High Field Science with Lasers

S. V. Bulanov

ELI BEAMLINES, Za Radnicí 835, Dolní Břežany, 25241, Czech Republic

ELISS-2019

ELI-Beamlines

29 August 2019

Dolni Brezany, Czech Republic

ACKNOWLEDGMENTG. M. Grittani, Y. J. Gu, P. Hadjisolomou, T. M. Jeong, H. Kadlecova, D. R. Khikhlukha,

O. Klimo, G. Korn, M. Matys, A. Yu. Molodozhentsev, J. Mu, J. Nejdl, P. V. Sasorov, P. Valenta,

W. Yan

ELI-Beamlines, Dolni Brezany, Czech Republic

F. Pegoraro

University of Pisa and National Institute of Optics, Pisa, Italy

K. V. Lezhnin

Princeton University, Princeton, NJ, USA

T. Esirkepov, A. Pirozhkov, J. Koga, M. Kando, H. Kiriyama, K. Kondo, T. Kawachi

Kansai Photon Science Institute, QST, Kizugawa, Kyoto, Japan

W. Leemans, E. Esarey, C. Schroeder, S. S. Bulanov

LBNL, Berkeley, CA, USA

J. Magnusson, A. Gonoskov, M. Marklund

Chalmers University of Technology, Gothenburg, Sweden

V. A. Gasilov, I. Tsygvintsev

Keldysh Institute of Applied Mathematics, RAS, Moscow, Russia

J.-X. Li

Xi’an Jiaotong University, Xi’an, China

K. Z. Hatsagortsyan

Max-Planck-Institut für Kernphysik, Heidelberg, Germany

2

4

EI c

G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006)

2 2 3

e eS

C

m c m cE

e e

Extreme Field Limits:

Nonlinear QED Vacuum

Towards studying of nonlinear QED effects with high power lasers.

Radiation dominated & QED regimes in the high intensity

electromagnetic wave interaction with charged particles & vacuum.

Schwinger (Sauter, Bohr,…) field

2 3 229

210

4

e SS S

m c E WE I c

e cm

Extreme field limits

The critical electric field of quantum electrodynamics, called also the Schwinger field, is so strong

that it produces electron-positron pairs from vacuum, converting the energy of light into matter.

Since the dawn of quantum electrodynamics there has been a dream on how to reach it on Earth.

With the rise of the laser technologies this field becomes feasible through construction of extreme

power lasers or/and with the sophisticated usage of nonlinear processes in relativistic plasmas. This

is one of the most attractive motivations for extreme power laser development, i.e. producing

matter from vacuum by pure light in fundamental process of quantum electrodynamics in the non-

perturbative regime.

The laser with intensity well below the Schwinger limit can create an avalanche of electron-

positron pairs similar to a discharge before attaining the Schwinger field. It was also realized that

the Schwinger limit can be reached using an appropriate configuration of laser beams.

The regimes of dominant radiation reaction, which completely changes the electromagnetic

wave-matter interaction, will be revealed. This will result in a new powerful source of high

brightness gamma-rays. A possibility of the demonstration of the electron-positron pair creation in

vacuum in a multi-photon processes can be realized. This will allow modeling under terrestrial

laboratory conditions the processes in various astrophysical objects.

An ultra-relativistic electron emits Cherenkov radiation in vacuum with an induced by strong

electromagnetic wave refraction index larger than unity. During the interaction with this wave the

electron also radiates photons via the Compton scattering. Synergic Cherenkov-Compton process

can be observed by colliding laser accelerated electrons with a high intensity electromagnetic pulse.

Extreme Field Limits

https://www.eli-beams.eu/projekty/hifi/

M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-

Wesley, Reading, MA.1995)

V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics

(Pergamon, New York, 1982)

=============================

W. Heisenberg and H. Euler, Zeit. fuer Phys. 98, 714 (1936)

=============================

V. I. Ritus, Sov. Phys. JETP 30, 1181 (1970)

E. B. Aleksandrov, A. A. Anselm, A. N. Moskalev, JETP 89, 1181 (1985)

I. M. Dremin, JETP Lett. 76, 151 (2002)

=============================

G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006)

M. Marklund and P. K. Shukla, Rev. Mod. Phys. 78, 591 (2006)

A. Di Piazza, C. Mueller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177

(2012)

FOR READING

K. V. Lezhnin, P. V. Sasorov, G. Korn, and S. V. Bulanov, High power gamma flare generation in

multi-petawatt laser interaction with tailored targets, Phys. Plasmas 25, 123105 (2018)

S. V. Bulanov, Dynamics of relativistic laser-produced plasmas, Rend. Fis. Acc. Lincei 2, 1 (2019)

J. Magnusson, A. Gonoskov, M. Marklund, T. Zh. Esirkepov, J. K. Koga, K. Kondo, M. Kando,

S. V. Bulanov, G. Korn, S. S. Bulanov, Laser-particle collider for multi-GeV photon production,

Phys. Rev. Lett. 25, 254801 (2019)

H. Kadlecova, G. Korn, S. V. Bulanov, Electromagnetic shocks in the quantum vacuum,

Phys. Rev. D 99, 036002 (2019)

H. Kadlecova, S. V. Bulanov, and G. Korn, Properties of finite amplitude electromagnetic waves

propagating in the quantum vacuum, Plasma Phys. Control. Fusion 61, 084002 (2019)

S. V. Bulanov, P. V. Sasorov, S. S. Bulanov, G. Korn, Synergic Cherenkov-Compton radiation,

Phys. Rev. D 100, 016012 (2019)

F. Pegoraro and S. V. Bulanov, Hodograph solutions of the wave equation of nonlinear

electrodynamics in the quantum vacuum, Phys. Rev. D 100, 036004 (2019)

References

Laser-matter interaction transition from the relativistic regime to dominant radiation

friction and nonlinear Thomson scattering through the limit when quantum electrodynamics

comes into play and to electron-positron avalanche/cascade development, towards

nonlinear vacuum with electron-positron creation and nonlinear vacuum polarization while

the EM field intensity increases.

