Characterizing ultra-low emittance electron beams using structured light fields
CCharacterizing ultra-low emittance electron beams using structured light fields
Andreas Seidel,
1, 2, ∗ Jens Osterhoff, and Matt Zepf
1, 2 Friedrich-Schiller-Universit¨at, F¨urstengraben 1, 07743 Jena, Germany Helmholtz-Institut Jena, Fr¨obelstieg 3, 07743 Jena, Germany Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany (Dated: August 5, 2020)Novel schemes for generating ultra-low emittance electron beams have been developed in the lastyears and promise compact particle sources with excellent beam quality suitable for future high-energy physics experiments and free-electron lasers. Current methods for the characterization oflow emittance electron beams such as pepperpot measurements or beam focus scanning are limitedin their capability to resolve emittances in the sub 0 . (cid:15) n = [0 . ,
1] mm mrad.
I. INTRODUCTION
The emittance of an electron beam is one of its mostimportant quality measures for many applications. Theemittance describes the volume occupied by the electronbeam in 6-dimensional phase-space [1]. The transverseemittance of the beam determines the smallest spot sizethat can be achieved for a given focusing arrangementand hence influences the luminosity in HEP experiments.Another area of importance is in determining the gain offree-electron lasers through the Pierce-parameter [2].In linear accelerators the injection of low emittanceelectron bunches into the accelerating structure is ofprime importance. Significant improvements in radio-frequency (RF) guns with laser photo-cathodes allow asmall initial source size and limit the emittance growthin the early stages using strong accelerating fields of sev-eral 10s of MV/m. Using this approach GeV level lin-ear accelerators (linacs) currently achieve a normalisedtransverse emittance (cid:15) n = β e γ e (cid:15) ( β e = v e /c , γ e - electrongamma factor and (cid:15) - geometric beam emittance) of 0 . .
89 mmmrad [4] for a bunch charge of 1 nC, with continuousstrides being made to achieve even lower emittances.Plasma wakefield accelerators [5, 6] feature very largeaccelerating fields reaching 100 GV/m or more. In princi-ple these large fields result in the electron beam becominghighly relativistic over a propagation length of a fractionof a mm and consequently provide a promising route toachieving extremely low emittances. As in the case ofRF accelerators, the injection volume is a key determi-nant of the final emittance. This is especially true forlow charge beams, where emittance growth due to spacecharge is negligible. To date most plasma wakefield ex-periments with a dedicated injection procedure have used ∗ [email protected] either ionisation-, downramp- or colliding pulse injection[7–9] and transverse emittances of similar magnitude toRF linear accelerators have been attained ( (cid:15) n < . µ m mradis feasible [13].The potential performance gains by such a markedreduction in emittance are highly desirable and exper-iments are underway to explore the development of loweremittance beams. Current measurement techniques suchas beam focus scanning have shown to be capable of mea-suring emittances down to (cid:15) n ≈ . µ m spatial scale using a laser beamand allows the size of the electron beam waist to bedetermined from the observed modulation on a simplebeam monitor screen downstream. The sensitivity canbe adjusted via the spatial frequency of the laser inter-ference pattern, allowing e-beam waists from 10s of nm a r X i v : . [ phy s i c s . acc - ph ] A ug FIG. 1. (a) Schematic of the measurement setup showing two laser beams crossing under an angle 2 · (90 − θ ) to form aninterference pattern in the interaction region(Fig. 2). After the laser-electron-interaction the imparted momentum modulationevolves into a transverse density modulation that can be observed on a downstream beam monitor screen and allows the e-beamwaist size to