1. Laser-Plasma Accelerators: Scaling Laws for Collider Applications
Eric Esarey
C. Schroeder, C. Benedetti, C. Geddes, W. Leemans
BELLA Center LBNL
FACET II Science Opportunities Workshop
SLAC, 14 Oct 2015
Supported by the U.S. DOE under Contract No. DE-AC02-05CH11231
Office of Science

2. Laser-Plasma Accelerator
Outline
LPA Basics
Scaling Laws
Zeroth-order picture of LPA collider assuming idealized physics
Real-world physics that further complicate LPA collider (Schroeder)
Other possible LPA collider approaches
Status and Required R&D for LPA collider (Leemans)
Laser-Plasma Accelerator

3. Conceptual LPA Collider
Based on 10 GeV modules
Quasi-linear wake: e- and e+
Driven by 30 J, 60 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)
Leemans & Esarey, Physics Today (2009); Esarey et al, Rev. Mod. Phys. (2009)

4. Laser-plasma accelerators (LPAs)
Tajima & Dawson, Phys. Rev. Lett. (1979); Esarey, Schroeder, Leemans, Rev. Mod. Phys. (2009)
Plasma wave: electron density perturbation
Laser ponderomotive force (radiation pressure)
λp = 2πc/ωp ~ np-1/2 ~ 30 μm
short pulse, ultra-intense laser: I~1018 W/cm2
Laser pulse duration
~ λp/c ~ tens fs
electron plasma density perturbation

5. Laser-plasma accelerators: >10 GV/m accelerating gradient
plasma wave (wakefield) E ~100 GV/m (for n~1018 cm-3)
>103 larger than conventional RF accelerators ⇒ “>km to <m”
Accelerating bucket ~ plasma wavelength
 ultrashort (fs) bunches (<λp /4)
plasma density
laser
bunch
beam charge (set by beam loading): ~100 pC (for n~1018 cm-3)
beam duration (set by trapping physics): <10 fs
 high peak current

6. *Shadwick, Schroeder, Esarey, Phys. Plasmas (2009)
Basic design of a laser-plasma accelerator: single-stage limited by laser energy
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+a02/2)-1/2 n1/2 ~ 10 GV/m
Energy gain determined by laser energy via depletion*
Laser: Present CPA technology 10’s J/pulse
laser
Ez wake
*Shadwick, Schroeder, Esarey, Phys. Plasmas (2009)

7. Wake structure depends on laser intensity
Blowout regime
a0 >> 1
very asymmetric
focuses e-
defocuses e+
self-trapping
self-guiding
Quasi-linear
a0 ~ 1
symmetric e+/e-
dark current free
channel required
tailor focusing forces via laser profile
a0=4
a0=1 e+
note - gain in E is no longer as a^2 above a=1; closer to linear in a but need numerics. This reduces the advantage of the bubble
not dealing here with emittance, etc - later slide
axial
axial
radial
radial

8. Limits to acceleration length: diffraction
DW = Ez . L
Limits to acceleration length: diffraction
“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)

9. Limits to Acceleration Length: dephasing
DW = Ez . L
Limits to Acceleration Length: dephasing
Dephasing: e- outrun wake,
Phase velocity: vp/c ≈ vg/c = 1- l02/2lp2
Ldephase (1-vg/c) = lp/2,
Ldephase = lp3/l02 ~ n-3/2 ~ 1.6 m
Solution: density tapering
vp<c
e- beam Ez
laser
vp<c
e- beam Ez
laser

10. Density Tapering: Phase Lock e-
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 n1
laser
e- beam Ez
n2>n1
e- beam
laser Ez
density
Katsouleas, PRA (1986); Rittershofer et al, PP (2010)

11. Limits to acceleration length: depletion
DW = Ez . L
Limits to acceleration length: depletion
Depletion: laser loses energy to wake
Energy balance: EL2 sz = Ez2 Ldeplete
Linear limit a02 << 1: Ldeplete = a0-2 Ldephase >> Ldephase
Nonlinear limit a02 >> 1: Ldeplete ~ Ldephase
laser
laser Ez Ez
Solution: staging

12. 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 GeV for I = 1018 W/cm2 and n = 1017 cm-3
Multiple-stages for controlled acceleration to high energy:
laser

