Su-Ming Weng
Theoretical Quantum Electronics (TQE),
Technische Universität Darmstadt, Germany
Joint work with Prof. P. Mulser, Prof. H. Ruhl
EMMI Workshop “ Particle dynamics under extreme matter conditions”
Speyer, Germany, 26-29 September 2010

“Open Sesame”,
Preface
“Ali baba and the forty thieves”
One Thousand and One Nights
Who open the door for ultrahigh intense laser into an overdense plasma?
1




Classical eletromagnetic (EM) wave propagation
Relativistic induced transparency



Relativistic critical density increase
Relativistic laser beam propagation




Fast ignition
Relativistic plasma shutter
Shortening of laser pulses

2

Classical EM wave propagation

Maxwell’s Equations
1

B c
 t
1


E

4 c t c

J current density
 For a high-frequency monochromatic laser beam in a plasma

B
 t
  i

B ,

E
 t
  i

E , J

Maxwell’s
Equations
 i

2 p
4

E , with

2 4
 p
 e 2 n / m e
, i

B c i
  c
E
(1)
( 2) plasma frequency
   pe
/
2
is the dielectric function of the plasm a.
Eq. (1)-(2)

E ( E )
 c 2
E

0 ( wave equati on )
3

Classical EM wave propagation
 Dispersion relation
0 holds for a uniform plasma in wave equation E ( E )
 c
2 and assume that E : exp( i k ) is a plane wave
E

0 ,
 k
2  
2  
2 p
 2 2 c k , （ disper sion relation ） c
2 plasma frequency
 p
is the minimum frequency for EM wave propagation in a plasma.
the electrons will shield the EM field when
  p
 Critical density the condition
 p
 
defines the socalled critica l d ensity , c n c
 m e

2
/ 4
 e
2   21
  m
2

3
1.1 10 / ( [ ]) cm
 Group velocity (or propagation velocity) v c g 
1



 

1
 p
2

2


1/ 2



 n n c
 1/ 2


4

Relativistic induced transparency


Dimensionless laser amplitude a :
I

2 

2 cA
2 


 18
W cm
2
μm 2

 a 2
Single particle’s 8-like motion for a ≥ 1 y y x a<<1 a
≥
1
T. C. Pesch and H. – J. Kull, Phys. Plasmas 14, 083103 (2007).
x-x d
5

Relativistic induced transparency
 If |v| ~ c, m e
  m e 0
  v
2
/ c
2

1/ 2
) m e 0
 Relativistic critical density n cr
 m e

2
/ 4
 e
2   n c the Lorentz factor averaged from the single particle‘s
8-like motion
   a
2 1/ 2
[1 / 2] ( Linear Polarizati on ),
   a
2 1/ 2
[1 ] ( Circular Polarization ).
 Group velocity (relativistic) v c g

 n n cr
 1/ 2



  n n c
 1/ 2

P. Mulser and D. Bauer, “ High Power Laser-Matter Interaction ” , Springer, 2010.
6

 Motion equations (particles)




 d dt m s r
 d u dt u
γ

,
 q s
( E

  u
2 c
2
1 /
+
B ) , s
 e , i
 PIC simulation cycle

Maxwell’s equations (fields)
 E

B t
B , E
1

0
 c
1
2 t
E
1

0 c
2 j , B 0
7
R. Lichters, et al.
, LPIC++, (1997).

 Laser and plasma parameters
 Cycle-averaged propagation appears very regular, and laser is mainly reflected at the e critical surface as that in nonrelativistic max max
,
 x

20) / L
 x x


5

20
L

L
, otherwise regime n e
 cos
2
 n cr
  e cos
2
 n c
, L is the scale length.
for a
 
Incident wave energy density
E in

( F

)
2 
( G

)
2
Reflected wave energy density
E re

( F

)
2 
( G

)
2 with
F

G


( E y

B z
) / 2

( E B z y
) / 2
8

Critical density VS laser intensity
 A very smoothed critical density can be achieved after being averaged over 10 cycles.
 In a normally incident and linearly polarized laser pulse, relativistic critical density increase is in a good agreement for a

10, motion n cr
/ n c
 
[1
2 a total
/ 2]
1/ 2   
R ) a
2
/ 2

9

Effect of plasma density profile
 For normal incident, if density scale length L> λ
L
,
  n /n is almost independent of density profile cr c
 For a very steep and highly overdense plasma.
  n /n is strongly suppressed cr c n cr
/n is only about 1.5 c for a

5, and n e 
0, x

5

L n c x

5

L
10 electric field at the surface and skin depth

1/ n
1/ 2 e

Effect of laser polarization
 For circular polarization, a density ridge prevents the laser from further propagation and restricts the critical density increase for a

5,
 
0
(a) Linear polari z a tion
Theoretically

1 R a 2 / 2] =4.98,
And fro m PIC,

is about 4.5
5 .
11
(b) Circular polar iza ti on
Theoretically
  
R a ] =6.91,
But from PIC

is a bout 5.33.

