Acceleration and Focusing by superstronglaser fields and their applications

to a laser micro-collider

Kazuhisa Nakajima

2nd ORION WORKSHOPSLAC, Feb. 18-20, 2003

KEK/JAERI

Outline

Laser acceleration and focusing mechanisms

Electron-positron micro-collider concept

Which is better for high energy acceleration,plasma wakefields or vacuum laser fields?

Necessity for high energy

acceleration mechanism

Effective constituent collision energy ofColliders (Physics Today, Jan. 2001)

1970 19801990 2000 20100

1

10

100

1,000

10,000

Colli

sion

Ene

rgy

(GeV

)Prin-Stan, VEPPII, ACO

ADONESPEAR, DORIS, VEPPIII

SPEARIIVEPPIVCESR

PETRA, PEP

TRISTANSLC, LEP

LEPIILC500

ISR

SppSTevatron I

LHC

e+e- colliders

Hadron colliders

RHIC

Energy frontier Accelerators

“The realization that we seem to be near the end of the road forconventional accelerators

has generated renewed interest in thepossibility of accelerators with

super-high accelerating fields.”

The Laser Acceleration of Particles Workshop, 1982, Los Alamos

(AIP Conference Proceedings No. 91, 1982)

To keep enregy frontier trend

Accelerating field limits

100cm 10cm 1cm 1mm 100m 10m 1m 0.1m1 MV/m

10 MV/m

100 MV/m

1 GV/m

10 GV/m

100 GV/m

1 TV/m

200 MHzproton linac

SLAC (SLC)

NLC

CLIC

3GHz 300 GHz30GHz300MHz 3THz 30THz 300THz 3PHz

W-band linac

LWFA

SM-LWFA

DLASWA

Wavelength

Frequency

Surface heating limit

Break

down

limit

Plasma growth limit

Plasma w

ave

breakin

g

Ele

ctri

c fi

eld

gra

die

nt

RF accelerators

Laser-Plasmaaccelerators

R.B. Palmer, AIP Conf. Proc. 91, 179-189 (1982)

1 EV/m Vacuum breakdown

Is plasma wakefield a high energyacceleration mechanism?

• The plasma wave-breaking causes background electron trapping. • The accelerating field should be less than the plasma wave-breaking limit.• The energy gain is limited by dephasing even ifthe optical guiding is accomplished.• Beam-plasma instabilities breaks high qualitybeam acceleration.• Field control is difficult.• High energy background particle and photon radiations

Many disadvantages kill a role of high energy accelerators.

The maximum accelerating energy

max w

1 p

eEmax

mc2 4 p2 eEmax

mc2kw

The maximum energy gain of particles trapped in the electricfield Emax of which phase velocity vp is given by

where w is the wavelength of the electric field,

kw 2 w p vp c p 1

1 p2

Assuming the maximum electrostatic potential eEmax

mc2kw

1

max 4 p2

max 1

eEmax

mc2 Lacc for the acceleration length Lacc cw

c vp

Super-high energy particle acceleration in plasma

The phase velocity of the electron plasma wave is given by

p vp

c 1

ne

nc

p nc

ne

1 2

where nc is the critical plasma density,nc cm 3

re0

2 1.11 10210 2 m

re e2

mc2 2.818 10 13 cm with the classical electron radius

The maximum energy gain is approximately given by

max 4 p2 4

nc

ne

4 1021

02 m ne cm 3

The acceleration length is Lacc p

1 p

2 p2p

p 2kp

reneWith the plasma wavelength

Lacc cm 2nc

ne

rene

2

02

rene

3 2

7.4 1027

02 m ne

3 2 cm 3

1015 1016 1017 1018 1019 1020 1021

106

102

104

10 m

11 cm

1 m

100 m

10 km

Ene

rgy

gain

Acce

lera

tion

leng

th L

acc

A single stage energy gain

Plasma density ne (cm-3)

max

Lacc

A naive estimate of 2.5 TeV electron linac

Consider a laser driver of

5 106

For 2.5 TeV electron acceleration,

Necessary plasma density is given by ne 8 1014 cm 3

Necessary acceleration length becomes Lacc ~ 3.3 km

the required energy gain is

0 1m

The required accelerating gradient is 0.76 GeV/m

The accelerator length will be the order of 1 km even if the multi-staging can be accomplished.