13. LPA Experiments (single stage)
LPA plasma density scalings - staging required
Laser-plasma interaction (depletion) length:
Accelerating gradient:
Energy gain:
For high-energy applications, laser depletion (and reasonable gradient) necessitates staging
Scalings verified with simulations
LPA Examples (single stage):
LPA Experiments (single stage)
LBNL 2014
Texas 2013
LBNL 2006
e-beam energy gain (MeV)
LLNL 2010
RAL 2009
MPQ 2010
U.Mich 2008
LOA 2006
APRI 2008
LOA 2004
LBNL 2004
RAL 2004
W ~ 10 GeV
n ~ 1017 cm-3
Lacc ~ 1 m
Ulaser ~ 40 J
Plaser ~ 1 PW (BELLA Laser)
W ~ 1 GeV
n ~ 1018 cm-3
Lacc ~ 3 cm
Ulaser ~ 1 J
Plaser ~ 100 TW
plasma density, np (cm-3)

14. LPA single-stage plasma density scalings
E0 (GV/m)
L (m)
Single-stage length
Accelerating gradient
n (cm-3)
n (cm-3)
Single-stage
energy gain
Coefficients determined from simulations of resonant laser with a=1.5
W (GeV)
n (cm-3)

15. Single-stage plasma density scalings verified with PIC simulations
Vay, Geddes, et al., Phys. Plasmas (2011)
Single-stage PIC simulations (using WARP code with boosted-frame technique) verifying plasma density scaling

16. Beam loading limits bunch charge: 300-500 pC for 10 GeV stages
Beam loading theoretical limit
e-bunch wake = laser wake
Linear theory , kp sz < 1, kp sr ~ 1
Nb ~ 9x9 (n0 16 cm-3)-1/2 (Ez/E0)
Nb ~ n0 -1/2
Ex.: Nb = 3x109 (0.5 nC) for n0 17 cm-3 and Ez/E0=1
plasma wake
Figures are 66% loaded
black: r = 0.085e-6 (kpr = 0.05)
3D equivalent charge = 3d charge/14
red: r = 0.085e-6 at n0 = 1018 cm-3
green: r = 0.085e-6 with kpLlaser=1
magenta: r = 1.68e-6 (kpr = 1)
3D equivalent charge = 3d charge/1.6
blue: r = 3.0e-6 (kpr = 1.8)
3D equivalent charge = 3d charge/1.2
(r=0.085 m)
3d eq charge 0.5pC
bunch
laser
VORPAL PIC simulations
500 pC at 1017 cm-3 for kpL=2, kpsr~ 2
10% of laser energy to electrons
* Cormier-Michel et al, Proc. AAC 2008, **Katsouleas PRA 1986

17. Collider Requirements: Luminosity
Rate of events: (luminosity) x (collision cross-section)
Luminosity: cross-section 
For fixed beam power, Pb=2f Nb(γmc2), transverse beam density must be increased
Limitations:
Achievable beam emittance
Final focus optics to IP: adiabatic plasma lens
Beam-beam interaction (beamstrahlung)
Emittance growth in main linacs (beam scattering in plasma)

18. Collider scalings (for fixed luminosity):
1 TeV Collider Parameters:
Plasma density and laser wavelength scalings
Collider scalings (for fixed luminosity):
Schroeder et al., Phys. Rev. ST AB (2010)

19. Collider scalings (for fixed luminosity):
1 TeV Collider Parameters:
Plasma density and laser wavelength scalings
Requires laser development
15 kHz
480 kW average power
Collider scalings (for fixed luminosity):
Schroeder et al., Phys. Rev. ST AB (2010)

20. Collider scalings (for fixed luminosity):
1 TeV Collider Parameters:
Plasma density and laser wavelength scalings
Present technology
32 J, 56 fs
a0 = 1.5
Collider scalings (for fixed luminosity):
Schroeder et al., Phys. Rev. ST AB (2010)

21. Collider scalings (for fixed luminosity):
1 TeV Collider Parameters:
Plasma density and laser wavelength scalings
Present technology
32 J, 56 fs
a0 = 1.5
Collider scalings (for fixed luminosity):
BELLA PW LBNL
Explore physics of 10 GeV stage
Schroeder et al., Phys. Rev. ST AB (2010)

22. Beamstrahlung-limited regime:
modifies LPA collider scalings at low density
Constraint of fixed beamstrahlung: fixed nγ (photons per electron)
Collider wall-plug power
beamstrahlung-limited regime
beam-loading limited regime
accelerating field:
linac length:
Ecm = 1 TeV
L = 2x1034 s-2 cm-2
η = 6%
σ2 = 100 nm2
laser energy/stage:
multi-bunch operation
Schroeder et al., PR ST-AB (2012)

