric permittivity (cid:15) r ) with its outer surface metallized [12]. Introducingthe cylindrical-coordinate system ( r, φ, z ), where r is ref-erenced w.r.t. the DLW axis along the ˆ z direction, anddenoting the inner and outer radii as respectively a , and b , the electromagnetic field ( EEE, BBB ) associated to the ac-celerating (TM ) has the following non-vanishing com-ponents: E z = E I ( rk ) sin( ωt − k z z + ψ ) ,E r = E k z k I ( rk ) cos( ωt − k z z + ψ ) ,B φ = ω(cid:15) µ E k I ( rk ) cos( ωt − k z z + ψ ) , (1)where k ≡ ω (cid:113) v p − c , k ≡ ω (cid:113) (cid:15) r c − v p , k z = ωv p , I m ( ... ) are the modified m -th order Bessel’s function ofthe first kind, E is the peak axial field amplitude, v p isthe phase velocity, ω ≡ πf and ψ is a phase constant.In the limit v p → c , i.e. lim k → I ( k r ) /k = r/ k → I ( k r ) = 1, thus the transverse fields becomecompletely linear and the longitudinal field becomes inde-pendent of the transverse coordinate. Conversely smallervalues of v p result in increasingly nonlinear transversefields and a strong dependence of E z on the transversecoordinate. This is a general feature of phase velocitymatched modes and not restricted to the specific casediscussed here. For high frequency structures the effectof nonlinearities is however exacerbated because the ratioof typical transverse beam dimensions to the wavelengthis larger than in conventional rf structures.Solutions of the characteristic equation [12] yield theallowed ( ω, k z ) for propagating modes and depend onthe DLW structure parameters ( a, b, (cid:15) r ). A propagatingmode must have a real-valued longitudinal component forthe wavevector k z . Correspondingly, k sets a limit onthe phase velocity of a propagating mode via k < ωnc , or v p > cn (where n ≡ √ (cid:15) r is the dielectric’s index of refrac-tion). Finally, we note that the field amplitudes reducewith decreasing phase velocities ( v p → c/n ).We now turn to modify Eq. 1 to describe the fieldsassociated to a tapered DLW. Specifically, we hypothesizethat the transverse dimensions ( a, b ) at a longitudinal coordinate z locally determine v p and E ( z ). In addition,the phase at a position z should depend on the integratedphase velocity upstream of the structure. Given theseconjectures, we make the following ansatz for the non-vanishing ( EEE, BBB ) fields E z = E ( z ) I ( rk ( z )) sin( ωt − (cid:90) z dzk z ( z ) + ψ ) E r = E ( z ) k z ( z ) k ( z ) I ( rk ( z )) cos( ωt − (cid:90) z dzk z ( z ) + ψ ) B φ = ω(cid:15) µ E ( z ) k ( z ) I ( rk ( z )) cos( ωt − (cid:90) z dzk z ( z ) + ψ ) , (2)where now k z ( z ) is integrated from the structure entrance( z = 0) to the longitudinal coordinate z . The latter setof equations also introduce an explicit z dependence for E ( z ), k ( z ), k z ( z ). For convenience we define Ψ( t, z ) ≡ ωt − (cid:82) z dzk z ( z ) + ψ .In order to validate our ansatz, we check that it sat-isfies Maxwell’s equations, starting with the Amp`ere-Maxwell law ∂EEE∂t = −∇∇∇ × BBB which yields1 c ∂E z ∂t = − r ∂∂r ( rB φ ) . (3)Computing the rhs and lhs of the equation given the fieldcomponents listed in Eq. 2 and making use of the identity ∂∂r rI ( k r ) = k rI ( k r ) confirms that Eq. 3 is fulfilledas both sides equal ωc E I ( k r ) cos(Ψ( t, z )).Next, we consider Gauss’ law ∇∇∇ · EEE = 0 which yields,for the fields proposed in Eq. 2, ∂∂z E z = − r ∂∂r ( rE r ) . (4)The rhs of the latter equation gives ∂∂z E z = E [ − k z I cos(Ψ( t, z )) + rk (cid:48) I sin(Ψ( t, z ))]+ E (cid:48) I sin(Ψ( t, z )) (5)while its lhs results in − r ∂∂r ( rE r ) = − E k z cos(Ψ( t, z )) k r ∂∂r ( rI ( k r ))= − k z E I ( k r ) cos(Ψ( t, z )) . (6)Equations 5 and 6 are generally not equal unless the ta-pering of the DLW is sufficiently slow or adiabatic, (cid:12)(cid:12)(cid:12)(cid:12) rk (cid:48) I ( k r ) k