Longitudinal Beam Shaping and Compression
Scheme for the UCLA Neptune Laboratory
R. J. England, J. B. Rosenzweig, M. C. Thompson
Department of Physics and Astronomy
University of California Los Angeles
Los Angeles, CA 90095
Abstract. We are developing a new beamline which will serve as a venue for future beamplasma interaction experiments using the 14MeV electron beam produced by the UCLA Neptune
1.625-cell photoinjector and PWT linac. An examination of the first and second-order optics
indicates that when certain nonlinear effects are minimized through the use of sextupole magnets,
the longitudinal dispersion is dominated by a negative R56- Simulations using the matrix
transport code ELEGANT indicate that for an appropriately chirped initial beam, this beamline
can be used to create a ramped picosecond to sub-picosecond beam that is ideal for driving large
amplitude wake fields in a plasma and producing high transformer ratios.
INTRODUCTION
Due to their capacity to support large electric fields, plasmas have been considered
in recent years as a means for acceleration of charged particles capable of producing
field gradients larger than those achievable with traditional radio-frequency linear
accelerating cavities by several orders of magnitude. Longitudinal field gradients in
excess of 1 GeV/m can be obtained by the excitation of large-amplitude relativistic
waves in a plasma. Various acceleration schemes have been proposed which rely
upon driving such plasma waves, using either a short intense laser beam (laser wake
field accelerator, LWFA) or a short relativistic electron beam (plasma wake field
accelerator, PWFA) [1-4]. In the case of the PWFA, the transformer ratio (the
maximum longitudinal accelerating electric field in the wake of the driving beam
divided by the maximum decelerating field in the tail of the beam) is a figure of merit
which provides a measure of the maximum energy gain of a test charge injected
behind the driving bunch. For a driving bunch with a uniform or gaussian current
profile and finite length, the value of the transformer ratio can be shown to always be
less than two [5,6]. Various methods have been proposed to overcome this limitation,
the most promising of which include the use of a single asymmetric drive bunch [7] or
a ramped bunch train (RBT) [8].
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
884
(b)
(b)
(a)
(c)
(c)
6
5
(a)
I (kA)
4
δp / p (a.u.)
Optimum profile
profile :
Optimum
3
2
1
(a.u.)
δ&z (a.u.)
0
0.8
0.8
11
1.2
1.2
1.4
1.4
(mm)
zz (mm)
1.6
1.6
1.8
1.8
Figure 1.
1. Plots
Plots showing
showing the
the longitudinal
longitudinal phase
phase space
space (a)
(a) and
and density
density profile
profile (b)
(b) of
of aa ramped
ramped beam
beam
Figure
produced
by negative
negative R
R5566 compression,
compression, as
as well
well as
as aa PIC
PIC simulation
simulation (c)
(c) of
of the
the wake
wake field
field produced
producedby
by
produced by
16
such aa beam
beam in
in aa plasma
plasma of
of density
density 2×10
2xl016
cm"-33..
such
cm
(i.e. aa square
square
In the case of a single asymmetric drive bunch a "doorstep" profile (i.e.
pulse for the first quarter of the beam duration followed by a triangular ramp)
approximates the optimal asymmetric current distribution that maximizes
maximizes the
the
transformer ratio and forces the retarding potential to be constant
constant within
within the bunch.
bunch.
The analytically derived transformer ratio of such
such aa driving
driving beam
beam is
is found
found to
to be
be
R=
k
L,
where
L
is
the
length
of
the
bunch
and
k
is
the
inverse
plasma
skin
depth
pp
= pp L ,
[9]. For such a profile the value of R may therefore exceed two so long as the bunch is
longer than two plasma skin depths.
an
We propose a scheme for the creation of a driving beam which approximates an
first and second
second order
order beam
beam optics.
optics. The
The
asymmetrical "doorstep" current profile using first
proposed method takes advantage of the RF curvature in the longitudinal phase space
space
distribution of a positively chirped (i.e. back-of-crest) driving
driving beam
beam due
due to
to its
its
longitudinal spread upon injection into the linac. Under aa pure (i.e.