23. LPA collider requires laser technology with high efficiency (~30%)
fL [Hz]
Laser average power requirements
UL [J]
Different operational plasma densities require different laser parameters (laser technologies) with varying efficiency
n [cm-3]
Laser duration (bandwidth) requirements
TL [fs]
n [cm-3]

24. Collider density scalings: total power required assuming fiber laser efficiency
Laser to plasma wave efficiency: ~50%
Plasma wave to beam efficiency: ~40%
Independent of plasma density
For fixed luminosity (e.g., 2x1034 s-1 cm-2 for Ecm = 1 TeV) and IP size, based on fiber laser technology
J. Dawson: “30% at 50kHz”; ”10% at 2kHz”; “falls rapidly to 0% below kHz”
wall-plug power
P (MW)
n (cm-3)

25. “International Coherent Amplification Network”
Coherent laser combining: new laser technology provides a path for high average power
Coherent combination of diode-pumped fiber lasers: path to high-peak power, high- average power, high-efficiency lasers:
Fiber lasers: sub-ps pulses, ~mJ energy, ~10 kHz, ~10% wall-plug efficiency
Coherent combination of fiber lasers is proposed to achieve high peak power (energy)
Challenge: Requires combining (control of all laser phases, group velocity delays, dispersion) ~104 fiber lasers
ICAN =
“International Coherent Amplification Network”
G. Mourou et al., Nature Photonics (2013)

26. Incoherent summation easier than coherent (requires no phase control)
Wakefield excitation by incoherently combined lasers: path to high-average power
Wakefield driven by time-integrated gradient of electromagnetic energy density: depends on the average properties of the radiation in the volume (~λp3)
Wakefield excitation does not require coherence, only energy density
Incoherent combination (of many low energy) lasers for wakefield excitation:
Require only sufficient energy deposited in ~λp3 volume
Benedetti et al., PoP (2014)
Incoherent summation easier than coherent (requires no phase control)

27. Laser-plasma accelerator (LPA) based collider: challenges and solutions
Laser diffraction
guiding in plasma channel
dephasing / depletion
LPA staging
positron focusing + acceleration
quasi-linear regime / control focusing
ion motion
scattering in plasma
near-hollow plasma channels
high-beam quality
shaped witness bunches
beamstrahlung
ultra-short beams / beam trains
efficiency
laser energy recovery
laser average power
laser beam combining

28. Future R&D to address challenges for
laser-plasma-based HEP applications
Plasma-based acceleration techniques: tremendous progress over the last decade
Beam quality preservation: emittance and energy spread
Plasma source devlopment
Plasma channels: mitigate scattering; independent control of focusing/acceleration
Plasma longitudinal shape: mitigate dephasing, control instabilities
Particle beam injection techiniques
Colliding laser pulses, ionization injection, down-ramp injection
Shaped beam currents enables high efficiency without induced energy spread
Charge limits imposed by beam loading, ion motion, instabilities, beamstrahlung, etc.
Staging: coupling of drive (laser/particle) and witness beams:
Compact transport with emittance preservation
Compact coupling to perserve high average gradient, e.g., plasma mirrors
LPA-based collider requires laser technology R&D (high average power, high efficiency)
Fiber-based lasers; incoherent laser drivers
Laser energy recovery
Lasers and plasmas for other collider systems:
Rapid methods for cooling electron/positron beams (replace damping rings?)
Novel final focus concepts
Size of plasma-based collider dominated by final focus/beam delivery (few km at TeV)
Plasma lens (adiabatic) to overcome Oide limit
As LPA R&D advances, collider concept and design will continue to evolve
Realization of “low energy” applications are critical to validate and develop laser-plasma accelerator technology: many applications to most areas of Office of Science

29. Additional Information: Report for HEPAP Subcommittee
R&D and facility investment needs towards a laser-plasma accelerator driven linear collider
Input for the HEPAP subcommittee on Accelerator R&D
W. Leemans, E. Esarey, C. Schroeder, C. Geddes
Lawrence Berkeley National Laboratory
December 1, 2014
Introduction
This document is in response to the request from the HEPAP subcommittee on accelerator R&D for the DOE-HEP office to provide input to the following questions in an from Prof. James Rosenzweig, dated November 17th, 2014.
For a 1 TeV collider with a luminosity of 10^34 and a 3 TeV collider with a luminosity of 10^35
       1. What are the milestones to get there?
       2. What R&D resources will be needed and when?
       3. How does the roadmap depend on funding level and in particular how do the R&D costs change over the next five to ten years?  Please consider a constrained funding scenario with low levels as well as an “unconstrained” funding scenario that would allow technically limited progress.