It is not so attractive compared to the conventional scheme.

Laser acceleration fieldThe laser fields are written as E

A

ct, B A

in terms of the vector potential A of the laser field.

For a linearly polarized laser field, A A0 cos kz t e

Defining the normalized vector potential as a eA

mec2

the laser strength parameter is given by

where I is the laser peak intensity, and

is the laser wavelength with frequency

a0 2e20

2 I

me2c5

1 2

0.855 10 9 I1 2 W/cm2 0 m

0 2c 0 0

The amplitude of the transverse electric field of linealy polarized laser is

EL TV m mec2k

ea0 3.21

a0

m 2.7 10 9 I1 2 W cm 2

e.g. I =1×1018 W/cm2 gives EL =2.7 TV/m

Lawson-Woodward TheoremThe net energy gain of a relativistic electron interacting with an electromagnetic field in vacuum is zero.

The theorem assumes that (i) the laser field is in vacuum with no walls or boundaries present, (ii) the electron is highly relativistic (v ≈ c) along the acceleration path,(iii) no static electric or magnetic fields are present,(iv) the region of interaction is infinite,(v) ponderomotive effects (nonlinear forces, e.g. v x B force) are neglected.

(J.D. Lawson, IEEE Trans. Nucl. Sci. NS-26, 4217, 1979)

Acceleration mechanism must violate the Lawson-Woodward theorem.Usually it is not difficult!

Acceleration by Higher-Order Gaussian ModesThe x-polarized Hermite-Gaussian TEM10 mode is given by

Ex 4E0w0 x

2w 2 z exp

r2

w2 z

sin

Ez 4E0w0

2kw2 z exp

r2

w 2 z

1

2x2

w2 z

cos

2zx 2

Z Rw2 z sin

-3 -2 -1 1 2 3

-1

-0.5

0.5

1 Ex

Ez

x w0

TEM10

w0 z 0

Ex,z

E0

kz t zr 2

Z Rw2 z 2tan 1 z

ZR

0where w z w0 1

z2

ZR2

1 2

E0 2

w0

P

cP

c

4E0

2w02

On the z-axis Ex=0 and the axial field accelerating an electron with vz ≈ cEz

4E0w0

2kw2 z cos z

z kdz

2z2 2 tan 1 z

ZR

0 z 1

vz2

c 2

1 2where

The phase velocity of the accelerating field is

v ph c 1 d2 1 z 2 1

d 2 kZR 1 2 w0where is the divergence angle.

Near the focal point z Z R v ph 1d2 11 2c

2 c 2d2 1 2

The slippage distance is given by Zs ZR / 2

1c2 z

2

For z c Zs ZR

2The particle injection energy for a finite interaction length is given by

Einj m0c2 m0 c2

4eEz0 k

eEz0

kTinj m0c 2 m0c 2

4 d2 Z0 P 2

1

d

2 Z0 P

2

Z0 377where

The maximum energy gain is given by

W MeV eEz0 ZR 2e 2 P c 2 Z0 P TW 15.5 P TW

1m,w0 10m, P 10TWe.g.

Tinj 2.1MeV

For

W 49MeV

Vacuum Beat Wave Accelerator

(E. Esarey, PRE 52, 5443,1995)

Two laser beams of different frequencies cause an axial acceleration from the beat term in the v×B force.

dp

dt e E v B / c E A

c tB A

Using u p mc a eA / mc2

1 u2 1 2 1 2 1 2

dudt

a t

cu a d

dt u

a t

u a

0.5 1 1.5 2 2.5 3

-1

-0.5

0.5

1Beat-Wave

t

A A1 A2

FzFz Fz

The axial component of the nonlinear ponderomotive force is

Fz e

cv B z

mc 2

2z

a a

The total laser field is A z,r, t A 1 A 2 A1, A2; the vector potential of laser 1 and 2.

a2 a a a12 a2

2 2a1a2 cos 2 1 az2 ai a0iw0 i / wi exp r2 / wi

2

The energy equation along the axis isddz

a01a02

z

ksin 02 01

Along the axis r=0 and near the focus of the two lasers

v ph c 1 1 Z R1 ZR2 / kZ R1

v ph c for ZR2 ZR1

In an infinite interaction region zI , zF

F2 I

2 2a01a02kZR

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

z ZR1

w

w0w1

w2

1 2 2

The accelerated electron energy for the injection energy and WIP1 P2

WF MeV WI MeV 2 750 1 2 1 P1 TW 1 2

e.g.WI 20MeV

1 2 2 1 m

P 10TW

w01 1

WF 89MeV

W 70MeV

Energy gain

Effect of Radiative Losses

An attainable maximum energy is limited by radiative losses due to transverse quiver oscillations.

