Wake Fields Excited in a Micron-Scale Dielectric
Rectangular Structure by a Train of
Femtosecond Bunches
T. C. Marshall1, J-M. Fang1, J. L. Hirshfield2'3, Changbiao Wang2,
V. P. Tarakanov4, and S. Y. Park5
Applied Physics Department, Columbia University, New York, New York 10027, USA
2
Physics Department, Yale University, New Haven, CT 06520 USA
f
Omega-P, Inc., New Haven, CT 06520 USA
4
ITES, RAS, Moscow, Russia
5
Postech, Korea
Abstract. We study the longitudinal wake field components which are induced in a rectangular,
dielectric-lined structure having micron-scale dimensions by the passage of one or more charge
bunches having femtosecond duration. The bunches would be obtained from a 500 MeV
LACARA "chopper" which uses a TW optical wave from a CO2 laser [1]; the bunches are
chopped from a macrobunch having duration ~1 psec obtained from a high brightness 500 MeV rf
linac. The high intensity laser wave accomplishes the chopping of the macrobunch into slices
which are roughly 10% of the 10.6 um radiation wavelength. These microbunches can be shaped
into a rectangular cross section, approximately 10 um x 150 um in dimension, and will excite
wake fields when injected into a rectangular dielectric wake field accelerating structure. We
compute sample 3D wake fields, using the PIC code KARAT, as well as by means of an analytic
method. The passage of just one pC bunch will set up a longitudinal wake field ~ 40 MeV/m, and
a train of ten properly-timed such bunches can produce a cumulative wake field ~ 600 MeV/m.
The choice of dimensions causes the wave solutions to approximate a single-mode excited by an
infinitely-tall bunch in a 2D structure; a highly uniform longitudinal wake field in the crosssectional plane of the structure results, suitable for accelerating a correctly positioned "test
bunch". KARAT includes the effect of interference between the Cerenkov radiation of the bunch
with the transition radiation emitted as the bunch enters the structure. The wake field structure is
several cm in length, and is both rigid and capable of microfabrication accuracy; it could
accordingly be a reproducible module in a staged array. The stability of the bunches and the
analytic formulation are dealt with in a companion paper [2].
INTRODUCTION
It is possible that narrow, femtosecond duration, sheet-like bunches can be created
and injected into an optical-scale dielectric-slab accelerator structure, which will allow
generation therein of a very strong longitudinal accelerating electric field (~ 1 GV/m)
[1]. In this paper we describe the wake field set up in a planar dielectric slab structure
energized by a train of high energy microbunches spaced by 10.6 jim. The bunches are
to be approximately 3.5 fsec in duration (-1 jim) and form a train of up to 30 bunches,
each containing -1 pC. This dielectric wake field accelerator structure is a vacuum
device that will pass a sheet beam having energy -500 MeV, approximately 10 jim x
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
361
150 jim in transverse dimensions. The slab dielectric which lines the parallel planar
walls of the structure would be a prepared microstructure a few microns in thickness.
The small transverse dimension of the device permits a buildup of very high
accelerating fields, limited only by the vacuum breakdown of dielectrics when exposed
to ultra-short (fsec) pulses of electromagnetic energy. A slab structure has the
advantage that it can pass more beam charge and has better stability to transverse beam
deflection than a cylindrical structure of comparable dimension [2, 3], particularly if
the ratio of the height to width is > 10. A schematic of a rectangular wake field
accelerator structure is shown in Fig. 1. The dielectric-lined waveguide supports a
longitudinal electric wake field induced by the passage of an electron bunch (the "drive
bunch"). Phase velocities for the modes of a dielectric-lined waveguide can be less
than the speed of light, so that Cerenkov radiation occurs, manifesting itself as a wake
field that reflects periodically from the conducting wall and fills the waveguide behind
the drive bunch. If a "test bunch" of lesser charge is injected at a suitable interval
following the drive bunch, it can move in synchronism with the wake field and
experience net acceleration. Thus no external radiation is required for acceleration.
In a study of a high energy accelerator based on these principles [1], femtosecond
bunches obtained from a SOOMeV rf linac followed by a LACARA chopper were used
to excite the dielectric structure with drive bunches and provide the energy for
accelerating test bunches. The LACARA system requires a TW-level CO2 laser, but
uses it only for chopping the bunch train, not for acceleration. A schematic of this
concept is shown in Fig. 2. A solenoidal magnetic field of 1.7 T is used to set up a
gyro-resonant interaction between the particles and the laser wave. The laser fields
cause the bunch of electrons to spiral around the longitudinal axis, so that the electrons
fall onto an annular pattern at the beamstop. The electrons intercepting the top half of
the beamstop in one laser period are shown in Fig. 3. Thus a small hole in the
beamstop (also shown) will transmit a pulse of charge having duration of a few fsec,
repeated every laser period (35.3 fsec). We estimate the quantity of charge in this
pulse to be approximately 1 pC, derived from a InC macrobunch; approximately thirty
microbunches are generated from each macrobunch. A microbunch transmitted
through the hole in the beamstop is distorted into a rectangular cross section shape ~35
cm downstream by a quadrupole (Fig. 4). This rectangular cross section profile is
maintained for several cm of axial travel [1] and determines the location of the
dielectric wake field structure. The longitudinal spreading of the fsec bunch due to
space charge or finite emittance is not expected to broaden the bunch for a distance of
at least 1m from the beamstop.
