Field-reversed bubble in deep plasma channels for high quality electron acceleration
A. Pukhov, O. Jansen, T. Tueckmantel, J. Thomas, I. Yu. Kostyukov
FField-reversed bubble in deep plasma channels for high quality electron acceleration
A. Pukhov , O. Jansen , T.Tueckmantel , J. Thomas and I. Yu. Kostyukov , Institut fuer Theoretische Physik I, Universitaet Duesseldorf, 40225 Germany Lobachevsky National Research University of Nizhni Novgorod, 603950, Nizhny Novgorod, Russia and Institute of Applied Physics RAS, Nizhny Novgorod 603950, Russia
We study hollow plasma channels with smooth boundaries for laser-driven electron accelerationin the bubble regime. Contrary to the uniform plasma case, the laser forms no optical shock andno etching at the front. This increases the effective bubble phase velocity and energy gain. Thelongitudinal field has a plateau that allows for mono-energetic acceleration. We observe as low as − r.m.s. relative witness beam energy uncertainty in each cross-section and 0.3% total energyspread. By varying plasma density profile inside a deep channel, the bubble fields can be adjustedto balance the laser depletion and dephasing lengths. Bubble scaling laws for the deep channel arederived. Ultra-short pancake-like laser pulses lead to the highest energies of accelerated electronsper Joule of laser pulse energy. PACS numbers: PACS1
The laser wake field acceleration (LWFA) [1] in un-derdense plasmas provides an option for high gradientparticle acceleration [2]. Especially efficient is the bubbleregime [3] when the laser expels background plasma elec-trons from the first half of the plasma wave. The advan-tage of the cavitated region is the transverse uniformityof its axial accelerating field [4] so that the energy gain ofthe accelerated electrons is not affected by their lateralmotions in the bubble and the bunch can remain mo-noenergetic. The quasi-monoenergetic electron bunchesare readily registered in experiments [5].Despite various theoretical approaches: the phe-nomenological model of the bubble [4], the nonlinear the-ory of blowout regime [6, 7] and the similarity theory[8], - a self-consistent analytical description of the bub-ble regime is still absent.The bubble theories exist for homogeneous plasmasonly. Similarity theory together with energy conserva-tion arguments resulted in the “optimal” scaling laws forhomogeneous plasma [9]. These were tested in 3d PICsimulations [10]. The scalings assume that the laser en-ergy is converted into the bubble fields first and thenharvested by the electron bunch. The leading similarityparameters for the bubble in homogeneous plasmas arethe S − number S = n e /an c (cid:28) and the pulse aspectratio Π = cτ /R ≤ . Here, n e is the plasma electrondensity and n c = π/r e λ is the critical plasma densityfor a laser pulse with the wavelength λ , a = eE /mcω is the dimensionless laser amplitude, ω = 2 πc/λ , and r e = e /mc is the classical electron radius, τ is the pulseduration and R is its radius.In a uniform plasma, the bubble regime has some draw-backs. First, the laser energy depletion length is shorterthan the electron dephasing length. This limits themaximum electron energy gain. Second, the transversebubble fields acting on the accelerated electron bunchare strongly focusing. Thus, electrons running forwardwith the relativistic factor γ oscillate about the bub- ble axis at the betatron frequency ω β = ω p / √ γ , where ω p = (cid:112) πe n e /m is the background plasma frequency.A relativistic electron can easily come into betatron res-onance with the Doppler-shifted laser that may result inenergy exchange [11]. This betatron resonance broadensthe electron bunch energy distribution and deterioratesthe beam quality [12, 13].Here, we consider the bubble regime of electron accel-eration, but in a deep plasma channel rather than in auniform plasma. This allows for a significant improve-ment of the acceleration . We demonstrate that (i) theeffective bubble phase velocity and energy gain increase inthe channel; (ii) the longitudinal field has a plateau thatallows for mono-energetic acceleration; (iii) the focusingforce acting on the accelerated bunch is strongly reduced;(iv) the bubble scaling laws and the