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, 2. Lecture: Basic plasma equations, self-focusing, direct laser acceleration 3. Lecture: Laser Wake Field Acceleration (LWFA) 4. Lecture: Bubble acceleration 5. Lecture: High harmonics and attosecond pulses from relativistic mirrors 1

Relativistic Laser Electron Interaction and Particle Acceleration

J. Meyer-ter-Vehn, MPQ Garching a = eA/mc 2 non-relativistic: a < 1

laser electron

relativistic: a > 1 beam generation I (W/cm 2 ) 10 25 10 20 GeV protons GeV electrons 10 18 a = 1 10 15 CPA 1960 1985 2000 2015 2

Relativistic plasma channels and electron beams at MPQ C. Gahn et al. Phys. Rev. Lett. 83, 4772 (1999) laser 6×10 19 W/cm 2 gas jet plasma 1- 4 × 10 20 cm -3 electron spectrum observed channel 3

Laser-induced nuclear and particle physics

10 7 positrons/shot 4

Neutrons From Deuterium Targets

5

Graphik IOQ Jena 2004 Ewald Schwörer 6

Relativistic protons: 5 GeV at 10

23

W/cm

2 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 Simulations: 1 kJ , 15 fs laser pulse focussed on 10 m m spot of 10 22 /cm 3 plasma 7

IPP Summer University, Garching 2006

Inertial Confinement Fusion (ICF)

J. Meyer-ter-Vehn Max-Planck-Institute for Quantum Optics, Garching few mg DT few mm imploded core 100 m m 8

Simulation

D

2

burn fast-ignited from DT seed

Atzeni, Ciampi, Nucl. Fus. 37, 1665 (1997) 15 ps 5 ps DT seed (0.1 mg T) 1000 g/cc

5 kJ

20 mg D 2 beam heated region 25 ps 35 ps bulk fuel 45 ps D 2 burn produces more tritium than in seed: breeding ratio: 1.37

55 ps yield 1.3 GJ 9

Nature Physics 2, 456 (2006)

Laser

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

L=3.3 cm, f =312 m m Plasma filled capillary Density: 4x10 18 /cm 3

1 GeV electrons

Divergence(rms): 2.0 mrad Energy spread (rms): 2.5% Charge: > 30.0 pC 10

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) 11

Observation of high harmonics from plasma surfaces acting as relativistic mirrors

B. Dromey, M. Zepf et al., Nature Physics 2, 698 (2006) w -5/2 12

Plane Laser Wave

 Re{

A e

0 

k

 w /

c

 w

t

w

t

) }

E

 

B S

   Re{(

i

w / ) 0

i

 } 4  ) Re{

ik



A e

0

i

 } lin. pol. (LP):

A

0 

A

0

e y

circ. pol. (CP):

A

0 

A

0 (

e y



ie z

)

I



S

 ( w

k

/ 8  )

A

0 2

I

0  2   2  ) , for LP , 2 , for CP . 8  2

A

0 2    2

c A

2 0   1 (2) for lin. (circ.) polarization 13

Relativistic Intensity Threshold

/    )  

eE

(non-relativistic v/c << 1)

v c

 Re   

eE imc

w    

eA

0

mc

2

e y

 for LP

e y

cos 

e z

s in  f or CP valid for

a

0 

eA mc

0 2 < < 1 , relativistic regim e :

a

0  1 Average intensity: Power unit:

I

0  2    2

P a

0 0 2    

P

0 

e e

3   18 W cm 2  

a

0 2 14

1. Problem: Normalized light amplitude a

0 = eA 0 /mc 2

Show that the time averaged light intensity

I 0

amplitude

a 0

by

I

0  2    2    18 is related to the normalized light W cm 2  

a

0 2 where

l

is the wavelength,  equals 1 (2) for linear (circular) polarization, and

P 0

is the natural power unit 2

mc mc

3

P

0  

e e

Confirm that the laser fields are

E

L   12

B

L  8 10 gauss 

a

0 

a

0 Use that, in cgs units, the elementary charge is

e =

4.8 10 10 1 gauss = 1 statC/cm 2 .

statC and 15

Special relativity

mechanics Galilei: t´= t x´= x - vt electrodynamics Lorentz: t´= g (t - vx/c 2 ) x´= g ( x - vt ) g  ( 1 v 2 /c 2 ) -1/2 Einstein (1905): Also laws of mechanics have to follow Lorentz invariance Relativistic Lagrange function: L = - mc 2 (1 v 2 /c 2 ) 1/2 - q F + (q/c) v•A d A = d  L d t  0 L = g L = -mc 2 (q/c) p m A m 16

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)  

mc

2 1 

v

2 /

c

2 

e c

f Use Euler-Lagrange equation to derive equation of motion

d



L

 

L



r

 

eE

  0

v



B

electron momentum

p

 g

mc

,   f g  1/ 1   2 .