Schematic of nonlinear QED processes

Dimensionless parameters

2

22 3 2 218

2 2 2

8/3 5 4/3-1

1

/3

1. Normalized EM wave amplitude

2. Radiation friction becomes important for

11.37 10

2 4

4

wi3

3th

rad

erel

e e

e erad rad rad R

a

a a

e A m c aeE m Wa I

m c m c e cm

r m ca a I

4/3

23

10/3 2

2 2

2 22 2

0 -

03 4 3 4 2 4

where - classical electron radius

3. QED parameters

and

4. QED li

12.65 10

8

/

mit :

e e

ee

e S e e e

m W

e cm

r e m c

e F p e F kpEa N e e

m c E m c m c m c

3 5

23

2 2

2 3 2 4 729

2 2 2

11 5.75

108

2 . 6 104

3S

ee Q

e e eS S

a a

S

m c m WI

e cm

m c m c m c WE a I

e e cm

H. R. Reiss, J. Math. Phys. (N.Y.) 3, 59 (1962)

A.I. Nikishov and V. I. Ritus, Sov. Phys. Uspekhi 13, 303(1970)

R. Ruffini, G. Vereshchagin, S. S. Xue, Physics Reports 487, 1 (2010)

V. I. Ritus, in: Issues in Intense-Field Quantum Electrodynamics (Nova Science, Commack, 1987)

Electron Motion in High Intensity EM Wave

Dimensionless amplitude

At a=arad emitted energy becomes equal to the

energy received from EM wave.

When the recoil of the emitted photon is significant, the emission probability is

characterized by e parameter (Lorentz and gauge invariant)

At the QED effects come into play

e

arad

00

a

I II

III IV

e *

4 regimes

Q

RR RR,Q

1/3 2

2

3

4rad ee e

ea r

r m c

* 1e e

Curves IR(w), IQ(w) and IR-Q(w) , IQ-R(w) subdivide

(I, w) plane to 4 domains:

I) Relativistic electron - EM field

interaction with neither radiation

friction nor QED effects

II) Electron - EM wave interaction is

dominated by radiation friction

III) QED effects important with

insignificant radiation friction effects

IV) Both QED and radiation friction

determine radiating charged particle

dynamics in the EM field

Four Interaction Domains

4 5 224

4 2

45 8 4

21

14 2

1

4/3 4/34 5 2

23

4 2

1 1

4 5 223

4 2

1 1

W3.8 10

144 cm

W3.8 10

W5.6 10

9 cm

W87 8.2 10

c

14

m

4 cm

e

eR

R

Q

Q R

eQ

m c eI

m c e

e

I

m c eI

cI

4

1 3

1

18

820 nm

ee m

!!!!!

SVB, T. Zh. Esirkepov, M. Kando, J. Koga, K. Kondo, and G. Korn,

Plasma Phys. Rep. 41, 1-51 (2015)

Radiation Friction Force:

Lorentz-Abraham-Dirac,

Pomeranchuk,

Landau-Lifshitz, …

Basic Equations: Minkovski & Maxwell Equations

Minkovski equations: where e

du eF u

ds m c

EM field tensor:

Maxwell equations: and

In 3D notations:

F A A

, dx dt

u dsds

0F

4

F jc

1/ 2

2 2

1= , = , 1

4 1

0

1

4

e e

t

t

p pe c

c m m c

c c

c

pp E v B x

B j E

B

E B

E

Intensity of radiation emitted by electron is given by

In circularly polarized EM wave (in plasma), whose

amplitude is equal to electron energy

losses are

For linearly polarized wave we have

Radiation Losses

2

i i

2 3

e

2e dp dpI =

3m c ds ds

00

e 0

eEa =

m c2

2( ) 0 0

0

14

2 3

e e

2e E eE=

3m c m c

22

( ) 0 0

0

31

8

4

2 3

e e

e E eE=

3m c m c

L.D.Landau & E.M.Lifshitz

“The Classical Theory of Fields”

e

Pattern of field emitted by

electron. T.Shintake, 2003

LAD-form of radiation friction force

Equations of electron motion are:

Radiation friction force is given by

Here , is proper time:

4-velocity is

and is 4-tensor of EM field

2

e

du em c = F u + g

ds c

2 2 2

2 2

2e d u d ug = - u u

3c ds ds

0,1,2,3

,i

i

e

dxu

ds m c

p

/ds cdt s

AAF

x x

Covariant and 3D forms of L-L expression

Radiation reaction force in the Landau-Lifshitz approximation (L-L II, §76):

3

3 2

2

3

2

e

e e

du e e em c = F u + F u u F F u F u F u u

ds c m c m c

3

3

4

2 4

2422

2 5 2

2 1

3

2 1 1

3

2 1

3

2

e

t t

e

e

e

dm c = e +

dt c

e+

m c c

e

m c c c

e

m c c c

p vE B

v E v v B

E B B B v E v E

vv E B v E

0rad a

2

0rad a

Dimensionless parameter:2

8

3

2 4 110

3 3e

rad

e

e r m

m c

2 2

0rad a

2 4 2

2 3

1 1

137

ecl S

e

m ce eE E E

r e c

Weak field approximation:

“Pomeranchuk Theorem”

We consider the electron colliding with the EM wave given by

Retaining the main order terms in the L-L radiation friction force,

we obtain equation for the x-component of the electron momentum

Its solution is

2 2

0

2(2 ) ,

3e

rad rad

d ra t

dt

0

22

00 0

0

8

0

1 ( ) ,

1 (2 ') '

For 10 =100 300 the electron gamma - factor becomes equal to

tt

rad lasrad

rad las

taa t dt

a

10

Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields, Pergamon, 1975

Pomeranchuk, I.: ‘Maximum Energy that Primary Cosmic-ray Electrons Can Acquire on the Surface of the Earth as a Result of Radiation in the

Earth's Magnetic Field’, J. Phys. USSR, 2, 65 – 69, 1940

0 0( / ) (2 )x ct

a t x c a t

Radiation friction effects on charged particle motion

2 22rad

2

1

Equations of electron motion

with radiation friction force

EM wave is modelled by rotating f

( )

,

ield

, , , 1

2

e

e e

e

e

cr e e

du em c = F u + g

ds c

g

E

G

n en t q

n m c m c

q q q

p Eq

a a a a q a a

a

1/ 22 2

2 3

3/ 2 2 3

43/ 2

0

QED effects incorporated with the form - factor, , equal to the ratio

of the full radiation intensity to the intensity emitted by a classi

( )

4 43( )

4

cal

electro

1

n

e e

e ee e

e

q q

G

x xG

x

( )

( )where is the Airy function

x xdx

x

J. Schwinger, Proc. Natl. Acad. Sci. U.S.A. 40, 132 (1954)

A.A. Sokolov, et al., Sov. Phys. JETP 24, 249 (1954)

A.R. Bell and J. G. Kirk, Phys. Rev. Lett. 101, 200403 (2008)

C. P. Ridgers, et al., Phys. Rev. Lett. 108, 165006 (2012)

I.V. Sokolov, et al, Phys. Rev. E 81, 036412 (2010)

J. G. Kirk, et al., Plasma Phys. Contr. Fusion 51, 085008 (2009)

R. Duclous, et al., Plasma Phys. Contr. Fusion 53, 015009 (2011)

A. Di Piazza, et al., Rev. Mod. Phys. 84, 1177 (2012)

Radiation friction effect on electron dynamics

In order to describe the electron motion we write the electron momentum as

Here and are the components of the electron momentum

parallel and perpendicular to the electric field.