30. Extras

31. Point designs: 10 and 100 GeV 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
1.71017
38 cm 10
1.5
1.1
0.30
130 63 99
1.11017
79 cm 1
0.45
0.22
170 82
140
6.01016
2.4 m
3.8
1.3 kJ
310
250
1.71016
12 m
100
3.0
390
200
1.11016
25 m
550
260
430
6.01015
78 m
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

32. Beam loading simulations predicts 300-500 pC for 10 GeV stages
Beam loading theoretical limit
e-bunch wake = laser wake
Linear theory , kp sz < 1, kp sr ~ 1
Nb ~ 9x9 (n0 16 cm-3)-1/2 (Ez/E0)
Ex.: Nb = 3x109 (0.5 nC) for n0 17 cm-3 and Ez/E0=1
~constant field inside bunch
Quasi-linear beam loading
matches linear theory
+ 2D
* 3D
-- theory
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
Figures are 66% loaded
black: r = 0.085e-6 (kpr = 0.05)
3D equivalent charge = 3d charge/14
red: r = 0.085e-6 at n0 = 1018 cm-3
green: r = 0.085e-6 with kpLlaser=1
magenta: r = 1.68e-6 (kpr = 1)
3D equivalent charge = 3d charge/1.6
blue: r = 3.0e-6 (kpr = 1.8)
3D equivalent charge = 3d charge/1.2
(r=0.085 m)
3d eq charge 0.5pC
density & kpL:
kpr = 0.3 1
1.8
kpL =2, a0=1
n0 = 1018 cm-3 +*
n0 = 1019 cm-3
kpL =1, a0=1.4 +
* Cormier-Michel et al, Proc. AAC 2008, **Katsouleas PRA 1986

33. Transverse beam stability in hollow plasma channels
Transverse wake fields can lead to beam break-up instabilities (growth or transverse displacement from channel axis)
Higher-order channel modes and growth rates
for the single-bunch transverse instability have been calculated
Assuming weak/no focusing, growth rate:
Moderate focusing suppresses instability:
Sufficient focusing for electron beams can be provided in near-hollow plasma channel (with channel density << wall density)
Higher-order mode detuning can be applied to suppress undesirable modes
C. B. Schroeder et al., Phys. Rev. Lett. (1999)
C. B. Schroeder et al., Phys. Plasmas (2013)

34. Length of staged-LPA linac [m]
Staged LPAs: average gradient determined by driver in-coupling distance
Number of stages:
Compact laser in-coupling distance (enables high average gradient)
Conventional optics: requires many Rayleigh ranges to reduce fluence on optic (avoid damage)
Plasma mirror: relies on critical density plasma production (high laser intensity): coupling <1 m
Length of 1 TeV staged-LPA linac
Length of staged-LPA linac [m]
coupling distance:
0.5 m
1 m
5 m
10 m
Plasma density [cm-3]

35. Summary Physics considerations for a collider based on LPA:
Laser depletion - staging required (with tapered plasma channels)
Operation in quasi-linear regime (a~1)
Allows positron acceleration
Independent control of transverse (focusing) fields
Plasma density scalings - determine required laser technology
Beamstrahlung at IP and total power requirements - operation at 1016 <n<1018
Total linac length and collider power requirements independent of laser wavelength
Estimates for a TeV-scale laser-plasma collider:
10 GeV stages (at n=1017 cm-3): requires laser pulse (at 1 micron) ~30 J, a~1, ~100 fs, ~15 kHz, ~30% efficiency
for additional details on LPA-based collider design see
Schroeder et al., PR ST-AB (2010)
Schroeder et al., PR ST-AB (2012)

36. electron plasma density
Focusing determined by background ion density in nonlinear regime: e+ focusing problematic
Nonlinear (bubble/blow-out) regime: accelerating/focusing wakefields determined by background plasma density:
Plasma electrons expelled (positron acceleration challenging)
Strong focusing of beam suppresses scattering for e-beam
Strong focusing results in electron dense beams:
emittance growth from ion motion
e+ focus
e- focus
electron plasma density
a=4
ion cavity
blow-out, expel electrons
ion motion
0.1 um emittance 100 pC charge 1017/cc density kpLb=1
(positron focusing challenging)
(emittance growth)
Rosenzweig et al., PRL (2005)
beam-driver, Rosenzweig et al., PRA (1991)
laser-driver, Pukhov&Meyer-ter-Vehn, APB (2002)

