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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

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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

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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

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Neutrons From Deuterium Targets

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GraphikIOQ Jena

2004Ewald

Schwörer

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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)

Experiments:Multi-10 MeV

ion beams from thin foils

observed

1 kJ , 15 fs laser pulsefocussed on 10 µm spot

of 1022/cm3 plasma

Simulations:

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Inertial Confinement Fusion (ICF)J. Meyer-ter-Vehn

Max-Planck-Institute for Quantum Optics, Garching

IPP Summer University, Garching 2006

few mmimploded core

100 µm

few mg DT

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D2 burn fast-ignited from DT seedAtzeni, Ciampi, Nucl. Fus. 37, 1665 (1997)

bulk fuel

DT seed (0.1 mg T)

beam heated region

20 mg D2

15 ps

25 ps

55 ps

35 ps

5 ps

45 ps

1000 g/cc

5 kJ

yield 1.3 GJ

D2 burn produces more tritium than in seed:

breeding ratio: 1.37

Simulation

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Nature Physics 2, 456 (2006)

L=3.3 cm, φ=312 µmLaser

1.5 J, 38 TW, 40 fs, a = 1.5

Plasma filled capillary

Density: 4x1018/cm3

Divergence(rms): 2.0 mradEnergy spread (rms): 2.5%Charge: > 30.0 pC

1 GeV electrons

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Design considerations for table-top, laser-based VUV and X-ray free electron lasers

F. Grüner, S. Becker, U. Schramm, T. Eichner, M. Fuchs, R. Weingartner, D. Habs, J. Meyer-ter-Vehn, M. Geissler, M. Ferrario, L. Serafini,

B. van der Geer, H. Backe, W. Lauth, S. Reiche

http://arxiv.org/abs/physics/0612125 (Dec 2006)See also from DESY: Arxiv:physics/0612077 (8 Dec 2006)

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ω-5/2

B. Dromey, M. Zepf et al., Nature Physics 2, 698 (2006)

Observation of high harmonics from plasma surfacesacting as relativistic mirrors

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Plane Laser Wave

0/ Re{( / ) }iE A c t i c A e ψω= − =r rr

0Re{ }iB A ik A e ψ= =rr rr

4( / )S c E Bπ=r r r

2

08( / )AI kS πω= =r for LP

for CP

2 , ,

(1 cos )

2 , .

ψ−

2 2

0 0

22

08 2

A Ak

I cζ ζπ

ω λ πλ = = for lin. (circ.) polarization1 (2) ζ =

00 yAA e=r r

00 ( )y zAA e ie=r r r

lin. pol. (LP): circ. pol. (CP):

( )0( , ) Re{ }i k r tA r t A e ω−=

r rr r

k r tψ ω= −r r

/ 2 /k cω π λ= =

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Relativistic Intensity Threshold

/ ( / )m dv dt e E v c B eE= − + −r r rr r

02

ReeAv eE

c imc mcω=

rr forcos

cos

LP

fs or CPin

y

y z

e

e e

ψψ ψm

rr r

20 0 0valid for , relativistic regim< e< : 1 1a eA mc a= >

(non-relativistic v/c << 1)

2 2 18 2 20 0 0 022

W 1.37 10μm

cmI P a aζ ζ

πλ = =

2 3

0 511 kV 17 kA = 8.67 GWmc mc

Pe e

= =

Average intensity:

Power unit:

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1. Problem: Normalized light amplitude a0 = eA0/mc2

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

=

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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)

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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.

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ICF target implosion