be determined. Inclusion of a magnet behind interaction allows the emittance of monochromatic beam slices to bemeasured even for beams with a large projected energy spread. (b) Schematic of the beam momentum distribution at the waist,the interaction point and after free space propagation (from left to right). (c) Transverse electron phase space before (solid)and after laser-electron interaction (dashed). The red line indicates the intensity profile of the two interfering laser beams atthe corresponding transverse position x at the interaction point. The ponderomotive force shifts the transverse momentum ofthe electrons. For an optimum laser intensity the momentum transfer is matched to the gradient of the phase-space and shiftsparticles from regions of high laser intensity to regions of low intensity highlighted by the dashed ellipses. to µ m-scale to be determined. The emittance range thatcan be covered by this technique is also determined bythe divergence of the e-beam and is compatible with alarge emittance parameter range from the mm mrad ofcurrent linacs to the µ m mrad anticipated for novel in-jection schemes. Measurements of small spatial scaleswith laser interference structures is a viable approach toassess the quality of electron beams and has been demon-strated in beam-size monitors, where in contrast to wirescanners the fine wire is replaced by a ’laser-wire’ formedby interfering lasers in the so called Shintake-monitor[14]. It can resolve nm-spatial scale electron beam fociusing a scanning technique. While this method requires ameasurement of the electron bunches at focus, which canbe an immense effort for poly-chromatic electron beams,our proposal is is compatible with broadband divergingbeams and also allows a determination of the slice emit-tance. II. THEORETICAL MODEL
The proposed measurement scheme is shown schemat-ically in Figure 1. Two laser beams with wavelength λ cross under an angle 2 · (90 − θ ) to form an interferencepattern with a periodicity d = λ/ (2 cos θ ) at a distance z from the electron beam waist (Fig. 2). In the absence ofthe laser beam the unperturbed electron beam profile isvisible on a measurement screen at a convenient location downstream of the interaction region. When the laser isswitched on, the transverse momenta of the electrons aremodulated by the field of the laser with the periodicityof the interference pattern allowing a modulated electronbeam pattern to be observed on the screen.The working principle is shown in greater detail inFig. 1(c). The electron beam transverse dimension x and divergence angle x (cid:48) can be described as an el-lipse [15] parameterized by TWISS parameters α, β, γ so that (cid:15) = γx + 2 αxx (cid:48) + βx (cid:48) . In this description
10 5 0 5 10 z [ m ] x [ m ] FIG. 2. Intensity pattern of the interference of the collidinglaser pulses. the beam width σ x and angular spread σ x (cid:48) are givenby σ x ( z ) = (cid:112) β ( z ) (cid:15) and σ x (cid:48) ( z ) = (cid:112) γ ( z ) (cid:15) as a functionof the distance z from the beam waist respectively andtan Φ = dx (cid:48) /dx ∝ /z . Note that while the ellipse hasan overall width of σ x (cid:48) the slice width at a given trans-verse position x can be significantly less ∆ x (cid:48) = 2 (cid:113) (cid:15)β ( z ) and decreases with the propagation distance z from thewaist. This implies that a visible modulation of the elec-tron beam can be achieved by imparting a transversekick to the electrons of ∼ ∆ x (cid:48) , in which case the electronbeam will exhibit areas of increased phase space densityat certain values x (cid:48) and reduced at others. As can be seenfrom Fig. 1(c), the separation of adjacent intensity peaks d must be be large enough to prevent the two adjacentregions with slice width ∆ x (cid:48) = 2 (cid:112) (cid:15)/β ( z ) from overlap-ping in x (cid:48) -space. At large values of α (i.e. far from thewaist) the lower limit for the laser interference