Bubble regime: condition

€

γemax =a02 /2=γg ≈

ω

ωp

1+a02 /2 /2

€

ω <<ωp

Then electrons can be trapped and move with pulse

a) cτ<λ

b) electron velocity exceeds the laser pulse group velocity

pλp is plasma wavelength

τ Is the pulse length

Zhidkov PRE 2004, Pukhov App Phys 2002

Simulation Parameters:I=10 20

W/cm 2

n=10 cm-3

τ=20 fs

19

: 600*80 simulation box μm2

λ =11 μ mp

DLA electrons

Bubble regime

Contour plots ofelectron density2D PIC in units of [n |e|]cr

wake

wave breaking

accelerating field

laser pulse

Blue:electron density

green: laser fieldRed: longitudinal electric field

Bubble regime

bubbletrapped electrons

DLA

bubble at later time

accelerated electrons

accelerated electrons

pulse erosion

Self Focusing,Channeling

€

ω pe−1 << τ <<ωpi

−1( )

€

eA

mc 2= a(x)(ex ± iey )exp[i(hz −ωt)]

Underdense homogeneous plasmaFixed ions Maxwell’s equations+Equation of motionAssuming

€

∇2a + 1−αn

γ

⎛

⎝ ⎜

⎞

⎠ ⎟a = 0

∇ 2Φ = α (n −1)

γ = 1+ a2

α =n0

1− h2 /k 2

These equations can be solved analytically

in 2D (F. Cattani et al, PRE, 2001).

Complete evacuationPartial electron evacuation

€

a(x) =2 α α −1( ) cosh( ε x)

α cosh2( ε x) −ε

ε = α −1

n = 3(1+ a2) + 21+ a2

α(−α − a2)

nmin = 0

€

xd = −1

α

ad ′ a d1+ ad

2

€

tanad [2α ( 1+ ad

2 −1) − ad2 ]1/ 2

α

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟=

1+ ad2

ad

2α 1+ ad2 −1( ) − ad

2[ ]

1/ 2

half of channel width

Numerical Results

€

density =1019cm−3

maximum laser intensity =1020 w /cm2

spot size = 5μm

pulse duration = 270 fs

Red points correspond toanalytical solution

Greens and blue points correspond tonumerical results

€

×1016 w /cm

€

xdhalf of channel width

€

μm€

Ptotal

Comparison with theoretical results:

channel evacuated of electrons

contour plot of electron density

Plot of electron density vs. y

evacuated channel

Strong electrostatic wake

Ex

Contour plot ofelectron density2D PIC (SP)in units of[n |e|]

cr

Contour plots ofLaser intensity 2D PIC (SP)a in unit of(eE/mωc)

In 2D PIC, data taken from a cut along x in the middle of the box. Laser pulse enters from left and propagates along x.

Laser pulse as it enters the plasma

Laser pulse after propagating 240 micrometers

electron density

Wave-breaking and electron acceleration

€

px

mcAccelerated electrons

electrons in front

€

Ex

€

Ey

n

accelerating field

pulse erosioncavity

25 Mev

d N/ d

E [A

r b. U

nit s

]

Energy in ev

Phase space at time= 1031 fs

After wave-breaking electrons are accelerated and injected into the pulse and the accelerating field.4.8% of total number of electrons are accelerated.

energy of accelerated electrons

time=1395 fs

Theoretical studies and particle-in-cell (PIC) simulations of nonlinear processes related to short pulse laser propagation in underdense plasmas. For the laser power above critical power for relativistic self-focusing in two spatial dimensions PIC simulation results converge to stationary laser filaments. Conditions for the formation of multifilament structures are discussed and demonstrated in simulations for relatively long pulses. For short laser pulses nonlinear propagation at relativistic intensities involves pulse erosion, frequency shift and characteristic steepening at the front of the pulse. Different mechanisms of particle acceleration are described including particle trapping at the front of the pulse, acceleration by the plasma wake fields and by the electromagnetic wave. These processes are simulated and discussed in the context of recent experiments with gas jet targets on the ALLS facility.

