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Physics of a 10 GeV laser-plasma accelerator stage Eric Esarey HBEB Workshop, Nov , C. Schroeder, C. Geddes, E. Cormier-Michel, W. Leemans LOASIS Program Lawrence Berkeley National Laboratory

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Regimes of laser-plasma accelerators Quasi-linear and highly nonlinear (blowout) Limits to single-stage energy gain in a LPA Diffraction, dephasing, depletion Scaling laws for single-stage energy gain Analytic theory and fluid simulations Conceptual design of a laser-plasma collider at 1 TeV Based on 10 GeV stages Requires tens of J laser pulses at tens of kHz Plasma and laser tailoring to improve performance Longitudinal density tapering to eliminate dephasing Higher-order laser modes to control transverse fields Outline Ref: Esarey, Schroeder, Leemans, Reviews of Modern Physics (2009)

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Laser Wakefield Accelerator (LWFA) B.A. Shadwick et al., IEEE PS Standard regime (LWFA): pulse duration matches plasma period Ultrahigh axial electric fields => Compact electron accelerators Plasma wakefields E z > 10 GV/m, fast waves (Conventional RF accelerators E z ~ 10 MV/m) Plasma channel: Guides laser pulse and supports plasma wave Tajima, Dawson (79); Gorbunov, Kirsanov (87); Sprangle, Esarey et al. (88)

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Conceptual LPA Collider Leemans & Esarey, Physics Today, March 2009 Based on 10 GeV modules Quasi-linear wake: e- and e+ Driven by 40 J, 130 fs pulses 80 cm plasma channels (10 17 cm -3 ) Staging & coupling modules Requires high rep-rate (10’s kHz) Requires development of high average power lasers (100’s kW)

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Basic design of a laser-plasma accelerator: single-stage limited by laser energy laser E z wake Laser pulse length determined by plasma density –k p z ≤ 1, z ~ p ~ n -1/2 Wakefield regime determined by laser intensity –Linear (a 0 1) –Determines bunch parameters via beam loading –Ex: a 0 = 1 for I 0 = 2x10 18 W/cm 2 and 0 = 0.8 m Accelerating field determined by density and laser intensity –E z ~ (a 0 2 /4)(1+a 0 2 /2) -1/2 n 1/2 ~ 10 GV/m Energy gain determined by laser energy via depletion* –Laser: Present CPA technology 10’s J/pulse * Shadwick, Schroeder, Esarey, Phys. Plasmas (2009)

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Linear & blowout regimes: e+/e- acceleration run 405 Blowout regime high field very asymmetric focuses e- defocuses e+ self-trapping Quasilinear linear: symmetric e+/e- high a 0 desired for gradient too high enters bubble a 0 ~1-2 good compromise dark current free e- accel e- focus e+ focus e+ accel a 0 =4 e- accel e- focus e+ focus e+ accel a 0 =1 Axial field Transverse field Plasma density

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“3D”: Diffraction, Dephasing, Depletion Diffraction of laser pulse Z R = r 0 2 / 0 ~ 2 cm, Z R << L dephase < L deplete Solution: Density channels Parabolic channel guides gaussian modes Channel depth: n [cm -3 ] = / (r 0 [ m]) 2 ~ 2x10 16 cm -3 W.P. Leemans et al, IEEE Trans. Plasmas Sci. (1996); Esarey et al., Phys. Fluids (1993) W = E z. L Limits to acceleration length: diffraction

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Dephasing: e - outrun wake, Phase velocity: v p /c ≈ v g /c = /2 p 2 L dephase (1-v g /c) = p /2, L dephase = p 3 / 0 2 ~ n -3/2 ~ 1.6 m Solution: density tapering W = E z. L Limits to Acceleration Length: dephasing e- beam laser EzEz v p

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For a 0 ~ 1, L dephase may be < L deplete Phase velocity depends on density Phase position ~ p ~ n -1/2 Taper density to tune wake velocity Depletion then limits e - energy gain Density Tapering: Phase Lock e- Katsouleas, PRA (1986); Sprangle et al, PRE (2001) e- beam laser EzEz EzEz n1n1 n 2 >n 1 density

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Alternative tapering options: Step density transition (1) (2) (3) (1) (2) (3) Maintains near-resonance of plasma response with laser Experimental realization: staged accelerator sections C. Schroeder et al.