37. Quasi-linear regime: e+ focusing and independent control of acceleration/focusing
e+ accel.
e- accel.
Operate in “quasi-linear” regime:
Quiver momentum weakly-relativistic a ~ 1
(Intensity ~ 1018 W/cm2)
Region of acceleration/focusing for both electrons and positrons
Stable laser propagation in plasma channel
Accelerating field
e- accel+focus
Transverse position
Plasma density
e+ accel+focus
e+ focus
e- focus
Quasi-linear regime: control of focusing force by laser driver shape (transverse intensity profile)
Shape driver profile to control focusing forces:
Provide weak focusing to prevent blow-out
enables positron acceleration
increases scattering
Cormier-Michel et al., PRST-AB (2011)
Focusing field
Longitudinal position

38. Shaping the transverse profile of plasma channel: independent control of accelerating and focusing
Schroeder et al., PoP (2013)
plasma density
radius
Near-hollow plasma channels:
Laser guided in channel (set by channel depth, not on-axis density)
Accelerating field (set by wall density):
Focusing field (set by channel density):
accelerating wakefield
distance behind driver
kw (laser spot) =2.3
kw (rms length) = 1
a0 = 1
channel size:
kw rc = 1.5
radial position
matched beam:
focusing wakefield
PIC simulation (INF&RNO)

39. Near-hollow plasma channels: ultra-low emittance preservation
Independent control over accelerating and focusing forces:
Acceleration uniform in radial position
Focusing uniform in longitudinal position (no head-to-tail variation in focusing) and linear in radial position
Near-hollow plasma channel geometry provides emittance preservation:
Mitigates Coulomb scattering (reduce plasma density near bunch):
Control of focusing force and beam density: prevents ion motion
for relevant (1 TeV) parameters:
Ion motion in channel negligible if ratio of beam density to wall density less than ion-electron mass ratio:
Schroeder et al., PoP (2013)

40. Beam loading in near-hollow plasma channels:
shaped bunches eliminate energy spread
Energy spread minimized using shaped beams
Katsouleas et al., PA (1987)
Ramped triangular current distribution :
normalized field amplitude
normalized distance behind driver, kw (z-ct)
constant acceleration
Example:
Trade-off between gradient and efficiency (for no induced energy spread)
fraction of peak accel. field
wake to beam efficiency
beam charge:
lower plasma density, higher beam charge

41. Positron beams accelerated in linear regime in hollow plasma channel
In quasi-linear regime, acceleration of positrons is symmetric to electrons
normalized field amplitude
normalized distance behind driver, kw (z-ct)
Focusing for positrons may be provided from external magnets

42. Ultra-short beams can be accelerated without energy spread growth
Ultra-short beams are desirable, e.g., beamstrahlung suppression in colliders.
beamstrahlung photons/electron:
Trapezoidal current distribution:
normalized field amplitude
normalized distance behind driver, kw (z-ct)
Improved efficiency achieved using bunch trains

43. Bunch trains allows ultra-short bunches with high efficiency, gradient, and no energy spread
Example:
with 1 bunches:
with 6 bunches:
normalized field amplitude
normalized distance behind driver, kw (z-ct)
Improved efficiency achieved using bunch trains:
Each bunch has same charge
Experiences same (constant across bunch) accelerating gradient
Efficiency increased by number of bunches in train
Trade-off between gradient and efficiency, with no energy spread growth
Transverse bunch stability requires analysis

44. Improved efficiency using additional laser pulses to absorb remaining plasma wave energy
Drive laser depositing energy into wave
(frequency red-shifting)
“Energy-recovery” laser absorbing wave energy
(frequency blue-shifting)
Additional laser pulse allows for no energy to remain in coherent plasma oscillations after energy transferred to particle beam
η(plasma to beam)= 0.75
gradient = 0.5(peak field)
laser
bunch
laser
normalized field amplitude
plasma
wave
normalized distance behind driver, kw(z-ct)

45. (fraction of laser energy to plasma wave)
Energy recovery: Improved efficiency (>50%) using additional laser pulses
Lasers
η(plasma to beam)= 0.75
η(energy recovery) = 0.9
plasma
with 2nd “energy-recovery” laser
beam
single (pump) laser
laser energy recovery
Efficiency of energy transfer: laser to beam
Re-use in another LPA stage
Sent to photovoltaic
(energy back to grid)
without energy recovery
total efficiency
(driver to beam):
beam energy gain
Pump laser depletion
(fraction of laser energy to plasma wave)
input laser energy
recovered laser energy