Physics of a 10 Ge V laserplasma accelerator

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

Outline § 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 Te. V § Based on 10 Ge. V stages § Requires tens of J laser pulses at tens of k. Hz § Plasma and laser tailoring to improve performance § Longitudinal density tapering to eliminate dephasing § Higher-order laser modes to control transverse fields Ref: Esarey, Schroeder, Leemans, Reviews of Modern Physics (2009)

Laser Wakefield Accelerator (LWFA) Standard regime (LWFA): pulse duration matches plasma period Ultrahigh axial electric fields => Compact electron accelerators Plasma wakefields Ez > 10 GV/m, fast waves (Conventional RF accelerators Ez ~ 10 MV/m) Plasma channel: Guides laser pulse and supports plasma wave B. A. Shadwick et al. , IEEE PS. 2002 Tajima, Dawson (79); Gorbunov, Kirsanov (87); Sprangle, Esarey et al. (88)

Conceptual LPA Collider § Based on 10 Ge. V modules § Quasi-linear wake: e- and e+ § Driven by 40 J, 130 fs pulses § 80 cm plasma channels (1017 cm-3) § Staging & coupling modules § Requires high rep-rate (10’s k. Hz) § Requires development of high average power lasers (100’s k. W) Leemans & Esarey, Physics Today, March 2009

Basic design of a laser-plasma accelerator: single-stage limited by laser energy • Laser pulse length determined by plasma density – kp z ≤ 1, z ~ p ~ n-1/2 • Wakefield regime determined by laser intensity laser – Linear (a 0<1) or blowout (a 0>1) – Determines bunch parameters via beam loading Ez wake – Ex: a 0 = 1 for I 0 = 2 x 1018 W/cm 2 and 0 = 0. 8 m • Accelerating field determined by density and laser intensity – Ez ~ (a 02/4)(1+a 02/2)-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)

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

DW = Ez. L Limits to acceleration length: diffraction § “ 3 D”: Diffraction, Dephasing, Depletion § Diffraction of laser pulse § ZR = p r 02/ 0 ~ 2 cm, ZR<< Ldephase < Ldeplete § Solution: Density channels § Parabolic channel guides gaussian modes § Channel depth: Dn [cm-3] = 1020 / (r 0[ m])2 ~ 2 x 1016 cm-3 W. P. Leemans et al, IEEE Trans. Plasmas Sci. (1996); Esarey et al. , Phys. Fluids (1993)

DW = Ez. L Limits to Acceleration Length: dephasing § Dephasing: e- outrun wake, vp<c § Phase velocity: vp/c ≈ vg/c = 1 - 02/2 p 2 § Ldephase (1 -vg/c) = p/2, e- beam Ez laser § Ldephase = p 3/ 02 ~ n-3/2 ~ 1. 6 m § Solution: density tapering vp<c e- beam Ez laser

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

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

DW = Ez. L Limits to acceleration length: depletion § Depletion: laser loses energy to wake § Energy balance: EL 2 z = Ez 2 Ldeplete § Linear limit a 02 << 1: Ldeplete = a 0 -2 Ldephase >> Ldephase § Nonlinear limit a 02 >> 1: Ldeplete ~ Ldephase laser Ez Solution: staging laser Ez

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 characteristic accelerating field: § Apply quasi-static approximation (laser slowly varying compared to plasma response): Shadwick, Schroeder, Esarey, Physics of Plasmas (2009)

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

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

Depletion: necessitates multiple stages § 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 Length: Accelerating field: Energy gain (linear regime): § Ex: Wstage = 10 Ge. V for I = 1018 W/cm 2 and n = 1017 cm-3 § Multiple-stages for controlled acceleration to high energy: laser + channel …

Scaling laws: analytic theory 16

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

Longitudinal e-bunch dynamics: energy spread minimum near dephasing Fluid plasma + e-bunch described by moments (includes beam loading) Laser Wake Momentum e-bunch Energy spread Position, kp(z-ct) Energy spread Initial: / = 0. 3% at = 100 Final: / = 0. 01% at = 3000 B. A. Shadwick et al. Time, pt 18