bubble field distribu-tion for the deep channel are derived; (v) according tothe new scaling laws, ultra-short pancake-like laser pulsesmatch the dephasing and depletion length in the channeland thus lead to the highest energy gains of acceleratedelectrons per Joule of laser pulse energy . In simulations,we observe as low as − r.m.s. relative witness beamenergy uncertainty in each cross-section and 0.3% totalenergy spread. The lack of focusing in the channel elim-inates the very possibility of a betatron resonance andleads to much sharper beam energy distributions.Normally, plasma channels are used to guide weaklyrelativistic pulses over distances much larger than theRayleigh length Z R = πR /λ . These channels are shal-low as a rule, i.e., the relative on-axis plasma density de-pletion is slight. Schroeder et al. [14] suggested recentlythe use of nearly hollow plasma channels to provide inde-pendent control over the focusing and accelerating forces.Chiou et al. [15] revealed that the transverse electricfield of the wake is not monotonic in the hollow channel.However, these papers were limited only to a quasilinearregime of the plasma wake generation and a rectangularchannel density profile. a r X i v : . [ phy s i c s . p l a s m - ph ] N ov W L , J 2.2 17.6 141 τ , fs 4 8 16 R , µ m 8 16 32 n , 1/cm · . · . · R ch , µ m 6 12 24 l i , µ m 0.8 1.6 3.2 N i , pC 10 20 40 E i , GeV 0.1 0.4 1.6 L A , cm 1.6 12.8 100 E dcmax , GeV 1.5 6 (6) 23.5 (24)Table I: Laser-plasma parameters used in the simulations andthe observed electron energies. Expected energies from thescaling law are in brackets. All the lasers had a = 10 ; thechannel parameter δ = 0 . . A plasma channel is not required for laser guiding inthe bubble regime, since the laser pulse is self-guided bythe cavitated region of the bubble. Thus, a preformedshallow plasma channel is not expected to change thebubble dynamics significantly. However, as will be seen,a deep plasma channel i.e., one that is (nearly) empty on-axis, can strongly modify the bubble fields, the nonlinearlaser dynamics, and the trapping. We are looking forlaser-plasma parameters that maximize energy of the ac-celerated electron bunch and improve its quality: mainly,reduce the energy spread.The laser pulse energy W L is the technologically impor-tant characteristics of a laser pulse. It is limited, e.g. bysize of the active crystal or by compression gratings, etc.We are comparing the energy scalings for the two cases:the uniform plasma case and the deep channel case.In uniform plasma, the similarity scalings for electronbunch energy E max from a bubble [9] can be expressed interms of the laser pulse power P L , duration τ and energy W L = P L τ : E uniformmax ∝ (cid:112) P L e cτ λ − = λ − (cid:112) W L e cτ (1)The scaling of Eq. (1) favors longer laser pulses. This re-flects the mechanism of laser pulse depletion in the stan-dard bubble. The laser pulse interacts with plasma elec-trons at the very front only. The major part of the pulsepropagates freely in the cavitated region. Thus, the pulsetail slowly overtakes its head, where an “optical shock”is formed and the pulse “etches”. This “etching” leadsto the pulse shortening [16] and faster dephasing that inturn lowers the maximum energy gain. The highest elec-tron energies are achieved with “spherical” laser pulses,whose duration cτ equals their radius R [10]. Such pulsesfill the cavitated region completely and interact with theaccelerated bunch affecting its quality.Also, in the uniform plasma case, the bubble may con-tinuously trap electrons causing strong beam loading andthus large energy spread. Also, the accelerating field Figure 1: Simulation results for the case R = 32 µ m. a) Thecross-section of the plasma electron density and the longitu-dinal bubble field; b) on-axis accelerating field of the bubble,GV/m. Flattening of E z at the bubble rear part is visible.The broken lines in frame (a) mark the vacuum part of theplasma channel. The purple rippling of the E z at the bubblehead is caused by the short laser pulse. in the standard bubble is a linear function of the lon-gitudinal coordinate [4, 6]. If one wants a truly mono-energetic acceleration of externally injected particles, thewitness