PR2

3remc3 2a0

2 k 2 1

2z2

1

kZR

2

ree2 / mc 2; the classical electron radius

The electron energy equation with radiative loss is

ddz

a01

a02

k 2

3re

2 a012

ZR12

a 02

2

ZR22

Assuming k k1, a01 a02 , ZR1 Z R2

the maximum electron energy is given by

Wmax mc2 w01 1 3w01 / re 1 3

Wmax GeV 1.3 w014 3 m 1 m

e.g.

Wmax 1.3GeV for w01 1 22 1m

50 100 150 200

500

1000

1500

2000

2500

z cm

Electron Energy vs Distance

WI 100MeV

a0 1

w01 1 22 1m

Ponderomotive acceleration

E ˆ x E0 sin, B ˆ y E0

csin t t

z t c

d

dt x

eE0

mc1 z sin

d

dt z

eE0

mcx sin

d

dt y 0

d

dt

eE0

mcx sin

y const 0 1 z const 0 1 0

0 1 a02 cos 1 2 1 0

2

x 2a0 cos 1

0 2 a02 1 0 cos 1 2

z 1 2 1 0

0 2 a02 10 cos 1 2

a0 eE0

mc

eE02mc 2

(F.V. Hartemann et al.,PR E,51,4833,1995)

Energy-momentum equations of electrons in a linearly polarized plane waveddt

e

mc2 vE

where v

c,

1

1 2, p mc

dp

dt e E v B

z ca0

1 0 0 3

4a0

2 1 0

a0

2 10 sin28

sin

x ca0 sin 0 1 0

The maximum energy at

max 0 1 2a02 10 0 1

e2 I2

20 m2c5 1 0

E0

2 I

0c

2n 1

The transverse and longitudinal positions are

Electron energy Electron trajectory

0.5 1 1.5 2 2.5 3

1000

2000

3000

4000

2

a0 10 0 10

20000 40000 60000 80000

-600

-500

-400

-300

-200

-100

z

x

a0 10 0 10

Energy and trajectory of an electron

with

100 200 300 400 500

2.5

5

7.5

10

12.5

15

17.5

20

0 1

0 2

0 5 0 10

Scattering angle

•The quiver amplitude becomes comparable to the beam waist in the high field.•The electron can be scattered away with a high energy as the interaction is terminated within a wavelength.

Threshold power given by

Pmax 5

02

0 m2c5

e2

w0

4

0.21

02

w0

4

TW

Scattering angle

x

z

2

2 1

z 2 1 1 z 0 1 0

1 2 1 x2 z

2

arctan

2

1 0

0

1

1 2

0 1 0

x

arctan x

zz

0

arctan2

1 0 10 0

Ponderomotive scattering

x

0a0 w0

a0th w0

0

0At 0 No acceleration

Is vacuum laser acceleration possible to accelerate particles up to super high energy?

2 2a0w0

0

Consider energy gain of laser acceleration in vacuum.

Higher Order Gaussian Mode Acceleration

Laser Beat Wave Acceleration

2 3 2a0w0

0

assuming 0 1 22

max w0

0

3w0

re

1 3

Radiation loss limit

Poderomotive scattering for 0 ~1

40a02

with a0 w0

001 10 100 1000

103

105

107

101

a0

RadiationLoss limit

HGMA

LBWA

PMSA

Energy gain as a function oflaser intensity

(HGMA)

(LBWA)

(PMSA)

w0 0 10

0 100

Direct laser-particle interaction

A

A

a eA

mc2p e

cA

P ~ mcaa 1

a 1Pz ~ mca2

~1

a2

Non-relativistic interaction

Relativistic interaction

is the normalized vector potential.