(i.e. with negligible
negligible
higher order contributions) negative R
Rs6
longitudinal phase
phase space,
space,
56 compression of the longitudinal
such a phase space distribution results in a ramp-shaped current profile of picosecond
picosecond
to sub-picosecond duration, which is ideal for
for the generation of large
large amplitude
amplitude
plasma wake-fields and high transformer ratios. The longitudinal phase
phase space
space of
of such
such
shown in Fig. 1(a).
l(a). The
The current
current profile associated
associated with
with the
the phase
phase space
space in
in
a beam is shown
l(a), and a comparison of it to the optimized
optimized "doorstep"
"doorstep" current
current profile,
profile, are
are shown
shown
Fig. 1(a),
l(b). The wake fields produced by this beam distribution, shown
shown in
in Fig.
Fig. 1(c),
l(c),
in Fig. 1(b).
proposed wake
wake field
field accelerator
accelerator
were obtained from a particle-in-cell simulation of aa proposed
S-Bahn, aa new section
section of
of beamline
beamline scheduled
scheduled for
for
experiment at ORION. The S-Bahn,
2002 has
has been
been designed,
designed, using
using
installation at the UCLA Neptune laboratory in 2002
produce aa pure
pure negative
negative RRs6
compression
sextupoles to cancel nonlinear effects, to produce
56 compression
shown in Fig. 1.
1. A
A diagram
diagram of
of this
this
capable of creating a ramped beam of the sort shown
beamline is shown in Fig. 2.
885
S-BAHN
S
S
B1
Q1
Q2
Q2
Q1
B2
PLASMA CHAMBER
GUN
PWT
CHICANE
PBWA
PBWA
FIGURE 2. Schematic
Schematic of
of UCLA
UCLA Neptune
Neptune beamline
beamline with
with S-Bahn.
S-Bahn.
S-BAHN
S-BAHN OPTICS
OPTICS
1.
1. System
System Constraints
Constraints and
and Notations
Notations
The
The S-Bahn
S-Bahn optics,
optics, shown
shown in
in Fig.
Fig. 2,
2, form
form aa nondispersive
nondispersive translating
translating system
system
incorporating aa widely
incorporating
widely used
used geometry
geometry known
known as
as aa dogleg.
dogleg. The
The last
last two
two dipoles
dipoles of
of aa
chicane compressor
chicane
compressor (B1)
(Bl) on
on the
the main
main beamline
beamline are
are used
used to
to effect
effect aa 45
45 degree
degree bend
bend
which diverts
beam off
the main
which
diverts the
the beam
off of
of the
main beamline
beamline immediately
immediately before
before the
the plasma
plasma beat
beat
wave experimental
wave
experimental section.
section. A
A second
second pair
pair of
of identical
identical dipoles
dipoles (B2)
(B2) at
at the
the end
end of
of the
the SSBahn produce
produce another
Bahn
another 45
45 degree
degree bend
bend to
to divert
divert the
the beam
beam onto
onto an
an alternate
alternate beamline,
beamline,
parallel to
to the
the original,
parallel
original, which
which will
will serve
serve as
as the
the venue
venue for
for future
future beam-plasma
beam-plasma
interaction experiments.
interaction
experiments. Two
Two pairs
pairs of
of quadrupoles
quadrupoles (Q1
(Ql and
and Q2)
Q2) form
form aa FODODOF
FODODOF
lattice, which
which is
lattice,
is symmetric
symmetric about
about the
the midpoint
midpoint of
of the
the beamline.
beamline. Two
Two sextupoles
sextupoles (S)
(S)
are
performing second-order
are included
included for
for performing
second-order corrections.
corrections.