A high acceleration field (~1 GeV/m) can be built up in this dielectric structure by
superimposing the wake field radiation of several bunches; such a large field in a
dielectric structure is thought to be possible [4] because the dielectric is exposed to an
intense field for only ~ 0.3 nsec. It is interesting to comment that similar acceleration
gradients are expected [5] in a practical plasma acceleration scheme that could achieve
GeV energy, yet the method we describe will enjoy the higher efficiency typical of the
rf linac as compared with a laser power source.
362
FIGURE 1. Schematic of slab bunch within a planar optical wake field structure.
Gaussian envelope
X
quadrupole
m r
beam envelope
^_-I-----I n
typical trajectory
——— 5 m ———
superconducting magnet
beam
stop
wakefield
element
FIGURE 2. Schematic of a LACARA-type accelerator used as a chopper [1] for bunches obtained from
a 500 MeV rf linac. The magnetic field is 1.7 T, and it uses a 5 TW circularly polarized CO2 laser in a
Gaussian beam with 266.7 cm Rayleigh range. The superconducting magnet lies between z = 60 cm to
560 cm; the beam stop extends from z = 620 cm to 625 cm; the quadrupole is located between z = 630
cm to 635 cm.
0.03
0.02 _
transmitted
electrons
aperture
E
o
0.01
-0.015
0
0.015
x (cm)
FIGURE 3. Distribution of electrons hitting the LACARA beamstop (upper half plane shown). The
aperture will transmit a pulse of current.
0.002
at z = 666 cm
E
o
at z = 671 cm
-0.002
-0.01
-0.005
0
_L
0.005
0.01
x (cm)
FIGURE 4. Sheet-like transverse profile of a bunch ~35 cm downstream from the quadrupole. Two
views are shown at different locations (note different scales for the x and y axes). The dielectric wake
field accelerator module is to be located here. The quadrupole has a pole-face field of 1.05 T and a pole
radius of 1 cm.
Wake field structures are attractive because no external source of energy is used in
the structure itself. The breakdown limit of the wake field structure is not determined
by the slow filling time of the structure by electromagnetic energy, but rather by the
much shorter time of the passing field pulses set up by the short microbunches. The
proposed vacuum-based wake field structure is rigid geometrically and therefore
should be a more reproducible and controllable element in a staged system than would
be an array of pulsed plasma elements. Also, the structure is capable of
microfabrication accuracy, an important consideration when staging a large number of
modules. The use of rectangular cross section bunches and structures may result in a
tolerably-weak transverse wake field instability which would overcome one of the
chief obstacles to realization of a practical dielectric wake field accelerator [3]. A laser
wake field accelerator such as we have described may combine the high efficiency of a
microwave rf system (which produces the drive bunches) with the advantages of short
bunch operation (large wake fields) permitted by the high power laser. Use of several
holes in the beamstop will result in several beams of drive bunches, improving the
utilization of energy provided by the rf linac. Because the fields travel at the same
speed as the highly relativistic charges, slippage is not an important factor within the
spatial scale of the wake field structure.
We emphasize the difference between chopping and bunching. Chopping, as we
have it here, consists of chopping out a slice of current from the bunch, leaving zero
current on either side. The axial definition of the chopped bunch remains crisp for a
considerable distance of travel. Bunching, such as is done in an optical IFEL (e.g.
STELLA at ATF [6]), is a spatially-transient affair that occurs as a result of an energy
modulation imposed upon the particles. Spatial bunching forms at a certain
downstream position and is transient; thus it may be suitable for use in a single wake
field module, but perhaps not an extended interaction such as staged wake field
acceleration.
