The scaling of LWFA in the ultra-relativistic blowout regime: Generation of Gev to TeV monoenergetic

electron beams

W.Lu, M.Tzoufras, F.S.Tsung, C. Joshi, W.B.MoriUCLA, USA

L.O. Silva, R.A.FonsecaIST, Portugal

Outline

•Motivation.

•Physical picture : Illustration of what the ultrarelativistic blowout regime looks like, what the fields are, how the electrons behave and evolution in time.

•Theory : Ideas behind the theory. Description of how the characteristic quantities of this regime relate to each other.

•Scaling laws : Scaling of beam energy, beam charge and energy conversion efficiency with laser and plasma parameters. Comparison between the theory and published (as well as unpublished) results, both experimental and simulation.

Extrapolation to exotic cases. The possibilities of building single stage 10Gev, 100Gev and even TeV laser electron accelerator and additional issues need to be addressed.

•Conclusion.

MotivationRecent results

3 Nature papers (September 2004) where monoenergetic electron beams with energy 70~170MeV by using 10TW 30fs class lasers were measured.

Phys. Rev. Lett. by Tsung et al. (September 2004) where monoenergetic beam with energy 260 MeV by using a 13TW 50fs laser were observed.

How can we scale this regime to higher energy and better beam quality?

Questions we try to answer ……

•Is there a consistent physical picture behind all the experiments and simulations?

What is the condition for self-injection of the electron beam?

•What are the energy , charge and efficiency scaling and their scalabilities?

•What is the condition for self-guided laser propagation?

•What is the optimal conditions to choose the parameters?

•What determines the beam quality ( energy spread, spot size and emittance) ?

Physical pictureGeometry - fields

• The ponderomotive force of the laser pushes the electrons out of the laser’s way.

• The particles return on axis after the laser has passed.

• The region behind the pulse is void of electrons but full of ions (ion channel).

• The resulting structure moves with the speed of laser’s group velocity, supporting huge accelerating fields and strong focusing force.

Physical pictureEvolution of the nonlinear structure

• The front of the laser pulse interacts with the plasma. As a result it loses energy (Local pump depletion) and etches back.

• The shape and size of the accelerating structure slightly change.

• Electrons are self-injected in the ion channel at the tail of the ion channel due to the accelerating and focusing fields.

• The trapped electrons slightly elongate the back of the spheroid.

Physical pictureEvolution of the nonlinear structure

• The blowout radius remains nearly constant as long as the laser power doesn’t vary much. Small oscillations due to the slow laser envelope evolution have been observed.

• Beam loading eventually shuts down the self injection.

• The laser energy is depleted as the accelerating bunch dephases. The laser can be chosen long enough so that the pump depletion length is matched with the dephasing length.

QuickTime™ and aDV/DVCPRO - NTSC decompressor

are needed to see this picture.

Theorythe spherical ion channel and the constant wake slope

•A spherical ion channel for ultra-relativistic blowout A fully nonlinear theory for the blowout regime for both beam and laser driver can show that for large blowout radius (ultra-relativistic blowout ),the ion channel will become a sphere .

5~4>bpRk

•A constant wakefield slope (1/2) The wakefield depends linearly on the distance from the center of the ion channel, and has a deep spike near the tail.

bpp

M

p

z

RkmceE

mceE

21

21

≈

≈

ω

ξω

TheoryChoosing the laser parameters - matched profile

bpp Rkwk ≈0

•Matched laser spot sizeFor given laser power P, there is a matched laser spot size W0, which is approximately equal the blowout radius Rb

Laser

31

0 2 ⎟⎟⎠⎞

⎜⎜⎝⎛

≈cPPaFor given laser power P and given plasma density np,

this matching condition gives:

00 2 aWkRk pbp ≈≈

€

∇a02

γ~ a0

2

kpRbγ~ a0

kpRb

E r ~ ∂φ∂r

~ kpRb

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

⇒ kpRb ~ a0

Ponderomotive :

Ion channel :

Balance of forces :

€

⎧ ⎨ ⎩

Approximately :

TheoryCondition for self injection

• The condition for self injection1. In the ultra-relativistic blowout regime ( kpRb>>1 and spherical ion channel), the

plasma electrons will get parallel speed close to c when they reach the axis near the tail of ion channel.