Effect of laser polarization
12
 For normal incident, the relativistic critical density increase can be well fitted by
 a
0.62
 a
2 1/ 2
0.76
] (linear polarization)

[1 a
1.2

0.84
a
2 1/ 2
] (circular polarization )

Critical density VS incident angle
 The highest critical density coincides with the lowest reflectivity in the case of 30 o incidence .
conflicts with n n cr
/ c
    
R a
2 1/ 2
[1 (1 ) / 2] .
13 for a

2, for a

2,
 
30 o
This phenomenon is due to the formation of semiblack density cavity and resonance.

14
Relativistic laser beam propagation (LP) linear polarization
 
0, a n e

10, at t=35

L
, and
 

0, x

5 n c x

5

L

L
Theoretically v prop
 v g

/ cr

1/ 2 c

0.66c, but from PIC v prop



L
L

0.35c,
 Sakagami and Mima attributed the inhibition of the propagation velocity to the oscillation of the ponderomotive force and hence the oscillation of electron density at the laser front.
H. Sakagami, K. Mima, Phys. Rev. E 54, 1870 (1996).

15
Relativistic laser beam propagation (CP)
 Ponderomotive force for circular polarized laser f p
  m
4

 x v
2 os
ˆ v os
( )
 
, without oscillatio n
Theoretically v prop
 v g

/ cr

1/ 2 c

0.7c, but from PIC v prop



L
L

0.1c.
CP pulse propagates even more slowly than LP pulse.
 Inhibition of propagation velocity is not attributed to the oscillation of ponderomotive force.

Relativistic laser beam propagation
 Laser pulse can not propagate freely into an overdense plasma, only a part of field penetrates into the plasma by skin effect.
 If this field is big enough, it can accelerate the electrons into relativistic motion and hence induce the relativistic critical density increase for further propagation
 Relativistic induced transparency is guided by the skin penetration.
 Propagation velocity depends on both the group velocity and the efficiency of skin penetration.
Open Sesame!
Open, skin penetration!
16

Relativistic propagation velocity for a

10, an d

= 0
(a) linear polariza ti o n v prop  e xp

  c

2 n
0 n c



1

 n
0 n c
 1/ 2

17 v prop  exp

  c

4 n
0 n c



1

 n
0 n c
 1/ 2
 the different heights of density ridge formed before the front of laser
 different reduced factors exp


2 n
0
/ exp


4 n
0
/

 n c n c


(linear polarized)
(circualr polarized )

Application (a): Fast ignition
 The configuration of fast ignition n e about 10 5 n c
18
 Relativistic induced transparency, together with coneguiding, channeling and hole boring may make the deeper penetration into an overdense target possible, and make the fast ignition easier.
J. J. Honrubia et al, Nucl. Fusion 46, L25 (2006)

Application (b): Relativistic plasma shutter
 A relativistic plasma shutter can remove the pre-pulse and produce a clean ultrahigh intensity pulse
19
This shutter is overdense but relativistic underdense.
S. A. Reed et al.
, Appl. Phys. Lett. 94, 201117 (2009).

Application (c): Shortening of laser pulses
 A quasi-single-cycle relativistic pulse can be produced by ultrahigh laser-foil interaction
Initial thin overdense foil evolves into a thick but rare plasma, which is relativistically transparent.
20
L. L. Ji et al.
, Phys. Rev. Lett. 103, 215005 (2009).

 Relativistic induced transpancy makes the penetration of a laser pulse into a overdense plasma possible.
 We proposed a method for determining the critical surface and the relativistic critical density increase in PIC simulations.
 For normal incident linearly polarized pulse, the critical density increase follows approximately the theory from a single particle orbit.
 Due to density profile steepening before the laser front, the critical density increase is much lower than predicted for a circular polarized pulse.
 Relativistic induced transparency is guided by the skin penetration. The propagation velocity depends upon both, the group velocity and the efficiency of skin penetration.
 Relativistic induced transparency finds wide applications in fast ignition scheme, relativistic plasma shutter, and shortening of laser pulses.
21

谢 谢!
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