patternperiod d is therefore: d > (cid:112) (cid:15)β ( z ) α ( z ) ≈ (cid:15)σ x (cid:48) = 2 (cid:15)σ x (cid:48) = 2 σ x (0) = const (1)The condition for the minimum periodicity is thereforeconnected to the beam waist size and this condition is in-dependent of the distance z from the electron beam waistwhere the measurement is taken. As we shall see in thefollowing, the waist size can be determined from the mod-ulation depth d without the need for a direct measure-ment at the location of the beam waist itself. Combinedwith a simple measurement of the unperturbed beam di-vergence we can therefore use this method to accuratelydetermine the emittance of the electron beam.The laser imparts momentum to the electrons via theponderomotive force F p ∝ ∇ E of the laser which tendsto push electrons from regions of high intensity to re-gions of low intensity. As can be seen from Fig. 1(c)the optimal modulation depth is achieved when electronsare shifted in transverse momentum by an amount thatmatches the gradient of the ellipse in phase space. Inmomentum terms this requirement can be expressed by∆ p x ∆ x = dp x dx = p z tan Φ ∝ p z z (2)Assuming that the ponderomotive force is the dominantforce in changing the particle momentum, we can derivethe requirement dp x dtdx ! = F pond dx For a given gradient of the transverse momentum acrossthe electron beam, there is therefore an optimal laser in-tensity that leads to the desired modulation of transversemomentum and hence maximum modulation visibility.The change in transverse momentum can be expressed interms of the interaction time t int and laser intensity I as dp x dx = (cid:104) F pond (cid:105) t int ( γ e ) dx ∝ d (cid:104) I (cid:105) ω dx t int ( γ e ) dx , (3) where (cid:104)(cid:105) is a time average over the duration of the inter-action and ω the laser frequency. Since the intensity ofthe interference pattern drops from its maximum to zeroover a distance of d/ p x ∆ x ∝ (cid:104) I (cid:105) · t int ( γ e ) , where (cid:104) I (cid:105) is the time averaged intensity required toachieve the optimum beam modulation. Inserting in toEq.(2) we find the average interaction intensity to achievethe optimum modulation depth with: (cid:104) I (cid:105) ∝ p z z · t int ( γ e )As one would expect, the laser intensity increases forhigher energy electron beams and decreases with increas-ing distance from the beam waist. The latter scaling canbe simply understood, as the increased beam size with in-creasing z reduces the gradient of transverse momentum.This corresponds to a shallower gradient of the emittanceellipse and a reduced momentum modulation required tobe imparted by the laser. Note that the required laserenergy also depends on the interaction length (ignoringbeam size effects) s i = cτ / (1 − sin( θ ), favouring shallowcrossing angles if a sufficiently small d is maintained. Therequired intensity can thus be kept sub relativistic, whichdiminishes the demands on the laser system.Assuming an electron bunch with Gaussian transversebeam waist σ x and corresponding momentum σ px weobtain n ( x, p x ) = n e · exp (cid:32) − p x σ px − x σ x (cid:33) (4)for the density distribution at waist position. WithEq.(2) and (3) we obtain n ( p x ) = n e (cid:90) ∞−∞ exp ( − (( p x − ∆ p x ) / tan Φ − x ) σ px − ( x/ tan Φ + p x + ∆ p x ) σ x ) dx, (5)for the momentum space after the laser-electron interac-tion, where ∆ p x = (cid:104) F pond (cid:105) · t int ( γ e ), corresponding to asharply peaked intensity distribution in transverse mo-mentum space with the peak height increasing sharplywith decreasing values of σ x . III. SIMULATION RESULTS
