NNano-scale electron bunching in laser-triggered ionization injection in plasmaaccelerators
X. L. Xu,
1, 2
C. J. Zhang, F. Li, Y. Wan, Y. P. Wu, J. F. Hua, C.-H. Pai, W. Lu, ∗ W. An, P. Yu, W. B. Mori, and C. Joshi Department of Engineering Physics, Tsinghua University, Beijing 100084, China University of California, Los Angeles, California 90095, USA (Dated: July 9, 2018)Ionization injection is attractive as a controllable injection scheme for generating high qualityelectron beams using plasma-based wakefield acceleration. Due to the phase dependent tunnelingionization rate and the trapping dynamics within a nonlinear wake, the discrete injection of electronswithin the wake is nonlinearly mapped to discrete final phase space structure of the beam at thelocation where the electrons are trapped. This phenomenon is theoretically analyzed and examinedby three-dimensional particle-in-cell simulations which show that three dimensional effects limit thewave number of the modulation to between > k and about 5 k , where k is the wavenumber ofthe injection laser. Such a nano-scale bunched beam can be diagnosed through coherent transitionradiation upon its exit from the plasma and may find use in generating high-power ultravioletradiation upon passage through a resonant undulator. Due to the ability to sustain ultra-high accelerationgradients (GV / cm), the field of plasma-based wakefieldacceleration has attracted much attention in the pasttwo decades [1]. Recently, ionization injection has beenproposed and demonstrated [2–9] as a viable injectionscheme and investigated for generating high brightness( ∼ A / rad / m ), stable, and tunable electron beams[10–15]. The basic idea is that the trapping threshold ofan electron is reduced when it is born inside the wake-field near the maximum of wake’s potential compared toan electron from a pre-ionized plasma. Such high bright-ness beams are needed for future free-electron-laser andcollider applications.The key to generating a high brightness beam is tolimit the volume within the wake where injection of elec-trons occurs [16]. In the case where injection is from fieldionization due to a laser pulse, the ionization volume islimited by choosing the intensity of the injection pulse(s)close the ionization threshold of bound electrons. There-fore, electrons are mostly born near the peaks and thetroughs of the oscillating laser electric field. The phase-dependent ionization leads to an intrinsic initial phasespace discretization at twice the optical frequency, whichis known to produce third harmonic generation in tunnelionized plasma [17, 18].We show in this Letter using theory and fully three-dimensional (3D) particle-in-cell (PIC) simulations, thatwhen ionization occurs on either side of the peak of thewake potential the electron bunch can be strongly mod-ulated in space on the nano-meter scale when it becomestrapped. In the 1D limit the spacing of the modulationscan be made arbitrarily small. However, we show thatthree-dimensional effects limit the discretization patternto less than one-fifth the laser wavelength. The concept isrobust and has the potential to provide lower overall en-ergy spread, lower emittances, shorter modulation wave-lengths, and more nano bunches than another recently proposed scheme [19]. Such an ultra-short and micro-bunched electron beam can be diagnosed via the coher-ent transition radiation upon exiting the plasma [20] andmay be used to produce high power coherent EUV radi-ation in a short resonant undulator.To illustrate the concept, we first consider ionizationinjection using a single laser pulse as shown in Fig. 1(a).An 800 nm laser pulse polarized in the x direction withnormalized vector potential a = 2 , w = 14 µ m and apulse length (fwhm of energy) of 26 fs, propagates intoa mixture of pre-ionized plasma and N ions. The pre-ionized electrons form a nonlinear wake. As has beenobserved previously [4], the K-shell electrons of nitro-gen with high IPs are released during the rising edgeof the wake potential [Fig. 1(a)], then slip to the backof the wake where some of these electrons are trapped[4]. This process is examined using the 3D PIC codeOSIRIS [21] using a moving window [22]. We definethe z axis to be the laser propagating direction. Thecode uses the Ammosov-Delone-Krainov (ADK) tunnel-ing ionization model [23]. The IPs of the sixth and sev-enth nitrogen electrons are I p ≈ . , . γ K = (cid:112) I p / U p ≈ . , . (cid:28)