We assume here that the wave is given.

Neglecting the change of the –component (near-critical plasma density),

we obtain

with the energy balance equation

The QED parameter is given by

1 1

2

3

1 0 0

0 cos( ) sin( )

0 sin( ) cos( )

q q

q q

q q

q q

1q

2 2

rad

22

rad

2 2

rad

2

( ) 1

( )

( )

1

e e e

e

e e

e

e e e

e

S

qq q G a a q

qq q a G a q

a u G aq a q

aq

a

S. V. Bulanov, T. Zh. Esirkepov, M. Kando, J. K. Koga, S. S. Bulanov, “Lorentz-Abraham-Dirac vs Landau-Lifshitz radiation friction force in the

ultrarelativistic electron interaction with electromagnetic wave (exact solutions)”, Phys. Rev. E, 84, 056605 (2011).

Electron trajectories

Dependence of and (a,b), and (c,d) and and (e,f)

on time for and .

(a,c) , (d,e) .

2q 3q q ||qeG e

8

rad 10 54.1 10Sa

1/30.25 rada 1/30.75 rada

Integral scattering cross section

Stationary solutions

The energy flux reemitted by the electron is equal to ,

which is .

The integral scattering cross section by definition equals the ratio

of the reemitted energy flux to the Poynting vector magnitude:

Here is the Thomson scattering cross section .

More convenient formula is

e v E

2 2 2

rad ( )e e e eG m c aq a q

2( )T e e

qG q

a

T2 25 28 / 3 6.65 10 cmT er

2

2 2 61

e eT

rad e e

G

G

Ya. B. Zel'dovich, Sov. Phys. Usp. 18, 97 (1975)

S.V. Bulanov, T. Zh. Esirkepov, J. Koga, and T. Tajima, Plasma Phys. Rep. 30, 196 (2004)

T. Nakamura, J. K. Koga, T. Zh. Esirkepov, M. Kando, G. Korn, S. V. Bulanov, Phys. Rev. Lett. 108, 195001(2012)

Cross section and electron energy e v.s. EM wave amplitude

1

2

33

1

2

Dependences of and of on

1) , 2) , 3) . lg / T lg e lg a

1 /12.5 1

112.5

Nonlinear Thomson - Compton Scattering

Cross Section and Electron Gamma Factor

vs EM Field Amplitude.

High Power Gamma Flare

Generation

High Power Gamma-Ray Source

T. Nakamura et al. Phys. Rev. Lett. 108, 195001 (2012)

K. V. Lezhnin, et al., Phys. Plasmas 25, 123105 (2018)

Applications

Photo-fission cross-section and pair

production cross-section in uranium [J. Galy, et al., New J. Phys. 9 (2007) 23]

• Photo-nuclear reactions

• Electron-positron pair creation

• Medicine

• Material sciences

•……….

• Radiation safety

Concept of high power gamma-flash generation:

3

2

2

22 2

Energy of photons emitted via NTS

scales as .

Electron energy is of the order of

Photon energy is in the γ - ray range if

with intensity

1

a

0

10 W/cbove .m

e

e

h

m c a

a

.

T. Nakamura, et al., PRL 108,195001 (2012)

C. P. Ridgers, et al., PRL 108, 165006 (2012)

……………………………………………………..

S. V. Bulanov, et al., PPR 41, 1 (2015)

Z. Gong, et al., Phys. Rev. E 95, 013210 (2017)

R. Capdepssus, et al., Sci. Rep. 8, 9155 (2018)

K. V. Lezhnin, et al., Phys. Plasmas 25, 123105 (2018)

3

0 0

0

0.3

3

4

a

N a

Energy of emitted photon is

Number of emitted photons/wave period

J. Koga, et al, Phys. Plasmas 12, 093106 (2005)

S.-Y. Chen, et al, Nature 396, 653 (1998)

G. Sarri, et al., Phys. Rev. Lett. 113, 224801 (2014)

W. Yan, et al, Nat. Photonics 11, 514520 (2017)

Simulation setup

EPOCH: T. D. Arber et al., PPCF 57, 113001 (2015)

Preplasma

T. Zh. Esirkepov, et al.” Prepulse and amplified spontaneous emission effects on the Interaction of a petawatt class laser with thin solid targets”,

Nuclear Instruments and Methods in Physics Research A 745, 150 (2014)

Density of aluminum target (g/cm3) after 1ns of

irradiation by prepulse with a 2 μm spot FWHM

size, and intensity 2x1011 W/cm2. Below is density

along x-axis (solid curve) and initial target profile

(dashed curve).

Density of iron target (g/cm3) after 3ns of irradiation

by prepulse with a 2 μm spot FWHM size, and

intensity 2x1011 W/cm2. Below is density along x-

axis (solid curve) and initial target profile (dashed

curve).