0

a2

where W

mc2 for the energy gain W.

Reflection of electromagnetic wave on an electron plasma

2 k 2c2 p2

p 4nee

2

mp

Photon

ne

p

The reflection condition of the electromagnetic wave (photon) is given by

Electromagnetic wave

m ph

p

c2

Photon-’Electron Plasma’ Interaction

Electron Plasma

The dispersion relation of the electromagnetic wave in plasmawith electron density ne is

This means that a photon inside plasma has an effective mass:

where

a

Electron

de Broglie wave

ph ,p

a mec2

a

1 a 2 1a0

2

2

1 2

Reflection of an electron on an intense photon field

Electron-’Photon Plasma’ Interaction

Photon plasma

In an analogy to reflection of electromagnetic wave fromthe electron plasma, an electron in an intense laser field has a mass shift

Considering a laser pulse as a ‘plasma’ of photons, the photon plasma frequency is

me2 a 2me

2 1

2a0

2me2

An electron de Broglie wave will be reflected for its frequency

e

mec2

ph, p

a mec2

i.e.

An electrom with momentum is expelled in the condition:

Particle acceleration in super-strong laser fields in plasma

Laser field and electron space charge field can be given by the vector potential A and the scaler potential

E 1

c

A

t B A

The equation of a particle motion is written as

dp

dt eE

v

cB

e

1

c

A

t

v

c A d

dt

t

v

The equation of energy isd

dtmc2 ev

1

c

A

t

with

Basic equations

The Maxwell’s equations are1

c2 2

t2 2

A

4c

j 1

c

t

A

1

c

A

t

4

is the charge density, en

j envwhere j is the current density,

The continuity equation of plasma density n isn

tnv 0

assuming the plasma is an ensemble of cold fluids.

1D field-electron interaction in plasma

A Axe x Aye yA Axe x

for circular polarization,

for linear polarization.

In a 1D laser field according to the Coulomb gauge Az 0

A 0letting

the vector potential is expressed as

Consider the laser pulse frame propagating at a group velocity in plasma

vg k

c 1 p

2

02 p

4nee2

mwhere

is the plasma frequency for the ambient electron density ne.

Transforming the momentum equation to a new variable,

z vgt

d

dp

e

cA

0

p e

cA

This gives conservation of canonical transverse momentum.

The electron quiver velocity is given by

a eA

mc2

v pm

e

mcA

a

From the transverse component of the equation,

Defining the normalized vector potential

1 2 1 2with the Lorentz factor

d

dpz

mc2

vg

e

vg

0

g vg

c

emc2

assuming that prior to laser interactiona 0 0 , z 0

gz const. 0 1 g0 where

The longitudinal component of the energy equation becomes

This gives

the solutions on and are given by

z g

2 H0 H02 1 g

2 2 1 2

H02 g

22

a

g2 H0 gg g

2H02

2 1 2

with H0 0 1 g0

1 a2Using 1 u

2 uz2 z where z

1

1 z2

H02 g

22

H0 g H02 1 g

2 2 1 2

The space-charge potential is obtained from

(Continuity equation)1

c

n

t n 0

2 kp2 n ne 1 (Poisson's equation)

where n is the electron density, k p p c

In an initial equilibrium (prior to the laser pulse), the space-charge potential is negligible, i.e. =0

Space-charge potential

g 1For

d2d 2

ˆ k p2

2

1 a2

1 2 1

0 1 0 ˆ k p

kp

03 2 1 0

letting

(Nonlinear wake equation)

Assuming n n d

dn g z 0 n ne

g 0

g z

In the short-pulse limit L ckp 1

where L is a laser pulse duration.

1

In the long-pulse limit L ckp 1

1 a2 s

1 2 1

1 a2 s

1 2 1 a0

2 2 1 2

is nearly constant.