1; therefore the ADKmodel should be valid. The simulation window has a di-mension of 63 . × . × . µ m with 500 × × x, y and z directions, respectively. This cor-responds to cell sizes of k − in the x and y directions and0.2 k − in the z direction, where k is the wavenumberof the laser pulse.The ( ξ i , x i ) space distribution of the trapped electronswhen they are ionized is shown in Fig. 1(b), where ξ ≡ v φ t − z is the relative longitudinal position and v φ is the phase velocity of the wake. Due to the laserphase-dependent ionization probability, the initial elec-tron distribution has a strong modulation at 2 k . Af-ter being released, the electrons slip to the back of the a r X i v : . [ phy s i c s . p l a s m - ph ] O c t wake and are accelerated by the longitudinal electricfield in the wake. Under the quasi-static approximation, γ − ( v φ /c ) p z − ψ = Const [24], where p z is normalizedto mc , ψ ≡ ( e/mc )[ φ − ( v φ /c ) A z ] is the pseudo poten-tial, ψ in the fully blown-out wake can be expressed as ψ ≈ [ r b ( ξ ) − r ] / r b ( ξ ) is the normalizedradius of the ion channel that has a spherical shape fora sufficiently large maximum blowout radius r m given by r b ( ξ ) = r m − ξ [25][26]. Note that all parameters withunits of length are normalized to the background plasmaskin depth. Using the constant of motion given above,the relative longitudinal position of the injected electroncan be expressed as ξ ≈ (cid:113) ξ i + r i − r − γ − ( v φ /c ) p z ] (1) p z [ m c ] [c/ω p ] t i [ / ω p ]
25 ξ i [c/ω p ] -2 -1 x i [ c / ω p ] t i [ / ω p ] x [ µ m ] electronslaser driver(a) (b)(d)(f) p z [ m c ] p z [ m c ] z [µm]6010080 472.4 475x [µm] 0-2.542.54(e) 02-2 z [fs] c u rr en t [ A ] ] b ( k ) L inj = 90 µm L inj = 190 µm-5 0 50 4 8 L inj = 90 µm L inj = 90 µm L inj = 190 µm FIG. 1: The nano-scale bunching of injected charge in aPWFA. The laser is focused at z = 0 mm plane. (a) Snap-shot of the charge density distribution of the background elec-trons, the K-shell electrons of nitrogen, and the laser electricfield in x direction. The red line is the on-axis ( x = y = 0)pseudo potential ψ . (b) The distribution of the ionization in-jected electrons in ( ξ i , x i ) space. The color represents the timewhen the electrons are ionized. The red peaks are integratedinjected charge in x i whereas the black is the integrated in-jected charge for all ξ i . (c) The ( ξ, p z ) space at z = 0 . z = 0 . z, x, p z ) phase space at z = 0 . L inj = 90 µ m case, the N ions are distributed from z = 0 .
038 mm to z = 0 .
128 mm and n N = 10 − n p ;for L inj = 190 µ m case, the N ions are distributed from z = 0 .
038 mm to z = 0 . n N = 5 × − n p . The electron conducts betatron oscillations in x and y with a decreasing amplitude under the focusing andacceleration fields [16][27]. An initial isolated slicein ( ξ i , x i , y i ) will be mapped to an isolated slice in( ξ, x, y, p z ) space. If r i (cid:28) γ − ( v φ /c ) p z (cid:28) ξ is mainly determinedby the initial ξ i as ξ ≈ (cid:112) ξ i , which means the initialmodulation in ξ i can be nonlinearly mapped to ξ . Thismeans that an initial slice (electrons with the same ξ i )will be mapped to the same final slice (same ξ ).Any spread in r i , r and γ − ( v φ /c ) p z will broaden the ξ distribution for an initial slice. This can be seen inthe simulation results shown in Fig. 1. In Fig. 1(c)we show the ( ξ, p z ) space at z = 0 . ξ due to the spread oftransverse motion is ∆ ξ ⊥ ≈ .
