1 International Max Planck Research School on Advanced Photonics Lectures on Relativistic Laser Plasma Interaction J. Meyer-ter-Vehn, Max-Planck-Institute for Quantum Optics, Garching, Germany April 16 – 21, 2007 1. Lecture: Overview, Electron in strong laser field, 3. Lecture: Basic plasma equations, self-focusing, direct laser acceleration 5. Lecture: Laser Wake Field Acceleration (LWFA) 4. Lecture: Bubble acceleration 9. Lecture: High harmonics and attosecond pulses from relativistic mirrors
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International Max Planck Research School on Advanced Photonics
Lectures onRelativistic Laser Plasma Interaction
J. Meyer-ter-Vehn, Max-Planck-Institute for Quantum Optics, Garching, Germany
April 16 – 21, 2007
1. Lecture: Overview, Electron in strong laser field,
3. Lecture: Basic plasma equations, self-focusing, direct laser acceleration
5. Lecture: Laser Wake Field Acceleration (LWFA)
4. Lecture: Bubble acceleration
9. Lecture: High harmonics and attosecond pulses from relativistic mirrors
2
Relativistic Laser Electron Interaction and Particle Acceleration
J. Meyer-ter-Vehn, MPQ Garching
a = eA/mc2
1025
1015
1020
200019851960
1018
I (W/cm2)
2015
GeV electrons
GeV protons
CPA
a = 1
non-relativistic: a < 1
laserelectron
a > 1relativistic:
beam generation
3
Relativistic plasma channels and electron beams at MPQ
C. Gahn et al. Phys. Rev. Lett. 83, 4772 (1999)
gas jet laser
6×1019W/cm2
observed channel
electron spectrum plasma 1- 4 × 1020 cm-3
4
Laser-induced nuclear and particle physics
107 positrons/shot
5
Neutrons From Deuterium Targets
6
GraphikIOQ Jena
2004Ewald
Schwörer
7
Relativistic protons: 5 GeV at 1023 W/cm2
D. Habs, G. Pretzler, A. Pukhov, J. Meyer-ter-Vehn, Prog. Part. Nucl. Physics 46, 375 (2001)
Show that the time averaged light intensity I0 is related to the normalized lightamplitude a0 by
where l is the wavelength, ζ equals 1 (2) for linear (circular) polarization,and P0 is the natural power unit
2 3
0 511 kV 17 kA = 8.67 GWmc mc
Pe e
= =
2 2 18 2 20 0 0 022
W 1.37 10μm
cmI P a aζ ζ
πλ = =
Confirm that the laser fields are
12L 03 10 V/m E a
8L 010 gauss B a
Use that, in cgs units, the elementary charge is e = 4.8 1010 statC and 1 gauss = 1 statC/cm2.
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Special relativity
Relativistic Lagrange function:
L = - mc2 (1− v2/c2)1/2 - qΦ + (q/c) v•A
Galilei:
t´= t x´= x - vt
Lorentz:
t´= γ (t - vx/c2) x´= γ ( x - vt )
γ = (1− v2/c2) -1/2
mechanics electrodynamics
Einstein (1905): Also laws of mechanicshave to follow Lorentz invariance
δA = δ Ldτ = 0
L = γL = -mc2 - (q/c) pµAµ
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2. Problem: Relativistic equation of motion
The Lagrange function of a relativistic electron is (c velocity of light, e and m electron charge and rest mass, f and A electric and magnetic potential)
Use Euler-Lagrange equation
to derive equation of motion
with electric field , magnetic field , and
electron momentum
2 2 2( , , ) 1 /e
L r v t mc v c v A ec
φ= − − − +rr r r
0d L L
dt v r− =r r
( / )/ e cdp dt eE v B= − −r rr r
/E A c t φ= − −rr
B A=rr
2, / , and 1/ 1 .p mc v cγβ β γ β= = = −r rr r
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Symmetries and Invariants for planar propagating wave
symmetry: invariant: ( / ) const , / 0 / e cL r L v p A⊥ ⊥ ⊥ ⊥= = − =
symmetry:
invariant: const
,
( ) / /
/ /
E
x
x
L x ct dH dt L t c L x c dp dt
p c
− = − = =− =
2 2 2 ( / )( , ) 1 / ( ) e cL v x ct mc v c v A x ct− = − − − −
1/ γ
0d L L
dt v r− =
( /c)/ eL v m v Aγ ⊥= −p
Relativistic Electron Lagrangian
2 2 2 2kin 2E ( 1) /2 /xmc p c p m mc aγ ⊥= − = = =
For electron initially at rest:
(relativistically exact !)