Optimal conditions for gamma flare generation (Hydrogen)

10 150 1 2.5 80 (ELI-BL L4)las plPW fs m m L m

a) Electron density and

laser intensity

b) electron x-px phase

c) ion x-px phase.

t =

18

0 fs,

< 1

% in g

am

ma

fla

re

t =

40

0 fs,

36

% in g

am

ma

fla

re

E < 10MeV E > 10MeV E > 100MeV

Pulse self-focusing, hole boring, electron heating, and ion acceleration

t = 400 fs Solid hydrogen

(Z=1)

10 150 1 2.5 80 (ELI-BL L4)las plPW fs m m L m

a) Electron density and

laser intensity

b) electron x-px phase

c) ion x-px phase.

t =

18

0 fs,

< 1

% in g

am

ma

fla

re

t =

40

0 fs,

36

% in g

am

ma

fla

re

E < 10MeV E > 10MeV E > 100MeV

Pulse self-focusing, hole boring, electron heating, and ion acceleration

t = 400 fsGold target

(Z=67)

Optimal conditions for gamma flare generation (Gold target)

10 150 1 2.5 80 (ELI-BL L4)las plPW fs m m L m

a) Electron density and

laser intensity

b) electron x-px phase

c) ion x-px phase.

t =

18

0 fs,

< 1

% in g

am

ma

fla

re

t =

40

0 fs,

36

% in g

am

ma

fla

re

E < 10MeV E > 10MeV E > 100MeV

Pulse self-focusing, hole boring, electron heating, and ion acceleration

t = 400 fsIron target

(Z=24)

Optimal conditions for gamma flare generation (Iron target)

10 150 1 2.5 80 (ELI-BL L4)las plPW fs m m L m

a) Electron density and

laser intensity

b) electron x-px phase

c) ion x-px phase.

t =

18

0 fs,

< 1

% in g

am

ma

fla

re

t =

40

0 fs,

36

% in g

am

ma

fla

re

E < 10MeV E > 10MeV E > 100MeV

Pulse self-focusing, hole boring, electron heating, and ion acceleration

t = 400 fsTilted iron

target (Z=24)

Optimal conditions for gamma flare generation (Tilted Iron target)

Photon generation efficiency

10 150 1 2.5 80 (ELI-BL L4)las plPW fs m m L m

Multi-parametric analysis of target & laser pulse

3D simulations of gamma flare generation

10 PW, 50 fs, 40 micron corona

Kinematics of inverse multi-photon Compton scattering in vacuum

i.e.

Recoil at imposes constraint on the photon energy :

The energy - momentum conservation equates the sum of e

Nonlinear Thomson scattering

3 30 0 0

20

20 0

0.3

/

ph

m e

a N a

mc a

a a m c

lectron

and photon energy - momentum before and after scattering

In vacuum It yields.

2 20 0 0 0

20 ,0 0

20 0 ,0 0 ,0

2 2 2

,

(

s n

)

i ( )

e e ph ph

ph e

ph e

m c m c N N

N p c m c

N m c p c p c

c

p k p k

k

Maximum photon energy energy in the head - on collision.

In the receding configuration, for co - propagating electron and laser pulse

the scattered photon energy is well bellow the incident

2,0

c

os

em c p

photon energy.

F. Ehlotzky, K. Krajewska, and J. Z. Kaminski, Rep. Prog. Phys. 72, 046401 (2009)

C. Bula, et al., Phys. Rev. Lett. 76, 3116 (1996)

Inverse multi-photon Compton scattering in collisionless plasma

The energy - momentum conservation equates the sum of electron

and photon energy - momentum before and after scattering

whereIn collisionless plasma

2 20

2

0 0

2

0

2 2

,e e ph p

e p

h

p

m

c

c m c N N

k

p k p k

Assuming (electron - e.m. wave interaction at critical surface) we obtain

Maximum photon energy

Elect o

.

r

0

20 0 0

2

2 2 20

0 0 ,0 ,0

20

(

4 /

)

sin o

1

c s

pe

ph e

e

h

e

p e

e

ne

N m c

N m c p c p c

m c

m a

a

n momentum :

Gamma photons are emitted at the angle :

being confined within the a e

,

θ

ngl

2,0 0 ,0 0

0 ,0 0 ,0

2 2 20 ||,

/2

/ /

(

1 ) /

e e

a

a e

p m ca p m ca

a p a p

a m c p 2 20 0 /ph eN m c

Nonlinear Electromagnetic Waves

in the QED Vacuum

QFT: Quantum Electrodynamics

Vacuum is not a void medium

Creation and annihilation of electron-positron pairs

Casimir effect, Delbruck scattering, photon-photon-scattering, …

Fluctuations

Vacuum Polarization

B. Green, The Elegant Universe (2003)

In the QED, photon-photon scattering occurs via creation-annihilation of virtual

electron-positron pairs by two initial photons followed by annihilation of the

pairs into final photons

Dependence of the photon-photon scattering cross section on the photon

frequency (assume that )

Photon–Photon Scattering

R. Karplus and M. Neuman, Phys. Rev. 83, 776 (1951)

6

2 2 2

2

22 22 2 2

973

10125

3

12

e e

e

ee e

r m cm c

m cr m c

for

for

ATLAS Collaboration, Nature Physics 13, 852 (2017)

1 2

Electron-positron pair creation via the Breit-Wheeler process has the cross

section

where

is Lorentz invariant

Near the threshold when

the cross section is given by

Pair Creation via the Breit-Wheeler Process

G. Breit and J. A. Wheeler, Physical Review 46, 1087 (1934)

X Ribeyre, E d’Humi`eres, S Jequier and V T Tikhonchuk, PPCF 60, 104001 (2018)

2 2 4 211(1 ) (3 ) ln 2 (2 )

2 1

eep e e e e e

e

r

2 4 2

1 21 2 / (1 cos )e em c

2 4 2

1 2 (1 cos ) 2 /em c

2 11ep e

e

r

1e

Photon-Photon Scattering: the Number of Scattered Photons

The number of photons in the electromagnetic pulse with amplitude

in the 4-volume

In optimal regime (maximal luminosity) the number of scattering events per

4-volume is proportional to the scattering cross section, to the product of the

photon numbers in colliding photon bunches, and it is inverse proportional to

the square of the wavelength

This yields

Here is the Schwinger (Sauter, Bohr, …) field equal to

where : e.m. field intensityQED VACUUM IS DISPERSIONLESS MEDIUM

E 3 / 2 3

4

EN

N NN

2

4 2

2973

10125 S S

E IN

E I

2 3

eS

m cE

e

SE

2 / 1/137e c 2 29 2/ 4 10 /S SI cE W cm

Heisenberg-Euler Lagrangian

0

| | / |

'

A A

-1

CIn the long wavelength and low frequency approximation ( |<< )

the Lagrangian describing the electromagnetic field in vacuum is

whe 0

1

16F F

F A A

re

with

gives the Maxwell equations.

The Heisenberg-Euler te

rm

4 2

2 2

2 3

0

exp( )' 1 cot coth

8 3

emd

Here the invariants and can be expressed in terms the Poincare invariants

a a a b b

b

a b

1

F F F F

F F

c

and

(dual tensor equals = ) as

and

Here we use the units , and the e.m. field is normalized on the

F G

a F G F b F G F2 2 2 2

SEQED critical field .

