Low Energy Spread Attosecond Bunching and Coherent Electron Acceleration in Dielectric Nanostructures
Uwe Niedermayer, Dylan S. Black, Kenneth J. Leedle, Yu Miao, Robert L. Byer, Olav Solgaard
LLow Energy Spread Attosecond Bunching and Coherent Electron Acceleration inDielectric Nanostructures
Uwe Niedermayer, ∗ Dylan S. Black, † Kenneth J. Leedle, Yu Miao, Robert L. Byer, and Olav Solgaard Technische Universit¨at Darmstadt, Institut f¨ur Teilchenbeschleunigung und Elektromagnetische Felder (TEMF),Schlossgartenstrasse 8, D-64289 Darmstadt, Germany Department of Electrical Engineering, Stanford University,350 Serra Mall, Stanford, California 94305-9505, USA Department of Applied Physics, Stanford University,348 Via Pueblo Mall, Stanford, California 94305-4090, USA (Dated: August 6, 2020)We demonstrate a compact technique to compress electron pulses to attosecond length, whilekeeping the energy spread reasonably small. The technique is based on Dielectric Laser Acceler-ation (DLA) in nanophotonic silicon structures. Unlike previous ballistic optical microbunchingdemonstrations, we use a modulator-demodulator scheme to compress phase space in the time andenergy coordinates. With a second stage, we show that these pulses can be coherently accelerated,producing a net energy gain of 1 . ± . . +0 . − . keV FWHM. We show that by linearly sweeping the phase between thetwo stages, the energy spectrum can be coherently moved in a periodic manner, while keeping theenergy spread roughly constant. After leaving the buncher, the electron pulse is also transverselyfocused, and can be matched into a following accelerator lattice. Thus, this setup is the prototypeinjector into a scalable DLA based on Alternating Phase Focusing (APF). Dielectric Laser Acceleration (DLA) provides the high-est gradients among structure based particle accelera-tors by utilizing the GV/m femtosecond laser damagethresholds of nanostructured dielectric materials. Af-ter the first proposals [1, 2], it took 50 years for thefirst experimental DLA demonstrations to be realized[3, 4]. Recently, the gradients have been further pushedto 690 MeV/m [5] and 850 MeV/m [6] for relativisticelectrons and to 133 MeV/m [7] and 370 MeV/m [8] fortheir low-energy counterparts. While the relativistic ex-periments use an RF photoinjector and sometimes mul-tiple RF-preaccelerator stages, the subrelativistic experi-ments require an ultra-low emittance nanotip-emitter [9–13] and an electrostatic preaccelerator to obtain suitableelectron beams for injection into DLAs. Especially atthese low injection energies, a beam confinement andbunching scheme [14–16] is required to scale DLAs upto MeV energy gain for various applications [17–19].With unbunched electron beam injection, the DLA en-ergy spectra show the typical symmetric shoulder modu-lation [3, 5, 7, 20], which can be analytically modeled us-ing probability theory [21]. Combined with on-chip bal-listic bunching, net-acceleration and steering in a down-stream DLA stage was recently demonstrated [22, 23].However, the large energy spread created in the buncherstage quickly dissolves the microbunch phase coherence.In this Letter, we demonstrate an optical modulator-demodulator that compresses first the bunch length andthen the energy spread. The resulting simultaneouslyultra-short and low energy spread electron pulses can becaptured in a potential well (”bucket”) and acceleratedin a lossless and scalable fashion.At low injection energies from typical nanotip elec- tron sources [9, 10, 13], the Alternating Phase Focusing(APF) scheme [15] is well suited to confine and acceleratethe beam, since it is flexible in focusing cell design andeconomic in field strength to acceleration gradient con-version, which