L0 cL p

2

L0 cL p

2

The averaging over a laser wavelength,

In the short-pulse limit 0 g 1let for vacuum

0 11 0

2a2

s

0 11 0

4a0

2

1

20a0

2

for0 1

z 0

1 0

2a2

11 0

2a2

0

1 0

4a0

2

11 0

4a0

2

a0 1

1

In the short-pulse limit ≈ 0

g20 1 g0 gg g

202 1 g0 2

2 1 2

g20 1 g0 gg g

202 g 0 2

a2 1 2

the maximum and minimum energies are

g0 1 g0 For

max g20 1 g0 gg g

202 g 0 2

a02 1 2

min 0

max g20 g 0 2

0 2

min 0

g0 1 g0 For

Scattering by ponderomotive potential

a0th

a0th g0 g 0

Scattering conditiong0 1 g0 Lth

Lth 1 a0th2 cos2 k 1 2

Electron trajectory in phase space

a0 a0th g2 1Scattering

-1 -0.5 0.5 1

50

100

150

200

k

g 10,0 1,0 0

a0 5 a0thTrapping

Simulations of ponderomotive acceleration

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Group velocity

g vg

c0.8

g 1.67

a0 5Below threshold

Above threshold a0 10

Ponderomotive acceleration and focusing

High energy booster acceleration of a pair-beam can be accomplished by the relativistic ponderomotive acceleration with focusing in vacuum ortenuous plasma.

The final energy is obtained approximately as

f a02

for a particle initially at rest.

E f GeV 0.37 10 21 I W/cm2 02 m

I 1022 W/cm2 E f 2.4GeVE.g.

I 1023W/cm2 E f 24GeV

I 1024 W/cm2 E f 240GeV

I 1025W/cm2 E f 2.4TeV

Acceleration

0 0.8mAt

El,m Re E0l,mel ,me

iti l ,m r,z Hl

x

2

Hm

y

2

0

exp

r2

4 2

z ct 2

4 z2

The electric field of Hermite-Gaussian modes is given by

0is the rms beam size. 0 1 z2 ZR

2

ZR is the Rayleigh length. z is the beam pulse length.

Consider the lowest mode of the laser beam with l=m=0, and the amplitude electric field E0

a0 e2mc2 E0 c

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

1.2

1.4

U0U1

U=U0 +U1

r 0

The ponderomotive potential is

U0 r, z, t a02 0

2

2 exp

r2

22

z ct 2

2 z2

Focusing

The amplitude of the quiver oscillations is estimated to be

x

0

a00

2ZR z 1 2

ZR2 1 2 8 z

Off axis particles in a Gaussian laser beam are accelerated by the radial ponderomotive force and quickly expelled from the beam in the radial direction.To avoid transverse instability of motion, superposition of a Gaussian mode with l=m=0 and higher order modes, e.g. a linear combination of l=1, m=0 and l=0, m=1 with the same polarization vectors e1,0 =e0,1, and E1,0 =iE0,1 creates a potential well in the radial direction:

U1 r, z, t a12 r20

2

4 exp

r2 02

24

z ct 2

2 z2

where a1 is the dimensionless vector potential.

The total ponderomotive potential is given by

U U0 U1

The focusing force is given by

Fr

mc 2 U

r 2a1

2 a02 r0

2

4 a1

2 r3 02

6

exp r2

22

z ct 2

2 2

Focusing strength at r=0, and z-ct=0

KF Ft

mc2r

2a12 a0

2

02

The beam envelope equation on the rms beam radius rb isd2 rb

dz2 KF rb reN

2 23 zb rb

b

2

rb3 0

where N is the number of electrons in the bunch, zb is the rms bunch length, b is the geometric emittance, n the normalized emittance re is the classical electron radius.

b n

Space charge force Thermal emittance

The equilibrium radius is obtained fromd2 rb

dz2 0

Estimate of a focused beam size

1) The space-charge-force dependent beam size

rb reN

2 1 4 KF1 23 2 zb

1 2 reN

2 1 4 2a12 a0

2 1 4 zb

1 2

0

a1 a0Assuming 0 r0 2 zb 0

rb 2 1 4

2

r0

a05 2

reN

0

rb pm 2 1024 N

I5 4 W/cm2 r0 m 0

3 m

e.g. For 0 0.8mr0 10m

Np 1 1010

I 1.06 1025 W/cm2

rb 0.2 pm

Assuming a1 a0

rb n0

a0

1

2a02

0r0

b n n a02 a0

4 0 r0 2

rb pm 4 1023

I W/cm2 r0

0

rb b

KF1 4

1 4 b0

2a12 a0

2 1 4

2) The emittance dependent beam size

e.g. For 0 0.8mr0 10m

Np 1 1010

I 1.06 1025 W/cm2

rb 0.13 pm

If an electron-positron pair beam is focused, the space chrage forcewill be neglected.