05. For this examplewhere a laser driver with moderate a is used, we findthat, 1 − v φ /c ≈ ω p / (2 ω ) [28]. Therefore, the term γ − ( v φ /c ) p z ≈ γ (1 − v φ /c ) ≈ ω p / (2 ω ) γ contributesdifferently for electrons with different energy leading toa spread in ξ for electrons with the same ξ i . Specif-ically, electrons ionized earlier (at different z i ) but atthe same ξ i can have higher energy and smaller ξ . InFig. 1(c) the difference in ξ due to the energy differ-ence is seen to be ∆ ξ γ ≈ .
1. This spread dependson the spread in z i which can be controlled by limit-ing the duration (distance) of ionization, L inj . In thesimulations we increased L inj from 90 µ m ≈ c/ω p to190 µ m ≈ c/ω p , by varying the region where N ex-isted. In Figs. 1(d)-(f) it can be seen that the differenceof ξ due to this spread in energy is increased to ∆ ξ γ ≈ . b ( k ) = (cid:12)(cid:12)(cid:82) dzg ( z )exp( ikz ) (cid:12)(cid:12) , where g ( z ) is the normalizeddistribution of the trapped electrons) are shown in Fig.1(d). The modulation in the current profile is peaked at k ≈ k , and the modulation and the bunching factorare reduced when L inj is increased from 90 µ m to 190 µ m due to the larger ∆ ξ γ . The discretized phase spacestructure can be seen clearly in ( z, x, p z ) phase space at z = 0 . L inj , the slices are slanted in ( p z , z ) space in-dicating that within a narrow energy slice of the beamthe bunching factor can still be large.By using two pulses to separate the wake formation andthe electron injection, the initial and final phase space ofthe trapped electrons can be better controlled [10, 11, 13–16]. Throughout the rest of this Letter, we consider thedriver pulse to be a relativistic electron bunch and theinjection pulse to be a co-propagating low intensity laserpulse. The injection laser can be focused to a very smallspot size to decrease the transverse ionization region anddue to its shorter Rayleigh length it will have a shorter L inj . This leads to much reduced ∆ ξ ⊥ and ∆ ξ γ . In a rel-ativistic beam driver case, the phase velocity of the non-linear wake is equal to the velocity of the driver bunch,which is typically closer to the speed of light than thegroup velocity of the laser. Therefore the term ∆ ξ γ ismuch reduced. The electron is longitudinally frozen inthe wake after it is boosted to relativistic energy (it doesnot dephase). This longitudinal position can therefore bedefined as ξ f and this is insensitive when it was ionized.Under the assumption of no phase slippage in thewake, the effect of the nonlinear mapping between ξ i and ξ f and the finite r i on the bunching factor can bequantified as follows. The distribution of the final lon-gitudinal parameters ξ f is g ( ξ f ) dξ f = dξ i (cid:82) dr i f ( r i , ξ i )can be obtained from the distribution of the ini-tial parameters is f ( ξ i , r i ), where r f is neglectedwhich is also reasonable when the energy of theelectron is high. The bunching factor is b ( k ) = (cid:12)(cid:12)(cid:82) dξ f exp( ikξ f ) g ( ξ f ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:82) dξ i dr i exp( ikξ f ) f ( ξ i , r i ) (cid:12)(cid:12) .Assuming δξ i ≡ ξ i − ¯ ξ i (cid:28) ¯ ξ i and r i (cid:28) ¯ ξ i where ¯ ξ i is the mean value of ξ i , then after expanding ξ f tothe order of O ( r i ) and O ( δξ i ), ξ f can be expressed as ξ f ≈ (cid:113) ξ i (cid:104) δξ i h m ¯ ξ i + δξ i h m ¯ ξ i (cid:16) − h m (cid:17) + r i h m ¯ ξ i (cid:105) ,where h m = (cid:113) ξ i / ¯ ξ i is the wavenumber upshiftfactor obtained from the nonlinear mapping pro-cess. We assume the initial distribution is f ( ξ i , r i ) = r i σ r exp (cid:16) − r i σ r (cid:17) √ πσ e exp (cid:16) − δξ i σ e (cid:17) + ∞ (cid:80) n = −∞ F n exp( − i nk δξ i ),where F n = (cid:82) dξ i f b ( ξ i )exp( i nk δξ i ) and f b ( ξ i ) is theinitial ξ i distribution in a single slice. Substitute theexpression of f ( ξ i , r i ) into the bunching factor, then it is (a) (b) (c) (d) (e) (f) FIG. 2: Bunching of electrons when charges is injected byusing a relatively low intensity laser pulse in prescribed andnon evolving wakefield. (a), (b) and (c) The red lines are thebunching factor while scanning the initial ¯ ξ i and the dashedlines