z I ( k r ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , and (cid:12)(cid:12)(cid:12)(cid:12) E (cid:48) E k z (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) . (7)Equations 7 are independent of (cid:15) r and present generalconditions to the evolution of a traveling mode in a ta-pered waveguide which may also be of interest to plasmaacceleration with e.g. smaller field gradients and poten-tially higher repetition rates.Insofar, our discussion has solely concentrated on theelectromagnetic aspect of the problem. Let us now con-sider the beam dynamics of a charged particle accelerat-ing in a tapered DLW. The requirement for continuoussynchronous acceleration imposes v p ( z ) = β ( z ) c . Thelongitudinal phase space dynamics is described by thecoupled ordinary differential equations, ∂z∂t = βc,∂β∂t = eE ( z ) γ mc I ( k r ) sin(Ψ) . (8)In addition to synchronous acceleration, the transverse-dynamics plays a crucial role in the formation of brightelectron beams. The transverse force can be calculatedfrom the Lorentz force, F r = e ( E r − βcB φ )= eE (cid:18) β p − β (cid:19) k I ( k r ) k cos(Ψ) , (9)where β p ≡ v p /c is the normalized phase velocity. Forsynchronous acceleration ( β = β p ), the latter equationsimplifies to F r = eE k γ β I ( k r ) k cos(Ψ) , (10)implying transverse defocusing forces for Ψ ∈ [ − ,
0] deg(i.e. the compression phase where the bunch tail expe-riences a stronger longitudinal field than the head) andtransverse focusing forces for Ψ ∈ [ − , −
90] deg and notransverse force on-crest at Ψ = −
90 deg. We note thatthe amplitude of the force is increased by a combinationof the particle and the phase velocity (Eq. 9), while thenonlinearity of the field is a result of the matching to thephase velocity alone. For v p ∼ c , as e.g. in conventional rfguns, (Eq. 9) reduces to F r = eE k / ( γ (1 + β )) r/
2; thetransverse fields are linear in r and the longitudinal fieldis independent of the radial coordinate. In the matchedcase however, strong and nonlinear fields appear at lowenergies, which in combination with the r-dependence of E z , strongly affect the beam dynamics. The transverseemittance is an important figure of merit which charac-terizes the phase-space density of the beam defined as ε r (cid:39) ε u = mc [ (cid:104) u (cid:105)(cid:104) p u (cid:105) − (cid:104) up u (cid:105) ] / where (cid:104) ... (cid:105) is the sta-tistical averaging over the beam distribution. Thereforeat injection, the transverse beam size σ r should be mini-mized to mitigate emittance and energy spread dilutions.We now describe a start-to-end simulation consider-ing a driving field with λ = 1 mm ( f = 300 GHz) and E =100 MV/m corresponding to α (cid:39) .
03. A C++ pro-gram was developed to integrate the equations of motion(Eqs. 8) for one electron given the set of initial conditions:electron injection energy, wavelength, peak acceleratingfield, and DLW geomtery. In parallel, the characteristic
FIG. 1. Diagram of the accelerator concept (top) and corre-sponding evolution of the bunch’s transverse emittance ( ε r ),rms transverse beam size ( σ r ), longitudinal bunch length ( σ z )(all left axis) and the kinetic energy (right axis) along the ac-celerator beamline (bottom). The example corresponds to anoperating point ( φ , E )=(79.3 deg, 106.875 MV/m); see textfor details. equation is solved to derive the appropriate taper. Conse-quently, an electron injected on crest will not experienceany phase-slippage through the structure. Additionally,scaling the accelerating field E → ηE offsets the pointof zero phase-slippage by an amount δ Ψ = arcsin(1 /η ).We specialize our study to the case where the DLW has aconstant inner radius a and devise the outer radius b ( z )to ensure synchronous acceleration throughout the DLW.The DLW is taken to be made of quartz ( (cid:15) r = 4 .