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Depletion: laser loses energy to wake Energy balance: E L 2 z = E z 2 L deplete Linear limit a 0 2 > L dephase Nonlinear limit a 0 2 >> 1: L deplete ~ L dephase W = E z. L Limits to acceleration length: depletion EzEz laser EzEz Solution: staging

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Rate of laser energy deposition Developed theory of nonlinear short-pulse laser evolution. Derived general energy evolution equation valid for any laser intensity and pulse shape Scale separation (laser frequency >> plasma frequency) Neglect backward going waves (Raman backscatter) Model plasma as cold fluid Apply quasi-static approximation (laser slowly varying compared to plasma response): Shadwick, Schroeder, Esarey, Physics of Plasmas (2009) characteristic accelerating field:

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Nonlinear plasma wave excitation by a Gaussian laser pulse Peak plasma wave driven by Gaussian laser insensitive to pulse duration (broad resonance) over intensity regime of interest

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Pump depletion length independent of intensity for ultra-intense pulses Pump depletion length for near-resonant Gaussian laser pulse: Pump depletion length: Characteristic length scale independent of intensity for relativistically-intense laser pulses

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Single stage energy gain limited by laser energy depletion Diffraction limitation: mitigated by transverse plasma density tailoring Dephasing limitation: mitigated by longitudinal plasma density tailoring Depletion: necessitates multiple stages Multiple-stages for controlled acceleration to high energy: Depletion Length: Energy gain (linear regime): laser + channel … Ex: W stage = 10 GeV for I = W/cm 2 and n = cm -3 Accelerating field:

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16 Scaling laws: analytic theory

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Laser pulse evolution Laser energy evolution: Laser field plasma density accelerating field ω p t=500 ω p t=1500 ω p t=2500 ω p t=3500 Laser evolution interplay between laser intensity steepening, laser frequency red-shifting, energy depletion Shadwick, Schroeder, Esarey, Physics of Plasmas (2009)

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18 Longitudinal e-bunch dynamics: energy spread minimum near dephasing Laser Wake Position, k p (z-ct) Fluid plasma + e-bunch described by moments (includes beam loading) B.A. Shadwick et al. Time, p t Momentum Energy spread e-bunch Energy spread Initial: / = 0.3% at = 100 Final: / = 0.01% at = 3000

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Scaling laws from fluid code: dephasing/depletion lengths & energy gain Fraction of laser energy at dephasing length Independent of k/k p Fix laser parameters (a 0, k p L 0, k p r 0 ), increase (k/k p ) to increase energy Energy and dephasing length from 1D fluid simulations a 0 =1: max = 0.7(k/k p ) 2, k p L dp = 4(k/k p ) 2 a 0 =1.5: max = 1.3(k/k p ) 2, k p L dp = 3.5(k/k p ) 2 a 0 =2: max = 2(k/k p ) 2, k p L dp = 3(k/k p ) 2 Quasi-linear: a 0 ~ 1 Dephasing ~ depletion Good efficiency

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Point designs: 10 and 100 GeV Laser power: P[GW] = 21.5(a 0 r 0 / ) 2, Critical power: P c [GW] = 17(k/k p ) 2, P/P c = (a 0 k p r 0 ) 2 /32. All assume: k p L 0 = 2, m a0a0 P/P c P(PW)WLWL t 0 (fs) r 0 ( m) p ( m) n 0 (cm -3 )L dp W e (GeV) J cm J cm J m kJ m kJ m kJ m100

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Parameter design for GeV and beyond P(PW) (fs) n p (cm - 3 ) w 0 ( m) L(m)a0a0 ∆n c /n p Q(nC)E(GeV) % % % % % % % % % Note: Channel guiding: 60% and 40%; Self-guiding: 0%; external injection: 60%; self-injection: 40% and 0% P/Pc=0.7 for 60% case, and 2 for 40% case W. Lu et al., Phys Rev STAB (2007)

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Beam loading simulations predicts pC for 10 GeV stages Quasi-linear beam loading matches linear theory density & k p L: k p r = k p L =2, a 0 =1 n 0 = cm -3 +* k p L =2, a 0 =1 n 0 = cm -3 +* k p L =1, a 0 =1.4 n 0 = cm D * 3D -- theory VORPAL PIC simulations 500 pC at cm -3 for k p L=2, k p r ~ 2 10% of laser energy to electrons Bunch length & profile alters field inside bunch flatten field across bunch – reduces E focusing must be matched for emittance Ongoing: precise control w/shaped bunches ~constant field inside bunch * Cormier-Michel et al, Proc. AAC 2008, **Katsouleas PRA 1986 Beam loading theoretical limit e-bunch wake = laser wake Linear theory, k p z < 1, k p r ~ 1 N b ~ 9x 9 (n 0 16 cm -3 ) -1/2 (E z /E 0 ) Ex.: N b = 3x10 9 (0.5 nC) for n 0 17 cm -3 and E z /E 0 =1