are the analytical results from Eq. 2. (d), (e) and (f)The corresponding Wigner transform of the current profileshowing the spatial modulation. The white dashed lines are k/k = 2 (cid:112) ξ i /ξ i and the red lines are the current profileof the injected charge. Note that (d) is 1D simulation andothers are 3D simulations. straightforward to obtain b ( k ) = + ∞ (cid:88) n = −∞ | F n | R (ˆ σ r , ˆ σ e )e − ( k − nhmk σ e h m (1+ˆ σ e ) (2)where R (ˆ σ r , ˆ σ e ) = (cid:2) (1 + ˆ σ r )(1 + ˆ σ e ) (cid:3) − / is the 3D re-duction factor, ˆ σ e = σ e (cid:112) (1 − /h m ) k/ ( h m ¯ ξ i ) and ˆ σ r = σ r (cid:112) k/ ( h m ¯ ξ i ). The ratio of the strongest modulationwavenumber in the current profile over the wavenumberof the injection laser (the modulation factor) is h = 2 h m = 2 (cid:113) ξ i / (cid:12)(cid:12) ¯ ξ i (cid:12)(cid:12) (3)where the factor 2 is from the ionization process andthe factor h m is from the nonlinear mapping process.Eq. (3) shows that the wavelength of the modulation isshortest for ¯ ξ i near zero (near the maximum of the wakepotential). However, ˆ σ r becomes very large for ¯ ξ i nearzero, therefore, from Eq. (2), R will be small in this limit.For this reason the wave number of the modulation islimited and the modulation is only seen when ionizationoccurs off the maximum of the potential.These conclusions are verified numerically. We useOSIRIS with non-evolving forces from the nonlinearwakefields, i.e., F z = − ξ/ , F r = r/ − v z ) r/
2. An800 nm laser with a = 0 . , w = 2 µ m propagates x [ µ m ] electronsinjection laser driver beam 04-4 (a) ψ i1.6 1.8 22.2 ψ f t i [ c / ω p ] z [µm]x [µm]1410-0.510.51 0 10 10.5 p z [ m c ] z [fs] -10 0 10 c u rr en t [ A ] k [k ] b ( k ) (b)(c) (d) FIG. 3: An electron driver-beam followed by an injectionlaser propagate to the right in a mixture of pre-ionized plasmaand He ions ( n p = 1 . × cm − , n He = 0 . n p ).Driver-beam: E b = 1 GeV , σ r = 8 . µ m , σ z = 10 . µ m , I b =19 kA; The injection laser is as same as in the case of Fig. 2.(a). Snapshot of the charge density distribution of the back-ground electrons, the 2nd electron of helium, and the laserelectric field. The red line is the pseudo potential at the cen-ter. (b) The dependence of the final ψ f on the initial ψ i forionization injected electrons. The color represents the ioniza-tion time. The black line represents ψ f − ψ i = −
1. (c) The( z, x, p z ) phase space distribution of the trapped electrons at z = 0 . h ≈
3) current profile andthe bunching factor of the trapped electrons at z = 0 . through a plasma with n p = 1 . × cm − and a10 − n p He plasma (to minimize space charge effects)provides the ionized electrons. The longitudinal delay be-tween the laser pulse and the plane with ξ = 0 is scannedto generate electrons with different ¯ ξ i and the resultingbunching factors are shown in Figs. 2(a)-(c). Whenthe laser is strongly focused, γ − p z − ψ is not strictlyconserved and the variation leads to a reduction of thebunching factor which more serious when h is larger [seeFig. 2(b)]. Due to the nonlinear mapping process, themodulation factor depends on ξ f , which can be seen fromthe Wigner transform of the current profile as shown inFigs. 2(d)-(e). The modulation factor can be very hightheoretically when ¯ ξ i is very close to zero, but in thiscase R is very small so b is rather small. However in a1D simulation, h as high as 15 was observed as shown inFig. 2 (d).We next present results from a fully self-consistent 3DOSIRIS simulation. We use a relativistic electron beamto produce the wake for the ionization of the inner shellelectrons. A mixture of preionized plasma and He ionsis used. The electric field of the electron beam is lowenough to not doubly ionize the helium. The simulationwindow has a dimension of 127 × × µ m with1000 × × x, y and z directions,respectively. This corresponds to cell sizes of k − in the x and y directions and 0.2 k − in the z direction. Thereare 4 particles per cell to represent the He ions. An800 nm injection laser with the same amplitude, spot sizeand pulse length used above (Fig. 2) is focused into thewake as shown in Fig. 3(a). The laser is focused at z =0 mm while the plasma starts from z = − .