bunch to be accelerated must be extremely short.Demanding, e.g. 1% energy spread, the witness bunchshould not occupy more than 1% of the bubble length.The use of the deep plasma channel gives us severalnew control parameters to improve the acceleration. Weconsider channels with a zero plasma density on-axis andsome smoothly growing density at the borders. We havechosen the following parameterization for the plasma ra-dial profile: n e = n [ δ exp( r/R ch ) − , for r ≥ r and n e = 0 for r < r , where r = R ch ln(1 /δ ) . Such pro-files can be produced by ablation from walls of an emptycapillary [17]. The parameters n , δ, R ch provide enoughfreedom to adjust fields in the channel while ensuring asmooth guiding of the relativistically intense laser pulse.In simulations, we reduced the maximum plasma densityfar from the channel axis for numerical stability.The freedom on the bubble fields in the deep channelallows the achievement of much higher electron energieswith a laser of a given pulse energy. Let us assume thatthe accelerating bubble field E is completely at our dis-posal and that the laser pulse energy is converted pri-marily into that field. The laser pulse energy scales as W L ∝ R Icτ , where I is the laser intensity. We omitdimensionless numerical factors like π etc. as we are in-terested in parametric dependencies only. The numericalpre-factors can be obtained from gauging the scaling lawsagainst PIC simulations.We can find the pulse depletion length L d by com- Figure 2: Transverse electric field of the bubble for the laserradius R = 40 λ and a = 10 . a) Homogeneous plasma: thefield is focusing. b) Channel: the field reverses its sign in thechannel walls; the smooth curve gives the analytic solutionfor the field E y . paring the energy deposited in the wake field W wake ∝ E R L d and the initial pulse energy, W wake = W L : L d ∝ Icτ E − . (2)Here we assume a cylindrically symmetric bubble whoseradius scales together with the laser radius R .The particle acceleration is limited either by the laserpulse depletion (2), or by electron dephasing. In theuniform plasma case, it was the laser pulse depletionthat limited acceleration in the bubble regime. In thedeep channel case of interest here, however, the depletionlength can be adjusted by properly choosing the acceler-ating field E , as can be seen from (2). The maximumparticle energy is achieved when the depletion length L d equals the dephasing length L A . Otherwise, the maxi-mum energy gain is limited by the shortest of the twolengths and is lower. The dephasing length scales as L A ∝ Rγ , where γ L is the relativistic γ − factor asso-ciated with the laser group velocity. In a deep plasmachannel, γ L is not influenced by the plasma and is de-fined solely by the laser pulse radius so that we can write γ L ∝ k R , where k = 2 π/λ . For the dephasing lengthwe obtain L dcA ∝ k R (3)We reuire the depletion length of Eq. (2) to be equalthe dephasing length (3) and obtain for the bubble field E ∝ (cid:112) Icτ /k R in the deep plasma channel. This leadsto the maximum energy gain scaling E dcmax ∝ eL A E ∝ e (cid:113) Icτ k R = k (cid:112) mc W L r e R (4)The deep channel scaling (4) differs dramatically from theuniform plasma bubble scaling (1). While the uniformplasma case requires the maximum pulse duration τ , in the deep channel, pulses with radius R as large as possiblehave an advantage.A deeper insight in the bubble field configuration pro-vides the quasistatic theory, which implies that the bub-ble slowly evolves in time and the bubble fields depend on ξ = t − z [18–20]. The nonlinear theory of the bubble inhomogeneous plasma [6] can be generalized to transver-sally inhomogeneous plasma (details be published else-where). It follows from the theory that the bubble fieldnot too close to the edge of the bubble is defined by thesource function s ( r b ) ≡ (1 / r b ) ´ r b ρ ion ( r ) rdr : E ≈ e r (cid:20) r b s ( r ) r − rs ( r b ) (cid:21) − e z ξs ( r b ) , B ≈ − rs ( r b ) . (5)For homogeneous plasma ρ ion ( r ) = 1 we recoverknown expression for the bubble field [4] E ≈ e r r/ − e z ξ/ and B ≈ − e ϕ r/ . For the plasma channelwith exponential profile discussed above the source func-tion takes a form s ( r ) = | e | n (cid:0) r − r + 2 ηr ch δ (cid:1) / (cid:0) r b (cid:1) ,for r ≥ r and s ( r ) = 0 for r < r , where η =exp( r/r ch ) ( r − r ch ) + exp( r /r ch ) ( r ch − r ) . The ob-tained solution says that inside the vacuum part of theplasma channel r < r the bubble field is purely electro-magnetic B ϕ = E r = 2( r/ξ ) E z . In the channel walls, theion field is added and can reverse the sign of the transver-sal electric field (see Fig.2) . Another conclusion is thatthe plasma channel reduces the gradients of the accelerat-ing and focusing forces in comparision with homogeneousplasma.To check the quasi-static theory and the energy scal-ings, we perform a series of 3d PIC simulations usingthe code VLPL [21]. We run the code in laboratoryframe for simulations with the smallest pulse radius. Thelaser pulse is circularly polarized and has the envelope a ( t, r ) = a exp (cid:0) − r /R − t /τ (cid:1) . The envelope is cutto zero at t = 2 τ, r = 2 R . We assume laser wavelength λ = 800 nm.We observe very little or no self-injection in the bubblewhen an empty on-axis channel is used. Thus, we injectan external co-propagating witness electron bunch at theend of the bubble. It has the half-length l i , initial energy E i , and the total charge N i . The simulation parametersare collected in Table I.We selected an extremely short laser pulse with τ =4 fs, R = 8 µ m, and energy of W L = 2 . J as the firstmember for the sequence of three in our power of 2 scal-ing sequence for both pulse duration and radius, as givenin Table I. Although such short and energetic pulses donot exist yet, projects are under way to achieve similarparameters. According to the similarity rule, we scale si-multaneously the laser pulse radius and duration withthe same factor α = 2 from stage n to stage n + 1 : R ( n +1) = αR ( n ) and τ ( n +1) = ατ ( n ) so that the aspectratio Π =
R/cτ ≈ . remains fixed, i.e. the pulses are Figure 3: Simulation results for the case R = 32 µ m( R = 40 λ ). a) Energy spectra of the witness beam at differ-ent acceleration stages. b) Evolution of relative r.m.s. spec-tral width σ E i . The homogenous acceleration lasts for thefirst 30 cm and minumum spectral width of 0.3% is reached.Later, the bunch gains positive energy chirp due to dephas-ing and the spectrum widens. c) Longitudinal phase spaceof the witness bunch at different propagation distances. d)The final phase space of the bunch zoomed. The local energyuncertainty is − and is limited by numerical resolution.Figure 4: Nonlinear evolution of the laser pulse for the sim-ulation case R = 32 µ m. No typical for bubble regime pulseshortening or “optical shock” formation. pancake-like. This allows us to reproduce the wake field(the bubble) exactly without additional search for theoptimal plasma channel parameters. The other param-eters scale as R ch(n+1) = αR ch(n) , n e ( n +1) = α − n e ( n ) , E ( n +1) = α − E ( n ) , L A(n+1) = α L A( n ) , and the parti-cle energy scales as E max( n +1) = α E max( n ) . The threescaled cases are digested in Table I. The most ener-getic pulse with τ = 16 fs, radius R = 32 µ m and en-ergy of W L = 141 J would correspond to the plannedApollon laser [22]. The acceleration lengths range from L A = 2 cm for the shortest laser, to L A = 1 m for thelargest laser pulse. These distances can be simulated onlyusing the Lorentz boost [23].The bubble generated by the largest laser pulse radius R = 32 µ m is shown in Fig.1. The accelerating field, Fig.1(a), is transversely uniform. It allows for mono-energetic acceleration of wide electron bunches. Thistransverse field uniformity is also observed in homoge-neous plasmas. The on-axis profile of the acceleratingfield in the channel, Fig.1(b), however, differs from thatin the uniform plasma case. There is a region of flataccelerating field at the very back of the bubble. Thisregion can be used to accelerate reasonably long witnessbunches mono-energetically.Fig.2 shows the bubble transverse fields in the bubblecorresponding to the case R = 40 λ and a = 10 . Wecompare predictions of the quasi-static model with thenumerical simulations. In the bubble center, dE y /dy ≈ . , dE z /dξ ≈ . ≈ dE y /dy ) in the simulation and dE y /dy ≈ . , dE z /dξ ≈ .
