Transcript Document

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.Mori
UCLA, USA
L.O. Silva, R.A.Fonseca
IST, 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published (as well as unpublished) results, both experimental and simulation.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.
Motivation
Recent results


Phys. Rev. Lett. by Tsung et al. (September 2004) where
monoenergetic beam with energy 260 MeV by using a 13TW 50fs
laser were observed.
3 Nature papers (September 2004)
where monoenergetic electron beams
with energy 70~170MeV by using
10TW 30fs class lasers were
measured.
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 picture
Geometry - 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 picture
Evolution 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 picture
Evolution 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.
QuickTime™ and a
DV/DVCPRO - NTSC decompressor
are needed to see this picture.
• 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.
Theory
the spherical ion channel and the constant wake slope
•A spherical ion channel for ultrarelativistic 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 k p Rb  4 ~ 5 ),the ion channel
will become a sphere .
•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.
eEz
1
 
m c p 2
eEM
1
 k p Rb
m c p 2
Theory
Choosing the laser parameters - matched profile
•Matched laser spot size
For given laser power P, there is a matched
laser spot size W0, which is approximately
equal the blowout radius Rb
k p w0  k p Rb
Laser

Balance of forces : 

a02
a0 
Ponderomotive :  ~ k R  ~ k R 

p b
p b
 k p Rb ~


E
~
~
k
R
Ion channel : r
p b


r
Approximately k
: p Rb
 k pW0  2 a0
a02

For given laser power P and given plasma density np,
this matching condition gives:

13
P
a0  2 
 Pc 
a0
Theory
Condition for self injection
•
1.
2.
The condition for self injection
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.
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.
Both conditions can be satisfied if
a0>4~5:
the matched
P
 P  8 ~ 16  1
a0  4 ~ 5   c
k R  4 ~ 5  1
 p b
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
Theory
Local pump depletion
Two types of absorption:
By ponderomotive particles
1D like absorption
 a02
 a0
For a0 around 4~5, these two absorption are comparable. For a0 around
10 or larger, the 1D like absorption dominates.


Absorption by ponderomotive particles

(scaling arguments)

Lpd  
2
 k0 
 c
nc
L L  L

k 
np
etch
 p
2
 k0 
  a0 L
k 
 p
1D like absorption
Electron density
100 TW, 3 10-18 cm-3
Theory
Etching 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.
Due to the laser etching
Wake,
2


k


1
p
  g  c 1  (  1)  
2

 k0  
This yields the dephasing length and the
pump depletion length:
Ldp
2k
  0
3
 kp
L pd
k
 0
k
 p
2


 Rb  4  k0

3

 kp
2




3



 c  2 c    k 0
 R  k

 b  p

a0 k




1
0
3
a0 k01
100 TW, 3 1018 cm-3
The same scaling for Ldp and Lpd. Typically
we can choose c  Rb to match dephasing
and pump depletion.
Scaling laws
Energy gain, charge and energy conversion efficiency
Energy gain
E  eEav Ldp :
2
k 
2
E  m c2  0  a0
k 
3
 p
Total charge
P
E  mc 2  
 Pr 
or
1
3
eEav N bRb  Eav2 Rb2
3 m c2 23 1
Nb 
a0 k p
2 e2
or
Energy conversion efficiency
E  N b 2


ET
a0
or
Nb 
3
2k0 re
 nc 
 
n 
 p
2
3
or
:
P
Pr
  NbE / E T :

E  N b  P 
  
ET
 Pc 

1
3
P
E  0.32( MeV ) 
 Pc 

2
3
P
 
 Pr 
Scaling laws
Verification 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.
E  P
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:
 nc 
a  
n 
 p
1
5
c
0
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- W0 (μm) L(m)
3)
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, selffocused 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.
formulas
1
 P 3
a0  2 
 Pc 
Matched a0 and spot size :
Pump depletion length:
 c  k 
Lpd  2  0 
 
 Rm  k p 
Dephasing length:
4k 
Ldp   0 
3  k p 
2
Energy gain:
k 
2
E  m c2  0  a0
k 
3
 p
Charge:
Nb 
Efficiency:
E  N b 2


ET
a0
3 m c2 23 1
a0 k p
2 e2
k pW0  2 a0
3
a0 k01
3
a0 k01
2
1
 P 3  n 3
E  mc 2    c 
 
 Pr   n p 
Nb 
3
2k0 re
P
Pr
E  N b  P 

  
ET
 Pc 
1
Critical a0 for self-guiding:
Ldp 
 n 5
a0c   c 
n 
 p

1
3
2  k0  1
Zr
3  k p  a0
P
E  0.32( MeV ) 
 Pc 

2
3
P
 
 Pr 
Beam quality and X-ray loss
Energy spread
For higher laser power and lower plasma density ( longer dephasing length), the
uncertainty in the energy shot by shot will decrease.
X-Ray emission
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.