Scaling laws from fluid code: dephasing/depletion lengths & energy gain Fraction of laser energy at dephasing length § Quasi-linear: a 0 ~ 1 § Dephasing ~ depletion § Good efficiency Independent of k/kp § Fix laser parameters (a 0, kp. L 0, kpr 0), increase (k/kp) to increase energy § Energy and dephasing length from 1 D fluid simulations § a 0 =1: max = 0. 7(k/kp)2 , kp. Ldp= 4(k/kp)2 § a 0 =1. 5: max = 1. 3(k/kp)2 , kp. Ldp= 3. 5(k/kp)2 § a 0 =2: max = 2(k/kp)2 , kp. Ldp= 3(k/kp)2

Point designs: 10 and 100 Ge. V a 0 P/Pc P(PW) WL t 0(fs) r 0( m) p( m) n 0(cm-3) Ldp We (Ge. V) 2 2. 2 0. 38 40 J 98 53 80 1. 7 1017 38 cm 10 1. 5 1. 1 0. 30 40 J 130 63 99 1. 1 1017 79 cm 10 1 0. 45 0. 22 40 J 170 82 140 6. 0 1016 2. 4 m 10 2 2. 2 3. 8 1. 3 k. J 310 170 250 1. 7 1016 12 m 100 1. 5 1. 1 3. 0 1. 3 k. J 390 200 310 1. 1 1016 25 m 100 1 0. 45 2. 2 1. 3 k. J 550 260 430 6. 0 1015 78 m 100 Laser power: P[GW] = 21. 5(a 0 r 0/ )2 , Critical power: Pc[GW] = 17(k/kp)2, P/Pc = (a 0 kpr 0)2 /32. All assume: kp. L 0 = 2, m

Parameter design for Ge. V and beyond W. Lu et al. , Phys Rev STAB (2007) P(PW) (fs) np (cm-3) 0. 020 30 1 1018 14 0. 040 30 1. 5 1018 0. 100 30 0. 200 w 0 ( m) L(m) a 0 ∆nc/np Q(n. C) E(Ge. V) 0. 016 1. 76 60% 0. 18 0. 99 14 0. 011 2. 53 40% 0. 25 0. 95 2. 0 1018 15 0. 009 3. 78 0% 0. 40 1. 06 100 1. 0 1017 45 0. 52 1. 76 60% 0. 57 9. 9 2. 0 100 3. 0 1017 47 0. 18 5. 45 0% 1. 8 10. 2 2. 0 310 1. 0 1016 140 16. 3 1. 76 60% 1. 8 99 40 330 4. 0 1016 146 4. 2 7. 6 0% 8 106 20 1000 1. 0 1015 450 500 1. 76 60% 5. 7 999 1000 6. 5 1015 460 82 12. 1 0% 40 1040 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

Beam loading simulations predicts 300 -500 p. C for 10 Ge. V stages Ø Beam loading theoretical limit Ø e-bunch wake = laser wake Ø Linear theory , kp z < 1, kp r ~ 1 Ø Nb ~ 9 x 9 (n 0 16 cm-3)-1/2 (Ez/E 0) Ø Ex. : Nb = 3 x 109 (0. 5 n. C) for n 0 17 cm-3 and Ez/E 0=1 Ø VORPAL PIC simulations ~constant field inside bunch Quasi-linear beam loading matches linear theory + 2 D * 3 D -- theory Ø 500 p. C at 1017 cm-3 for kp. L=2, kp r~ 2 • 10% of laser energy to electrons Ø Bunch length & profile alters field inside bunch • flatten field across bunch – reduces DE • focusing must be matched for emittance density & kp. L: kp r = 0. 3 1 kp. L =2, a 0=1 n 0 = 1018 cm-3 1. 8 +* Ø Ongoing: precise control w/shaped bunches kp. L =2, a 0=1 n 0 = 1019 cm-3 +* * Cormier-Michel et al, Proc. AAC 2008, **Katsouleas PRA 1986 kp. L =1, a 0=1. 4 n 0 = 1019 cm-3 + +* +*

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

Adjusting length flattens field for minimum energy spread § Gaussian bunch § Length adjusts wake loading within bunch Beam loading versus bunch length § Bunch & laser wakefield nearly balanced even for symmetric bunches § Flattens field across bunch – reduces DE § Shaped bunch can further reduce DE no charge L = 0. 085 m L = 0. 51 m kp r = 0. 3 scaled charge 60 p. C

Axial density taper locks bunch phase: improves gain and reduces DE for e+, e- § Linear taper at kp. L=2 produces 4 x gain § Positron acceleration ~symmetric § Ongoing: § optimize taper, emittance matching § initial kp. L=1 results : 50% depletion, 10 Ge. V gain for 300 p. C, 2. 5%FWHM no taper __e- --e+ Taper 0 § Compensate dephasing by changing p ~ n 1/2 #/Ge. V/c [A. U. ] 1 Spectra at dephasing 0 Gain [Ge. V/c at 1019/cc] 0. 12 0 Scale Gain [Ge. V/c at 1017/cc] 12 gain in stages with kp. L=2 at 1019 cm-3 50% beam loaded -kp r = 1, kp L = 0. 5 3 D charge: 22. 5 p. C 225 p. C, 9 Ge. V gain, 4% FWHM, at 1017 cm-3

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 Ge. V), n 0 = 1017 cm-3, en = 1 mm mrad § Limits electron beam charge and quality § Increase y for higher charge, with nbpeak small § In linear regime kb 2 ∝ E⊥∝ ∇⊥a 2 § Reduce transverse field gradient to increase matched beam radius 26

Higher order laser mode to tailor transverse wakefield § Linear regime : E⊥/E 0 ~ ∇⊥a 2/2 § Add first order Hermite-Gaussian mode in 2 D a 2 = a 02 HG 02 + a 12 HG 12 a 1/a 0 0. 7 0. 5 0. 4 0 HG 1 y/r 0 gaussian first order hermitegaussian mode exact solutions of the paraxial wave equation Ey/E 0 y/r 0 analytic calculation (low a) no channel y/r 0

Higher order mode propagation in plasma channel Low intensity propagation in matched plasma channel § Hermite-Gaussian modes exact solution of the linear paraxial wave eq kpy § Guiding in plasma channel is the same for all modes integrated transverse intensity profile HG 02 (HG 0 + HG 1)2 § Dn = Dnc = 1/prer 02 HG 12 a 0=0. 1 a 1=0. 1 § Intensity modulation when modes copropagate kbeat = m/ZR (HG 02 + HG 12) kpy § Phase / group velocity different for each mode § a 0=0. 1 a 1=0. 5 a 0 Solution § Use cross-polarization § Use different frequencies kbeat >> kp

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 Y(µm) -30 1935 X(µm) 1980 1935 X(µm) 1965 integrated laser intensity profile 0. 03 225 1935 X(µm) 1965 0 X(mm) 4 -0. 03 0. 3 -0. 3 Ex X(µm) Y(µm) Ey 200 -30 laser envelope 200 X(µm) 225 30 Y(µm) 200 X(µm) 225 30 200 X(µm) 240 high order mode reduces Ey, @ y = -1 m (y/w 0 ~ 0. 1) ___ E /E x 0 ---- Ey/E 0 higher order mode …. . Ey/E 0 gaussian -30 Y(µm) -30 Ex -30 30 Ey 30 laser envelope 30 short pulse kp. L = 1 minimizes pulse variations shallower plasma channel compensates for self-focusing Y(µm) 30 § § §

Summary § Design considerations for a laser-plasma collider module § Diffraction, Dephasing, Depletion: necessitates staging § Conceptual design of laser-plasma collider at 1 Te. V § Quasi-linear wake (a 0 ~ 1), electrons and positrons § 10 Ge. V modules: Laser pulse 40 J, 130 fs, 10 k. Hz § Requires development of 100’s k. W average power (10 k. Hz) 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 Ge. V stages

