A conceptual design of the single stage multi-TeV electron-positron pair beam collider

Kazuhisa NAKAJIMAKEK

International Workshop onHigh Energy Electron Acceleration Using Plasmas 2005

HEEAUP 2005: 8-10 June 2005 -Institut Henri Poincaré, Paris, France

Outline

Electron-positron pair-beam productionin strong laser fields

Laser Plasma Collider-type INonlinear Laser Plasma acceleration andPlasma lens final focus

Laser Plasma Collider-type IILaser Ponderomotive Acceleration and Focus

Multi-stage or Single stage?

Multi-stage or Single stage?

Multi-staging technology of laser plasma accelerators

makes it possible to extend a short acceleration cell

toward a high energy accelerator in a complex, long system,

though spatial alignment and temporal synchronization

must be resolved from the viewpoint of accelerator physics.

Technologically less attractive!

Single-staging technology is based on violent acceleration

in a single interaction without complex system, though

an extremely high peak power laser will be required.

5 TeV e+e- LWFA Linear Collider consisting of staged plasma channels

T3

OPTICALDELAY

MARXGENERATOR

WATERCAPACITOR

MULTILTSG

YAG(Triger)

T3

OPTICALDELAY

MARXGENERATOR

WATERCAPACITOR

MULTILTSG

YAG(Triger)

T3

OPTICALDELAY

MARXGENERATOR

WATERCAPACITOR

MULTILTSG

YAG(Triger)

~1 m

2.5 TeV e- LWFA 2.5 TeV e+ LWFA

~1 km55GeV/m, 10GeV/stage

Parameters of 5TeV e+e- Linear Collider based on LWFA

Collider parameters

N 5x107/ bunch

CM Energy Ecm

Luminosity Lg

Emittance y

Beta at IP y

Beam size at IP y

Bunch length z

Number of particles

Collision frequency fc 50 kHz

Average beam power Pb 2 MW

Disruption parameter Dy 0.93

Beamstrahlung parameter 3485

0.32 m

0.1 nm22 m

2.2 nm

1035 cm-2s-1

5 TeV

LWFA parameters

Plasma density ne 3.5x1017cm-3

Acceleraion length Lac 20 cm

Accelerating gradient Ez 55GV/mEnergy gain/ stage W 10 GeV

Laser pulse energy

Laser power/ stage Pav 100 kW

EL 2 J

Laser pulse duration 100 fsLaser peak power P 20 TW

Number of stages 500Total laser power 50 MW

(M. Xie et al.,AIP CP398,AAC96,233,1997)

Total length ~ 1 km

Pair-beam production in nuclear fields The yield of pair production via trident process in plasma ions foris given by

γ >>3

Np ≈0.48π3

103 α 2a08Z3 nc

ne0

⎛

⎝ ⎜

⎞

⎠ ⎟

2r0λ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2 Δλ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

Np ≈2.8×10−45Z3I W/cm2[ ]

4ne0

−2 cm−3[ ]r0

2 μm[ ]Δ2 μm[ ]

where r0 is the laser spot radius and is the plasma thickness.

≈0.8×10−6a08Z3 nc

ne0

⎛

⎝ ⎜

⎞

⎠ ⎟

2r0λ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2 Δλ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

e.g. Z=54 (Xe)ne0 =1020cm−3

r0 =10μmΔ =100μmFor

I =1022W/cm2

Np ≈4.4×1014

Virtual photon

e-

e-

e- in the laser field

e+

Coulomb fieldof nuclear charge

Z

e+Z → ′ e e+e−Trident pair creation in plasma

γ >>3

€

It =2.6 ×1019

λ L μm( )[ ]2 W/cm2

Threshold intensity

A priori scaling for Nonlinear Wakefield Accelerator with self-channel guiding

€

Emax ≈ mec2a0

2 ncne

€

a0 = 6.8 P TW[ ]λR

The maximum energy by dephasing

Acceleration length

€

Lacc ≈ 0.6λa0ncne

⎛

⎝ ⎜

⎞

⎠ ⎟3 2

€

nc =mω2

4πe2 =π

reλ2 ≈

1.1×1021

λ μm[ ]cm−3

[ ]

Acceleration by driving laser pulse =30fs,=0.8m, R=10m

Using relativistic self-guiding condition

€

P TW[ ] ≥ 0.017ncne

P[TW] a0 nc/ne Lacc Emax

20 2.4 1176 7.3 cm 3.5 GeV

100 5.4 5882 2.7 m 87

300 9.4 17647 32.4 m 797

500 12.2 29412 103 m 2237

A priori scaling for NonlinearWakefield Accelerator with plasma channel

Maximum energy gainat the wave breaking limit:

€

γmax ≈ 4γg

3

Operating plasma density:

