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Relativistic induced transparency and laser propagation in an overdense plasma

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

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Preface

“Open Sesame”,

“Ali baba and the forty thieves”

One Thousand and One Nights

Who open the door for ultrahigh intense laser into an overdense plasma?

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OutlineTheoretical background

Classical eletromagnetic (EM) wave propagation Relativistic induced transparency

Numerical simulations Relativistic critical density increase Relativistic laser beam propagation

Applications Fast ignition Relativistic plasma shutter Shortening of laser pulses

Conclusion

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Classical EM wave propagation

Maxwell’s Equations1

1 4

c t

c t c

BE

EB J

2 2where =1- / is the dielectric function of the plasm a.pe 2

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( ) 0 (wave equati ) onc

E E E

For a high-frequency monochromatic laser beam in a plasma

2 22

, , , with 4 / , 4 p e

p ei

i it

nt

m

B EB E J E

(1)

( 2)

i

ci

c

E B

B E

Maxwell’s Equations

Eq. (1)-(2)

current density

plasma frequency

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Classical EM wave propagation Dispersion relation

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2( )0 holds for a uniform plasma in wave equation ,

and assume that exp( ) is a plane wave

0c

i i t

E E

x

E

E k

E

:2

2 2 2 2 22

, disper sion relationpk c kc

（ ）

the condition defines the criticaso- l dcalled ensity ,p cn

2 2 21 2 3/ 4 1.1 10 / ( [ ]) cm c en m e m

Group velocity (or propagation velocity)

1/ 2 1/ 22

2

11 1g p

c

v n

c c k n

plasma frequency is the minimum frequency for EM wave propagation in a plasma.

the electrons will shield the EM field when

p

p

Critical density

6Relativistic induced transparency

Single particle’s 8-like motion for a ≥ 1

x

y

a<<1

x-xd

y

a ≥1T. C. Pesch and H. –J. Kull, Phys. Plasmas 14, 083103 (2007).

2 2 18 22

W 21.37 10 μm2 cm

I cA a

Dimensionless laser amplitude a:

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Relativistic induced transparency

If |v| ~ c,2 2 1/2

0 0(1 / )e e em m v c m

2 1/ 2

2 1/ 2

( ),

( ).

[1 / 2]

[1 ]

Linear Polarizati

Circular Polarization

ona

a

2 2/ 4cr e cn m e n

Group velocity (relativistic)1/21/2

1 1g

cr c

v n n

c n n

Relativistic critical density

the Lorentz factor averaged from the single particle‘s 8-like motion

P. Mulser and D. Bauer, “High Power Laser-Matter Interaction”, Springer, 2010.

8Particle-in-Cell (PIC) simulation

2 2 ,

(

1 /

,, )s s

d

dt

dm

u c

idt

eq s

r u

γ

u uE B

Motion equations (particles)

0

2 20

1,

1 1, 0

t

tc c

E B E

B E j B

+

PIC simulation cycle

Maxwell’s equations (fields)

R. Lichters, et al., LPIC++, (1997).

9Determination of critical density

max

max

0, 5

, 20

exp ( 20) / , otherwise

is the scale length.

L

e L

L

x

n n x

n x L

Laser and plasma parameters

2 2 cos cos ,e cr e cn n n

Cycle-averaged propagation appears very regular, and laser is mainly reflected at the critical surface as that in nonrelativistic regime

for 10, incident angle =0a

2 2

2 2

Incident wave energy density

Reflected wave energy

( ) ( )

( ) (

density

with

( ) / 2

( ) / 2

)

in

re

y z

z y

E F G

E F G

F E B

G E B

10Critical density VS laser intensity

A very smoothed critical density can be achieved after being averaged over 10 cycles.

for 10,

incident angle =0

a

2 1/ 2 2 1/ 2 / [1 / 2] [1 (1 ) / 2 ( reflectivity)] , . cr c totaln n a R Ra

In a normally incident and linearly polarized laser pulse, relativistic critical density increase is in a good agreement with the averaged value from the single particle’s 8-like motion

11Effect of plasma density profile

For a very steep and highly overdense plasma.