2. When the electrons reach the axis, their initial velocities are typically smaller than the phase velocity of the wakefield. If they can get enough energy before dephaing through a narrow region near the tail of ion channel, which has both strong accelerating field and focusing force, they get trapped and keep gaining energy.

⎪⎩

⎪⎨⎧

>>≈

>>≈⇒≈

15~4

116~85~40

bp

c

RkPP

a

Both conditions can be satisfied if the matched a0>4~5:

Simulations show that for even very low plasma density like np=1*10^15 cm-3 (very high wake phase velocity ), trapping can be achieved by this condition

⎪⎪⎪⎩

⎪⎪⎪⎨

⎧

⎟⎟⎠⎞

⎜⎜⎝⎛

≈≈

⎟⎟⎠⎞

⎜⎜⎝⎛

→

≈

LkkL

nnLc

Lakk

L

pp

c

etch

p

pd 2

0

0

2

0arguments) (scaling

υ

Absorption by ponderomotive particles

1D like absorption Electron density100 TW, 3 10-18 cm-3

Two types of absorption:By ponderomotive particles

1D like absorption

€

∝ a0

€

∝ a02

Theory Local pump depletion

For a0 around 4~5, these two absorption are comparable. For a0 around 10 or larger, the 1D like absorption dominates.

TheoryEtching velocity, phase velocity of the wake and dephasing length

The laser front etches back by local pump depletion. After pump depletion, it diffracts.The etching back velocity Vetch is in principle depends on a0 ( for 1D, Vetch is independent of a0). More detail calculation can show that the 3d Vetch is close to 1D results even the energy loss mechanism changes when a0 gets large.

⎥⎥⎦

⎤⎢⎢⎣

⎡⎟⎟⎠⎞

⎜⎜⎝⎛

+−≈=2

0, )1

21(1

kk

c pgWake υυ φ

100

3

0

2

0

100

3

0

2

0

2

34

32

−

−

⎟⎟⎠⎞

⎜⎜⎝⎛

⎟⎟⎠⎞

⎜⎜⎝⎛

≈⎟⎟⎠⎞

⎜⎜⎝⎛

≈

⎟⎟⎠⎞

⎜⎜⎝⎛

≈⎟⎟⎠⎞

⎜⎜⎝⎛

≈

kakk

Rcc

kkL

kakkR

kkL

pbppd

pb

pdp

ττ

This yields the dephasing length and the pump depletion length:

Due to the laser etching

100 TW, 3 1018 cm-3

The same scaling for Ldp and Lpd. Typically we can choose to match dephasing and pump depletion.

bRc ≈τ

Scaling lawsEnergy gain, charge and energy conversion efficiency

Energy gain :

32

31

2

⎟⎟⎠⎞

⎜⎜⎝⎛

⎟⎟⎠⎞

⎜⎜⎝⎛

≈Δp

c

r nn

PPmcE ⎟⎟⎠

⎞⎜⎜⎝⎛

⎟⎟⎠⎞

⎜⎜⎝⎛

≈Δ−

rc PP

PPMeVE

32

)(32.00

2

02

32 a

kkmcEp⎟⎟⎠⎞

⎜⎜⎝⎛

≈Δ

123

02

2

23 −≈ pb ka

emcN

reb P

Prk

N023

≈

0

2aE

NE

T

b ≈•Δ

≡η 31

−

⎟⎟⎠⎞

⎜⎜⎝⎛

≈•Δ

≡cT

b

PP

ENEη

dpavLEeE ≈Δ

or or

Total charge :

or

Energy conversion efficiency :

Tb EEN /Δ≡η

22bavbbav RERNEe ≈

or

Scaling lawsVerification of the scaling through simulations

As long as the laser can be guided ( either by itself or using shallow plasma density channel), one can increase the laser power and decrease the plasma density to achieve a linear scaling on power.