Abstract

Relativistic self-focusing, electron acceleration and ultra-short laser pulse propagation in underdense plasmas

Neda Naseri , Paul-Edouard Masson-Laborde , Valery Bychenkov, Wojciech Rozmus , University of Alberta , Lebedev Physics Institute 1 1 2 1 1 2

Filamentation of intense laser beam in plasma By using transversely flat modulated laser pulse, filamentation instability is being studied.

Contour plot ofLaser intensity 2D PIC a in unit of(eE/mωc)

€

density =1×1020cm−3

maximum laser intensity = 5 ×1019 w /cm2

FWHM of laser intensity = 40μm

pulse duration = 300 fs

Wave breaking, acceleration (SP)Contour plot of electron density 2D PIC (SP). The picture on the left shows the wave breaking and the picture on the right shows injected electrons.

injected electronswave breaking

Relativistic self-focusing

Maximum intensity is 3 timesbigger than maximum initial Intensity (SP).

€

Pcr =16.2 ×ncr

nGW

€

Pcr = 0.5TW

P = 7.6TW

P =15.2Pcr

parameters as

SP

Contour plot ofLaser intensity 2D PIC (SP)

Snapshots of laser intensity cross section

Pulse erosion, 1D in hydro

pulse erosion

Pulse Field

Density

Longitudinal Field

Strong steepening of longitudinal field

Relativistic fluid model

( )

( )

( )

22 2 2

20

2

2

20

0

pe et x

e e

e

e

e i

n ac a

c n

n n p

t x m

pm c e

t x x

en Zn

x

ω

γ

γ

γ φ

φε

∂ − ∂ = −

⎛ ⎞∂ ∂+ =⎜ ⎟

∂ ∂ ⎝ ⎠

∂ ∂ ∂+ =

∂ ∂ ∂

∂= −

∂

rr

ur

ur

Model equations in 1D:

Maxwell + Full Hydro + Poisson

Maxwell Equation

Hydro: continuity

+ motion equations

Poisson equation

€

px

mc

d N/ d

E [A

r b. U

nit s

]

Electron energy in ev

Electron energy in ev

180 Mev

Electron energy spectrum Phase space

Threshold power for bubble regimeGordienko, phys plasmas, 2005

Numerical models•Particle-in-cell code MANDOR (1D , 2D): Romanov et al. PRL, 2004•Relativistic cold plasma approximation and Maxwell equations in 1D-limited by the absence of kinetic effects•Standard parameters (SP) - consistent with experimental conditions: pulse duration, τ=30fs, spot size=13μm, intensity, I=4*1018 W/cm2, density, n=5* 1019 cm-3, p-polarized. The experiment carried out at the Advanced Laser Light Source (Z. L. Chen, Y. Y. Tsui, R. Fedosejves) • Homogeneous plasma slab with 40 microns linear ramp in the front. 400-800 microns in length - propagation distance is limited by laser pulse scattering and absorption.

ALLS

bubble

I=4 1018, =30fs, n=5 1019cm-3

I=1020, =20fs, n=1019cm-3

[fs]

P[TW]

€

Pthreshold = 0.03× (τ ( fs)

λ (μm))2 TW

n1 < ne < n2

n1

nc

≈8 ×10−3

P(TW)

n2

nc

≈1.6 × (λ (μm)

τ ( fs))3 P(TW)

•From laser plasma accelerators, quasi monoenergetic 70 – 170 MeV, Mangles et al. Nature 2004, Geddes et al. ibid 2004, Faure et al. ibid 2004.

longitudinal filed DLA electrons

100 Mev

DLA electrons

electrons in front part of bubble

electrons in back of bubbleelectrons in front part of bubble

electrons in back of bubble

Input modulated laser pulse

filaments

Lase

r in

tens

ity (

eE/m

ωc)

Lase

r in

tens

ity (

eE/m

ωc)

filaments

Simulation parameters: