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. LeemansLOASIS Program

Lawrence Berkeley National Laboratory

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)

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 acceleratorsPlasma wakefields Ez > 10 GV/m, fast waves(Conventional RF accelerators Ez ~ 10 MV/m)Plasma channel: Guides laser pulse and supports plasma wave

Tajima, Dawson (79); Gorbunov, Kirsanov (87); Sprangle, Esarey et al. (88)

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 (1017 cm-3) Staging & coupling modules

Requires high rep-rate (10’s kHz) Requires development of high

average power lasers (100’s kW)

Basic design of a laser-plasma accelerator: single-stage limited by laser energy

laser

Ez wake

• Laser pulse length determined by plasma density

– kp sz ≤ 1, sz ~ lp ~ n-1/2

• Wakefield regime determined by laser intensity

– Linear (a0<1) or blowout (a0>1)

– Determines bunch parameters via beam loading

– Ex: a0 = 1 for I0 = 2x1018 W/cm2 and 0 = 0.8 m

• Accelerating field determined by density and laser intensity

– Ez ~ (a02/4)(1+a0

2/2)-1/2 n1/2 ~ 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

run 405

Blowout regime high field very asymmetric

focuses e- defocuses e+

self-trapping

Quasilinear linear: symmetric e+/e- high a0 desired for gradient

too high enters bubble

a0 ~1-2 good compromise

dark current free

e- accel

e- focus

e+ focus

e+ accel

a0=4e-

accele- focus

e+ focus

e+ accel

a0=1

Axial field

Transverse field

Plasma density

“3D”: Diffraction, Dephasing, Depletion

Diffraction of laser pulse

ZR = p r02/l0 ~ 2 cm, ZR<< Ldephase < Ldeplete

Solution: Density channels

Parabolic channel guides gaussian modes

Channel depth: Dn [cm-3] = 1020 / (r0[mm])2 ~ 2x1016 cm-3

W.P. Leemans et al, IEEE Trans. Plasmas Sci. (1996); Esarey et al., Phys. Fluids (1993)

DW = Ez . LLimits to acceleration length: diffraction

Dephasing: e- outrun wake,

Phase velocity: vp/c ≈ vg/c = 1- l02/2lp2

Ldephase (1-vg/c) = lp/2,

Ldephase = lp3/l0

2 ~ n-3/2 ~ 1.6 m

Solution: density tapering

DW = Ez . LLimits to Acceleration Length: dephasing

e- beam

laser

Ez

vp<c

e- beam

laser

Ez

vp<c

For a0 ~ 1, Ldephase may be < Ldeplete

Phase velocity depends on density Phase position ~ lp ~ 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

e- beam laser

laser

Ez

Ez

n1

n2>n1

€

z Ld

€

n

n0

density

Alternative tapering options:Step density transition

€

vb ≈ c

€

v ≈ vg < c

€

n(z)

n(0)

€

kp3z k0

2

(1)

(2)

(3)

(1)

(2)

(3) • Maintains near-resonance of plasma response with laser

• Experimental realization: staged accelerator sections

C. Schroeder et al.

Depletion: laser loses energy to wake

Energy balance: EL2 sz = Ez

2 Ldeplete

Linear limit a02 << 1: Ldeplete = a0

-2 Ldephase >> Ldephase

Nonlinear limit a02 >> 1: Ldeplete ~ Ldephase

DW = Ez . LLimits to acceleration length: depletion

Ez

laser

Ez

laser

Solution: staging

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):

€

∂∂ct

dζ1

8πE⊥

2 + B⊥2

( )∫ =E0

2

16πdζ | a |2 ∂ζ n n0γ( )∫

€

∂εL

∂ωp t= −

kp2

k02

Emax E0( )2

Shadwick, Schroeder, Esarey, Physics of Plasmas (2009)

€

E0 = mcωp q∝ n1/ 2

characteristic accelerating field:

€

E z

€

a

Nonlinear plasma wave excitation by a Gaussian laser pulse

€

Emax E0 ≈π

2e

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2a2

2(1+ a2 /2)≈

0.38a2

(1+ a2 /2)

• 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

€

kpLpd =k0

2

kp2

25 / 2e

πkpL 1+

2

a2

⎛

⎝ ⎜

⎞

⎠ ⎟≈ 8.7

k02

kp2

2a−2

1

⎧ ⎨ ⎩

for a2 <<1

a2 >>1

Pump depletion length for near-resonant Gaussian laser pulse:Pump depletion length:

€

∂εL

∂ωp t= −

kp2

k02

Emax

E0

⎛

⎝ ⎜

⎞

⎠ ⎟

2

= −εL

Lpd

• Characteristic length scale independent of intensity for relativistically-intense laser pulses

€

a0

€

kp3Lpd k0

2

€

∝1/a2

€

∝const

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: LD ∝1

ne3/2

Energy gain (linear regime):

laser

€

Wstage[GeV] ≈I[W/cm2]

n[cm-3]

+ channel …

Ex: Wstage = 10 GeV for I = 1018 W/cm2 and n = 1017 cm-3

Accelerating field:

€

E0[V/m] ≈100 n[cm-3]

16

Scaling laws: analytic theory

Laser pulse evolution

Laser energy evolution:

Laser field

plasma density

accelerating field

ωpt=500

ωpt=1500

ωpt=2500

ωpt=3500

• Laser evolution interplay between laser intensity steepening, laser frequency red-shifting, energy depletion

€

a0 =1

€

k0 /kp = 20

€

kpL =1

€

a0 = 8.6 ×10−10 λ[μm]I1/ 2[W/cm-2]

Shadwick, Schroeder, Esarey, Physics of Plasmas (2009)

18

Longitudinal e-bunch dynamics:energy spread minimum near dephasing

LaserWake

Position, kp(z-ct)

Fluid plasma + e-bunch described by moments (includes beam loading)

B.A. Shadwick et al. Time, pt

Momentum

Energy spread

e-bunch

Energy spread Initial: / = 0.3% at = 100 Final: / = 0.01% at = 3000

Scaling laws from fluid code: dephasing/depletion lengths & energy gain

Fraction of laser energy at dephasing length

Independent of k/kp

Fix laser parameters (a0, kpL0, kpr0), increase (k/kp) to increase energy

Energy and dephasing length from 1D fluid simulations a0 =1: max = 0.7(k/kp)2 , kpLdp= 4(k/kp)2

a0 =1.5: max = 1.3(k/kp)2 , kpLdp= 3.5(k/kp)2

a0 =2: max = 2(k/kp)2 , kpLdp= 3(k/kp)2

Quasi-linear: a0 ~ 1

Dephasing ~ depletion

Good efficiency

Point designs: 10 and 100 GeV

Laser power: P[GW] = 21.5(a0r0/)2 , Critical power: Pc[GW] = 17(k/kp)2, P/Pc = (a0kpr0)2 /32.All assume: kpL0 = 2, m

a0 P/Pc P(PW) WL t0(fs) r0(m) p(m) n0(cm-3) Ldp We

(GeV)

2 2.2 0.38 40 J 98 53 80 1.71017 38 cm 10

1.5 1.1 0.30 40 J 130 63 99 1.11017 79 cm 10

1 0.45 0.22 40 J 170 82 140 6.01016 2.4 m 10

2 2.2 3.8 1.3 kJ 310 170 250 1.71016 12 m 100

1.5 1.1 3.0 1.3 kJ 390 200 310 1.11016 25 m 100

1 0.45 2.2 1.3 kJ 550 260 430 6.01015 78 m 100

Parameter design for GeV and beyond

P(PW) (fs) np (cm-3) w0 (m) L(m) a0 ∆nc/np Q(nC) E(GeV)

0.020 30 11018 14 0.016 1.76 60% 0.18 0.99

0.040 30 1.51018 14 0.011 2.53 40% 0.25 0.95

0.100 30 2.01018 15 0.009 3.78 0% 0.40 1.06

0.200 100 1.01017 45 0.52 1.76 60% 0.57 9.9

2.0 100 3.01017 47 0.18 5.45 0% 1.8 10.2

2.0 310 1.01016 140 16.3 1.76 60% 1.8 99

40 330 4.01016 146 4.2 7.6 0% 8 106

20 1000 1.01015 450 500 1.76 60% 5.7 999

1000 1000 6.51015 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

W. Lu et al., Phys Rev STAB (2007)

Beam loading simulations predicts 300-500 pC for 10 GeV stages

Quasi-linear beam loadingmatches linear theory

density & kpL: kpr = 0.3 1 1.8

kpL =2, a0=1 n0 = 1018 cm-3

+*

kpL =2, a0=1 n0 = 1019 cm-3

+* +* +*

kpL =1, a0=1.4n0 = 1019 cm-3

+

+ 2D* 3D-- theory

€

Q pC[ ]n0 1017

Ex E0

HR

kp2σ r

2

VORPAL PIC simulations

500 pC at 1017 cm-3 for kpL=2, kpsr~ 2• 10% of laser energy to electrons

Bunch length & profile alters field inside bunch• flatten field across bunch – reduces DE• 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 , kp sz < 1, kp sr ~ 1

Nb ~ 9x9 (n0 16 cm-3)-1/2 (Ez/E0)

Ex.: Nb = 3x109 (0.5 nC) for n0 17 cm-3 and Ez/E0=1

Linear theory

Symmetric bunches

Energy spread ~ N/Nmax

Efficiency ~ N/Nmax (2 - N/Nmax)

Ex: Spread100%, Effic100% as NNmax

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

Beam loading: tailored bunches for high efficiency

T. Katsouleas et al., Part. Accel. 22, 81 (1987)

Blowout regime:M. Tzoufras, et al., PRL (2008)

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 DE

Shaped bunch can further reduce DE

Beam loading versusbunch length

no chargeL = 0.085 mL = 0.85 mL = 0.51 m

kpr = 0.3 scaled charge 60pC

Axial density taper locks bunch phase:improves gain and reduces DE for e+,e-

Compensate dephasing by changing lp ~ n1/2

Linear taper at kpL=2 produces 4x gain

Positron acceleration ~symmetric

Ongoing: optimize taper, emittance matching initial kpL=1 results : 50% depletion,

10 GeV gain for 300 pC, 2.5%FWHM

Spectra at dephasing

gain in stages with kpL=2 at 1019 cm-3

50% beam loaded -kp r = 1, kp L = 0.5

3D charge: 22.5pC 225pC, 9 GeV gain, 4% FWHM, at 1017 cm-3

Taper

no taper

0 Gain [GeV/c at 1019/cc] 0.120 Scale Gain [GeV/c at 1017/cc] 12

0

#/G

eV/c

[A

.U.]

1

__e---e+

Matched electron beam spot size is small

Matched beam spot size

linear regime

bubble regime

matched beam < 1 mm (<< lp ~ 100 mm) for g = 20,000 (10 GeV), n0 = 1017 cm-3, en = 1 mm mrad

Limits electron beam charge and quality

Increase sy for higher charge, with nbpeak small

In linear regime kb2 E∝ ⊥ ∝ ∇⊥a2

Reduce transverse field gradient to increase matched beam radius

26

€

σ y2 =

εn

γkβ

€

kβ2 y = kp E y E0γ

kβ2 = kp

2 /2γ

Higher order laser mode to tailor transverse wakefield

Linear regime : E⊥/E0 ~ ∇⊥a2/2 Add first order Hermite-Gaussian mode in 2D

y/r0

€

e−y 2 / r02

gaussian

first order hermite-gaussian mode

exact solutions of the paraxial wave equation

€

e−y 2 / r02

2 2 y r0

HG0

HG1

y/r0

a2

0.70.50.40

a1/a0

a2 = a02 HG0

2 + a12 HG1

2

Ey/E0

y/r0

analytic calculation (low a) no channel

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 Dn = Dnc = 1/prer0

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(HG0 + HG1)2

(HG02 + HG1

2)

HG02

HG12

k pyk py

Solution Use cross-polarization Use different frequencies kbeat = m/ZR

kbeat >> kp

a0=0.1 a0=0.1a1=0.1 a1=0.5a0

Transverse field tailoring in the quasi linear regime

Wakefield driven by higher order modes in the quasi linear regime a0=1 Transverse field flattened by flat top laser profile Mode propagation to depletion

short pulse kpL = 1 minimizes pulse variations shallower plasma channel compensates for self-focusing

200 X(µm) 225

Y(µ

m)

30

X(µm)

-30

Y(µ

m)

30

200 X(µm) 240

-30

Y(µ

m)

30

X(µm)-0.3

0.3

-0.0

30.

03

-30

200 225 200 225

@ y = -1 mm(y/w0 ~ 0.1)___ Ex/E0

---- Ey/E0 higher order mode….. Ey/E0

gaussian

1935 X(µm) 1965

Y(µ

m)

30

X(µm)

-30

Y(µ

m)

30

1935 X(µm) 1980

-30

Y(µ

m)

30

-30

1935 1965

-30

Y(µ

m)

30

X(mm)0 4

integrated laser intensity profile

laser envelope Ey Exhigh order mode

reduces Ey,

laser envelope ExEy

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 (a0 ~ 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

Additional information

Linac length will be determined by staging technology

Lstage

LPA

Laser

LaccLc

• Conventional optics (~10 m)• Plasma mirror (~10 cm)

€

Nstage ∝ n• Number of stages:

Proper choice of plasma density and staging minimizes main linac length

0.5 TeV γ-γ Collider Example

€

E0 ∝ n1/ 2

€

Wstage ∝ n−1

€

Lstage ∝ n−3 / 2

€

Nb ∝ n−1/ 2

Plasma density scalings:

Stage density scalings:

€

Nstage ∝ n

€

Pb ∝ n1/ 2

€

Plaser ∝ n−1/ 2

€

Pwall ∝ n1/ 2

€

f ∝ n

Collider density scalings (for fixed luminosity):

ne (1/cm^3) 2.0e18a0 1lambda_p(um) 24kp*L_laser 2tau (fs) 25w0 (µm) 20kp*w0 5.3P(TW) 14P/Pc 0.9

40 J 10 GeV~300pC

10 GeV gain with efficient loading accessible on BELLA

ne (1/cm^3) 1.0E+17a0 1.4lambda_p(um) 108kp*L_laser 1tau (fs) 57w0 (µm) 90kp*w0 5.3P(TW) 563P/Pc 1.8

0.5 J 0.4 GeV~50pC

300 pC 10 GeV stage with taper@kpL=1

Demonstrated control by shaping laser, plasma, ebunch

• Initial efforts reduced DE10%2.5%• shaped bunches & taper in progress• matching bunch emittance, shape to

structure

Laser mode controls transverse field, controls bunch emittance matching

Ey @ 5.1018

scale 20GV/m

1070 X(µm) 1095-30

Y(µ

m)

30

Ex@ 5.1018

scale 60GV/m

1070 X(µm) 1095-30

Y(µ

m)

30

Laser EnvelopeScale 1

1070 X(µm) 1110-30

Y(µ

m)

30

* Cormier-Michel et al, in prep.

Emittance matched bunch radius << lp 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

0 ct(ZR) 5

--10

10--

k px

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 n0 = 5x1018 cm-3

matched emittance 0.014 mm mrad varies < 1%

scaled parameters at 1017 cm-3

sy = 2 mm en = 0.1 mm mrad

s y (m

m)

0.0

0.1

0.2

0.3

1 2 30ct (mm)

—·— gaussian + hermite gaussian with delay —— gaussian + hermite gaussian______ gaussian pulse…….. gaussian pulse (unmatched)

Energy depletion: Analytic result in good agreement with numerical solution

€

εL εL (0) =1− z Lpd( ) − z Lpd( )2

€

εL εL (0)

€

εL εL (0)

€

kp3z k0

2

€

kp3z k0

2

€

a =1

€

a = 0.75

Analytic result (- - - - - -) :

€

z = Ldephasing

€

z = Ldephasing

38

Axial wakefield

€

E z

E0

Energy gain

Fluid simulations: verify and quantify scaling laws

Laser pulse

1D fluid code (B.A. Shadwick)

- Standard LWFA regime

- a0 = 1.5, k0/kp = 40, kp L =2

- Laser: 0.8 mm, 5x1018 W/cm2, 30 fs

- Plasma: 1018 cm-3, 3 cm

1 GeV

=z-ctE. Esarey et al., AAC Proc 2004

GeV-class example:

39

distance

monemtum

Fluid simulation of scaled BELLA point design

Scaled point design example: 1D fluid code (B.A. Shadwick)

- Quasi-linear LWFA regime

- a0 = 1.0, k0/kp = 40, kp L =2

- Laser: 0.8 mm, 2x1018 W/cm2, 40 fs

- Plasma: 1018 cm-3,

- Bunch: kpsz = 0.5, /Dg g = 0.9% (initial), 0.05% (final@ 0.5 GeV)

n

Ez

bunch

energy spread

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[µ

m]

X[µm]

5-5

800 2000

Transverse motion