1/ 2

c/n is only about 1.5

for 5, and

0, 5

100 ,

electric field at the surface

and skin depth 1/

5

cr

Le

c

e

L

nn

n

n

a

x

x

For normal incident, if density scale length L>λL,

/n is almost independent of density profilecr cn

/n is strongly suppressedcr cn

12Effect of laser polarization

for 5, 0a

For circular polarization, a density ridge prevents the laser from further propagation and restricts the critical density increase

2 1/2

Theoretically

=[1 (

(a)

1 ) / 2] =4.98,

Linear

And fro

polari

m PIC, is about 4.5

a

5

tion

.

z

R a

2 1/2

Theoretically

=[1 (1 ) ] =6.91,

(b) Circular polar

But from PIC

iza

is

ti

a

on

bout 5.33.

R a

13Effect of laser polarization

For normal incident, the relativistic critical density increase can be well fitted by

0.62 2 1/ 2

1.2 2 1/ 2

(linear polarization)

(circular polarization

[1 0.4 0.76 ]

[1 0.84 ] )

a a

a a

14Critical density VS incident angle

The highest critical density coincides with the lowest reflectivity in the case of 30o incidence.

for 2, a

2 1/ 2conflicts with / [1 (1 ) / 2] .cr cn n R a

This phenomenon is due to the formation of semi-black density cavity and resonance.

ofor 2, 30a

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Relativistic laser beam propagation (LP)

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.

L

linear polarization 0,

10, at t=35 , and

0, 5

5 , 5L

ec L

a

xn

n x

1/2

Theoretically

1 / 0.66c,

but from PIC

(15.5 5)0.35c,

(35 5)

prop g cr

Lprop

L

v v n n c

v

H. Sakagami, K. Mima, Phys. Rev. E 54, 1870 (1996).

16Relativistic laser beam propagation (CP)

Inhibition of propagation velocity is not attributed to the oscillation of ponderomotive force.

2 ˆ( ) , without oscillatio( ) / 4

n, osp os v xm

f v x xx

eE m

1/2

Theoretically

1 / 0.7c,

but from PIC

(8.0 5)0.1c.

(35 5)

prop g cr

Lprop

L

v v n n c

v

Ponderomotive force for circular polarized laser

CP pulse propagates even more slowly than LP pulse.

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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.

Relativistic induced transparency is guided by the skin penetration.

Propagation velocity depends on both the group velocity and the efficiency of skin penetration.

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

Open Sesame!

Open, skin penetration!

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Relativistic propagation velocity

1/2

0 0

(a) linear polariza

2e

ti

for 10, an

xp 1

=

o

d 0

n

prop

c c

v n n

c n

a

n

1/2

0 04exp

(b) circular polari a

1

z tion

prop

c c

v n n

c n n

0

0

exp 2 / (linear polarized)

exp 4 / (circualr polarized

different reduced factors

)

c

c

n n

n n

the different heights of density ridge formed before the front of laser

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Application (a): Fast ignition The configuration of fast ignition

J. J. Honrubia et al, Nucl. Fusion 46, L25 (2006)

ne about 105nc

Relativistic induced transparency, together with cone-guiding, channeling and hole boring may make the deeper penetration into an overdense target possible, and make the fast ignition easier.

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Application (b): Relativistic plasma shutter

A relativistic plasma shutter can remove the pre-pulse and produce a clean ultrahigh intensity pulse

This shutter is overdense but relativistic underdense.

S. A. Reed et al., Appl. Phys. Lett. 94, 201117 (2009).

21Application (c): Shortening of laser pulses

A quasi-single-cycle relativistic pulse can be produced by ultrahigh laser-foil interaction

L. L. Ji et al., Phys. Rev. Lett. 103, 215005 (2009).

Initial thin overdense foil evolves into

a thick but rare plasma, which is relativistically transparent.

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Conclusion Relativistic induced transpancy makes the penetration of a laser

pulse into a overdense plasma possible.

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.

We proposed a method for determining the critical surface and the relativistic critical density increase in PIC simulations.

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Vielen Dank!

谢 谢 !

Thank You!