Laser micro colliderTwo counter propagating laser-accelerated beams make a micro collider.

The space charge limited luminosity

L Np

2 frep

4 rb2

a050Np frep

2 3 2rer02

L cm 2s 1 2 10 30 I 5 2 W/cm2 06 m r0

2 m N p frep

0 0.8m

r0 10m

Np 1 1010

I 1.06 1025 W/cm2 EC.M. 5TeVe.g.

L 2 1040 frep cm 2s 1

P = 17 EW EL > 2×4 kJ

Required peak power and pulse energy

e+e- pair-beam micro-collider

Two counter-propagating laser-accelerated pair beams will createa new e+e-, e-e-, e+e+ micr-size collider without beam disruption at collision.

L Np

2 frep

4 rb2

a04 Np

2 frep

0r0

The emittance-limited luminosity is

where Np is the number of accelerated e+e- pairs and frep is the repetition rate of laser pulses.

L cm 2s 1 5.3 10 27 I2 W/cm2 03 m r0 1 m Np

2 frep[Hz]

E.g. For

0 0.8m

r0 10m

Np 1 1010

L 3 1042 frep cm 2s 1

I 1.06 1025 W/cm2

Laser Micro Collider Concept

e- beam e+ beam

Pair beam production

Collision

Plasma Pulse shaping

0

Pair-beam micro-collider concept

EC.M.[GeV]

1

I [W/cm2]

2.11021

P [PW/pulse]

3.3

EL [J]

1.6

L [cm-2s-1] at 1 Hz

1.2 1035

4 8.4 1021 13 6.4 2 1036

10 2.11022 33 16 1.2 1037

500 1.1 1024 1700 800 3 1040

1000 2.110243300 1600 1.2 1041

5000 1.1 102517000 8000 3 1042

Vacuum or tenuous plasma

factory

J/ factory

B factory

Luminosity of laser micro-colliders

1042

1039

1036

1033

1030

1027

Lum

inos

ity a

t 1 H

z [c

m-2s-1

]

100

110

C.M

. Ene

rgy

[GeV

]

100010000

1020 1021 1022 1023 1024 1025

Laser Intensity [W/cm2]

C.M. energy

Emittance-limitedluminosity

Space charge-limitedluminosity

0 0.8m r0 10m Np 1 1010

Summary

The proper shaped superstrong laser fields are useful for acceleratingand focusing electron and/or positron beams.

Two counter-propagating electron and positron beams result ina micro-collider with extremely high luminosity.

The plasma acceleration concept has difficulties in high energy gainand high beam quality.

A single stage energy gain < 1 GeV

I 1025W/cm2 E f 2.4TeV

rb 0.13 pm

L 3 1042 frep cm 2s 1

I 1025W/cm2

Np 1 1010

Np 1 1010

L 2 1040 frep cm 2s 1

The beam injector,The e+e- pair beam generator

Advanced Accelerator Research & Development at KEK

There are great interests in advanced accelerator researches, growing in the national and industry institutes, andthe universities around Japan.

The C.O.E. on A.A. R&D is conceived in KEKto organize developments of the future high energy acceleratortechnologies, holding long term visions of the post-LC, or the post-B factory.

We are eager to make collaboration and a word-wide network for A.A. R&D.

The 1st Workshop on Advanced Accelerator R&DApril 24, 25, 2003 at KEK, Tsukuba, Japan

The 1st Workshop on Advanced Accelerator R&DApril 24, 25, 2003 at KEK, Tsukuba, Japan

KEKJAERICRIEPAIST・・・

U. TokyoTokyo I. Tech.Tokyo M. U.Waseda U.U. UtsunomiyaU. KyotoU. OsakaU. Hiroshima・・・・・

KoreaKAER

I

ChinaIHEP

Fudan U.

Japan

Taiwan

India

U.S.A.SLAC ORION

BNLLBLNRLANL

US Univ.

EuropeCERN

RAL (UK)LULI, LOA

(France)GSI (Germany)

CNR (Italy)IGP (Russia)

Advanced Accelerator R&D NETWORK