254 mm. Bytracking particles, we confirm that the trapping condition[4] ψ f ≈ ψ i − z = 0 . ξ i ≈ .
87. For thiscase, based on Eq. 3 the predicted modulator factor is h ≈ .
9. The current profile of the electron beam and the (a) z [fs] -5 0 5 c u rr en t [ k A ] (b) FIG. 4: The current profile (a) and bunching factor (b)of the trapped electrons by using 200 nm injection laser at z = 0 . n He =0 . n p and the black dashed line is the result when n He =3 × − n p . bunching factor b ( k ) at this distance are shown in Fig.3(d).By replacing the 800 nm injection laser by its 4th har-monic - 200 nm injection laser, an electron bunch with astrong UV frequency modulation is generated. We simu-late this using OSIRIS with the external wakefield modeldescribed above for the same plasma density. The He density was either 3 × − n p or 0 . n p . A 200 nm in-jection laser with a = 0 . , w = 2 µ m are used to re-lease the 2nd electron of helium. The current profile andthe bunching factor at z = 0 . γ = 1000). The radiated spec-trum will contain the fundamental and the second har-monic of the nano-structured beam at 65.6 nm and 32.8nm respectively. Detection of this coherent radiation atwavelengths shorter than the ionizing laser wavelengthis a good diagnostic of this self-bunching in the wake.Space charge interaction between the injected electronswill blur the modulation at nhk and thus reduce themodulation and the bunching factor at nhk which canbe seen from the comparison between the dashed line( n He = 3 × − n p ) and the solid line ( n He = 0 . n p )in Fig. 4(b).Due to the small spot size and low intensity of theinjection laser, the emittance and energy spread of thetrapped beam are both very small, e.g, for the 200 nminjection laser case discussed above (cid:15) nx = 10 . , (cid:15) ny =10 . σ γ = 3 .
2. If such an electron beam canbe accelerated further, extracted from the plasma andcoupled into a short, resonant undulator without degrad-ing its emittance [30] it will produce intense coherent ra-diation. For example, consider an electron beam with¯ γ = 1068 . σ γ = 3 . λ u = 3 cm and normalizedundulator parameter K = 2. The undulator is reso-nant at the modulation wavelength of the electron beam, λ r = 65 . P sat = 400 MW in 3 m undulator when by simulatingthis process with 3D GENSIS 1.3 code [31].In conclusion, we have shown that the discrete injec-tion of the electrons due to the laser ionization injectionprocess is mapped to the final phase space of the ac-celerated beam in a plasma accelerator operating in theblowout regime. Theoretical analysis and 3D PIC simula-tions are presented. This intrinsic phase space discretiza-tion phenomenon leads to nano-scale micro bunching ofthe accelerated beam that can be diagnosed through co-herent transition radiation upon the beam’s exit from theplasma and may find use in generating high-power ultra-violet radiation upon passage through a resonant undu-lator.Work supported by NSFC grants 11175102, 11005063,thousand young talents program, DOE grants de-sc0010064, de-sc0008491, de-sc0008316, and NSF grantsACI-1339893, PHY-1415386, PHY-0960344. Simulationsare performed on the UCLA Hoffman 2 and Dawson 2Clusters, and the resources of the National Energy Re-search Scientific Computing Center. ∗ [email protected][1] C. Joshi and T. Katsouleas, Physics Today , 47 (2003).