“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