W. Heisenberg and H. Euler, Zeit. fuer Phys. 98, 714 (1936)

Weak Field Approximation

2 22 27 90 13

'4 4 315 16

F F F F F F F F F F

The Heisenberg-Euler term in the weak field approximation we have

With the constant

4

4

2

0

4 360

'

e

em

The first two terms describe four interacting photons and the last two terms

correspond to six photon interaction.

gives equations of the nonlinear electrodynamics

in the dispersionless medium with refraction index depends

on the electromagnetic field.

( ) / 2, ( ) / 2

1

( , ) ( ', ')

x x t x x t

c

x t x t

Below we use the light cone coordinates

Here .

The variables in the lab. frame of reference are related to

in the boosted frame mov

' cosh sinh , ' cosh sinh

(1 ) /(1 )

' ( ' ') / 2 ( ) / 2 , '

x x t t t x

x x t e x t e x x

ing with the velocity as

where

The Lorentz transform of the light-cone variables is

( ' ') / 2 ( ) / 2

( ) '

(

t x

x t e x t e x

e

u a w a

e a b a

u

For derivatives we have

Introducing the fields and

related to the electric and magnetic field by

2 2

) / 2 ( ) / 2

' '

( ) / 2

' '

e b w e b

u e u w e w

uw e b

u w uw

and

we obtain

and

The field product

is Lorentz invariant.

Dirac’s Light Cone Coordinates

( ) / 2, ( ) / 2

zA

x x t x x t

x

The e.m. waves have the same polarization ( )

We use the light cone coordinates

Propagation along the axis,

A e

2

( , )

( , )

1

4

A Wx a x x

Wx

a x x

W w u W

where is the crossed e.m. field

and is the 4 potential for counter-propagatying e.m. wave.

H-E Lagrangian can be cast to

2 32 3

3

= , =

/ / 0

w u W w u

u a w a

L u L w

with the field components

Lagrangian variation

yields system of nonlinear equations for electromagnetic wave

2 2 32

2 3 2 3

2 3

2 3

2

0

1- 4 9 3

3 0

(2 / 45 )

w u

W w u W w u u W w W w u u

u W w u w

with a 3 (32 / 315 ) nd

Finite Amplitude Counter-Propagating EM Waves

Phase and Group Velocity

222

22

2

11 2

1W

x

Wv W

W

The e.m. wave propagates along the direction with velocity

The phase velocity of the wave is equal to the group velocity,

i.e. the vacuum is dis

/ph g Wv c

persionless medium.

The normalized = are less than unity, i.e.,

the wave phase (group) velocity is below the speed of light in vacuum.

Corresponding dispersion equation for the e.m. wave

fre

2

2

2

2

2

2

( ) ( 2 ) 0

1 2

2

kc kc kc W

n W

n W

Bialynicka - Birula & Bialynicki - B

quency and vave number has a form

Refraction index of the QED vacuum

is greater than unity with

irula (1970)

Dittrich & Gies (2000)

Marklund & Shukla (2006)Electromagnetic waves

in the (x,t) plane

Electromagnetic Riemann Wave in Vacuum

2 2 32 3 2 2

2 3 2 3 2 3

( ),

3 1- 4 9 3 0

w w u

u W w u J W w u W w u J W w W w u

dwJ

du

Assuming the solution of nonlinear wave equations in the Riemann wave form:

we obtain

-

Here is the Jacobian.

For small

2 2 3

2 2 34 3

u

J W W u

but finite it can be found to be equal to

We arrive to the equation for nonlinear e.m. wave in the form

2 2 3

2 2 3

2 3

2 3

4 3 _ 0

,

2 4 3 0t W x

u W W u u

x t

u v W u u

In the variables it can be rewritten as

Wave Breaking (Rarefaction Wavebreak)

2 3

2 3

0

2 4 3

t W xu v u u

u W u

This is the Hopf equation. We rewrite it as

with

For g

(

ive

*)

n

ini

0

0 W

u x

u u x v u t

tial condition its solution in implicit form is given by

It can be obtaine

d by integrating

al

ong the ch

(** )

aract

0 0 0 0 0

, 0

( )

W

W

x

dx duv u

dt dt

u u x x x v u x t

u

eristics:

with the solution ,

Calculating the field gradient we find that

0

0

0

0 0

2 3

2 3 0 0

2 3

2 3 0 0

( )

1 2 4 3 ( )

1

2 4 3 ( )

x

x

br

br

x

u x

W u x t

t t

tW u x

At

the denominator vanishes - the wave breaks ( )gradient catastrophe

Lutzky , Toll, Phys. Rev. 113, 1649 (1959); Böhl, King, Ruhl, Phys. Rev. A 92, 032115 (2015);

H. Kadlecova, G. Korn, SVB, Phys. Rev. D 99, 036002 (2019)

EM Shock Wave in QED Vacuum

Expression describes high order harmonics generation and wave steepening in a vacuum.

Perhaps one of the simplest mechanisms. We choose the initial electromagnetic wave as

(**)

0 1

2

1 1

cos

1( , ) cos ( sin 2 (

2W W W

u a kx

u x t a k x v t a kv t k x v t

and find from that

When a wave approaches the wave breaking point, wave steepening is equival

(**)

ent

( , )u x t

to

harmonics generation with higher numbers leading to the gradient catastrophe,

i.e., at the shock wave.

The e.m. shock wave separates two regions, I and II, where the function takes

the values

.

I I II

SW SW SW

u u

u x v t

and . The shock wave front (it is an interface between regions I and II)

moves with a velocity equal to , i.e. shock wave front is localized at

Integrating Eq. over an i(*) nfini

2 5 2

2 3

( , ) 0

(4 3 ) 0

{ } ( ) ( ) 0 ( )

sw

SW SW

SW W x x

x

x x

v u v u W u

f f x f x f x

tely small interval , where , we obtain

where at denotes the discontinuity of function at

2 5

2 3(4 3 ) ( )

/

SW W I II

C e

x

v v W u u

m

point .

This yields for the shock wave front velocity

Shock wave front width should be of the order of the Compton wavelength, c.

