W.B.Mori
Departments of Physics and Astronomy and of Electrical Engineering
Plasma Simulation Group
Departments of Physics and Astronomy and of Electrical Engineering
Los Angeles, CA
http://exodus.ucla.edu
W.B.Mori AAC Workshop, July 30, 2008 Invited Talk
Particle beam dynamics in plasma
accelerators: Beam Loading
Collaborators:
Michail Tzoufras (Oxford)
Chengkun Huang
Wei Lu
Viktor Decyk
Frank S. Tsung
Miaomiao Zhou
T. Katsouleas (Duke)
Reza Gholizadeh (USC)
Luis O. Silva (IST Portugal)
Ricardo .A. Fonseca
Samuel Martins
Jorge Vieira
E167 Collaboration
W.B.Mori NNSA, June 25, 2008 Science Talk
Outline
What is beam loading?
Review consequences of linear beam loading.
Nonlinear beam loading.
Other witness beam dynamics issues
Synchrotron radiation
Hosing
ion motion
W.B.Mori AAC Workshop, July 30, 2008 Invited Talk
Transverse Dynamics and Beam Quality
•
Emittance
ε
n
= phase space area:
x
p
x
σ
•
Matching
: Plasma focusing (~2
π
n
o
e
2
σ
) = Thermal pressure (grad p
∼
ε
/
σ
3
)
•
No spot size oscillations (phase space rotations)
•
No emittance growth
€
σ
2
ε
n
2
γ
c
ω
p
F
p
F
th
Transverse Dynamics and Beam Quality
Plasma focusing causes
beam to rotate in phase
space
•
Emittance
ε
n
= phase space area:
1/4 betatron period
(tails from nonlinear F
p
)
x
p
x
σ
•
Matching
: Plasma focusing (~2
π
n
o
e
2
σ
) = Thermal pressure (grad p
∼
ε
/
σ
3
)
•
No spot size oscillations (phase space rotations)
•
No emittance growth
€
σ
2
ε
n
2
γ
c
ω
p
F
p
F
th
Transverse Dynamics and Beam Quality
Plasma focusing causes
beam to rotate in phase
space
•
Emittance
ε
n
= phase space area:
1/4 betatron period
(tails from nonlinear F
p
)
Several betatron periods
(effective area increased)
x
p
x
σ
•
Matching
: Plasma focusing (~2
π
n
o
e
2
σ
) = Thermal pressure (grad p
∼
ε
/
σ
3
)
•
No spot size oscillations (phase space rotations)
•
No emittance growth
€
σ
2
ε
n
2
γ
c
ω
p
F
p
F
th
What is beam loading?
•
The placing of a bunch charge on an accelerating structure to extract energy
with high efficiency, small energy spread, and emittance preservation.
•
This involves understanding:
 how much charge can be loaded.
 where to place the charge.
 and how to shape the charge.
•
There is not much in the literature on how to do this for plasma
wakes
Beam Loading
linear regime
[1] T. Katsouleas, S. Wilks, P. Chen, J. M. Dawson, J. J. Su  Part. Accel, 1987
•
In the linear regime one
superimposes the wake by trailer
to the wake by the driver to find
the total wakefield [1].
•
Bunch shaping for no energy
spread: Triangular.
•
The total charge can be found by
requiring that all of the energy in
the wake is absorbed.
•
Works for electron and positron
loads.
Original simulation results: 1987
•
In linear theory just use superposition:
Add wakes =>
Drive beam (laser or particle beam)
Note that wedge gives nearly constant
decelerating field
Properly phased trailing beam of
particles: Loads wake
Katsouleas, Wilks et al., 1987
€
N
max
≈
10
11
n
1
n
o
10
16
cm
−
3
n
o
•
100% energy extraction (though V
gr
=0)
•
100% energy spread
(for spot size c/
ω
p
)
•
•
What does linear theory tell us?
Our interpretation
•
Emittance can only be preserved in two limits:
a. very narrow beam loads
b. very wide drive beams
•
Wide beams (multiple wavelengths) are ruled out for stability reasons
•
Narrow witness beams can only absorb energy of the wake out to a skin
depth so driver must be ~a skin depth or efficiency is low.
•
The trailing beam’s profile effects the accelerating and transverse fields
 nonlinear focusing fields
 transverse variation of E
z
 phase slippage can be an issue
•
But, narrow beams with high charge themselves make nonlinear
wakes.
Nonlinear physics is unavoidable for either PWFA or LWFA
Why?
Trailing beam density
Beam load efficiency
Matching
Energy spread=>
bunch length
Gives
For possible
collider parameters
€
n
b
N
(
2
π
)
3
/
2
σ
r
2
σ
z
€
N
~
1x10
10
€
σ
r
2
2
γ
k
p
−
1
ε
N
−
1
€
σ
z
~
α
c
ω
p
€
n
b
n
0
1.4
x
10
4
N
1
x
10
10
m
−
rad
ε
Nx
ε
Ny
Energy
250
GeV
1
α
€
n
b
n
0
~
10
4
−
10
5
Plasma response in the nonlinear regime
Rosenzweig et al. 1990, Puhkov and Meyerdervehn 2002, Lu et al. 2006
Driven by an electron beam
Driven by a laser pulse
Ion channel formed by trajectory crossing
Ideal linear focusing force for electrons
Uniform acceleration
Fluid model breaks down!
3D and electromagnetic in nature!
Wake excitation for given drive beam ……
Evolution of drive beam, e.g, instabilities…
Transformer ratio, shaped bunches, train of bunches
Beam loading, beam quality ……
How to put these all together in a design?
What about positrons?
What do we want to know and predict?
Positrons loading linear wake
Field structure in blowout or bubble regime
Relativistic blowout regime: Lu et al.PRL 16, 16500 [2006]
€
eE
z
mc
ω
p
r
b
2
dr
b
d
ξ
≈
1
2
ξ
eE
M
mc
ω
p
≈
1
2
k
p
R
b
≈
Λ
a
0
Bubble radius :
€
k
p
R
b
≈
2
Λ
or 2
a
0
for
k
p
σ
z
≈
1
€
Λ
n
b
n
0
k
p
2
σ
r
2
≈
5
N
2
x
10
10
10
m
σ
z
1
Uniform accelerating field
Linear focusing field
Nonlinear wakeﬁelds are completely characterized by
the trajectory of the blowout radius
W. Lu et al., Phys. Rev. Lett. 16, 16500 (2006)
The trajectory of the inner most particle is
Space charge of beam
Ponderomotive force
of laser
Two distinct limits:
Nonrelativistic blowout
Ultrarelativistic blowout
(behind the driver)
Nearly a circle!
This provides a physical picture and theory for
beam loading of nonlinear laser driven or electron
driven wakes from by electron beams
M. Tzoufras, submitted
•
The behavior in Regions I and
III is identical.
•
Region II:

The innermost particle
stops returns to the axis
more slowly.

This reduces Ez and is how
the wake is loaded.

The trajectory can actually
turn around: wakefield
goes to zero inside the
bunch

Focusing fields are not
perturbed by the beam!
Analytical solutions are possible
Assume that the blowout radius is large
For large blowout radius:
This can be integrated once to yield:
In Region I (and Region III):
The electric field must be continuous so solutions for each region need to be
matched.
Optimum bunch profile
Condition for constant wakefield:
The optimal profile is a trapezoid.
The wakefield and the trajectory for
this profile are:
Veriﬁed in fully nonlinear PIC
simulations using OSIRIS
High efﬁciencies are possible
Superposition does not work
Efficiency
•
There is a tradeoff between maximum
charge and maximum energy
•
In the blowout regime the energy gain
per unit length is orders of magnitude
higher than that in the linear regime.
•
For nonlinear beam loading the
efficiency approaches 100%, while E
z
is constant in z and r.
Efficiency
•
There is a tradeoff between maximum
charge and maximum energy
•
In the blowout regime the energy gain
per unit length is orders of magnitude
higher than that in the linear regime.
•
For nonlinear beam loading the
efficiency approaches 100%, while E
z
is constant in z and r.
s
Phase slippage / Gaussian bunches
Flattop bunch:
•
Analytical solutions have been found for
any charge per unit length.
•
The efficiency is the same as for the
trapezoidal bunch.
•
For such a bunch,
phase slippage will
not damage the beam quality
.
•
A Gaussian bunch behaves similarly.
Nominal 25 GeV stage
Preionized
n
p
= 1
×
10
17
cm
3
N
driver
= 2.9
×
10
10
,
σ
r
= 3
σ
z
= 30
, Energy = 25 GeV
N
trailing
= 1.0
×
10
10
,
σ
r
= 3
,
σ
z
= 10
, Energy = 25 GeV
Spacing= 110
R
trans
= E
acc
/E
dec
> 1 (Energy gain exceeds 25 GeV per stage)
1% Energy spread
Efficiency from drive to trailing bunch ~48%!
These equations are integrated for a
trapezoidal
λ
(
ξ
) to obtain E
z
(
ξ
) and r
b
(
ξ
).
This allows us to design accelerators with
100% beamloading efficiency that
conserve the energy spread.
Nonlinear beam loading:
Solve equation for R
b
(
ξ
)
M. Tzoufras, C.K.Huang et al., in preparation
25 GeV Driver
25 GeV Trailing beam
25 GeV Driver
475 GeV Trailing beam
Highly optimized 25 GeV Stages
Less than 1% energy spread
n
p
=5.66
×
10
16
cm
3
N
driver
= 4.42
×
10
10
,
σ
r
= 3
L = 58
, Energy = 25 GeV
N
trailing
= 1.7
×
10
10
,
σ
r
= 3
, L = 22
, Energy = 25 GeV or 475 GeV
R
trans
= E
acc
/E
dec
= 1
25 GeV Driver
25 GeV Trailing beam
25 GeV Driver
475 GeV Trailing beam
Highly optimized 25 GeV Stages
Less than 1% energy spread
n
p
=5.66
×
10
16
cm
3
N
driver
= 4.42
×
10
10
,
σ
r
= 3
L = 58
, Energy = 25 GeV
N
trailing
= 1.7
×
10
10
,
σ
r
= 3
, L = 22
, Energy = 25 GeV or 475 GeV