€

ne ≈ ncγ g−2 = nc

γ4

⎛ ⎝ ⎜

⎞ ⎠ ⎟−2 3

€

n0 cm−3[ ] ≈

1.1×1021

λ 0 μm[ ]

γ4

⎛ ⎝ ⎜

⎞ ⎠ ⎟−2 3

Required laser intensity:

€

a02 ≈ 4γg = 4

γ4

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1 3

€

P TW[ ] ≈ 0.1γ4

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1 3 r0λ 0

⎛

⎝ ⎜

⎞

⎠ ⎟2

Accelerating length:

€

Ld ≈ γg2γ⊥λ p ≈

24γλ 0

€

γ≈2 ⋅106

€

n0 ≈ 2.2 ⋅1017 cm−3

€

a0 ≈ 18

P ~ 1.2 PW

€

r0 =10μm,λ 0 = 0.8μmfor

€

Ld ≈ 71cm

E = 1 TeV

€

γg =ncne

Plasma lens final focusBoth electron and positron beams self-focus by a self-pinching effect in plasma.

Ft =−2πremec2nbr

Beam density: nb =N

2π( )3 2σbr2σbz

Focusing strength:

KF =Ft

γmec2r

=reN

2πγσbr2 σbz

=reN

2πεnβ0σbz

re =2.818×10−13cm : classical electron radius

for a Gaussian density profile: n r,z( ) =nb exp−r2

2σ br2 −

z2

2σbz2

⎡

⎣ ⎢ ⎤

⎦ ⎥

Self-focusing force for an overdense plasma

ne >nb

where

εn : Normalized beam emittance

β0 : Beta function at the plasma lens

Focal length: f =1

K Fl=

2πεnβ0σbz

reNl

σbr : rms beam radius σbz : rms bunch length

β0 =γσ br2 εn

l : Length of plasma lens

σbr

ne

nb

l

σbr∗

f

Particle bunch

Plasma

cτb =2 2ln2σbz ≈2.35σbz

FWHM bunch length

(P. Chen, PRD, 39, 2039, 1989)

Luminosity by plasma lens final focus

The beta function at the collision point β∗

β0 =β∗+s2 β∗≈ f 2 β∗

The spot size at the collision point

σ ∗2 =εβ∗≈εn f 2

γβ0

=2πεnσbrσ bz

reNl

⎛

⎝ ⎜

⎞

⎠ ⎟

2

The luminosity for a Gaussian beam

L =N2

4πσ 2 =1

8π2

reN2l

εnσ brσbz

⎛

⎝ ⎜

⎞

⎠ ⎟

2

=γ

8π2εnβ0

reN2l

εnσbz

⎛

⎝ ⎜

⎞

⎠ ⎟

2

εn ≈λL π β0 ≈πrL2 λL

€

rL ≈ 10μm

L =γ

8rL2

reN2l

λLσbz

⎛

⎝ ⎜

⎞

⎠ ⎟

2

Assuming

C.M. Energy 2x1TeV

Number of particles 1.4x1010 e-

Plasma lens length ~5mm

Laser spot size

Laser wavelength

€

L = 0.8μm

Luminosity 5x1035 cm-2/s-1

Repetition rate 10 Hz

Laser intensity distributions of Hermite-Gaussian modes

Radius

ElectronElectron

Intensity

Fpond Fpond

Scattering of the electrons

x-z plane

x-y plane

z

x

y

x

xz

x

Intensity

Radius

FpondFpond

Electrons confinement

TEM(0,0)

y

TEM(1,0)

xy

TEM(1,0)+TEM(0,1)

by S. Miyazaki, Utsunomiya Univ.

The momentum in the x, y and z direction

-50

0

50

100

150

200

250

300

350

400

0 10000 20000 30000 40000 50000

PxPy

Pz

t[λ/c]

P[m

c］

Electron acceleration by TEM01+TEM10

by S. Kawata & S. Miyazaki

Laser intensity : I = 1.23×1018[W/cm2] 　　　　　　　　　⇒ a0 = 0.5

Wave length:λ ～ 1.053[μm]Minimal spot size: w0=35λ

Pulse length: Lz=10λ

a)

-100

0

100

200

300

400

500

-10000 0 10000 20000 30000 40000 50000

Transverse

Sum

Longitudinal

Δγ

t[λ/c]

z

x

y

t=0

0

8 Lz

2w0

2w0

Electron bunch

Laser pulse

Simulation model

~200 MeV/cm

Ponderomotive acceleration energy for the laser intensity

1 10 100

1

10

100

1000

10000

100000

a0

γ f

Initial Velocity : 0Minimal spot size : 20λPulse length : 10λ

With radiationWithout radiation

by S. Miyazaki & S. Kawata

Ponderomotive acceleration and focusing in vacuum

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

γ f ≈a02

for a particle initially at rest.