Date: Page:

L’ Oreal – UNESCO award for women in science 2019, awarded

to Hedvika Kadlecová

10.8.2018 53

“Hedvika Kadlecová who works in the ELI Beamlines Research Center in Dolní Břežany

became one of the three winners of the prestigious award L’Oréal-UNESCO for women in

science for her theoretical work on nonlinear electromagnetic waves in quantum vacuum”

https://www.eli-beams.eu/en/media-en/hedvika-kadlecova-was-awarded-loreal-unesco-prize-for-women-in-

science/

H. Kadlecová, et al.,

Electromagnetic shocks in the quantum vacuum

Phys. Rev. D 99, 036002 (2019)

Hodograph Transform

F.Pegoraro, SVB, Phys. Rev. D 100, 036004 (2019)

+

2 2 2 3 2 3

2 3 + 2 3 2 3

0

1- 4 9 3 3

w u

uw u w u w w u u u u w

Nonlinear equations for electromagnetic wave in 1D case can be written as

+

2 3

+

(2 / 45 ) (32 / 315 )

w

u w

with and

System is linear with respect to highest order terms in partial derivatives and

with coefficients nonlinearly dependent on and , admits hodograph transfor

u w

2 2 2 3 2 3

2 3 w 2 3 w 2 3

0

1 4 9 3 3

x x u w

x x

uw u w x w w u x u u w

mation.

We consider and as functions of and . This yields

u

2 2 2 2

2 3 2 3

0

1- 2 3 1- 2 3

x

C Cx x

u uw u w w uw u w

It possesses reach family of particular solutions including Lorentz invariant solutions:

and

Discussions

Measuring the phase difference between the phase of the electromagnetic pulse

colliding with the counterpropagating wave and the phase of the pulse which does

not interact with high intens

Experiment :

2

2

ity wave, it is equal to

4

where is the wavelength of high frequency pulse and d is the interaction length,

plays a centr

dW

al role in discussion of experimental verification of the QED vacuum birefringence.

For 10

B. King and T. Heinzl, High Power Laser Sci. Eng. 4, 1 (2016); T. Heinzl, et al., Opt. Express 267, 318 (2006).

24 2 2 -5

4

PW laser the radiation intensity can reach 10 W/cm , for which W = 10 .

Taking the ratio / equal to 10 , i.e. equal to the ratio between the

optical and x - ray radiation wavelength, we find that

d

-4

21 2

= 10 .

Assuming 2 / and the intensity of the x - ray pulse of 10 W/cm ,

we obtain that the ratio of the second harmonic amplitude to

the amplitude of fundamental harmonic is approximate

ly

Wkv t d HOHG :

-11equal to 10 .

Synergic

Cherenkov Radiation –

Compton Scattering

1ArcCosC

c

vn

Synergic Cherenkov Radiation - Compton Scattering

2 3

0 10a 24At the focus of 10 PW laser the field intensity can reach 10 W/cm , i.e.

Vacuum polarization changes the refraction index: radiation correction to the "photon mass" results

in the dispersion eq

22 2 2 2

, 2

2

2 2

,

0

11 3 3 3 1 8exp

90 2 16 3e

ck c

i

m

ua

tion for the e.m. wave frequency and wave vector

where (Ritus 19

7

0)

2/3

4

2

2/3

5 1 23 (1 3) 3

28 3

1 e

i

m

When the "photon mass" tends

for 1

f

to .

or 1

1

In the limit the difference between the vacuum refraction ind

2

2 3

11 3

45

/

S

S e

En

E

E m c e

with i. e. the norma

lized phase velocity of the e.m. wave

equals

ex and unity is

2

4

1

(11 3) / 45 10 .

S

E

E

w

he

re

T. Erber, Rev. Mod. Phys. Rev. Mod. Phys. 38, 626 (1966); V. I. Ritus, Sov. Phys. JETP, 30, 1181 (1970); J. Schwinger, W. Y. Tsai, and T. Erber,

Ann. Phys. 96, 303 (1976); W. Bekker, Physica 87A, 601 (1977); V. L. Ginzburg and V. N. Tsytovich, Phys. Reports 49, 1 (1979); I.M. Dremin,

JETP Lett. 76, 151 (2002); S. V. Bulanov, P. V. Sasorov, S. S. Bulanov, G. Korn, Phys. Rev. D 100, 016012 (2019)

Kinematics of SCRCS

2 2

0 0 0 ,0

2

0 0 /

,

1

e em

p

m c c

Following to we describe the kinematics of electron-photon interaction in QED vacuum as

with

V. L. Ginzburg (1940)

p k p k

2 2 2 2 2, 1 / , p p , ( )

1 .

e e x y x ym c p m c cos sin

n nc

and

In medium with the wave frequency and wav

e vector are related to each other as

The photon

energy is

p = e + e k =| k | e + e

kk k =

| k |

2

0 ,02

0 2

2

,0 0 0 0

2

21

cos

1

e

e

m c p cg g s

n

s

p c s n m c sg

n

where is the number of photons

and

Wh 2

0 0

2

0 ,0

0 2

/

22( 1)

e

e

Ch

g s m c

m c p cg s

n

en the function is positive and the photon energy can

be found to be

This correspo

nds t

2

0 0

0

0 /e

C

g s m c

s

o the Cherenkov radiation.

In the opposite limit, when , in the limit the Compton scat

tering mode f

requency is give by

n

2

0 ,0

2

,0 0 0 0

2 2 2 2

0 0 0 0 0

cos

, / 4 4

e

e

m m e C m C e

m c p c

p c s n m c s

s s s m c s s s m c

In the limit In the limit , where the photon energy equals ; at , we have= = < > .

Cherenkov radiation

2 4 2 2

,0 ,0

0

cos

1 cos

em c p c p cns

n

Condition of Cherenkov radiation,

The electron energy should be large enough to

have

2

0 2

00

2

0 0

2 29

45130

(11 3)2

/ 4

10

S S

Ch

S S

E I

IEn

I cE

I cE

24 2

2

Here the laser intensity in the focus region of 10 PW laser is approximately equal

/4

t

W/cm ,

o 10 W/cm ;

i. e. the Ch

02 /Ch SI I -5

erenkov radiation threshold is exceeded for the electron energy above 10 GeV.

The Cherenkov cone with the angle n the focus of 10 PW laser it is approximately equal to i

2 10 .

Angle distribution of the energy logarithm for photon radiated

by the SCCRS mechanism. Blue color used for Compton

scattering and red color for the Cherenkov radiation.

Magenta lines show the Cherenkov cone.

