“European Extreme Light Infrastructure:Scientific Prospects”

ENSTAParis

December, 10, 2005

S. V. BulanovAdvanced Photon Research Centre, JAEA,

Kizu, Kyoto, Japan

Electrodynamics of Continuous Mediain the Extreme Relativistic Regime

M. Borghesi (QUB, Belfast, UK)T. Esirkepov (APRC-JAEA, Kizu, Japan)Y. Kato (JAEA, Japan)J. Koga (APRC-JAEA, Kizu, Japan)G. Mourou (ENSTA, Paris, France)F. Pegoraro (Pisa University, Pisa, Italy)T. Tajima (APRC-JAERI, Kizu, Japan)

Laser Intensity vs. years

Maiman, T. H. "Stimulated Optical Radiation in Ruby." Nature 187, 493-494, 1960

Mourou, G. A., Barty, C. P. J., and Perry, M. D., 1998, Phys. Today 51, 22 Bahk, et al., Opt. Lett. 29, 2837 (2004)

Ultra-Relativistic Effects in Laser – Plasma Interaction

Quiver electron energy becomes larger than mec2 when the dimensionless amplitude of the laser pulse is greater than unity: a0=eE0/mewc>1 . The electron energy scales as E = mec2 a0

2/2.This corresponds for 1μm laser wavelength to the intensity above 1.35ä1018W/cm2. Recently Bahk, et al., Opt. Lett. 29, 2837 (2004), reported the experimental demonstration of I=1022W/cm2.The ELI will achieve even high intensity. For such intense laser the nonlinear plasma electrodynamics becomes of the key importance with charged particle (electron and ion) acceleration, laser pulse shortening and its frequency upshifting.

Laser Accelerators of Charged Particles

A. Electron acceleration in the wake wave left in a tenuous plasma behind the ultra-short laser pulse, by the Laser Wake Field Acceleration (LWFA) mechanism or/and direct electron acceleration by the laser field.

B. Ion acceleration in the regimes of strong electric charge separation (when the electrons accelerated by the laser radiation leave the irradiated by the laser pulse region) and by the Radiation Pressure Dominated Acceleration (RPDA)mechanism when the ions are trapped inside the plasma cloud, which is accelerated by the light pressure.

Laser Pulse Shortening and Inensificationduring Nonlinear Laser-Plasma Interaction

A. Laser pulse shortening with its intensification and the frequency upshifting during interaction with nonlinear Langmuir waves in the Flying Mirror Light Intensification (FMLI) process.

Extreme Intensity and Power Laser Radiation for Nonlinear Vacuum Probing

A. Electron-positron pair creation in vacuum.

B. Nonlinear refraction index due to vacuum polarization.

QED processes

Electron-positron pair generation; Bremsstrahlung; Inverse Compton Scattering; Trident process; Bethe-Heitler process

Laser Pulse

e+e-

e-γ

γe+

e-

e-

e-e+

e-

e-

γ e+

e-

e-

e-

e-

e-

laser pulsetarget

We reach a limit when the nonlinear QED with the electron-positron pair creation in the vacuum comes into play, at the critical QED electric field, which corresponds to so strong electric field that produces a work on the Compton length equal to mec2, i.e.

2 3 /eQ EDE m c e=

29 210 /W cm≈

Upper Limit on the Electric Field Amplitude

eEx2mc−

e+e− 22 / | |mc eE

x

E 2mc

0

Sub-barrier tunneling

2

2 4

1 exp4

QED

QC ED

Ec EwE E

ππ

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

W.Heisenberg, H.Euler (1936)J. Schwinger (1951)E. Brezin & C. Itzykson (1970)V. S. Popov (1971)

It corresponds to the intensity

ELECTRON-POSITRON PAIR PRODUCTION BY FOCUSED ELECTROMAGNETIC PULSES

Pair production by single focused pulse: N. B. Narozhny, et al., Phys. Lett. A 330, 1 (2004). Electron-positron pairs produced by focused laser pulse intensity two orders of magnitude less than critical ~1027 W/cm2