constitutes the performance bottleneck ofa scalable accelerator [16]. However, as compared to con-stant longitudinal focusing [24], which does not provideany transverse confinement, the temporal acceptance ofAPF is slightly smaller. The structure we present hereis an APF-based buncher, which is suitable to inject intoan APF-based accelerator, due to matching of both thesub-fs bunch length and the low energy spread.Attosecond pulses of subrelativistic electron beams canbe created by ballistic bunching [19, 22–24], i.e., a sinu-soidal energy modulation ∆ W is turned to bunching afterreaching the longitudinal focal length of L = λ g β γ m e c π ∆ W , (1)where λ g is the period length of the modulation, β isthe injection velocity in units of c , γ is the mass factor,and m e c is the electron rest energy. Note that if themodulation is produced by a laser of wavelength λ in aDLA grating, the Wideroe resonance condition λ g = βλ has to be fulfilled, where λ g is the grating period.The motion towards the focus is linear only in thevicinity of the fixed point(s) and strongly nonlinear else-where. Thus, the longer the dispersive drift after themodulation, the more irreversible longitudinal emittancegrowth is produced. Moreover, since the modulationis essentially longitudinal focusing, according to Earn-shaw’s theorem [25] transverse defocusing comes alongwith it. The idea of the setup presented here is to in-troduce a grating segment which removes these draw- a r X i v : . [ phy s i c s . acc - ph ] A ug FIG. 1. Scanning electron microscope image of the structure.The inset shows an enlarged top-view of the buncher stagewith the designed distribution of the synchronous phase. backs by removing the modulation after a certain lengthof drift, which is shorter than the focal length. The fi-nal longitudinal focus is then reached with a significantlysmaller energy spread and transversely focused.The phase acceptance of a scalable APF-DLA dependson the choice of the synchronous phase of the accelerator,i.e., in an overall design one has to trade off temporalacceptance and resulting average gradient. At ± ◦ off-crest, where the gradient is half the peak gradient, thefull temporal acceptance in a longitudinal focus is about5% of an optical cycle [15], which is δt = 330 as for the1980 nm driver laser we use here. The matched energyspread in an APF lattice is (see [15] supplemental Eq. 18) δW = m e c β γ c ˆ β L δt, (2)where ˆ β L is the longitudinal Courant-Snyder beta-function at the beginning of the accelerator. The minimaof ˆ β L can reach down to 10 − µ m, depending on thelaser amplitude on which the APF accelerator is driven.At a reference energy of 57 keV, the resulting full energyspread acceptance is 286-572 eV and can be filled by aninjector as presented here.Numerical simulations in DLAtrack6D [26] are per-formed to describe the nonlinear dynamics of the exper-iment. The design principles can however be understoodfrom simple analytic considerations. In each DLA cell,the energy gain is given by∆ W = qe λ g cosh (cid:18) ωyβγc (cid:19) sin( ϕ P − ϕ S ) (3)where ω = 2 πc/λ , q = − e is the electron charge, ϕ P = ω ∆ t is the particle phase and ϕ S is the synchronousphase. The laser field amplitude is characterized by thesynchronous mode coefficient e , which we design to be50 MV/m. Note that qe is the on-crest gradient in thecenter of the channel. During n DLA cells, the phase ofan off-energy particle will slip as [26]∆ ϕ P = 2 πnβγ m e c ∆ W. (4) FIG. 2. Longitudinal phase space after each part of the struc-ture a)-i) as indicated in Fig. 1. Panel j) shows a slightlylonger accelerator stage. The insets show the projected en-ergy spectra in arb. linear units scaled to their maximum.