Ef GeV[ ] ≈0.37×10−21I W/cm2[ ]λ0

2 μm[ ]

I =1022W/cm2 Ef ≈2.4GeVe.g.

I =1023W/cm2 Ef ≈24GeV

I =1024W/cm2 Ef ≈240GeV

I =1025W/cm2 Ef ≈2.4TeV

Acceleration

λ0 =0.8μmAt

for final energy scaling

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

γmc2r=

2a12 −a0

2

γσ⊥02

The beam envelope equation on the rms beam radius rb is

d2σ rb

dz2+KFσ rb −

reN2πβ2γ3σ zbσrb

−εb

2

σ 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.

εb =εn γβ

Space charge force Thermal emittance

The focusing force is given by

Fr

mc2=

∂U∂r

= 2a12 −a0

2( )rσ⊥0

2

σ⊥4 −a1

2 r3σ⊥02

σ⊥6

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ exp−

r2

2σ⊥2 −

z−ct( )2

2σ⊥2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Focusing by TEM00 + TEM01 +TEM10 Ponderomotive Potential

The equilibrium radius is obtained fromd2σrb

dz2=0

Focused beam size The space-charge-force dependent beam size

σrb ≅reN

2π( )14KF12βγ32σ zb

12 ≅reN

2π( )14 2a12 −a0

2( )

14σ zb

12

σ⊥0

γ

a1 =a0Assuming σ⊥0 =r0 2 σ zb ≈λ 0

σrb ≈2π( )14

2r0

a05 2

reNλ0

σrb pm[ ] ≈2×1024 NI5 4 W/cm2

[ ]

r0 μm[ ]λ0

3 μm[ ]

e.g. For λ0 =0.8μm r0 =10μm Np =1×1010

€

I =1.0 ×1022

W/cm2

€

rb ≈ 1.2 nm

€

I =1.06 ×1025W/cm2

€

rb ≈ 0.2 pm

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

The space charge limited luminosity is given by

L =Np

2frep

4πσ rb2 ≅

a05λ0Np frep

2π3 2rer02

L cm−2s−1[ ] ≈2×10−30I 52 W/cm2

[ ]λ06 μm[ ]r0

−2 μm[ ]Np frep

λ0 =0.8μm

r0 =10μm

Np =1×1010

I =1.06×1025W/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-colliderTwo counter-propagating laser-accelerated pair beams will createa new e+e-, e-e-, e+e+ micro-size collider without beam disruption at collision.

L =Np

2frep

4πσ rb2 ≅

a04Np

2 frep

λ0r0

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[ ] ≈5.3×10−27I2 W/cm2

[ ]λ03 μm[ ]r0

−1 μm[ ]Np2 frep[Hz]

E.g. For

λ0 =0.8μmr0 =10μm

Np =1×1010

L ≈3×1042 frep cm−2s−1

I =1.06×1025W/cm2

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

λ0 =0.8μm r0 =10μm Np =1×1010

A conceptual design of Laser Micro Collider

C. M. Collision energy 1 TeVInitial beam energy 50 MeVNumber of particles per bunch 1010 Laser wavelength 0.8 mLaser spot size 10 mRepetition frequency 10 HzRequired peak intensity 4.2×1022 W/cm2

Required peak power 660 PWRequired pulse energy 8 kJ for =10%Space charge limited luminosity 2×1035 cm-2s-1 Emittance limited luminosity 5×1038 cm-2s-1

Parameters of LMC

(K. Nakajima, High Energy Accelerator Seminar OHO’03)

A conceptual design of 2TeV Advanced Colliders

Linear LWFA

Non LinearLWFA

LaserPonderomotive

ILC+EnergyDoubler

PBGLaserCollider

Total length~200m ~2m ~2m 30+0.2km 2kmAcceleratingGradient 55GV/m 1TV/m 1TV/m

35MV/m+4GV/m 1GV/m

Number ofparticles 1.4x1010 1.4x1010 1.4x10101.5x1010 105

CollisionFrequency 10Hz 10Hz 10Hz 14.1kHz 433MHz

Luminosity(cm-2s-1) 5x1035 5x1035 5x1035 1x1034 3x1035

LaserPeak Power/Duration

100x20TW/100fs

2x1.4PW/16ps

2x115PW/200fs

The key issue is luminosity for beam collisions.

Road Map toward TeV

1~10 MeV Proof-of-Principle

experiments

100~350 MeVdemonstration

Mono energetichigh quality beam

>1 GeV Channel Guided LWFA

10~100GeVSingle stage

1 TeV demonstration

1993

2003

2004

2006

2008

2010

2015

During the last decadehigh-quality beam up to near 1GeVwas achieved.

In the next decade worldwide Advanced Accelerator Communitywill aim at realizing1 TeV electron acceleration.

Multi TeV collider

Select single stageor multi-stage