Pair production by oppositely directed focused laser pulses: N. B. Narozhny, et al., JETP Lett., 80, 434 (2004). Pair production at intensities one-two orders of magnitude less than single focused pulse ~1026 W/cm2

Nonlinear QED Vacuum

In a strong EM field vacuum behaves similarly to a birefracting, i.e. anisotropic medium. This fact is known since papers published by Halpern (1933), and by Heisenberg & Euler (1936). After discovering the pulsars and with the emerging of the lasers able to generate relativistically strong EM fields, it becomes clear that the effects of vacuum polarization can be observed in cosmos and under laboratory conditions .

One of the most beautiful effects predicted by QED is photon-photon scattering due to vacuum polarization. This process is described by the diagram:

The cross section of photon-photon scattering in the limit

is equal to

for we have

it reaches its maximum

at

622

2

97310125 e

e

rm c

ωσ απ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

2em cω 2

44.7 cσ αω

⎛ ⎞= ⎜ ⎟⎝ ⎠

30 2max 10 cmσ − −≈

21.5 em cω ≈

Berestetskii, Lifshitz, Pitaevskii, Quantum Electrodynamics (1982)

+ -e e pair creation

2em cω

Heisenberg-Euler Lagrangian

The intense laser light utilization for studies of nonlinear QEDvacuum were discussed by Aleksandrov, et al., (1985), Rozanov (1993,1998), Marklund and Shukla (2005) who considered theoretically a number of nonlinear processes: 4-wave interactions, induced focusing, etc.

Theoretical description of nonlinear QED vacuum in the limitand

is based on the Heisenberg-Euler Lagrangian

with and

QEDE E 2em cω

( )L21 5 14

16 64F F F F F F F Fαβ αβ βγ δμαβ αβ αβ γδ

κπ π

⎡ ⎤= − −⎢ ⎥⎣ ⎦

4 74/ 45 ee m cκ π= F A Aα ααβ β β= ∂ −∂

Interaction of two counter-propagating pulses is described by 2 2

1 1 2 1 2 1 2 2 2 2 1 2| | ; | |z zt tE E in E E E E in E Eω ω∂ +∂ = ∂ −∂ =

with 4 742 7( /45 )en e m cπ=

These equations yield for exp( ) 1,2j j jE I i j= Φ =

0 0

1 1 2 2

0 01 1 2 1 2 2 2 2 2 1

( ), ( ), ,

( , ) ( , ) ( /2) ( ) ; ( , ) ( , ) ( /2) ( )v u

v u

I I u I I v u z t v z t

u v u v n I s ds u v u v n I s dsω ω

= = = − = +

Φ = Φ + Φ = Φ −∫ ∫

The pulse profiles transport without any distortion and the phases undergo nonlinear shifts (N. N. Rozanov, 1993)

1

2

for it is equal to

In order to approach the “nonlinear vacuum frontier” we must increase either the EM wave power or decrease its wavelength.

There is no self-focusing of the plane EM wave because both the

invariants, and , vanish

Two counter-propagating EM waves undergo induced focusing because in this case

2 2( )/2F B E= − ( )G E B= ⋅

P2 22454QED

cr

cE λπα π

=

The critical power for transverse nonlinear effects is

P 242.5 10cr W≈ ×

0F ≠

1 mλ μ=

20

1'' 4

1 ph

cc

+ω = ω ≈ γ ω

−ph

ph

vv

EM Pulse Intensification and Shortening in Flying Mirror Light Intensification (FMLI) process

(RELATIVISTIC ENGINEERING)