By means of a fractional period drift of l d , the syn-chronous phase can be changed as∆ ϕ S = 2 π l d λ g , (5)while the change in the particle phase is negligible dueto the shortness of this drift. We use l d = λ g / π -shift of ϕ S ; see the inset of Fig. 1for the design of the synchronous phase and Fig. 2 forthe longitudinal phase space evolution as computed byDLAtrack6D. The bunch profiles and energy spectra inidentical arbitrary units are plotted in Fig. 3. The com-bination of modulation and demodulation enables thedevice design as shown in Fig. 1, which modulates thebeam first, then transports it from (b) to (f), where al-ternating modulation/demodulation is employed for thepurpose of transverse confinement. Then the segment(f)-(g) demodulates the beam, such that only a smallresidual energy spread remains. This remaining smallenergy spread finally compresses the already prebunchedbeam to a minimum bunch length roughly at the begin-ning of the second stage (h). Simulation predicts thatthe bunch at point (h) is on the order of 230 as in length,with an energy spread of about 237 eV, i.e., this bunchwould be suitable to inject into a scalable APF accel-erator (cf. Eq. 2). We only implement one segment of28 µ m length as the second stage here, which extends upto point (i). A slightly longer second stage would reduce FIG. 3. Simulated bunch lengths (top panel) and energyspreads (bottom panel) along the structure. The arrivaltime difference ∆ t = ϕ P /ω is plotted for one laser period T = λ/c = 6 . the output energy spread to 165 eV at point (j), but thiswas not implemented in the experiment.The transverse focus of the bunched electrons isroughly in the center of stage 2. The 35 µ m drift be-tween the stages is chosen as sufficiently large to spatiallyseparate the laser beams (1) and (2) to avoid cross-talk.The phase difference between stage (1) and (2) can bechosen arbitrarily, thus the synchronous phase of stage(2) is arbitrary but held constant during measurement.The stage 2 synchronous phase in Fig. 2 was chosen suchthat the electron bunch drifts over the crest, producinga net energy gain of about 1.3 keV. Without this phaseslippage, the energy gain would be | qe | L = 1 . −
10 Ω-cm B:Si. The elliptical pillars have dimensionsof roughly r z = 690 nm, r y = 830 nm, and height h = 2 . µ m, and the channel gap between the pillarsis 300 nm. The electron macro-bunch is produced by il-lumination of a silicon nanotip cathode [12] with a laserof 1 µ m wavelength and roughly 300 fs pulse length. Therepetition rate is 100 kHz, which produces electron pulsesat the same repetition rate and 730 ±
100 fs in length,measured by cross-correlation with the DLA drive laser.The electron beam is focused to a circular gaussian spotof width 230 ±
30 nm RMS at the beginning of the struc-ture (a).The outcoming electron beam is analyzed by a sectormagnet spectrometer with a roughly 100 eV point spreadfunction. The injection reference energy can be set in arange between 56 keV and 60 keV, with an energy spreadless than 10 eV. Since the electron beam is operated with ∼
300 electrons/sec, i.e., less than one electron per laserpulse, space charge is negligible. The energy spread isprimarily limited by the power supply ripple of roughly1 V at 60 kV.The DLA structures are pumped with a commercialOPA system, driven by the same 1 µ m regenerative am-plifier that drives the cathode (also at 100 kHz). Thepulse length is 605 ± λ = 1980 nm. The four DLA drive beams (seeFig. 1) are focused to 1 /e intensity radius of 22 ± µ m.Each branch provides pulses ranging from 0 to ∼
50 nJ,depending on the desired acceleration gradient. Thephase of each branch is differentially controlled by (free-running, not locked) piezo delay stages. The phasestability is maintained to better than λ/