PE ∝Δ

Self-guiding condition

The laser self-guiding is based on two effects:1. The main part of the laser is inside a index of refraction channel

made by the laser blowout. 2. The laser front keeps etching back, which prevents the leading

front from diffraction before pump depletion.

A fully nonlinear theoretical analysis based on the index of refraction gives the following critical a0 for guiding:

51

0 ⎟⎟⎠⎞

⎜⎜⎝⎛

≈p

cc

nna

For all the 3D simulations we have done ( np>1*10^18cm-3), a0~4 is enough for guiding. For density like np= 2*10^17cm-3, this gives a0 aroud 5~6. In the future, 3D simulations will be used to test this condition for low density.

Parameter designs for Gev,10Gev,100Gev,1Tev

P(PW) τ (fs) np (cm-3)

W0 (μm)

L(m) a0 Δnc/np Q(nC) E(Gev)

0.12 30 2e18 15 0.009 4 0% 1.3 1.12

1.2 100 2e17 47 0.28 4 <20% 4 11.2

12 300 2e16 150 9 4 <20% 13 112

120 1000 2e15 470 280 4 <20% 40 1120

1 80 5e17 35 0.08 5.1 0% 4 5.8

10 180 1.2e17 80 0.8 6.8 0% 12 33

100 430 2.8e16 190 8 9.1 0% 40 182

1000 1000 6.5e15 450 80 12.1 0% 120 1012

Conclusions

• We have developed a theory that allows us to design laser plasma accelerators operating in the ultrarelativistic blowout regime.

• We have found that a laser with ”matched” profile achieves stable, self-focused propagation for the entire interaction length.

• Given the power of a laser we can:1. Pick the density for self-focused propagation .2. Choose the rest of the laser parameters. 3. Predict the energy of the monoenergetic beam.

• For these accelerators, since the energy is proportional to the laser power:- we have shown via numerical simulations that nC, GeV electron

bunches can be generated by 100-200 TW lasers.- According to the scaling, TeV laser plasma accelerators will become

possible for 100-200PW lasers.

32

31

2

⎟⎟⎠⎞

⎜⎜⎝⎛

⎟⎟⎠⎞

⎜⎜⎝⎛

≈Δp

c

r nn

PPmcE ⎟⎟⎠

⎞⎜⎜⎝⎛

⎟⎟⎠⎞

⎜⎜⎝⎛

≈Δ−

rc PP

PPMeVE

32

)(32.0

100

3

0

34 −

⎟⎟⎠⎞

⎜⎜⎝⎛

≈ kakkLp

dp rp

dp Zak

kL0

0 132

⎟⎟⎠⎞

⎜⎜⎝⎛

≈

100

3

02 −

⎟⎟⎠⎞

⎜⎜⎝⎛

⎟⎟⎠⎞

⎜⎜⎝⎛

≈ kakk

RcL

pmpd

τ

0

2

02

32 a

kkmcEp⎟⎟⎠⎞

⎜⎜⎝⎛

≈Δ

31

0 2 ⎟⎟⎠⎞

⎜⎜⎝⎛

≈cPPa

123

02

2

23 −≈ pb ka

emcN

reb P

Prk

N023

≈

0

2aE

NE

T

b ≈•Δ

≡η 31

−

⎟⎟⎠⎞

⎜⎜⎝⎛

≈•Δ

≡cT

b

PP

ENEη

51

0 ⎟⎟⎠⎞

⎜⎜⎝⎛

≈p

cc

nna

formulas

Matched a0 and spot size : 00 2 aWk p ≈

Pump depletion length:

Dephasing length:

Energy gain:

Charge:

Efficiency:

Critical a0 for self-guiding:

Beam quality and X-ray loss

Except the Tev designs, the X ray losses are small comparing with the beam energy. For the Tev designs, the X ray losses are less than 200Gev.

X-Ray emission

Energy spread

For higher laser power and lower plasma density ( longer dephasing length), the uncertainty in the energy shot by shot will decrease.