6max 0'' ( )ph phI R I≈ γ γ

seedlaserpulse

Wake Wave

driverlaserpulse

g c≈v

0 0, Iω

reflectede.m. pulse

'', ''Iω

S. V. Bulanov, T. Esirkepov, T. Tajima, Phys. Rev. Lett. 91, 085001 (2003)

3( )ph phR −γ γ∼

Wake Waves

Kelvin Ship Waves

21

21

1cos 1 cos2

cos sin/ 2 / 2

x X

X

k

y

g

θ θ

θ θ

ω

π θ π

⎛ ⎞= −⎜ ⎟⎝ ⎠

=− < <

=

( )1 / 22

2 /

4 /

p p p p h p e

p e e

k k

n e m

λ π ω

ω π

= =

=

v

Wake Plasma Waves Wave Break

Wake-Wave-Breaking can be destructive or it can develop in a gradual (gentle), i.e. in a controllable way, which, in the case of the wake wave, provides a mechanism for the electron injection into the acceleration phase.

gvphv

ph g=v v

pλ

← − − − →pλ

←→

wake wave

3D relativistically strong wake wave has a paraboloidal form

laser pulse

Paraboloidal Form of the Wake Wave

/ 5

212

pe

al

ω ω

λ

=

==

Transverse Wake-wave Breaking

sL

sD

,s sIωph phcβ=v

2'' / 4s s phL L γ≈

'λ

''λ

phcβ=ev

'sL

sD

', 's sIω0ph =v

'λ

'λ

L

L

M

LaboratoryFrame

MovingFrame

LaboratoryFrame

θ

e

eEa invm eω

= =

2

1

1ph

ph

γβ

=−

e ph=v v

1'

1 2sph

sph ph

λβλ λ

β γ−

= ≈+

Focal spot diameter 'λ≈

EM Pulse Length: '2

ss

ph

LLγ

≈

Incident Pulse

Reflected Pulse 2

1''

1 4sph

sph ph

λβλ λ

β γ−

= ≈+

Focal region:

|| ''l λ≈

'l λ⊥ ≈

Wake

Wave

Counter-propagation interaction

sL

sD

,s sIωph phcβ=v

'λ

''λ

phcβ=ev

'sL

sD

', 's sIω0ph =v

'λ

'λ

L

L

M

LaboratoryFrame

MovingFrame

LaboratoryFrame

θ

e

eEa invm eω

= =

2

1

1ph

ph

γβ

=−

2 2, 1 / !ph p se ph c ω ω> = −= gv v vv

21'

/ 1ph

s sph g

βλ λ λ

−= >

−v v

Focal spot diameter ' sλ λ≈ >

EM Pulse

Reflected Pulse

2

2

1''

1 2ph

sph ph g

βλ λ

β β β−

≈+ −

Focal region:

|| ''l λ≈

' sl λ λ⊥ ≈ >

Wake

Wave

Co-propagation interaction: photon accelerator

S.V. Bulanov and A

. S. Sakharov, JETPLett. 54, 203

(1991)S. V. B

ulanov, F. Pegoraro, A. M

. Pukhov, Phys. Rev. Lett. 74, 710 (1995)

Z-M.Sheng, Y. Sentoku, K

. Mim

a, K. N

ishihara,Phys. Rev. E 62, 7258 (2000)

0

1''( )

1 cosph

ph

βω θ ω

β θ+

=−

Reflected E.M. Beam

Frequency:

1/ phθ γ:

Almost all theenergy isemitted

into narrowangle

e ph=v v Focus

Collimation:

Intensity: 3 20( / )ref ph s sI D Iγ λ≈

Power: 0ref phP Pγ≈

Energy: 1

0ref phγ −≈E E

WakeWake

WaveWave

Critical QED Electric Field

Intensity: 3 20( / )ref ph s sI D Iγ λ≈

4 refref QED

IE E

cπ

= →

The electric field can not be larger than the Schwinger field in the co-moving with the mirror reference frame, M.

In the laboratory reference frame, L, we have for the EM field invariant

i.e. the upper limit for the electric field is

22 2

22 2QED

ph

EB EFγ

−= ≈ −

ref ph QEDE Eγ≤

Take an example of the wakefield excitation in 1018cm− 3plasma by the EM wave with a0= 15. This means the Lorentz factor associated with the phase velocity of the wakefield is related to ω pe / ω , as a0

1/2(ω pe/ω ), ≈125. Thus the laser pulse intensification of the order of 465 may be realized. The counter-propagating 1μm, 2× 1019 W /cm2 laser pulse is partially reflected and focused by the wakefield cusp. If the reflected beam diameter is 40µm, the final intensity is 5× 1028 W /cm2.