R
trans
= E
acc
/E
dec
= 1
25 GeV Driver
25 GeV Trailing beam
25 GeV Driver
475 GeV Trailing beam
Highly optimized 25 GeV Stages
Less than 1% energy spread
n
p
=5.66
×
10
16
cm
3
N
driver
= 4.42
×
10
10
,
σ
r
= 3
L = 58
, Energy = 25 GeV
N
trailing
= 1.7
×
10
10
,
σ
r
= 3
, L = 22
, Energy = 25 GeV or 475 GeV
R
trans
= E
acc
/E
dec
= 1
25 GeV Driver
25 GeV Trailing beam
25 GeV Driver
475 GeV Trailing beam
Highly optimized 25 GeV Stages
Minimal hosing and emittance growth!
n
p
=5.66
×
10
16
cm
3
N
driver
= 4.42
×
10
10
,
σ
r
= 3
L = 58
, Energy = 25 GeV
N
trailing
= 1.7
×
10
10
,
σ
r
= 3
, L = 22
, Energy = 25 GeV or 475 GeV
R
trans
= E
acc
/E
dec
= 1
25 GeV Driver
25 GeV Trailing beam
25 GeV Driver
475 GeV Trailing beam
Highly optimized 25 GeV Stages
Minimal hosing and emittance growth!
n
p
=5.66
×
10
16
cm
3
N
driver
= 4.42
×
10
10
,
σ
r
= 3
L = 58
, Energy = 25 GeV
N
trailing
= 1.7
×
10
10
,
σ
r
= 3
, L = 22
, Energy = 25 GeV or 475 GeV
R
trans
= E
acc
/E
dec
= 1
Hosing in the blowout regime
Parameters:
Huang et al., Phys. Rev. Lett.
Phys. Rev. Lett. 99, 255001 (2007)
I
peak
= 7.7 kA
€
A
(
s
,
ξ
)
A
old
(
c
r
c
ψ
)
1
/
3
≈
A
old
(
1
4
(
1
.2
Λ
)
)
1
/
3
€
For the paramters on the previous slide the new
A
(
L
pd
,
2
σ
z
)
5
and
e
5
5
3
/
2
≈
13
Ion Motion
•
Ion motion when
•
Matched beam spot size shrinks at large
γ
, low
ε
n
•
For future collider
−
ε
ny
down by 10
2
(e.g., 10nmrad)
−
γ
up by 10+

n
b
up by 10
2

Ion motion must be included in design/models
€
n
b
∝
σ
x
−
1
σ
y
−
1
1
ε
nx
ε
ny
γ
2
ω
p
c
€
n
b
n
o
M
i
m
e
Ref. S. Lee et al., AAC Proc (2000); J. Rosenzweig et al., PRL (2006)
Density wake of
scale beams
R. Gholizadeh USC
Conclusions and perspectives
A new theory for beam loading of electrons in nonlinear wakes was developed
Key differences between beam loading in linear and nonlinear wakes:
The volume of the wake energy left behind is smaller than ahead in nonlinear
theory
100 % beam load efﬁciency with high gradient is therefore possible with shaped
bunches in nonlinear theory
The selfwake of the beam load in linear theory effects the focusing force and
the radial dependence of the accelerating forces on later particles in the witness
bunch. In nonlinear theory the selfwake does not because blowout is complete.
Properly phased Gaussian bunches ﬂatten the wake better in nonlinear theory
leading to smaller energy spread.
Optimal shape is almost square in nonlinear theory and a reverse triangle in
linear theory.
Background:
Beam loading, Linear regime
•
For ultrashort beams the maximum particle number is [1]:
•
Issues affecting the energy spread:
 phase slippage (dephasing).
 transverse variation of the accelerating field:
•
Low emittance requires narrow beams.
•
Blowout will happen unless the charge is very low.
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