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Linear theory Symmetric bunches Energy spread ~ N/N max Efficiency ~ N/N max (2 - N/N max ) Ex: Spread 100%, Effic 100% as N N max Triangle bunches (high density in front) Load wake with constant E z inside bunch Can minimize energy spread with high efficiency (at reduced E z ) Requires density tapering to phase lock bunches Beam loading: tailored bunches for high efficiency T. Katsouleas et al., Part. Accel. 22, 81 (1987) Blowout regime: M. Tzoufras, et al., PRL (2008 )

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Adjusting length flattens field for minimum energy spread Gaussian bunch Length adjusts wake loading within bunch Bunch & laser wakefield nearly balanced even for symmetric bunches Flattens field across bunch – reduces E Shaped bunch can further reduce E Beam loading versus bunch length no charge L = m L = 0.85 m L = 0.51 m k p r = 0.3 scaled charge 60pC

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Axial density taper locks bunch phase: improves gain and reduces E for e +,e - Compensate dephasing by changing p ~ n 1/2 Linear taper at k p L=2 produces 4x gain Positron acceleration ~symmetric Ongoing: optimize taper, emittance matching initial k p L=1 results : 50% depletion, 10 GeV gain for 300 pC, 2.5%FWHM Spectra at dephasing gain in stages with k p L=2 at cm -3 50% beam loaded -k p r = 1, k p L = 0.5 3D charge: 22.5pC 225pC, 9 GeV gain, 4% FWHM, at cm -3 Taper no taper 0 Gain [GeV/c at /cc] Scale Gain [GeV/c at /cc] 12 0 #/GeV/c [A.U.] 1 __ e- --e+

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Matched electron beam spot size is small Matched beam spot size linear regime bubble regime matched beam < 1 m (<< p ~ 100 m) for = 20,000 (10 GeV), n 0 = cm -3, n = 1 mm mrad Limits electron beam charge and quality Increase y for higher charge, with n bpeak small In linear regime k 2 ∝ E ⊥ ∝ ∇ ⊥ a 2 Reduce transverse field gradient to increase matched beam radius 26

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Higher order laser mode to tailor transverse wakefield Linear regime : E ⊥ /E 0 ~ ∇ ⊥ a 2 /2 Add first order Hermite-Gaussian mode in 2D y/r 0 gaussian first order hermite- gaussian mode exact solutions of the paraxial wave equation HG 0 HG 1 y/r 0 a2a a 1 /a 0 a 2 = a 0 2 HG a 1 2 HG 1 2 E y /E 0 y/r 0 analytic calculation (low a) no channel

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Higher order mode propagation in plasma channel Hermite-Gaussian modes exact solution of the linear paraxial wave eq Guiding in plasma channel is the same for all modes n = n c = 1/ r e r 0 2 Phase / group velocity different for each mode Intensity modulation when modes co- propagate Low intensity propagation in matched plasma channel integrated transverse intensity profile (HG 0 + HG 1 ) 2 (HG HG 1 2 ) HG 0 2 HG 1 2 kpykpy kpykpy Solution Use cross-polarization Use different frequencies k beat = m/Z R k beat >> k p a 0 =0.1a 0 =0.1 a 1 =0.1 a 1 =0.5a 0

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Transverse field tailoring in the quasi linear regime Wakefield driven by higher order modes in the quasi linear regime a 0 =1 Transverse field flattened by flat top laser profile Mode propagation to depletion short pulse k p L = 1 minimizes pulse variations shallower plasma channel compensates for self-focusing 200 X(µm ) 225 Y(µm) 30 X(µm ) -30 Y(µm) X(µm ) Y(µm) 30 X(µm) y = -1 m (y/w 0 ~ 0.1) ___ E x /E E y /E 0 higher order mode ….. E y /E 0 gaussian 1935 X(µm ) 1965 Y(µm) 30 X(µm ) -30 Y(µm) X(µm ) Y(µm) Y(µm) 30 X(mm)0 4 integrated laser intensity profile laser envelope EyEy ExEx high order mode reduces Ey, laser envelope ExEx EyEy