We used the wavebreaking condition: γe ≈ a02. The driver pulse intensity

should be sufficiently high and its beam diameter should be enough to give such a wide mirror, i.e. to be 4× 1020 W /cm2 with the diameter 40µm. Thus, the driver and source must carry 6 kJ and 30 J, respectively.

Reflected intensity can approach the Schwinger limit. In this range of the electromagnetic field intensity it becomes possible to investigate such the fundamental problems of nowadays physics using already available laser, as e.g. the electron-positron pair creation in vacuum and the photon-photon scattering WITH the ELI PARAMETERS.

Needed Laser Pulse Energies

Laser pulse

plasma(electron density)

wake wave

(3D PIC simulation)

t=14 λd/c

paraboloidal

relativistic

flying

mirrors

3D Particle-In-Cell simulation

3D Particle-In-Cell simulation

Collimated

bursts of

polarized

EUV or X-ray

radiation

114

1f ph

i ph

ω βω β

+≈ =

−

0.87phβ ≈

2phγ ≈

max

0

16fEE

≈

parabolidal mirror velocity

γph-factor

frequency upshift

reflected EM wave amplitude

reflected EM wave intensity max

0

256fII

≈

3D Particle-In-Cell simulation gives:

Laboratory Astrophysics Laboratory Astrophysics with the High Power Lasers

Paticle Physics Astro-Particle Physics

Astrophysics Laboratory Laser Physics

What can we learn from the Astrophysics of Cosmic Rays?

V. S. Berezinskii, S. V. Bulanov, V. L. Ginzburg, V. A. Dogiel, V. S. Ptuskin, Astrophysics of cosmic rays.

(North Holland Publ.Co. Elsevier Sci. Publ. Amsterdam, 1990)

Laser-Plasma Interaction in the Radiation Dominated Regime

(a0>316)

The Crab Pulsar, lies at the center of the Crab Nebula. The picture combines optical data (red) from the Hubble Space Telescope and x-ray images (blue) from the Chandra Observatory. The pulsar powers the x-ray and optical emission, accelerating charged particles and producing the x-rays.

However these high energy electrons can not reach the Earth!

PeV γ from Crab Nebula

Gamma-ray image, HESS telescopes, of the SNR RX J1713.7 2 3946, and the gamma-ray spectrum (F. A. Aharonian, et al., Nature, 432, 75 (2004)).

Supernova Shock Wave Acceleration of Charged Particles

In solar flares the synchrotron losses become dominant for relativistic electrons with the energy above 1 GeV

Solar Cosmic Ray Acceleration during Magnetic Field Line Reconnection

In the circularly polarized EM the charged particle moves along a circle trajectory. We may borrow the expressions for the properties of the radiation emitted by the particle from the theory of synchrotron radiation. Equations of the electron motion are:

Where the radiation force is given by

iik i

e kdu em c F u gds c

= +

22 2

2 2

23

ii i k kd ue d ug u u

c ds ds

⎛ ⎞⎟⎜ ⎟= −⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

Radiation Losses of Ultrarelativistic Electrons in the EM Wave – Plasma Interaction

In the relativistic limit the radiation losses are described by the formula,

which gives

For synchrotron losses we have and

2

2 3

23 e

dp dpeWd dm c

μ μ

τ τ⎛ ⎞

= ⎜ ⎟⎝ ⎠

24 4

2

23e cWR

β γ=

/R pc eB=4 2

22 3

163 8e

e c BWm cπ γ

π=

In the case of the charged particle interaction with the electromagnetic wave (circular polarization) the orbit radius and the particle momentum are