Additional information

Proper choice of plasma density and staging minimizes main linac length • Number of stages: § Linac length will be determined by staging technology Laser LPA Lc Lacc Lstage • • Conventional optics (~10 m) Plasma mirror (~10 cm)

0. 5 Te. V γ-γ Collider Example Plasma density scalings: Stage density scalings: Collider density scalings (for fixed luminosity):

10 Ge. V gain with efficient loading accessible on BELLA 40 J 0. 5 J 10 Ge. V 0. 4 Ge. V ~300 p. C ~50 p. C ne (1/cm^3) 1. 0 E+17 ne (1/cm^3) a 0 1. 4 a 0 lambda_p(um) 108 lambda_p(um) kp*L_laser 1 tau (fs) 57 tau (fs) w 0 (µm) 90 kp*w 0 5. 3 kp*w 0 P(TW) 563 P(TW) P/Pc 1. 8 P/Pc 2. 0 e 18 1 24 Ø 300 p. C 10 Ge. V stage with taper@kp. L=1 2 25 Ø Demonstrated control by shaping laser, plasma, ebunch 20 • Initial efforts reduced DE 10% 2. 5% 5. 3 • shaped bunches & taper in progress 14 • matching bunch emittance, shape to 0. 9 structure

Laser mode controls transverse field, controls bunch emittance matching Ø Emittance matched bunch radius << p for Gaussian-laser linear, nonlinear regimes • can reduce loading efficiency and/or cause ion motion Ø Linear regime: Fields shaped via laser mode to compensate emittance* • demonstrated propagation, channel compensation 30 30 Y(µm) -30 --10 --kpx 10 30 Propagation to depletion Ex@ 5. 1018 scale 60 GV/m -30 Laser Envelope Ey @ 5. 1018 Scale 1 scale 20 GV/m 1070 X(µm) 111 0 1070 X(µm) 1095 0 ct(ZR) 5 Ø Ongoing: compensation of beam loaded fields * Cormier-Michel et al, in prep.

2 D PIC simulations demonstrate a factor 3 in matched electron beam radius § With higher order mode and delay beam radius can be increased x 3 charge x 9 § Beam radius limited by linear region of focusing field § Can increase flat top region by using higher order modes y ( m) 0. 3 0. 2 0. 1 0. 0 0 2 1 3 ct (mm) —·— gaussian + hermite gaussian with delay —— gaussian + hermite gaussian ______ gaussian pulse ……. . gaussian pulse (unmatched) 36 § simulation at n 0 = 5 x 1018 cm-3 § matched emittance 0. 014 mm mrad varies < 1% § scaled parameters at 1017 cm-3 § y = 2 m § en = 0. 1 mm mrad

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

Fluid simulations: verify and quantify scaling laws Axial wakefield Ge. V-class example: 1 D fluid code (B. A. Shadwick) - Standard LWFA regime - a 0 = 1. 5, k 0/kp = 40, kp L =2 - Laser: 0. 8 m, 5 x 1018 W/cm 2, 30 fs - Plasma: 1018 cm-3, 3 cm Energy gain Laser pulse 1 Ge. V E. Esarey et al. , AAC Proc 2004 =z-ct 38

Fluid simulation of scaled BELLA point design n monemtum Ez energy spread bunch distance Scaled point design example: 1 D fluid code (B. A. Shadwick) - Quasi-linear LWFA regime - a 0 = 1. 0, k 0/kp = 40, kp L =2 - Laser: 0. 8 m, 2 x 1018 W/cm 2, 40 fs - Plasma: 1018 cm-3, - Bunch: kp z = 0. 5, D = 0. 9% (initial), 0. 05% (final@ 0. 5 Ge. V) 39

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 -5 Y[µm] 5 Transverse motion 800 X[µm] 2000 ⇒ Use injector based on controlled trapping at lower wake amplitude and separately tunable acceleration stage to reduce emittance and energy spread