(s)
Optimal
Optimal operation
operation of
of this
this device
device relies
relies upon
upon the
the horizontal
horizontal dispersion
dispersion function
function ηrj(s)
passing through
through zero
passing
zero at
at the
the midpoint
midpoint between
between bends.
bends. Under
Under an
an optically
optically symmetric
symmetric
geometry, this
this ensures
geometry,
ensures that
that the
the final
final dispersion
dispersion function
function and
and its
its derivative
derivative with
with respect
respect
to the
the path
path length
parameter ss are
to
length parameter
are both
both zero
zero at
at the
the exit
exit of
of the
the device.
device. In
In order
order to
to
further
utilize the
further utilize
the symmetry
symmetry of
of the
the beamline
beamline in
in controlling
controlling the
the beam
beam size
size and
and
eliminating
net emittance
eliminating net
emittance growth,
growth, we
we additionally
additionally require
require that
that aa waist
waist be
be formed
formed at
at the
the
midpoint.
This
ensures
that
the
beta
functions
have
mirror
symmetry
and
therefore
midpoint. This ensures that the beta functions have mirror symmetry and therefore
return to their original values at the exit of the beamline. In summary, then, the
constraints imposed upon the system are that it
(i) be optically symmetric,
(ii) be nondispersive (r\ = rj = Oat the exit),
(Hi) have a waist at the midpoint (ax = ay = 0).
For the optical analysis we will represent the phase space coordinates of a test
particle in the beam distribution using the transport 6-vector notation
X = (jc,jc* ,j,y ,z,<5), where <5 = Ap/p 0 is the momentum dispersion and primes denote
derivatives with respect to the path-length s along the design trajectory. In order to
avoid confusion between conflicting sign conventions, we note that the longitudinal
coordinate z is positive for particles at the head of the bunch and negative for particles
at the tail of the bunch. The transport equation from the S-Bahn entrance (s = 0) to an
arbitrary point s along the design trajectory expanded to second order in powers of the
(presumably small) phase space coordinates then takes the form
Xt(s) = 0(0,5) + ^(09s)Xj(0) + ^(0,^)X7.(0)X,(0) + ...
(1)
where R is the 6 x 6 linear transport matrix, T is the 6 x 6 x 6 second-order transport
tensor, and Q is a 6-vector representing any offset of the beam centroid due to
misalignments or edge effects. In addition we define the 6 x 6 matrix £ of second
moments of the beam distribution f(X, s) as follows:
It is customary to make the following connections between the elements of this matrix
and the Twiss parameters (£, /?, a, 7) of the beam:
E33 =
2. First Order Optics
Since we are concerned with the values of the various system parameters primarily
at three points (the entrance s = 0, the midpoint s = s, and the exit s = As), we adopt
the following simplified notation:
R = R(0,s), R = fl(0,As), i = Z(0), t =
887
The linear analysis then proceeds straightforwardly, and is further simplified by the
symmetry condition (i) which effectively reduces the problem to that of the halflattice, containing only a compound bend (radius p), two quadrupoles, and three drifts.
The free parameters are then the quadrupole focal lengths and the initial Twiss
parameters of the beam (which can be adjusted using the optics on the main beamline).
However, with the assumption of a round beam at the entrance the remaining
conditions, which we may write as
(IM) E12 = E21 = E34 = E43 = 0
form constraints upon the system which effectively reduce the number of free
parameters from four to one. Equations (4) produce a set of simultaneous algebraic
equations for the values of the focal lengths f\ and /2 (or equivalently, the field
strengths) of the quadrupoles and the initial Twiss parameters oco and (3o, which are
connected to the midpoint sigma-matrix elements by the linear transport relation
t = RTZR.
(5)
For a given initial beam, Eqs. (4) provide the values of the quadrupole strengths which
optimize the lattice. However, obtaining real-valued solutions requires that the initial
Twiss parameters describe a convergent beam (a0 > 0) at the entrance of the S-Bahn.
This requirement is easily accommodated using the existing optics on the main
Neptune beamline. Taking the optimal values to be those which minimize the
transverse beam size at the midpoint, we obtain the optimized parameter values
a0 = 2.99, j80 = 2.10m, /x = -0.269m, and /2 = 0.536m.
3. Second Order Optics and Beam Ramping
For considerations of beam shaping, we are concerned with the longitudinal (or / = 5)
component of the transformation of Eq. (1) from the entrance s = 0 to the exit of the
final bend s = As which is given, to second order, by
z = z0+ R568 + r561*0<5 + T562 V 8 + T56682 .
(6)
Due to the small initial emittance (PARMELA simulations indicate a normalized
emittance of £N = 0.5 mm mrad at the entrance of the S-Bahn) we have neglected in
Eq. (6) the contributions from R5i, R52, and all nonvanishing second order elements for
which the second index 7 ^ 6 , since those terms are multiplied only by the initial
coordinates JCG, jc 0 ', yQ9 y0' which are small under this condition. By the same
argument we also expect that among the remaining terms in Eq. (6) those containing
Rs6 and Ts66 will dominate. This prediction is supported by simulations using
PARMELA and the matrix-based transport code ELEGANT.
When the horizontal dispersion vanishes in accordance with the second of Eqs. (1),
the full S-Bahn lattice has an R56 given by
np
dz
(7)
where p is the bend radius in the dipoles, yo is the central momentum and As is the
total path-length of the S-Bahn. For the Neptune values (p = 32.5cm, y0 = 24, As =
2.46m), we obtain R56 = -0.05m. The value of T566 can be calculated analytically by
perturbing the expression for R56 in Eq. (7) about the design energy and then taking
the derivative with respect to the momentum dispersion 8 while holding all hardware
parameters (i.e. field strengths) fixed. This procedure yields an approximate value of
T566 ~ dR56198 = -1.5m. The strong Ts66 contribution can however be eliminated by
the utilization of higher order optics on the beam line, namely the sextupole magnets
(S) shown in Fig. 2. With a symmetric placement of the sextupoles adjacent to the
outermost pair of quadrupole magnets (Ql), the quadratic dependence of longitudinal
position upon momentum dispersion introduced at the quadrupoles can be
counteracted in second order without altering the linear optics of the beamline. In Fig.
3 the values of the second order elements T 56 i, T562, and T566 are plotted against
sextupole field strength K2. We see that all three second order contributions cannot be
made to vanish simultaneously. However, it is observed that when the dominant
contributor (T566) is made to vanish the other two elements change sign and are
reduced in magnitude by a factor of two. Under these conditions, the longitudinal
transformation of Eq. (6) becomes nearly linear and is dominated by the negative Rs6.
This dominant negative Rs6 may be utilized to create the sort of compressed ramped
beam shown in Fig. 1, which was described as being ideal for generating strong wake
2
I
1.5
I
' T566
1
T 5 61
0.5
' T562
0
-0.5
-1
-1.5
200
400
600
K 2 /m 3
800
1000
1200
FIGURE 3. Plots of second order matrix elements as functions of sextupole field strength K2.
889
z(rrm
z(rrrn )
0.2
0.2
0.175
0.15
0.125
0.1
/L
' 0.075
0 . 05
0.025
0.175
0.15
0.175
0.15
0.125
0.1
0.075
0.05
' 0.075
0.05
0.125
0.1
0.025
- 1
0
z(rrm )
0.025
1
rnn)
1
~2
-1
^i ^)
z(rm
1
FIGURE 4. Phase space plots and density profiles showing: (a) a chirped uncompressed beam at the
entrance of the S-Bahn from a PARMELA simulation, (b) the same beam at the exit of the S-Bahn from
an ELEGANT simulation with sextupoles off, and (c) an ELEGANT simulation with sextupoles turned
on showing a ramped profile.
fields in a plasma. To see this, we note that with T566 =0and R56 = -0.05m, the
transformation of Eq. (6) corresponds to a counterclockwise rotation by an angle
0 = arctan(-^56/p) of the distribution in the longitudinal phase plane of z and 8.