0 2cR λ

ω π= = 0ep m ca=

420 0

43

ee

rW m c aπ= ω

λThe energy gain rate

is larger than the energy losses if

20 0em c a≈ ω

1/3

0 31643

era−π⎛ ⎞<< ≈⎜ ⎟λ⎝ ⎠

Zel’dovich, Ya. B., 1975, Sov. Phys. Usp. 18, 79

Zhidkov, A., Koga, J., Sasaki, A., and Uesaka, M., 2002, Phys, Rev. Lett. 88, 185002

S. V. Bulanov, T. Esirkepov, J. Koga, T. Tajima, Plasma Phys. Rep. 30, 221 (2004)

and

It yields the emitted intensity

22 422 2 2

2 20 02( )rade e e

p p pa am c m c p m c

ε⎡ ⎤⎛ ⎞ ⎛ ⎞⎟ ⎟⎢ ⎥⎜ ⎜⎟ ⎟= + +⎜ ⎜⎟ ⎢ ⎟ ⎥⎜ ⎜⎟ ⎟⎜ ⎜+⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

Here43e

rad

rπελ

= with the classical electron radius 2

2ee

erm c

=

For the transverse components of the momentum we have the equation which describes a balance between the particle acceleration and its slowing down due to the radiation damping force

In the limit of relatively low amplitude of the laser pulse, when1 / 3

01 r a d ra da a ε −= or 18 23 210 10 /I W cm< <

0ep m ca=In the limit or

the momentum dependence is0rada a 23 25 210 10 /I W cm< <

( )1/40/e radp mc a ε=

Cross section of nonlinear Thomson scattering

ELI

PIC Simulation Method

( , ) ( , ) 0f f fq t tt cα α α

α∂ ∂ ∂⎛ ⎞+ ⋅ + + × ⋅ =⎜ ⎟∂ ∂ ∂⎝ ⎠

vv E x B xx p

4

div 4div 0

t

t

cc

π

π ρ

∂ = ∇× −∂ = − ∇×

==

E B JB E

EB

Vlasov equation for collisionless plasma

Maxwell equations

( ) ( )( ) ( )( )

( ) ( )( )

, ,

0 , 0 , 0

f t t t

f

α α α αα

α α α α

δ δ= Γ ⋅ − ⋅ −

Γ =

∑x p x x p p

x p

α counts a point in the phase space (x,v)

– quasiparticle

t = 0 t

Γα

Γβ

( ))(),( tt αα vx

( ))0(),0( αα vx

( ))0(),0( ββ vx ( ))(),( tt ββ vx

where meet the ( ) ( )( )d , , 0

df f f f t t tt tα α α

α∂ ∂ ∂

+ ⋅ + ⋅ = =∂ ∂ ∂

x p x px p

2, 1 /

( , ) ( , )

mm

q t tc

αα

α

γγ

= = +

⎛ ⎞= + ×⎜ ⎟⎝ ⎠

px p

vp E x B x

сharacteristic equations( ) , ( )t tx p

Values Γ α are transported by quasiparticles

( )

( )

, ,

, ,

q f t d

q f t d

α αα

α αα

ρ =

=

∑ ∫

∑ ∫

x p p

J v x p p

( )( ) ( )( )t tα α αδ δΓ ⋅ − ⋅ −x x p p

one-particledistribution

function

PIC Simulation Method

( , ) ( , ) 0f f fq t tt cα α α

α∂ ∂ ∂⎛ ⎞+ ⋅ + + × ⋅ =⎜ ⎟∂ ∂ ∂⎝ ⎠

vv E x B xx p

4

div 4div 0

t

t

cc

π

π ρ

∂ = ∇× −∂ = − ∇×

==

E B JB E

EB

Vlasov equation for collisionless plasma

Maxwell equations

( ) ( )( ) ( )( )

( ) ( )( )

, ,

0 , 0 , 0

f t t t

f

α α α αα

α α α α

δ δ= Γ ⋅ − ⋅ −

Γ =

∑x p x x p p

x p

α counts a point in the phase space (x,v)