41) withlength L = 11 . a = 0 . v p > c/n ∼ . c thereby requiring an injection energy >
70 keV. Thegroup velocity of ∼ ∼
383 ps corresponding toa pulse energy of 7.2 mJ for E =100 MV/m.We consider a compact, low-power, field-enhanced S-band ( f = 3 GHz) gun [13] with a photocathode as ourelectron source. The gun is operated off-crest to gen-erate short σ z ∼ µ m, 205 keV electron bunches atthe DLW entrance with a total charge Q = 100 fC. Anelectromagnetic solenoid with variable peak axial mag-netic field B is located 5 cm downstream of the cathodeand focuses the bunch into the DLW structure positioned10 cm from the cathode. To control the strong defocusingforces (Eq. 9) during the early stages of acceleration, a 4-slab, ∼ B =1.5 T surrounds part ofthe DLW and is located at a distance z s from the cathode;see Fig. 1(top). This setup was not globally optimized.At the entrance of the structure the matched phase veloc-ity is v p = 0 . c and the accelerating gradient is reducedto E ∼
20 MV/m; the adiabatic condition from Eq. 7for r =100 µ m at z =0 gives 0.0017 and approaches 10 − (a) (b) (c)(a)(a) FIG. 2. (a) Geometry of the dielectric-layer tapering (green shaded area, right axis) over the entrance of the structure; theinitial dielectric thickness is 143 µ m ( v p = 0 . c ) and asymptotically approaches 91 µ m ( v p = c ). In addition we show thecomparison between our analytic field ansatz with FDTD code CST-MWS over the first 20 mm. (b) Final energy (solid traces,left column) and end phase (dashed lines, right column) as a function of injection phase for various accelerating gradientsand initial kinetic energies. The black diagonal dashed line shows φ e = φ i , intersections with the phase portraits indicate zerophase-slippage. (c) The compression ratio between the injection phase and end phase, ∆ φ i / ∆ φ e in log-scale as a function ofinjection energy and accelerating gradient for an input bunch length spanning 60 deg (∆ φ i =60 deg.) toward the end of the structure. For completeness we usethe FDTD program CST MWS [15] to simulate the fieldpropagation for the first 2 cm of the DLW where the ma-jority of the taper occurs; here we simulated a GaussianTHz pulse with 1% bandwidth, while our simulation in astra utilizes a flat-top pulse. The simulated fields arein excellent agreement with our semi-analytical field; seeFig. 2(a). Some discrepancies arise at the entrance andexit of the structure due to transient effects not includedin our model.We can gain some significant insight into the longitu-dinal dynamics with a single electron; we illustrate theenergy gain and end phase as a function of initial phasein Fig. 2(b). Generally, larger accelerating gradients andinjection energies than the matched conditions increasethe longitudinal acceptance of the structure. Plateaus inthe end phases suggests bunch compression across the flatinjection phase width. In Fig. 2(c) we show the resultingcompression ratio in log-scale, ∆ φ i ∆ φ e , as a function of ini-tial energy E i and field strength E z for an input bunchspanning 60 degrees. The phase trajectory through thestructure is determined by E i , E z and φ i ; changing theseparameters will alter the phase trajectory and impact theforces experienced by the bunch along the structure; thisimplies that a single structure has a very broad range ofoperational capabilities.The transverse matching into and through the struc-ture essentially depends on the balance between thetransverse defocusing forces from the DLW and focusingoptics from the solenoids. Different phase trajectorieswill generally have different transverse forces along thestructure. We illustrate the transmission through thestructure as a function of our matching optics B and z s in Fig. 3 associated to a local compression maximum( E i , E z )=(205 keV, 105.8 MV/m). To accommodate suchan injection energy, we accordingly minimize the bunchlength by choosing an appropriate field strength and in-jection phase in the gun, E gun =113.55 MV/m, φ gun =217 deg. Larger acceptances allow less stringent requirementson the beam matching and allows a larger operationalrange for the same matching point.Finally, we explore the beam dynamics of the struc-ture for a matching point in Fig. 3, ( B , z s )=(0.179 T,10.5 cm) and show the resulting final energy, energyspread, inverse bunch length, and transverse emittancefor scans over ( φ , E ) in Fig. 4. In each figure we includewhite contour lines representing the final bunch charge;in cases with large offsets to the originally matched con-ditions, e.g. large gradients, the larger defocusing forcesleads to internal collimation, which in some instancesleads to e.g. reduction in emittance. One should of courseinvestigate and optimize a structure based on the de-sired final bunch characteristics and injection constraints;however the large operational range of a single structureimplies broad and stable operation for a single match-ing point. Some notable operating points include, ( σ z , FIG. 3. Charge fractional transmission through the struc-ture as a function B and z s for injection parameters ( E i , E z )=(205 keV, 105.8 MV/m) corresponding to a maximumbunch compression point from Fig. 2(c). A black dashed lineencompasses 100% transmission. (cid:15) r , σ E , Q )=(1.2 µ m, 250 nm, 47.6 keV, 100 fC) for ( φ , E )=(79.3 deg, 106.875 MV/m). The shortest bunchlength acheived in our scans for the associated structurewas ( σ z , (cid:15) r , σ E , Q )=(730 nm, 158 nm, 83 keV, 80 fC) forthe operational point ( φ , E )=(48.8 deg, 123.75 MV/m);smaller energy spreads can be reached also, at the ex-pense of other final parameters. − − − φ i ( d e g r ee s ) . . . . . . . . . . . . . .