10 over shorttimescales ( < FIG. 4. Top panel: Data as recorded on the MCP screen fordifferent stage 2 synchronous phases. The horizontal deflec-tion axis is not significantly influenced by the spectrometermagnet. Bottom panel: Comparison of the respective spectrato simulation. We determine 1 . ± . . +0 . − . keV FWHM energy spread at maximum gain. accurate measurement of e in stage 1 is not possible, wemeasure e in stage 2 and assume the same power level atstage 1. Together with the phase error in stage 1 (laser1a vs. 1b) this is the main source of driver errors.Two example experimental spectra are shown in Fig. 4,where the electron spectrum is coherently accelerated ordecelerated depending on the injection phase into thesecond DLA stage. The increase of the transverse spotsize is small, since the captured electrons spend roughlythe same time on focusing and defocusing synchronousphases. The energy gain is 1 . ± . . +0 . − . keV FWHM at maximum accel-eration and 0 . +0 . − . keV FWHM at maximum decelera-tion.The main source of measurement error is the spec-trometer point spread function, which shows a broaderspectrum on the screen than reality. This is also visi-ble in Fig. 4 in the laser-off curve exhibiting a width ofabout 0.2 keV FWHM. The energy spread is still largerthan predicted by the simulations after accounting forthe spectrometer point spread function, however. Thisexcess energy spread is caused primarily by 3D geom-etry effects such as the finite height of the pillars andthe mesa structure, leading to a possibly strong e ( x )dependency, see [16]. Consequently, there can be under-bunching, over-bunching, and vertical deflection in thesame measurement. Additionally, there is non-uniformityin e due to minor cross-talk between the drive lasers andintra-stage phase errors. FIG. 5. The sinusoidal spectrogram shows excellent agree-ment with simulation, albeit with higher energy spread, whichcan be accounted to a vertical spread of e ( x ) and the spec-trometer point spread function. Bottom panel: Mean andstandard deviation of the measured spectrogram. A full synchronous phase sweep measurement is shownin Fig. 5 with e = 53 ± e and dueto the lower energy gain, there is significant backgroundfrom non-trapped electrons. The centroid and standarddeviation of the measurement data shown in the bottompanel indicate that the spread stays roughly constant at800 ±
200 eV, with slight increase at maximum deceler-ation, where some of the electrons were lost. Note thatmean and std. dev. were taken over each entire spectrumincluding the decelerated tail. This causes the mean to beonly about 0.6 keV above injection energy as comparedto the 1.3 keV of the edge of the spectrum.Together with the streaking experiment presentedin [22], we are now able to coherently move the elec-tron beam in both dimensions (energy and deflection) onthe microchannel plate (MCP) screen by changing therelative phase of the two stages. The observable smallangle spread indicates that the beam leaves the structurewithout a large increase in divergence. For streaking itis advantageous to have a shorter second stage, whichreduces phase slip errors. If only the energy spectrumis measured, a longer second stage (cf. Fig. 2 (j)) leadsto higher energy gain and thus better relative resolution.Moreover, after slipping over the crest, the energy spreadis additionally compressed at the expense of a slightlylonger bunch length. The bunch length can be inferredfrom the measured energy spectra via comparison to sim-ulations [23], which we do not attempt here due to thelarge uncertainty for that measurement with our experi-mental parameters.In conclusion, we have demonstrated small energyspread bunching in a DLA based on a modulation-demodulation APF scheme. The resulting sub-femtosecond electron pulses were injected into a sec-ond stage for coherent acceleration. Good