2D simulations to test the scheme were conducted us-ing the code EPOCH [16]. The simulations were per-formed in a moving box with the size of z = 320 µ m and x = 80 µ m with 20 cells/micron resolution in every di-mension. The electron bunch, sampled by 8 · macro p x / p z [ m r a d ] I = 0.1 I I = 0.32 I x [ m ] p x / p z [ m r a d ] I = 1.0 I x [ m ] I = 2 I FIG. 3. Momentum space of electron beam ( γ e = 100, σ x =100 nm, σ x (cid:48) = 2 mrad, σ z = 2 µ m) after interaction withdifferent laser intensities. As can be seen the modulationoptimizes for a specific intensity and reduces again for highervalues. White curve: transverse momentum space integratedalong x (measured signal on detector). particles, had a longitudinal size of σ z = 2 µ m and vari-able transverse size σ x and propagated along the z-axis.The electron bunch charge was chosen at 150 fC to ensurean interaction between laser and electron bunch withoutfurther charge related effects. The interaction laser wasmodelled as two pulses propagating linearly polarized iny-direction and with a duration τ F W HM = 17 fs, spotsize at interaction σ F W HM = 30 µ m at a wavelength of λ = 0 . µ m. The laser intensity distribution producedby the two interfering pulses is shown in Fig. 2 withthe lasers propagating from the upper left side and thelower left side. The angle between z-axis and lasers is90 − θ = 30 ◦ .Fig. 3 shows the electron beam modulations for dif-ferent laser intensities. The initial electron bunch pa-rameters are γ e = 100, σ x = 100 nm, σ x (cid:48) = 2 mrad, σ z = 2 µ m and I = 3 . · W/cm . The pondero-motive force imparts transverse momentum to the elec-trons and concentrates the electrons at certain propaga-tion angles corresponding to the nodes of the interferencepattern, resulting in an intensity modulation on the di-agnostic. Note that the electrons do not change positionappreciably during the interaction at these laser inten-sities, instead the laser imparts a change in transversemomentum which is visible after further propagation.The modulation is strongest for the matched intensity I and decreases above and below this. For intensities
500 1000 1500 2000 2500 3000 distance electron waist to interaction[ m ] I [ W / c m ] x -fitsimulation FIG. 4. Optimum peak Intensity of laser used, to getstrongest modulation signal in momentum space of electronbeam( γ e = 100, σ x = 150 nm, σ x (cid:48) = 2 mrad, σ z = 2 µ m) forvarious distances between electron beam being at waist andlaser-electron interaction. The vertical bars indicate the rangeover which there is no appreciable change in the modulationdepth. that are too low, the perturbation is much smaller thanthe local slice angular spread ∆ x (cid:48) (the ellipse-width in x (cid:48) -direction) leading to a negligible effect. At large in-tensities the electrons are displaced by much more thanthe local ellipse-width and the effect is ’smeared out’. Fora given set of experimental parameters, the peak laser in-tensity at which the optimal modulation depth occurs isshown in Figure 4). With P Laser ≈ I / · πσ the requiredlaser power for this particular electron bunch and inter-action point 1600 µ m behind the electron waist matchedto a laser spot size σ F W HM = 30 µ m is 300 GW.One might assume that this method places increas-ingly onerous requirements on the laser to make a mea- p x / p z [ mrad ] p a r t i c l e s / d [ a r b . U . ] I = 1 10 = 100, I = 5 10 = 200, I = 20 10 = 500, I = 125 10 = 1000, I = 500 10 FIG. 5. Transverse momentum space of electron beams withdifferent γ e ( σ x (cid:48) = 2 mrad, σ z = 2 µ m, σ x = 150nm) afterinteraction. Laser intensity is optimized for maximum mod-ulation depth. surement for beams with very high electron beam energy.Since the width of the ellipse and therefore the optimalintensity depends on the distance z from the beam waistas 1 /z , allowing the optimal intensity to be controlled bythe interaction geometry. Note that for limited availablelaser power there is no intrinsic requirement for the laserspot to be larger than the electron beam. In principlesmaller spots can be used with scanning measurements todetermine the emittance of the beam, therefore enablingmeasurements with lasers that are easily co-located withan electron