22 2

0

2 2

/ 1

1

e

e

e C Sv n c

d Ee c ed

dx c v n E

The rate of the energy loss due to the Cherenkov radiation

friction force along the electron trajectory is

Integration is done over the reg

11( / 3.8 10 )

e

Ch C e C e

v n

l m c

-5

ion where

The formation length is given by

It is approximately equal to 2 10 .

Traversing the laser focus region the electron emits 0.2 photon

/c>1.

s.

For th

cm

e el

4

ectric charge of the LWFA electron bunch of 100 pC

we obtain 10 phot ons. I. E. Tamm and I. M. Frank, Comptes Rendus de l’Accademie des Sciences de l’USSR 14, 109 (1937)

Probing the QED vacuum

2

2 2

,

4

2

11 3 3 3 1 8exp

90 2 16 3

5 1 23 (1 3) 3

28 3

e

i

m

i

for 1

As one may see from the expression for the photon invariant mass

2/3

at the high photon energy end, when the parameter becomes larger than unity, the vacuum

polarization effects weaken. In this limit the Cherenkov radiation

for 1

does

not o

2

0/e Sm c E E

ccur.

As a result the photons with the energy above are not present in the high frequency spectrum

of the radiation. For 10 PW laser parameters this energy is approximately equal

to 100 MeV.

Towards Schwinger Limit

Extreme

power

lasersRelativistic

construction

Optimal

configuration

of laser beams

QEDe–e+

plasma

Schwinger field

2 3

29 21 6

210 10 3 10

1 1S

WI W J

cm m m

Relativistic Flying Mirror Concept

Relativistic Flying Mirror: Concept & Proof-of-Principle

ExperimentsConcept:

SVB et al., PRL 91, 085001 (2003); M.Kando et al., PRL 99, 135001 (2007); PRL 103, 235003 (2009);

A. Pirozhkov, PoP 14, 123106 (2007); …… ; SVB et al., Phys. Uspekhi 56, 429 (2013), PSST (2016)

2

6

0 0

0

1,

1

Mr r ph

M

DI I

R

Space-Time Overlapping

of Driver and Source Pulses

tp = -750 fstp = -250 fstp = -150 fstp = -050 fstp = 0

Driver

200 mJ, 76 fs

Source

12 mJ

Vacuum

focus

Jet

center630 μm

Results

In a plane EM wave, both the invariants are zero,

therefore e–e+ pairs are not created for an arbitrary magnitude of the EM field.

Relativistic flying mirror can create field times greater than QED critical field.

F F F F

& F G

M

Towards Schwinger field ... and beyond

sL

sD

,s sIph phcv Incident Pulse

Wake

Wave

2'' / 4s s phL L

'

''

phcev

1/ M

Reflected Pulse

cosr s

с

с

v

v

The driver and source must carry 10 kJ and 30 J, respectively

Reflected intensity can approach the Schwinger limit

It becomes possible to investigate such the fundamental problems of nowadays physics, as e.g. the electron-positron

pair creation in vacuum and the photon-photon scattering

The critical power for nonlinear vacuum effects is

Light compression and focusing with the FLYING MIRRORS yields

for with the driver power

Laser Energy & Power to Achieve the Schwinger Field

242.5 10cr W

2

0 01 / 4

phm

0 ph

30ph

2 2 245

4S

cr

cE

2W/cm2910QEDI

J. K. Koga et al., Phys. Rev. A 86, 053823 (2012)

100cr PW

Laser Driven e-e+ Plasma

• Sub-MeV e+ from radioactive isotopes or accelerators are used in material science:

A. P. Mills, Science 218, 335 (1982); P. J. Schultz and K. G. Lynn, Rev. Mod. Phys. 60, 701 (1988);

A.W. Hunt et al., Nature (London) 402, 157 (1999); D.W. Gidley, et al., Ann. Rev. Mater. Res. 36,

49 (2006); B. Oberdorfer, et al., Phys. Rev. Lett. 105, 146101 (2010)

• e+ emission tomography:

M. E. Raichle, Nature (London) 317, 574 (1985)

• Basic antimatter science such as antihydrogen experiments:

M. Amoretti, et al., Nature (London) 419, 456 (2002);

G. Gabrielse et al., Phys. Rev. Lett. 89, 213401 (2002)

• Bose-Einstein condensed positronium:

P. M. Platzman and A. P. Mills, Phys. Rev. B 49, 454 (1994)

• Basic plasma physics:

C. M. Surko and R. G. Greaves, Phys. Plasmas 11, 2333 (2004)

e - - e+ Plasma in Terrestrial Laboratories

53

GRBE 10 erg

e - , e+ Plasma in Space:

Gamma Ray Bursts & the Leptonic Era of the Universe

First GRB detected

by Vela, 1967

(e - , e+ , p, ) - plasma

Meszaros, P., & Rees, M. J. 1992, MNRAS, 257, 29

Meszaros, P., & Rees, M. J. 1993, ApJ, 405, 278

In the Leptonic Era the Universe

consisted mainly of leptons and photons.

It began when the temperature dropped

below 1012 K some 10-4 seconds after the

Big Bang, and it lasted until the

temperature fell below 1010 K, at an era of

about 1 second.

J. Silk, ‘The Big Bang’, New York: W. H. Publishers

(1990)

• e - - e+ generation in laser-solid target interaction

(trident mechanism of the e - - e+ pair generation)

e - - e+ pairs via trident + Bethe-Heitler processes

2 2 2 2 2

2 2 2

(14 / 27 ) ( ) ln 1 /

/ , / 1/137

e ee Z e e e Z

e e

r Z p m c

r e m c e c

C.Gahn, et al., Phys. Plasmas 9, 987 (2002)

H.Chen, et al., Phys. Plasmas 16, 122702 (2009)

H.Chen, et al., Phys. Rev. Lett. 105, 015003 (2010)

a) b)

)

2

)

a trident

e Z e e Z

b bremsstrahlung

e Z e Z

pair production

Z e e Z

Breit-Wheeler process of e - - e+ generation with

if

Multiphoton inverse Compton

&

Multiphoton B-W processes

' e e 2 2

ee er

2 2 4' em c

0e N e

0N e e

2

( / ) 0.3

( / )( / ) 0.15

e S e

S e

E E

E E m c

01.6 0.3ps a 50 GeV

Breit-Wheeler process of e - - e+ generation

G. Breit, J. A. Wheeler, Phys. Rev. 46, 1087 (1934)

D. L. Burke, et al., Phys. Rev. Lett. 79, 1627 (1997)

Electromagnetic cascades induced by pairs

2

3 4

0

14

e e

e S

e F p E En n

m c E e

D.L.Burke et al., PRL 79, 1626 (1997)