agreementwith simulation results was achieved, with the increasedexperimental energy spread accounted for by three-dimensional field non-uniformity. This non-uniformitycan be quite strong but can also be exploited to con-fine the beam vertically in future experiments, enablinga scalable APF DLA accelerator [16]. Preliminary full 3Dparticle tracking simulations indicate that APF buncherstructures with vertical confinement would achieve sim-ilar energy spreads as predicted in the 2D simulationspresented here. Due to the more involved longitudinalphase space manipulations as compared to a simple bal-listic buncher, we call this device ”fancy buncher”.This work is funded by the Gordon and Betty MooreFoundation (Grant GBMF4744). ∗ [email protected] † Uwe Niedermayer and Dylan S. Black equaly contributedto this work.[1] K. Shimoda, Applied Optics , 33 (1962).[2] A. Lohmann, IBM Technical Note , 169 (1962).[3] E. A. Peralta, K. Soong, R. J. England, E. R. Colby,Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. J.Leedle, D. Walz, E. B. Sozer, B. Cowan, B. Schwartz,G. Travish, and R. L. Byer, Nature , 91 (2013).[4] J. Breuer and P. Hommelhoff, Nuclear Instruments andMethods in Physics Research, Section A: Accelerators,Spectrometers, Detectors and Associated Equipment , 114 (2013).[5] K. P. Wootton, Z. Wu, B. M. Cowan, A. Hanuka, I. V.Makasyuk, E. A. Peralta, K. Soong, R. L. Byer, andR. J. England, Opt. Lett. , 2696 (2016).[6] D. Cesar, S. Custodio, J. Maxson, P. Musumeci, X. Shen, E. Threlkeld, R. J. England, A. Hanuka, I. V. Makasyuk,E. A. Peralta, K. P. Wootton, and Z. Wu, Communica-tions Physics , 1 (2018).[7] P. Yousefi, N. Sch¨onenberger, J. Mcneur, M. Koz´ak,U. Niedermayer, and P. Hommelhoff, Optics Letters ,1520 (2019).[8] K. J. Leedle, R. Fabian Pease, R. L. Byer, and J. S.Harris, Optica , 158 (2015).[9] D. Ehberger, J. Hammer, M. Eisele, M. Kr¨uger, J. Noe,A. H¨ogele, and P. Hommelhoff, Physical Review Letters , 227601 (2015).[10] A. Feist, N. Bach, N. Rubiano da Silva, T. Danz,M. M¨oller, K. E. Priebe, T. Domr¨ose, J. G. Gatzmann,S. Rost, J. Schauss, S. Strauch, R. Bormann, M. Sivis,S. Sch¨afer, and C. Ropers, Ultramicroscopy , 63(2017).[11] A. Tafel, S. Meier, J. Ristein, and P. Hommelhoff, Phys-ical Review Letters , 146802 (2019).[12] Andrew Ceballos, Silicon Photocathodes for DielectricLaser Accelerators , Ph.D. thesis, Stanford University(2019).[13] T. Hirano, K. E. Urbanek, A. C. Ceballos, D. S. Black,Y. Miao, R. Joel England, R. L. Byer, and K. J. Leedle,Applied Physics Letters , 161106 (2020).[14] B. Naranjo, A. Valloni, S. Putterman, and J. B. Rosen-zweig, Physical Review Letters , 164803 (2012).[15] U. Niedermayer, T. Egenolf, O. Boine-Frankenheim, andP. Hommelhoff, Physical Review Letters , 214801(2018).[16] U. Niedermayer, T. Egenolf, and O. Boine-Frankenheim,submitted to PRL (arXiv:2004.05458) (2020).[17] A. H. Zewail and J. M. Thomas,
4D Electron Microscopy (Imperial College Press, 2010).[18] R. F. Egerton, Advanced Structural and Chemical Imag-ing (2015), 10.1186/s40679-014-0001-3.[19] Y. Morimoto and P. Baum, Nature Physics , 252(2018).[20] N. V. Sapra, K. Y. Yang, D. Vercruysse, K. J. Leedle,D. S. Black, R. J. England, L. Su, R. Trivedi, Y. Miao,O. Solgaard, R. L. Byer, and J. Vuckovic, Science ,79 (2020).[21] T. Egenolf and U. Niedermayer, in proc. EAAC (arXiv1911.08396) (2019).[22] D. S. Black, U. Niedermayer, Y. Miao, Z. Zhao, O. Sol-gaard, R. L. Byer, and K. J. Leedle, Physical ReviewLetters , 264802 (2019).[23] N. Sch¨onenberger, A. Mittelbach, P. Yousefi, J. McNeur,U. Niedermayer, and P. Hommelhoff, Physical ReviewLetters , 264803 (2019).[24] U. Niedermayer, O. Boine-Frankenheim, and T. Egenolf,Journal of Physics: Conference Series (2017).[25] S. Earnshaw, Trans. Camb. Phil. Soc. , 97 (1842).[26] U. Niedermayer, T. Egenolf, and O. Boine-Frankenheim,PRAB20