Electron acceleration by a tightly focused laser pulse

Konstantin Popov1

Valery Bychenkov1,2

Wojciech Rozmus1

Richard Sydora1

1University of Alberta2Lebedev Physics Institute

Charged particle in electromagnetic wave

An exact solution is known only for plane EMW

Maximum energy in plane EMW is proportional to the laser intensity

−⋅===2cm

W]μm[100

2

max_ 105.8 ,4

Icm

eEaaplane λ

ωγ

( ) BeEetm

×+= vd

vd γ

Equations of motion:

Tight focus Focusing is needed to get high intensity Field representation in our model: laser, tightly

focused by a perfect parabolic mirror

Stratton-Chu integrals

( ) ( ) ( ) ( )[ ] ( )

( ) ( ) ( ) ( )[ ] ( )( )

. ||

||exp

, d4

1d41

, d4

1d41

ps

ps

CA

CA

rrrrjk

G

lEGjk

AGHnGHnGnEjkPH

lHGjk

AGEnGEnGHnjkPE

−−

=

⋅∇−∇⋅+∇××+×=

⋅∇+∇⋅+∇××+×=

∫∫ ∫

∫∫ ∫

ππ

ππ

- inner normal to theintegrating surface A

Developed in 1939[1] and is used in many modern studies[2, 3].An exact field representation for given fields at the excitation surface A and its boundary C :

n

1. Stratton, Chu. “Diffraction Theory of Electromagnetic Waves”. Phys.Rev., vol. 56 (1939)2. Varga, Török. “Focusing of Electromagnetic Waves by Paraboloid Mirrors”. JOSA, vol.17, No.11 (2000)3. Bahk et all. Applied Phys. B, Lasers and Optics. 80, 823-832 (2005)

Application of Stratton-Chu integrals to the parabolic mirror

( ) [ ] ( )

( ) ( )[ ] ( ) .e42,2,44

,e2,2,44

2200222

0

0

20222

0

0

tkxj

tkxj

yfyzzfzyf

ESH

yzyyfzyf

ESE

ω

ω

−−

−−

⋅+−−⋅++

=

⋅−⋅++

=

. ' , ' , ' , ' ττττ HHHHEEEE nnnn

=−=−==B.C. at the metal surface:

( )zyfzyf

n −−++

= ,,24

10222

0

Normal to the mirror is

( ) ( )

( ) ( ).

otherwise , 0 , e,0,0

,otherwise , 0

, e0,,0

0

0

∈⋅

=

∈⋅

=

+−

+−

APEH

APEE

tkxj

inc

tkxj

inc

ω

ω

Geometrical approximation of the incident field at the surface of the mirror:

A plane electromagnetic wave incident onto a parabolic mirror. The mirror has radius and focal length . Mirror f-number is .mr 0f

mmn r

fDff

200 ==

Intensity distribution of a mirror focused laser (f#=0.9, Imax=1022)

Electromagnetic fields in the focus

Phase of the field

( )[ ] ( )tyxE

tyxEtyx

y

y

,,2

,,,,tan 0

+

=ωπ

ψ

Information on phases can be extracted from field using formula

Phase velocity( ) 0v

dd =∇+

∂∂= ψtψx,y,tψ

t ph

phv is given[1] by

The phase velocity in the direction of energy flow:

There are subluminous phase velocity regions!

1. Porras et all. “Pulsed light beams in vacuum with superluminal and negative group velocities”. Ph.Rev. E, 67 (2003)

Parameters

Laser pulse is Gaussian in time ( ) Laser energy gives if the beam is focused

into spot of 1 wavelength in diameter Pulse length is 30 fs for wavelength equal to 0.8 micron Particles are initially cold

222 W/cm10~I

Mirror f-number Best particles initial positions

To be optimized for maximum electrons energy:

22 /e~ τt−

Map of electrons energies: f#=0.9Particle gets some energy as a result of interaction with the laser pulse. This energy depends on the initial position of the particle.

The energies and deflection angles on this picture are averaged over the laser phase

1,2

1

2 types of acceleration: synchronization (1) and strong field interaction (2)

1

1

2

2t=100 fs t=133 fs

t=167 fs t=200 fs

Synchronization[1]

Synchronization happens when particle moves with phase velocity of the laser

Regions 1 on the energy map Synchronization gives slow but long rise of

energy

1. Pang et all. “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration”. Phys. Rev. E, 66 (2002)

Synchronization

Particle gets major part of energy only after synchronization!

Energy evolution and field phase at the particle location for a synchronized particle

SynchronizationPhase velocity in the direction of particle motion and the particle velocity:

After some time, the particle velocity becomes equal to the average wave phase velocity

Strong field interactionThis type of interaction happens with particles from region 2 on the energy map

Although synchronization might happen, it doesn’t play any role in energy rise. The main energy is obtained during very short interaction.

Jets formationAngular distribution of the time-averaged Poynting vector at 20 Rayleigh lengths of the best focus and angular distributions of the most energetic particles:

How to focus the laser to get the best energy?

Results of 3D PIC simulations: ultrathin foilMANDOR simulations for I=1022 W/cm2, tau=30 fs, f#=0.9Foil thickness: 50 nm, n=100 nc

Results of 3D PIC simulations: nanodiskMANDOR simulations for I=1022 W/cm2, tau=30 fs, f#=0.9Disk thickness: 100 nm, disk diameter = 1 micron, n=100 nc

Conclusions Exact field structure was analysed The electromagnetic field has subluminous

phase velocity regions Numerical calculation of particle trajectories 2 acceleration mechanisms were revealed:

synchronization and strong field interaction f#~2.7 was found to be the best for the given

laser energy and pulse length