– quasiparticle

t = 0 t

Γα

Γβ

( ))(),( tt αα vx

( ))0(),0( αα vx

( ))0(),0( ββ vx ( ))(),( tt ββ vx

where meet the ( ) ( )( )d , , 0

df f f f t t tt tα α α

α∂ ∂ ∂

+ ⋅ + ⋅ = =∂ ∂ ∂

x p x px p

2, 1 /

( , ) ( , )

mm

q t tc

αα

α

γγ

= = +

⎛ ⎞= + ×⎜ ⎟⎝ ⎠

px p

vp E x B x

сharacteristic equations( ) , ( )t tx p

Values Γ α are transported by quasiparticles

( )

( )

, ,

, ,

q f t d

q f t d

α αα

α αα

ρ =

=

∑ ∫

∑ ∫

x p p

J v x p p

( )( ) ( )( )t tα α αδ δΓ ⋅ − ⋅ −x x p p

one-particledistribution

function

become important at the electron energy, when the recoil due to the photon emission becomes of the order of the electron momentum, i.e. at

( )1/22 / q em cγ γ ω≥ =

The electron gamma factor ( )1/40/e radaγ ε=

That is why the quantum limit is2 2

0 2 /3 q ea a e m c ω> =

For the equivalent electric field of EM wave it yields22

2

2 2 1,3 3 137

QEDeq

Eem cEα

α= = ≈

2 822 2

0 I ( ) rade e

p pam c m c

ε⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟− = ϒ⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

In the quantum limit, when 2

2 1e e

pm c m cω⎛ ⎞⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟ϒ = ⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠⎝ ⎠

3/81/20

2

0. 4

3 e

e rad

ap m c

m cω

ε

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎟ ⎜≈ ⎜ ⎟⎟ ⎜⎜ ⎟⎟⎜ ⎟⎜⎜⎝ ⎠ ⎝ ⎠

Laser intensity:25 29 210 10 /I W cm< <

Equation

gives

Quantum Effects

Scaling of electron energy

Reflection of EM Wave at the Relativistic Mirror

20'' 4ω = ω≈ γ ω

c+vc-v

v

0ω

-v

The mirror “compresses” the EM wave The EM wave pushes the mirror

02

1''4

ω = ω≈ ωγ

c-vc+v

0ω

Counter-Propagation

Co-Propagation

High Order Harmonics from the Oscillating High Order Harmonics from the Oscillating Relativistic MirrorRelativistic Mirror

Incident pulse

electronsprotons

ω0

Reflected Wave

n ω0 n ω0

Transmitted Wave

±v||

±v

Oscillating/Sliding Mirror

In RPDA the Laser pulse is confined inside the relativistic cocoon. Long laser pulse almost completely transforms its energy into the fast ion energy.T. Esirkepov, M. Borghesi, S.V. Bulanov, G. Mourou, and T. Tajima, Phys. Rev. Lett. 92, 175003 (2004).

Ion momentum depends on time as

Muti-GeV Ion Generation via Radiation Pressure Dominated Acceleration (RPDA) Mechanism

1/32

2 00

0

32

ep

p pe

m ctp m c am l

ωω

⎛ ⎞⎛ ⎞⎜ ⎟≈ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Ion max energy vs time

~t 1/3

6543210

dN/dE

(107

μm

−2 M

eV−1

)

1000 2000 3000 E (MeV)

Final ion energy

/ toti las N=E E

Radiation pressure on the front part of the “cocoon” is equal to22 2

0 0

0

'' 12 2 1

M

M

E EP ω βπ ω π β

⎛ ⎞ −= =⎜ ⎟ +⎝ ⎠It yields

( )( )

1 / 22 2 220

1 / 22 2 20 02p

p

m c p pEd pd t n l m c p pπ

⎛ ⎞+ −⎜ ⎟= ⎜ ⎟⎜ ⎟+ +⎝ ⎠

Solution to this equation gives for energy balance2 ( 1)