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Relativistic side calculation
2 2 2 2
2 2
2 2 2
2
2
( )( ) ( )
( )
( ) ( )2
x
x
x
x
E mc p c
mc
p c
mc
p c
m ccc pp
⊥= + +
= +
= + +
2 /2xp c p m⊥=
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Relativistic electrons from laser focus observedC,L.Moore, J.P.Knauer, D.D.Meyerhofer, Phys. Rev. Lett. 74, 2439 (1995)
2 2kinE ( 1) /2xmc p c p mγ ⊥= − = =
( )
2
2 kin2
kin
2 E 2tan
1/x
p m
p E cθ
γ⊥= = =
−
p⊥
xpθ
γ >>1 electrons emergein laser direction
(follows from L(x-ct) symmetry)
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Relativistic equations of motion
( )ˆ / 0, , y zp p mc a a a⊥ ⊥= = =r
2 2 2kin kin
ˆ ˆ ˆE E / 1 /2 /2xmc p p aγ ⊥= = − = = =
2( ) /a eA mcτ ⊥=
( ) ( )/t t x t cτ = −
2 211 1
2 2
d d d dx d a a d d
dt dt d c dt d d d
τγ γ γτ τ τ τ
= = − = + − =
ˆ ( )y y y
dyp a
c dt
γγβ τ= = =
2ˆ ( ) / 2x x
dxp a
c dt
γγβ τ= = =
ˆ ( )z z z
dzp a
c dt
γγβ τ= = =
( )y
dyca
dτ
τ=
( )z
dzca
dτ
τ=
2 ( )/2dx
c ad
ττ
=
22
Relativistic electron trajectories: linear polarization
0 cosdy
cad
ωττ
= 0( ) ( / )siny t ca ωτω=
220 cos
2
cadx
dωτ
τ=
2
0 12
4 2( ) sin
cax t ωτ
ωτ= +
20a:
0a:
Figure-8 motion in drifting frame (ω=kc)
20
0
2( ) ( /8)sin
sinD
ky a
k x x a ωτ
ωτ=
− =
x
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Relativistic electron trajectories: circular polarization
)0 0 0ˆ ˆ( ) Re{ ( ) } ( 0, cos , sin )i
y za a e ie e a aωτ ωτ ωττ −= =
2 2 2 20 const( ) + y za a a aτ = = = 2
0 const1 /2 aγ = + =
20( ) ( /2 ) x t a ctγ=
2
0
2
0
/2
1 /2( )/ 1 /
a
at x t c t tτ γ
+= − = − =
20ˆ /2x x
dxp a
c dt
γγβ= = =
0 ( / )sin tdz
ac dt
ω γγ =
0 ( / )cos tdy
ac dt
ω γγ = 0 ( / )( ) ( / )sin ty t ca ω γω=
0 ( / )( ) ( / ) cos tz t ca ω γω= m
x
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{ }0Re ( ) ( , , ) exp( )y za e ie a r z t ikz i tω= −r r r
2 20 0 0 0 0( , , ) , / / a a r z t a t a a z kaω=r = =
22
022
2
10 2 ( , ) 0ik a
c tra ζ
ζ⊥+ =− =r
3. Problem: Derive envelope equation
Consider circularly polarized light beam
Confirm that the squared amplitude depends only on the slowly varyingenvelope function a0(r,z,t), but not on the rapidly oscillating phase function
Derive under these conditions the envelope equation for propagation invacuum (use comoving coordinate ζ=z-ct, neglect second derivatives):
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0
1 2 ( , ) 0r ik a r z
r r r z+ =
4. Problem: Verify Gaussian focus solution
Show that the Gaussian envelope ansatz2
0 0 ( , ) exp( ( ) ( )( / ) )a r z P z Q z r r= +
2 2 2 20
2/[ (1 / )]
0 2 22 20
/ ( , ) exp arctan
1 /1 /
Rr r z LR
R RR
z Le z ra r z i i
L r z Lz L
− +
= − +++
inserted into the envelope equation
leads to
Where is the Rayleigh length giving the length of the focal region.
2 20 0/ 2 /RL kr rπ λ= =
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New physics described in these lectures
At relativistic intensities, Iλ2 > 1018 W/cm2 µm2, laser light accelerates electrons to velocity of light in laser direction and generates very bright, collimated beams.
The laser light converts cold target matter (gas jets, solid foils) almost instantaneously into plasma and drives huge currents. The relativisticinteraction leads to selffocused magnetized plasma channels and directlaser acceleration of electrons (DLA).
In underdense plasma, the laser pulse excites wakefields with hugeelectric fields in which electrons are accelerated (LWFA). For ultra-shortpulses (<50 fs), wakefields occur as single bubbles which self-trap electronsand generate ultra-bright mono-energetic MeV-to-GeV electron beams.
At overdense plasma surfaces, the electron fluid acts as a relativistic mirror,generating high laser harmonics in the reflected light. This opens a new route to intense attosecond light pulses.