100 120 140 E (MV/m) − − − φ i ( d e g r ee s ) . . . . . . .
100 120 140 E (MV/m) . . . . . . . e n e r g y ( M e V ) . . . . . . . . i n v e r s e bun c h l e n g t h ( µ m ) e n e r g y s p r e a d ( k e V ) . . . . . . (cid:15) ⊥ ( π mmm r a d ) FIG. 4. Final bunch energy, inverse bunch length, energyspread, and normalized transverse emittance for the matchedcase, ( B , z s )=(0.179 T, 10.5 cm). In each case we overlaythe final bunch charge as white contour levels for 0.3, 0.6and 0.9 charge transmission. While all final energies are ap-proximately equal ( ∼
11 MeV), the structure allows for theproduction of widely-tunable electron beams.
In summary, we have proposed adiabatically-tapereddielectric-lined waveguides to accelerate and manipulatelow-energy charged particles with non-relativistic fieldstrengths i.e. in the low- α regime. We hypothesized anansatz for the transient field equations and support themwith Maxwell’s equations and computer simulation. Weimplemented the fields directly into astra and performbeam dynamics simulations for a low-energy electronbunch accelerating in a 1 mm field with 100 MV/m. Thederived structure supports non-phase-slipping trajecto-ries for various input powers; offsets in the initial param-eters leads to very similar final energies but with a widevariety of other bunch properties, notably small bunchlengths and energy spreads. We presented a very sim-ple beam matching scheme to accommodate the strong-defocusing forces in the early stages of acceleration. Ourderivation of Eq. 7 is independent of (cid:15) r and is thereforea more general relation to the evolution of a travelingmode and may appeal to e.g. plasma acceleration withsmaller field gradients and potentially higher repetitionrates via tapered density profiles.The proposed setup could be further optimized givena specific application. The beam matching could be op-timized via the positions and strengths between the ele-ments to further minimize emittance growth for a givencharge. Likewise, larger charges could be acceleratedwith higher injection energies, longer accelerating wave-lengths, and larger accelerating field strengths. Accel- eration with lower initial energies can be acheived withlarger dielectric permittivities, and could possibly realizea standalone relativistic electron source. A performanceanalysis given fabrication imperfections is also requiredand will especially be critical to the scaling of the conceptto optical wavelengths, e.g. as needed for DLAs. Finally,exploring alternative tapering profiles could open otherapplications of the proposed tapered accelerator includ-ing, e.g., the development of novel beam-manipulationtechniques.FL is thankful to DESY’s accelerator division forsupport to continue this ongoing work. This projectis funded by the European Union’s Horizon 2020 Re-search and Innovation programme under Grant Agree-ment No. 730871 and also supported by the EuropeanResearch Council (ERC) under the European Union’sSeventh Framework Programme (FP/2007-2013)/ERCAXSIS Grant agreement No. 609920. PP is sponsored byUS NSF grant PHY-1535401. We acknowledge the use ofDESY’s high-performance computing center Maxwell and are grateful to F. Schluenzen for support. We thankM. Fakhari for providing the S-band gun design. [1] K.-J. Kim, Nucl. Instr. Meth. Phys. Res. A , 201(1989).[2] J. Rosenzweig and E. Colby, AIP Conference Proceedings , 724 (1995).[3] P. Piot, L. Carr, W. S. Graves, and H. Loos, Phys. Rev.ST Accel. Beams , 033503 (2003).[4] K. Floettmann, Phys. Rev. ST Accel. Beams , 064801(2015).[5] T. Plettner, P. Lu, and R. L. Byer, Phys. Rev. ST Accel.Beams , 111301 (2006).[6] J. Breuer and P. Hommelhoff, Phys. Rev. Lett. ,134803 (2013).[7] I. M. Kapchinskii and V. A. Teplvakov, Prib. Tekh. Eksp. (1970).[8] F. Lemery and P. Piot, in Proc. 6th International Parti-cle Accelerator Conference (IPAC’15), Richmond, VA,USA, 2015 (JACoW, Geneva, Switzerland, 2015) pp.2664–2666.[9] F. Lemery, K. Floettmann, P. Piot, and F. X. Kaertner,in