16 = 2( dE y /dy ) in the mod-els that is in a very good agreement. Fig.2(b) clearlyshows the field reversal in the channel.To check the acceleration, we inject a witness bunchthat occupies about 10% of the bubble length. Theenergy spectra of the accelerated bunch are shown inFig.3(a) at different times. The bunch stays very mo-noenergetic for the first 30 cm of acceleration, because itwas injected in the flat accelerating field part of the bub-ble. The relative r.m.s. energy spread σ E / E , Fig.3(b), ofthe bunch is merely 0.3% when it gains 7.5 GeV energy.This is much better than the ratio of the bunch length tothe bubble length that is 10%. Later, the bunch slowlyleaves this flat E − field region and advances into the re-gion with linearly growing E z − field. Finally, it gains apositive energy chirp as seen in the bunch longitudinalphase space, Fig.3(c).However, the quality of acceleration is best understoodwhen we zoom in at the longitudinal phase space of thewitness bunch, Fig.3(d). The r.m.s relative width of thewitness bunch energy uncertainty taken at a particularlongitudinal position is merely − and is probably de-fined by numerical resolution of our code. This very nar-row energy spread is due to (i) the transverse uniformityof accelerating field in the bubble and (ii) the lack of abetatron resonance in the hollow channel.Fig. 4 shows the nonlinear evolution of the laserpulse. Very different from the uniform plasma case, weobserve no laser pulse shortening and no optical shockformation. The channel parameters were chosen to bal-ance the dephasing length and the laser depletion length: L A = L D = 1 m.In conclusion, we have shown that electron accelera-tion in deep plasma channels is scalable and the energyscalings favor ultra-short pancake-like laser pulses. Theaccelerating field has a nearly flat field region at the rear,where the accelerating field depends only slightly on thelongitudinal coordinate. This allows for nearly monoen-ergetic acceleration of a witness bunch.This work has been supported by the DeutscheForschungsgemeinschaft via GRK 1203 and SFB TR 18,by EU FP7 project EUCARD-2 and by the Governmentof the Russian Federation (Project No. 14.B25.31.0008)and by the Russian Foundation for Basic Research(Grants No. 13-02-00886, 13-02-97025). [1] E. Esarey, C. B. Schroeder, and W. P. Leemans, Rev.Mod. Phys. , 1229 (2009)[2] C. Joshi and A. Caldwell, “Plasma Accelerators,” Hand-book for Elementary Particle Physics, Volume III, editedby S. Myers and H. Schopper, Heidelberg, Springer(2013), 12-1.[3] A. Pukhov, J. Meyer-ter-Vehn, Applied Physics B 74 ,355 (2002)[4] I. Kostyukov, A. Pukhov, S. Kiselev, Phys. Plasmas ,5256 (2004)[5] V. Malka Laser Plasma Accelerators, in Laser-Plasma In-teractions and Applications, Scottish Graduate Series, pp281-301 (2013)[6] W. Lu et al. PRSTAB , 061301 (2007)[7] W. Lu et al., Phys. Rev. Lett. , 165002 (2006).[8] S. Gordienko, A. Pukhov Physics of Plasmas , 043109 (2004).[9] A. Pukhov, S. Gordienko, Phil Tr. R. Soc. A, , 623(2006).[10] O. Jansen, T. Tückmantel, and A. Pukhov, Eur. Phys.J. Special Topics , 1017–1030 (2014)[11] A Pukhov, ZM Sheng, J Meyer-ter-Vehn, Physics of Plas-mas , 2847 (1999)[12] S. Cipiccia et al Nature Physics
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