beam. As is clear from the discussion in theprevious section and can be seen in Fig. 5 higher en-ergy electron beams require a higher laser intensity foroptimal modulation for otherwise identical parameters.The increase is quadratic in the Lorentz-factor γ e due tothe relativistic contraction of the interaction length andthe higher transverse momentum required to achieve thesame ratio of p x /p z on the beam electrons. It is im-portant to note, that at the optimum laser intensity themaximum modulation depth is independent of the elec-tron bunch γ e .From the discussion above we can now formulate ameasurement approach for calculating the desired e-beamcharacteristics. As is clear from figure 6 the change inthe beam intensity modulation depends only on sourcesize for fixed laser intensity with the beam modulationvisible as long as the criterion from eq. 1 is met. The pro-posed measurement strategy is therefore as follows. Firsta location at some distance z from the e-beam waist is p x / p z [ m r a d ] x = 50 nm x = 100 nm x [ m ] p x / p z [ m r a d ] x = 200 nm x [ m ] x = 350 nm FIG. 6. Momentum space of electron beam( γ e = 100, σ x (cid:48) = 2mrad, σ z = 2 µ m) after interaction with laser ( I = 3 . · W/cm ) for different electron source sizes σ x . White curve:transverse momentum space integrated along x (measured sig-nal on detector). chosen depending on the beam and available laser pa-rameters. Varying the intensity of the laser allows theoptimal laser intensity to be set by optimising the mod-ulation depth, thereby eliminating any systematic effectssuch as small offsets in achieved actual laser intensityfrom nominal or distance z from the electron beam waist.It should be noted here that deviations from the opti-mal laser intensity of <
10% did not lead to measurablechanges in the simulations peak height (see Fig. 4). Atthe optimum intensity and for a known interference pe-riod d the modulation depth measured in a single shotallows the beam waist size to be directly inferred fromthe modulated beam profile by comparing with the pic-simulation/analytical solution. Our analysis has shownthis comparison also applies for a laser profile with mi- FIG. 7. (a) Transverse momentum space of electron beamswith different σ x ( σ x (cid:48) = 2 mrad, σ z = 2 µ m, γ e = 100 and d = 0 . µ m) after interaction. Solid line indicates the PICsimulation results and dashed the solution of Eq. 5 assum-ing a sinusoidal electrical laser field. (b) Normalized peakheight of modulation in transverse momentum space afterlaser-electron interaction calculated with Eq. 5. No inter-action results in a height of 1. nor deviations to the theoretically assumed Gauss profile.We compared versions of the analytical solution in whichnoise of up to 20% was added to the ponderomotive forceto the version without and observed no measurable dif-ference in modulation depth ( σ x > . · d ).Figure 7 compares the calculation of the modulationfrom the analytical considerations above to the PIC sim-ulations. Clearly the source size can be sensitively deter-mined over a large range with a single measurement con-figuration. The measurement sensitivity can be furtherincreased by reducing the interference period d . Notethat for the smallest waist-size (50 nm) in the simulation,which corresponds to an emittance of 10 µ m mrad, anelectron detection system with a resolution better than40 µ rad would be required.This method is capable of characterising very smallwaist size beams with unprecedented emittance accu-rately. The sensitivity of the emittance measurementincreases for lower divergence electron beams. Electronbeam waists down to 10s of nm can be resolved usingthis technique corresponding to normalised emittance ofthe order of µ m mrad. Using current laser technologyit is possible to characterise GeV e-beams, thereby pro-viding a precise technique to characterise ultra-low emit-tance electron beams under development. In principlethis technique can also accommodate beams with signif-icant energy spread, such as those from wakefield accel-erators by combining the beam modulation in one planewith a dipole magnet dispersing the electron beam inthe other plane. Simulations for a poly-chromatic elec-tron bunch with an energy spread of <
