1. Physics of a 10 GeV laser-plasma accelerator stage Eric Esarey HBEB Workshop, Nov 16 -19, 2009 http://loasis.lbl.gov/ C. Schroeder, C. Geddes, E. Cormier-Michel, W. Leemans LOASIS Program Lawrence Berkeley National Laboratory

2.  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)

3. Laser Wakefield Accelerator (LWFA) B.A. Shadwick et al., IEEE PS. 2002 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)

4. 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)

5. 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)

6. 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

7.  “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 ] = 10 20 / (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

8.  Dephasing: e - outrun wake,  Phase velocity: v p /c ≈ v g /c = 1- 0 2 /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 <c e- beam laser EzEz v p <c

9.  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

10. 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.

11.  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

12. 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:

13. 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

14. 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

15.  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 = 10 18 W/cm 2 and n = 10 17 cm -3 Accelerating field:

16. 16 Scaling laws: analytic theory

17. 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)

18. 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

19. 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

20. 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) 22.20.3840 J985380 1.7  10 17 38 cm10 1.51.10.3040 J1306399 1.1  10 17 79 cm10 10.450.2240 J17082140 6.0  10 16 2.4 m10 22.23.81.3 kJ310170250 1.7  10 16 12 m100 1.51.13.01.3 kJ390200310 1.1  10 16 25 m100 10.452.21.3 kJ550260430 6.0  10 15 78 m100

21. 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) 0.02030 1  10 18 140.0161.7660%0.180.99 0.04030 1.5  10 18 140.0112.5340%0.250.95 0.10030 2.0  10 18 150.0093.780%0.401.06 0.200100 1.0  10 17 450.521.7660%0.579.9 2.0100 3.0  10 17 470.185.450%1.810.2 2.0310 1.0  10 16 14016.31.7660%1.899 40330 4.0  10 16 1464.27.60%8106 201000 1.0  10 15 4505001.7660%5.7999 1000 6.5  10 15 4608212.10%401040 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)

22. Beam loading simulations predicts 300-500 pC for 10 GeV stages Quasi-linear beam loading matches linear theory density & k p L: k p  r = 0.3 11.8 k p L =2, a 0 =1 n 0 = 10 18 cm -3 +* k p L =2, a 0 =1 n 0 = 10 19 cm -3 +* k p L =1, a 0 =1.4 n 0 = 10 19 cm -3 + + 2D * 3D -- theory  VORPAL PIC simulations  500 pC at 10 17 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

23.  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 )

24. 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 = 0.085  m  L = 0.85  m  L = 0.51  m k p  r = 0.3 scaled charge 60pC

25. 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 10 19 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 10 17 cm -3 Taper no taper 0 Gain [GeV/c at 10 19 /cc] 0.12 0 Scale Gain [GeV/c at 10 17 /cc] 12 0 #/GeV/c [A.U.] 1 __ e- --e+

26. 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 = 10 17 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

27. 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 a2a2 0.7 0.5 0.4 0 a 1 /a 0 a 2 = a 0 2 HG 0 2 + a 1 2 HG 1 2 E y /E 0 y/r 0 analytic calculation (low a) no channel

28. 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 0 2 + 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

29. 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) 30 200 X(µm ) 240 -30 Y(µm) 30 X(µm) -0.3 0.3 -0.03 0.03 -30 200225 200 225 @ y = -1  m (y/w 0 ~ 0.1) ___ E x /E 0 ---- E y /E 0 higher order mode ….. E y /E 0 gaussian 1935 X(µm ) 1965 Y(µm) 30 X(µm ) -30 Y(µm) 30 1935 X(µm ) 1980 -30 Y(µm) 30 -30 19351965 -30 Y(µm) 30 X(mm)0 4 integrated laser intensity profile laser envelope EyEy ExEx high order mode reduces Ey, laser envelope ExEx EyEy

30.  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

31. Additional information

32.  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

33. 0.5 TeV γ-γ Collider Example Plasma density scalings: Stage density scalings: Collider density scalings (for fixed luminosity):

34. 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/Pc0.9 40 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/Pc1.8 0.5 J 0.4 GeV ~50pC  300 pC 10 GeV stage with taper@k 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

35. Laser mode controls transverse field, controls bunch emittance matching Ey @ 5.10 18 scale 20GV/m 1070 X(µm) 1095 -30 Y(µm) 30 Ex@ 5.10 18 scale 60GV/m 1070 X(µm) 1095 -30 Y(µm) 30 Laser Envelope Scale 1 1070 X(µm) 1110 -30 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 )5 --10 10 --k p x

36. 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 0.014 mm mrad varies < 1%  scaled parameters at 10 17 cm -3   y = 2  m   n = 0.1 mm mrad  y (  m) 0.0 0.1 0.2 0.3 1 23 0 ct (mm)

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

38. 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: 10 18 cm -3, 3 cm 1 GeV  =z-ct E. Esarey et al., AAC Proc 2004 GeV-class example:

39. 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: 10 18 cm -3, - Bunch: k p  z = 0.5,  = 0.9% (initial), 0.05% (final@ 0.5 GeV) n EzEz bunch energy spread

40. 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] 5 -5 8002000 Transverse motion