Lecture 3: Laser Wake Field Acceleration (LWFA)

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Lecture 3Laser Wake Field Acceleration (LWFA)
1D-Analytics

1. Nonlinear Plasma Waves
2. 1D Wave Breaking
3. Wake Field Acceleration

Bubble Regime (lecture 4)

1. 3D Wave Breaking and Self-Trapping
2. Bubble Movie (3D PIC)
3. Experimental Observation
4. Bubble Fields
5. Scaling Relations

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Direct Laser Acceleration versus Wakefield
Acceleration
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Laser pulse excites plasma wave of length lp c/wp
lp
z
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How do the electrons gain energy?
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Phase velocity and gph of Laser Wakefield
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1D Relativistic Plasma Equations (without laser)
Consider an electron plasma with density N(x,t),
velocity u(x,t), and electric field E(x,t), all
depending on one spatial coordinate x and time
t. Ions with density N0 are modelled as a
uniform, immobile, neutralizing background. This
plasma is described by the 1D equations
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10. Problem Normalized non-linear 1D plasma
equations
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Nonlinear 1D Relativistic Plasma Wave
1. integral energy conservation
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Wave Breaking
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11. Problem Derive non-linear wave shapes
Show that the non-linear velocity can
be obtained analytically in non-relativistic appro
ximation
from
with the implicit solution
Notice that this reproduces the linear
plasma wave for small wave amplitude bm.
Then discuss the non-linear shapes
qualitatively Verify that the extrema of b(t),
n(t), and the zeros of E(t) do not shift in t
when increasing bm, while the zeros of b(t),
n(t), and the extrema of E(t) are shifted such
that velocity and density develop sharp crests,
while the E-field acquires a sawtooth shape.
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Wakefield amplitude
The wake amplitude is given between laser
ponderomotive and electrostatic force
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Dephasing length
Acceleration phase
Time between injection and dephasing
Dephasing length
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Viewgraph taken from E. Esarey Talk at Dream Beam
Symposium www.map.uni-muenchen.de/events.en.html U
ID symposium PWD dream beams
PHASE-SPACE ANALYSIS FLUID VS. TRAPPED ORBITS
trapped orbit (e- kicked from fluid orbit)
1D case Trapped electrons require a sufficiently
high momentum to reside inside 1D separatrix
cold fluid orbit (e- initially at rest)
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Maximum electron energy gain Wmax in wakefield
Electron acceleration (norm. quantities)
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Wave Breaking
p/mc b
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Example
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GeV channeling over cm-scale

• Increasing beam energy requires increased
  dephasing length and power
• Scalings indicate cm-scale channel at 1018
  cm-3 and 50 TW laser for GeV
• Laser heated plasma channel formation is
  inefficient at low density
• Use capillary plasma channels for cm-scale, low
  density plasma channels

Capillary
Plasma channel technology Capillary
1 GeV
e- beam
40-100 TW, 40 fs 10 Hz
Laser
3 cm
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0.5 GeV Beam Generation
225 mm diameter and 33 mm length capillary
Density 3.2-3.8x1018/cm3 Laser 950(?15)
mJ/pulse (compression scan) Injection threshold
a0 0.65 (9TW, 105fs) Less injection at higher
power -Relativistic effects -Self modulation
a0
Stable operation
500 MeV Mono-energetic beams a0 0.75 (11 TW,
75 fs)
Peak energy 490 MeV Divergence(rms) 1.6
mrad Energy spread (rms) 5.6 Resolution
1.1 Charge 50 pC
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1.0 GeV Beam Generation
312 mm diameter and 33 mm length capillary
Laser 1500(?15) mJ/pulse Density
4x1018/cm3 Injection threshold a0 1.35 (35TW,
38fs) Less injection at higher
power Relativistic effect, self-modulation
1 GeV beam a0 1.46 (40 TW, 37 fs)
Peak energy 1000 MeV Divergence(rms) 2.0
mrad Energy spread (rms) 2.5 Resolution
2.4 Charge gt 30.0 pC
Less stable operation
Laser power fluctuation, discharge timing,
pointing stability
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Wake Evolution and Dephasing
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WAKE FORMING
Longitudinal Momentum
0
Propagation Distance
200
INJECTION
Longitudinal Momentum
0
Propagation Distance
200
DEPHASING
DEPHASING
Longitudinal Momentum
0
Propagation Distance
Geddes et al., Nature (2004) Phys. Plasmas
(2005)
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Bubble regime Ultra-relativistic laser, I1020
W/cm2
A.Pukhov J.Meyer-ter-Vehn, Appl. Phys. B, 74,
p.355 (2002)