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Design considerations for a laser-plasma collider module Diffraction, Dephasing, Depletion: necessitates staging Conceptual design of laser-plasma collider at 1 TeV Quasi-linear wake (a 0 ~ 1), electrons and positrons 10 GeV modules: Laser pulse 40 J, 130 fs, 10 kHz Requires development of 100’s kW average power (10 kHz) lasers Requires research on LWFA physics and staging technology Demonstrate low emittance, high charge, short e-bunches Plasma and laser tailoring to improve performance Longitudinal density tapering to eliminate dephasing Higher-order laser modes to control transverse fields BELLA will give us the capabilities to study 10 GeV stages Summary

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Additional information

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Linac length will be determined by staging technology L stage LPA Laser L acc LcLc Conventional optics (~10 m) Plasma mirror (~10 cm) Number of stages: Proper choice of plasma density and staging minimizes main linac length

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0.5 TeV γ-γ Collider Example Plasma density scalings: Stage density scalings: Collider density scalings (for fixed luminosity):

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ne (1/cm^3)2.0e18 a01 lambda_p(um)24 kp*L_laser2 tau (fs)25 w0 (µm)20 kp*w05.3 P(TW)14 P/Pc J 10 GeV ~300pC 10 GeV gain with efficient loading accessible on BELLA ne (1/cm^3)1.0E+17 a01.4 lambda_p(um)108 kp*L_laser1 tau (fs)57 w0 (µm)90 kp*w05.3 P(TW)563 P/Pc J 0.4 GeV ~50pC 300 pC 10 GeV stage with p L=1 Demonstrated control by shaping laser, plasma, ebunch Initial efforts reduced E10% 2.5% shaped bunches & taper in progress matching bunch emittance, shape to structure

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Laser mode controls transverse field, controls bunch emittance matching scale 20GV/m 1070 X(µm) Y(µm) scale 60GV/m 1070 X(µm) Y(µm) 30 Laser Envelope Scale X(µm) Y(µm) 30 * Cormier-Michel et al, in prep. Emittance matched bunch radius << p for Gaussian-laser linear, nonlinear regimes can reduce loading efficiency and/or cause ion motion Linear regime: Fields shaped via laser mode to compensate emittance* demonstrated propagation, channel compensation Ongoing: compensation of beam loaded fields Propagation to depletion 0ct(Z R ) k p x

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2D PIC simulations demonstrate a factor 3 in matched electron beam radius With higher order mode and delay beam radius can be increased x3 charge x9 Beam radius limited by linear region of focusing field Can increase flat top region by using higher order modes 36 simulation at n 0 = 5x10 18 cm -3 matched emittance mm mrad varies < 1% scaled parameters at cm -3 y = 2 m n = 0.1 mm mrad y ( m) ct (mm)

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Energy depletion: Analytic result in good agreement with numerical solution Analytic result ( ) :

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38 Axial wakefield Energy gain Fluid simulations: verify and quantify scaling laws Laser pulse 1D fluid code ( B.A. Shadwick) - Standard LWFA regime - a 0 = 1.5, k 0 /k p = 40, k p L =2 - Laser: 0.8 m, 5x10 18 W/cm 2, 30 fs - Plasma: cm -3, 3 cm 1 GeV =z-ct E. Esarey et al., AAC Proc 2004 GeV-class example:

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39 distance monemtum Fluid simulation of scaled BELLA point design Scaled point design example: 1D fluid code ( B.A. Shadwick) - Quasi-linear LWFA regime - a 0 = 1.0, k 0 /k p = 40, k p L =2 - Laser: 0.8 m, 2x10 18 W/cm 2, 40 fs - Plasma: cm -3, - Bunch: k p z = 0.5, = 0.9% (initial), 0.05% 0.5 GeV) n EzEz bunch energy spread

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Reducing energy spread and emittance requires controlled injection Self-injection experiments have been in bubble regime: Cannot tune injection and acceleration separately Emittance degraded due to off-axis injection and high transverse fields. Energy spread degraded due to lack of control over trapping ⇒ Use injector based on controlled trapping at lower wake amplitude and separately tunable acceleration stage to reduce emittance and energy spread Y[µm] X[µm] Transverse motion

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