Through the appropriate choice of injection phase, a momentum chirp may be put on
the beam such that the final phase space distribution after such a rotation produces a
compressed ramped longitudinal density profile. Figure 4(a) shows a plot of the
longitudinal phase space of a chirped beam at the entrance of the S-Bahn obtained
from a PARMELA simulation of the main beamline with 10,000 macroparticles. This
phase space was used as an input file for a series of simulations of the S-Bahn using
the matrix-based transport code ELEGANT. The results of these ELEGANT
simulations are shown graphically in Figs. 4(b) and 4(c), where we see the output
phase space at the end of the S-Bahn with sextupoles turned off and on respectively.
Below each phase space plot is a longitudinal density profile (in arbitrary units) plotted
against z. The S-shaped distribution in Fig. 4(b) is indicative of the presence of a
quadratic dependence on 8 due to the strong T566 contribution. The resulting density
profile shows good compression but no ramping. With the sextupoles turned on in
Fig. 4(c) we see the phase space of a pure negative Rs6 transformation. Due to the RF
curvature and the chosen injection phase of the initial beam in Fig. 4(a) the result of
this transformation is a "hook-shaped" distribution in phase space with a hard cutoff at
the tail of the beam. This produces a ramped, nearly triangular profile of 3ps duration.
CONCLUSIONS
Replicating experimentally the simulation results of Fig. 4 will require the
implementation of appropriate diagnostics on the S-Bahn beamline. To aid in
optimizing the quadrupole field strengths and killing the horizontal dispersion
890
function, phosphorescent YAG screens for imaging of the beam cross section will be
placed at the midpoint and adjacent to each bend magnet. A coherent transition
radiation (CTR) interferometer currently in use at the Neptune laboratory will be used
to perform measurements of the final beam length. Also, a preliminary measurement
of the beam's longitudinal profile can be obtained using a streak camera to measure the
emitted Cherenkov radiation produced at an aerogel. However, since streak cameras
have only ps resolution and the expected beam duration is 3ps, a successful
implementation of this scheme will require increasing the length of the beam by at
least a factor of two (e.g. by decompressing the laser pulse on the cathode). Also,
additional simulations will be required in order to assess the effect of space charge on
the optics of this beamline. In order to model both sextupole magnets and spacecharge simultaneously, the code for sextupole magnets must be incorporated into the
UCLA version of PARMELA. Judging from past simulations of the Neptune chicane
which showed that longitudinal self-forces tend to oppose the (forward of crest) chirp
needed in a positive Rs6 compressor, it is expected that for negative Rs6 systems the
collective longitudinal forces will actually add to the needed positive chirp, enhancing
the efficiency of compression.
One of the first beam-plasma experiments to take advantage of the S-Bahn's beam
shaping and compression capabilities will be a plasma density transition trapping
experiment conducted by M. Thompson [10]. This experiment is based upon a new
self-trapping scheme recently proposed by Suk, et. al. [11] for use in the blow-out
regime of the plasma wake field accelerator (PWFA) in the underdense case (n0 < nb).
In this scheme, a dense ultrashort beam is used to drive a large-amplitude plasma wake
field across a sharp downward transition in the background plasma density. Wave
breaking at the density transition results in phase mixing and trapping of background
plasma electrons in the wake. These trapped electrons can then be accelerated to high
energies by the strong electric field gradients which plasma wake-fields produce. The
optimum drive beam for plasma transition trapping is denser than the plasma in both
regions and has a radius and length comparable to a plasma skin depth k~l. At the
operating plasma density of 1013cm"3, a picosecond drive beam with a ramped profile
of the sort shown in Fig. 4(c), is ideal for driving the plasma wake-fields required for
the trapping experiment.
In addition, the use of sextupoles to linearize the transformation of the longitudinal
phase space is being considered for implementation at ATF for application to the
VISA experiment. On that beamline the positive Rs6 dogleg section relies upon a
strong nonlinear component (T566 = -10.5m) for longitudinal compression.
Preliminary simulations in ELEGANT indicate that by introducing a pair of sextupoles
and varying their field strengths, the value of T566 (and therefore the degree of
compression) in this system can be modified or reduced to zero.
ACKNOWLEDGEMENTS
The authors wish to thank H. Suk and C. E. Clayton for insightful discussions.
891
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