2 1p

p W Wm c W

+=

+

FLUENCE

Final ion energy depends on the laser pulse energy as Efficiency of the laser energy conversion into the fast ion energy can be formally up to 100%: 30 KJ laser pulse can accelerate 1012 protons up to the energy equal to 200 GeV

2/0

0 0

( )2

t x c

i

E dxW tc n l m c

ξ ξπ

−

−∞

⎛ ⎞− =⎜ ⎟⎝ ⎠ ∫

Laser heavy-ion collider

92U

multiGeV/n

Compare with RHIC/BNL collider (Au+Au,100 GeV/nucleon) per day

92U

multi

GeV/n

quark-gluonplasma?

2 12 2 236

2

(10 ) 10 3 10(10 )

pNS m

σπ μ

−

= = ≈ ×N events

Thanksfor

your attention

THE END

Thanksfor

your attention

THE END

k

E

3D Particle3D Particle--InIn--Cell simulation (I)Cell simulation (I)Driver pulse: a=1.7size=3λx6λx6λ, GaussianIpeak=4⋅1018 W/cm2×(1μm/λ)2

Plasma:ωpe/ ωd = 0.3ne=1020cm-3×(1μm/λ)2

λ= 100dx, Npart = 1010, grid: 2200x1950x1920

HP Alpha Server SC ES40/227 (720 CPU)

3D Particle3D Particle--InIn--Cell simulation (II)Cell simulation (II)

k

E

kE

Driver pulse: a=1.7size=3λx6λx6λ, GaussianIpeak=4⋅1018 W/cm2×(1μm/λ)2

Reflecting pulse: a=0.05size=6λx6λx6λ,Gaussian, λs = 2λIpeak=3.4⋅1015

W/cm2×(1μm/λ)2

XZ,color: Ey XY,contour: Ez XY,color: Ex at z=0

Ez

Ex

1 33

4rade

arλπ

⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠≈ 440 2.7×1023 W/cm2

equa

e m ca2

2

23 ω

= ≈ 2008 5.6×1024 W/cm2

2e

QED

m caω

= ≈ 4.1×105 2.4×1029 W/cm2

p ea m m= ≈ 43 2.5×1021 W/cm2

1.4×1018 W/cm2relativistic e−

radiation damping

quantum effects

e− e+ pairs

Laser intensityDimensionless amplitudeof EM wave (for λ = 1μm)

LaserLaser--Matter Interaction RegimesMatter Interaction Regimes

relativistic p+

Pres

ent-

day

Lase

rs

need Liénard-Wiechert poten-tials description

need QEDdescription

We cons i d er i n t e r ac t ions here

e

e Eam cω

=

Classical ↔ Quantum

a=1

n e/n

cr

Driver,ωd

''sωsignal, ωs

Here are ½ of all electrons(in the wake wave period)

Electron density cusp ∝ (x-xpeak )−2/3

In the moving frame we seek solution of this Eq.

( ) ( )( )2

ph20 ph

4 10, 12

ett z z z e p

e e

e n x tA c A A n n x t

mπ

λ δγ

−∂ − Δ + = ≈ + −

vv

A A Azz z

d q xdx

22

2 ( )χδ ′+ =′ with

2 22 2

2ph

02

pesq kc c

ωωγ⊥

′⎛ ⎞ ′= − − >⎜ ⎟⎝ ⎠

zA iqx q iqxexp( ') ( )exp( ')ρ= + −

It yields

22

3ph

1( ) | ( ) |2

d

s

R q q ωρω γ

⎛ ⎞= ≈ ⎜ ⎟

⎝ ⎠

In the strongly nonlinear wake:

We find the reflection coefficient

Wave equation for the vector-potential of EM pulse

qiq

( )2

χρχ

= −+

( ) ( )1/2 1/24 2 4 2pe

p ph phpe

ccω

λ γ χ γω

≈ ⇒ ≈

Reflection at the “Flying Mirror”Reflection at the “Flying Mirror”

zA

where