[2] M. Chen, Z.-M. Sheng, Y.-Y. Ma, and J. Zhang, Journalof applied physics , 056109 (2006).[3] E. Oz et al., Phys. Rev. Lett. , 084801 (2007).[4] A. Pak et al., Phys. Rev. Lett. , 025003 (2010).[5] C. McGuffey et al., Phys. Rev. Lett. , 025004 (2010).[6] C. E. Clayton et al., Phys. Rev. Lett. , 105003 (2010).[7] J. S. Liu, C. Q. Xia, W. T. Wang, H. Y. Lu, C. Wang,A. H. Deng, W. T. Li, H. Zhang, X. Y. Liang, Y. X.Leng, et al., Phys. Rev. Lett. , 035001 (2011).[8] B. B. Pollock et al., Phys. Rev. Lett. , 045001 (2011).[9] N. Vafaei-Najafabadi, K. A. Marsh, C. E. Clayton,W. An, W. B. Mori, C. Joshi, W. Lu, E. Adli, S. Corde,M. Litos, et al., Phys. Rev. Lett. , 025001 (2014).[10] B. Hidding et al., Phys. Rev. Lett. , 035001 (2012).[11] F. Li et al., Phys. Rev. Lett. , 015003 (2013).[12] N. Bourgeois, J. Cowley, and S. M. Hooker, Phys. Rev.Lett. , 155004 (2013).[13] A. Martinez de la Ossa, J. Grebenyuk, T. Mehrling,L. Schaper, and J. Osterhoff, Phys. Rev. Lett. ,245003 (2013).[14] L.-L. Yu, E. Esarey, C. B. Schroeder, J.-L. Vay,C. Benedetti, C. G. R. Geddes, M. Chen, and W. P. Leemans, Phys. Rev. Lett. , 125001 (2014).[15] X. L. Xu, Y. P. Wu, C. J. Zhang, F. Li, Y. Wan, J. F.Hua, C.-H. Pai, W. Lu, P. Yu, C. Joshi, et al., Phys. Rev.ST Accel. Beams , 061301 (2014).[16] X. L. Xu et al., Phys. Rev. Lett. , 035003 (2014).[17] W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh,A. Dyson, and C. Joshi, Phys. Rev. Lett. , 321 (1992).[18] W. P. Leemans, C. E. Clayton, W. B. Mori, K. A. Marsh,P. K. Kaw, A. Dyson, C. Joshi, and J. M. Wallace, Phys.Rev. A , 1091 (1992).[19] M. Zeng, M. Chen, L. L. Yu, W. B. Mori, Z. M. Sheng,B. Hidding, D. A. Jaroszynski, and J. Zhang, Phys. Rev.Lett. , 084801 (2015).[20] W. Leemans, C. Geddes, J. Faure, C. T´oth,J. Van Tilborg, C. Schroeder, E. Esarey, G. Fubiani,D. Auerbach, B. Marcelis, et al., Physical review letters , 074802 (2003).[21] R. Fonseca et al., Lecture notes in computer science , 342 (2002).[22] C. D. Decker and W. B. Mori, Phys. Rev. Lett. , 490(1994).[23] M. V. Ammosov, N. B. Delone, and V. P. Krainov, Sov.Phys. JETP , 1191 (1986).[24] P. Mora and T. M. Antonsen Jr, Physics of Plasmas(1994-present) , 217 (1997).[25] W. Lu et al., Phys. Rev. Lett. , 165002 (2006).[26] W. Lu et al., Phys. Plasma , 056709 (2006).[27] S. Wang et al., Phys. Rev. Lett. , 135004 (2002).[28] W. B. Mori, IEEE J. Quantum Electron. , 1942 (1997),and references therein.[29] L. D. Landau, J. Bell, M. Kearsley, L. Pitaevskii, E. Lif-shitz, and J. Sykes, Electrodynamics of continuous media ,vol. 8 (elsevier, 1984).[30] X. Xu, Y. Wu, C. Zhang, F. Li, Y. Wan, J. Hua, C.-H. Pai, W. Lu, P. Yu, W. An, et al., arXiv preprintarXiv:1411.4386 (2014).[31] S. Reiche, Nuclear Instruments and Methods in PhysicsResearch Section A: Accelerators, Spectrometers, Detec-tors and Associated Equipment429