24 22.5 10 /I W cm

•W.Heisenberg, H.Euler (1936)

•N. B. Narozhny, “Extreme Power Photonics: New Phenomena” ISTC-ELI/HiPER/PETAL Workshop, 2008

•A. R. Bell and J. G. Kirk, “Possibility of Prolific Pair Production with High-Power Lasers ” Phys. Rev. Lett. 101, 200403 (2008)

•A. M. Fedotov, N. B. Narozhny, G. Mourou, G. Korn, “Limitations on the Attainable Intensity of High Power Lasers” PRL 105, 080402 (2010)

0.3, 0.15e

+e

-e

-e

-e

3/ 22 2 3

3

2/32 2 37

5 3

3 8( ) exp for 1

32 2 3

327 (2 / 3)( ) for 1

56 2

The number of absorbed laser photons :

e

e

e m cW

e m cW

0 0. .

24

e

3 2 2

e

Photon mean - free - path before the pair is created :

2200.2

: 1 for and

1 gives 10 gives I > 10 /

Her

e S

S S

l

m f p

e

N a

la a

a aa W m

a a

a

a c

THRESHOLD

2e / /S S e ea eE m c m c

Probability of pair creation

A. M. Fedotov, N. B. Narozhny, G. Mourou, G. Korn, “Limitations on the Attainable Intensity of High Power Lasers” PRL 105, 080402 (2010)

Quasiclassical regime ( )

Pro

F. Sauter, 1931

W.Heisenberg, H.Eulba er, 19bi 3lity ( )6

22

21

/

/

12 32

0

3

2exp | ( ) |

4

1 |

4

exp 1| |

exp

e

e

z m c eE

z m c eE

e

SE

w A p z dz

m cA d

e E

E

w2 32 2 3

2 3

|exp

| |e e e

e

e E m c m c m c

m c e E

Energy is plotted against momentum

and coordinate (electric field is parallel z - axis).

Solution to Dirac equation shows that the

- function is large only in the regions I and III.

In the region II

, it decreases exponentially.

Therefore transmission coefficient through

region II, which gives probability of pair creation,

has the order of magnitude 2 3exp( / | | )em c e E

“Schwinger” mechanism of e - - e+ pair creation

22 2

2 2 2 2

2 2

*, ( )

2 4 4

4

S

e e

F F Finv inv

e EW

are the electric and magnetic fields in the frame where they are parallel

F G

E F G F, B F G F

E BE B

3 3

2 22

2 3

coth exp

,

0 0

0 exp4

S S

e e

Schw

e e

c

E E

W

e EW

c

for the rate

at the rate

The number of

bab

a a

b= B/ a E/

e

b aa

e 4

4 4exp

64

x y z

me e e eC m

c l l lN W dVdt

pairs a

a

+ -e

Rate of Pair Generation

Electron-positron pairs can be created before the laser field reaches the Schwinger limit, due to a large phase volume occupied by a high-intensity EM field.

S. S. Bulanov, N. B. Narozhny, V. D. Mur, V.S. Popov, “Electron-positron pair production by electromagnetic pulses”. JETP, 102, 9 (2006)

A. R. Bell & J. G. Kirk, “Possibility of Prolific Pair Production with High-Power Lasers”. Phys. Rev. Lett. 101, 200403 (2008)

A. M. Fedotov, N. B. Narozhny, G. Mourou, G. Korn, “Limitations on the Attainable Intensity of High Power Lasers”. Phys. Rev. Lett. 105, 080402 (2010)

S.S.Bulanov, V.D.Mur,

N.B.Narozhny, J.Nees,

V.S.Popov, Phys. Rev. Lett. 104,

220404 (2010)

Laser beams

Number of

pulses

Required energy (kJ) to

create one pair

2 910–19 40

4 310–9 20

8 4 10

16 1.8103 8

24 4.2106 5.1

Extreme

power

lasers Relativistic

constructionOptimal

configuration

of laser

beams

QEDe–e+ plasma

Schwinger field

Multiple Laser Beam Configuration

The vector field shows r- and z-components of the poloidal electric field in the plane (r,z):

The color density show toroidal magnetic field distribution.

The first Poincare invariant

( , )B r z

( , )r zE

3D EM configuration TM - mode

φB

1

0

1

2 2

TM

2 2

0 02a a

E BF

2

TM / aF

0 01/ 2 01/ 2

0

1

0

TM

( , )

si

configuration :

Magneti

n( )(

c fi

)

eld

Electric field

(cos )8

l

n n

R

B tJ k R L

k R

ik

B

e

E B

S.S.Bulanov, T. Zh.Esirkepov, A.Thomas, J.Koga, S.V.Bulanov, “On the Schwinger limit attainability with extreme power lasers”

Phys. Rev. Lett. 105, 220407 (2010)

Threshold for the e - - e+ avalanche

- +

0 0

0 0

0 0 0 0 013/ 2 3/ 2

0 0 0 0 0 0 0 01 0 23/ 2 3/ 2 3/ 2 3/ 2

0 0

e (e )trajectory in , - plane = 1, 1

( ) ,

( ) ,2 2

( )

INSTABILI

2 2 16 2 2

with (

:

.

TY

2

z e

r e

r z t a

p t m c a t

a k r t tp t m c I

a r t a r t t tr t I I I

k r

1/ 2 3/ 2

0 05 / ) ;growth rate : / 2Sa a

0

0

27 2

assuming 0.1 we find that the avalanche

will start at / 0.105

which corresponds to * 4 10 W

THRESHOLD :

/cm

S

t

a a

I

Here normalized Schwinger field is

Integrating over the 4-volume the probability of the pair creation

we obtain the number of pairs produced per wave period,

he first pairs can be observed for an one-micron wavelength laser intensity of the order of

, which corresponds to , i.e. a characteristic size, ,

is approximately equal to

TMF

23/ 2 4

2 0 00 0 0 5

5

16 S

ar z ct

a

2

5

0

5.1 10

eS

m ca

3/ 24

03

0

5exp

4

S

e e e e

aN W dVdt a

a

≈ 10 W /cm0 / 0.05Sa a 0r

00.04

2

TM / aF

0z

0r

The pairs are created in a small 4-volume near the electric field maximum

(maximum of ) with the characteristic size

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