17

Symmetries and Invariants for planar propagating wave

Relativistic Electron Lagrangian  )  

mc

2 1  2 / 2  1/ g  (  )

d



L

 

L



r

 0    

p A

 symmetry:

L r

 0 , invariant:

L v



p

  symmetry:  ) , invariant: E 

p c x

 const

x

/

A

  const For electron initially at rest: E kin 

mc

2 ( g

p c x



p

2  /2

m

 (relativistically exact !) 2 / 2 18

Relativistic side calculation

E

2  (

mc

2 2 )  (

p c x

) 2  (

mc

2 

p c x

) 2  (

p c

) 2  (

m c

2 2 )  2

mc

2

p x c

 (

p x c

) 2

p c x



p

2  /2

m

19

Relativistic electrons from laser focus observed

C,L.Moore, J.P.Knauer, D.D.Meyerhofer, Phys. Rev. Lett. 74, 2439 (1995) E kin 

mc

2 ( g

p c x



p

2  /2

m

tan 2    

p



p x

  2  (

E

kin /

c

kin  2  g 2  1 (follows from

L(x-ct)

symmetry)

p

 

p x

g >>1 electrons emerge in laser direction 20

Relativistic equations of motion

a

 t 2    ˆ kin  E / kin

mc

2 g ( 0,

a y

,

a z



x



p

2  /2 

a

2 /2 g

d dt

 g

d

t

d dt d

t  g  1  1

dx



d

t  1 

a

2 2 

a

2 2  

d d

t 

d d

t

y

 g

y z

 g

z

 g 

dy

g

c dt dz c dt



a y



a z x

 g

x

 g

dx c dt



a

2

dy d

t 

ca y dz d

t 

ca z dx d

t  2 21

a

0 Relativistic electron trajectories: linear polarization

dy d

t 

ca

0 cos wt

dx d

t 

ca

0 2 2 cos 2 wt  (

ca

0 wt 

ca

0 2 4   t  1 2 w sin 2 wt  

a

0 2 x Figure-8 motion in drifting frame ( w =kc)

ky

 

a

0

x D

)  sin wt (

a

0 2 /8)sin 2 wt 22

Relativistic electron trajectories: circular polarization

a

t

a

2  Re{ ( 0 ˆ

y

 ˆ

z



a

2

y

+

a z

2 

a

0 2  

i

wt ) 

a

0 cos wt , 

a

0 sin wt ) const g

a

0 2 /2  const

x

 g

x

 g

dx c dt



a

0 2 /2 t  1   (

a

0 2 g

ct a

0 2 /2 1 

a

0 2 /2   

t



t

/ g g

dy



a

0 cos ( w g

c dt

g

dz

 

a

0 sin ( w g

c dt

 (

ca

0  (

ca

0 ( w g ( w g 23 x

3. Problem: Derive envelope equation Consider circularly polarized light beam

a

 

e y



z

) 0 ( , , ) exp(  w )  Confirm that the squared amplitude depends only on the slowly varying envelope function

a 0 (r,z,t)

, but not on the rapidly oscillating phase function

a

2 

a r z t

0 2 ( , , ) , 

a

0 / 

t

w

a

0 

a

0 / 

z ka

0 Derive under these conditions the envelope equation for propagation in vacuum (use comoving coordinate 

=z-ct

, neglect second derivatives):  

c

1 2 

t

2  

a

 0    2

ik

    

a

0   0 24

4. Problem: Verify Gaussian focus solution Show that the Gaussian envelope ansatz

a r z

0   ( )( / ) ) 0 inserted into the envelope equation  1 

r

 

r

 2

ik

 

z



a r z

0 ( , )  0 leads to

a r z

0 

e



r

2 /[

r

0 2 (1 

z

2 /

L

2

R

)] 1 

z

2 /

L

2

R

exp   

i

arctan  

z L R

 

L R



kr

the focal region.

0 2 / 2  

r

0 2 /  2 1 

z

2 /

R L

2

R

  25

New physics described in these lectures

At relativistic intensities, I  2 > 10 18 W/cm 2 m m 2 , 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 relativistic interaction leads to selffocused magnetized plasma channels and direct laser acceleration of electrons (DLA).

In underdense plasma, the laser pulse excites wakefields with huge electric fields in which electrons are accelerated (LWFA). For ultra-short pulses (<50 fs), wakefields occur as single bubbles which self-trap electrons and 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.

26

ICF target implosion

27