5% resulted inno measurable changes in the peak height, which agreeswith the sensitivity of the modulation depth to devia-tions from the optimal intensity. We note that while thistechnique becomes more demanding in terms of laser in-tensity at higher electron beam γ e it can still be applied and required laser beam energy can be reduced by usingsufficiently small spots combined with scanning measure-ments across the beam spatial dimensions. IV. CONCLUSION
We have described a novel scheme for characterisingthe properties of a an electron beam. To our knowledge,the method is unique in allowing the measurement of ex-tremely small broadband electron beam source sizes andemittances in the µ m mrad regime predicted to be acces-sible using advanced accelerator techniques. Our simula-tions have shown that emittances as small as (cid:15) n = 0 . σ x < µ m. V. ACKNOWLEDGEMENTS
This research was funded by the Federal Ministry ofEducation and Research of Germany in the Verbund-forschungsframework (project number 05K16SJB). [1] J. Buon, in
Cas Cern Accelerator School-Basic Course 4 (Cern, 1990) pp. 30–52.[2] W. A. Barletta, J. Bisognano, J. N. Corlett, P. Emma,Z. Huang, K.-J. Kim, R. Lindberg, J. B. Murphy, G. R.Neil, D. C. Nguyen, C. Pellegrini, R. A. Rimmer, F. San-nibale, G. Stupakov, R. P. Walker, and A. A. Zholents,Nuclear Instruments and Methods in Physics ResearchSection A: Accelerators, Spectrometers, Detectors andAssociated Equipment , 69 (2010).[3] Y. Ding, A. Brachmann, F.-J. Decker, D. Dowell,P. Emma, J. Frisch, S. Gilevich, G. Hays, P. Hering,Z. Huang, R. Iverson, H. Loos, A. Miahnahri, H.-D.Nuhn, D. Ratner, J. Turner, J. Welch, W. White, andJ. Wu, Phys. Rev. Lett. , 254801 (2009).[4] S. Rimjaem, G. Asova, J. B¨ahr, H. J. Grabosch,L. Hakobyan, M. H¨anel, Y. Ivanisenko, M. Khojoyan,G. Klemz, M. Krasilnikov, and Others, Proceedings ofFEL2010 (2010).[5] E. Esarey, C. B. Schroeder, and W. P. Leemans, Rev.Mod. Phys. , 1229 (2009). [6] M. C. Downer, R. Zgadzaj, A. Debus, U. Schramm, andM. C. Kaluza, Rev. Mod. Phys. , 035002 (2018).[7] M. Chen, E. Esarey, C. B. Schroeder, C. G. R. Geddes,and W. P. Leemans, Physics of Plasmas , 33101 (2012).[8] T.-Y. Chien, C.-L. Chang, C.-H. Lee, J.-Y. Lin, J. Wang,and S.-Y. Chen, Phys. Rev. Lett. , 115003 (2005).[9] E. Esarey, R. F. Hubbard, W. P. Leemans, A. Ting, andP. Sprangle, Phys. Rev. Lett. , 2682 (1997).[10] R. Weingartner, S. Raith, A. Popp, S. Chou, J. Wenz,K. Khrennikov, M. Heigoldt, A. R. Maier, N. Ka-jumba, M. Fuchs, B. Zeitler, F. Krausz, S. Karsch, andF. Gr¨uner, Phys. Rev. ST Accel. Beams , 111302(2012).[11] G. R. Plateau, C. G. R. Geddes, D. B. Thorn, M. Chen,C. Benedetti, E. Esarey, A. J. Gonsalves, N. H. Matlis,K. Nakamura, C. B. Schroeder, S. Shiraishi, T. Sokol-lik, J. van Tilborg, C. Toth, S. Trotsenko, T. S. Kim,M. Battaglia, T. St¨ohlker, and W. P. Leemans, Phys.Rev. Lett. , 64802 (2012). [12] B. Hidding, G. Pretzler, J. B. Rosenzweig, T. K¨onigstein,D. Schiller, and D. L. Bruhwiler, Phys. Rev. Lett. ,035001 (2012).[13] G. G. Manahan, A. F. Habib, P. Scherkl, P. Delinikolas,A. Beaton, A. Knetsch, O. Karger, G. Wittig, T. Heine-mann, Z. M. Sheng, J. R. Cary, D. L. Bruhwiler, J. B.Rosenzweig, and B. Hidding, Nature Communications , 15705 (2017).[14] J. Yan, M. Oroku, Y. Yamaguchi, T. Yamanaka,Y. Kamiya, T. Suehara, S. Komamiya, T. Okugi, N. Terunuma, T. Tauchi, S. Araki, and J. Urakawa,Physics Procedia , 1989 (2012).[15] K. Floettmann, Phys. Rev. ST Accel. Beams , 034202(2003).[16] T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G.Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, PlasmaPhysics and Controlled Fusion57