364
NUMERICAL SIMULATION USING KARAT
Because of the good beam quality obtained using a 500 MeV bunch [1], as well as
the fact that higher bunch energy is more appropriate for a series of wake field
accelerator modules (fewer modules required), the theory of the wake field acceleration
in a rectangular dielectric-lined module is carried out at 500 MeV energy. However,
insofar as wake field acceleration itself is concerned, the choice is based upon beam
quality. In Figure 1, we standardize the following parameters: H (height of structure
and bunch) = 150 jim; 2a = 15 jim; 2b = 18.8 jim; dielectric constant = 3.0; the bunch
width is 10 jim, length = 1 jim; bunch charge = 1 pC and energy = 500 MeV. The
parameters are such that a micron-scale wake field period will be set up, but not
necessarily at 10.6 jim (this can be obtained by readjusting the dielectric thickness).
KARAT is a versatile, relativistic, fully electromagnetic ID, 2D, or 3D code based
on the PIC method, developed by Vladimir Tarakanov [7]. Although the code permits
a full computation of all the fields set up by the passing bunch, we shall show only a
few examples involving the axial ( E z ) component. The first of these (Fig. 5) shows
the wake field trailing a single 3D rectangular bunch charge, showing a powerful
wakefield ~40 MV/m set up behind the traveling bunch; the period of this wake field is
about 20 jim. (The color scale indicates the magnitude and sign of the Ez field; the
wake field reverses sign [color] periodically.) The wake field period depends on the
dimensions of the structure and the dielectric constant of the material. The fields are
shown when the bunch has reached the right-hand side of the figure at z = 60 jim; the
field is displayed on the mid-plane, and the two dielectric slabs are indicated by the
boundaries at the top and bottom of the figure. Figure 6 shows the Ez wake field at z =
51 jim in the x-y plane; the field is relatively uniform in the cross section of the
structure. We have found that smaller channel width is more desirable in terms of
improved wake field amplitude and definition.
The wake field patterns obtained here do not have the sharp localization found in a
previous study [8]. The reason for this has to do with two conditions: first, the bunch
length chosen here results from chopping and therefore is ~10% of the laser period;
secondly, the structure contains a large channel bounded by thin slabs of dielectric. As
a result, only the lower spatial modes are excited and there is no dispersion effect to
complicate the timing of multiple bunches. Another advantage is that a test bunch
undergoing acceleration in the wake field of a drive bunch can have a length ~ 1 fsec
without experiencing appreciable longitudinal energy spread. The test bunch need not
have a rectangular cross section; indeed it may be desirable that it is centered in the
structure and have a width of the order of its height [9]. Stability issues will affect
bunch transport if the injected bunch is not centered accurately [2].
An overall view of the 3D wake fields is shown in Figure 7 for two timed bunches,
located at z = 60 and z = 40 jim. The wake field is displayed as a color pattern on 2D
x-y sheets at selected axial locations. Qualitatively, from this figure it can be seen that
the additional bunch causes approximately a doubling of the wake field intensity,
showing that superposition of the separate bunch fields occurs. As one might expect,
the wake field is rather uniform along the long (y) dimension, and appears to be of
365
satisfactory quality.
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366
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as
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left
60
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m
.
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as
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lines
on
left
60 m . Note the uniformity of the field. The dielectric boundaries are shown as vertical lines on left
and
right.
and
right.
and right.
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located at
at various
various z,z, showing
showing the
the wake
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in the
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from two
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FIGURE 7.
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FIGURE
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fields
in
the
plane
resulting
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are located
located at
at zzz == 60
60\im
mand
and zzz === 40
40\im
m... Note
Notethe
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increase
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m
and
40
m
Note
the
increase
bunches,
moving
to
the
right.
ingray
grayscale
scaleat
atthe
thelocation
location of
of the
the second
second bunch
bunch (~
(~ 40
40\im
m)) compared
compared with
with the
theleading
leadingbunch
bunch(first
(firstatat
atthe
the
in
m)
compared
with
the
leading
bunch
(first
the
in
gray
scale
at
the
location
right). Moving
Moving from
from right
right to
to left,
left, relative
relative to
to the
the first
first panel,
panel, the
the second
second wake
wake field
field isis
is in
in the
the opposite
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left,
right).
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in
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the fourth
fourth panel
panel has
has the
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same wake
wake field
field direction
direction as
as the
the first
first one,
one, and
and the
the sixth
sixth has
has the
the
direction;
the
fourth
panel
has
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and
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inthe
thesame
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directionas
asthe
thesecond
secondpanel.
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in
the
same
direction
the
second
wakefield
367
RESULTS FROM AN ANALYTIC TREATMENT
In addition to finding analytic solutions for the wake fields themselves, analytic
formulas have been derived for the forces on a test particle moving in the wake of other
particles [2]. These solutions are needed to explore questions of stability. The
mathematical development is presented in the companion paper [2], which examines
stability issues. In this paper, we quote a few computational results which may be
compared with KARAT.
Although it is appealing to describe fields in dielectric-lined rectangular
waveguides in terms of well-known LSE and LSM modes [12], their use for
calculation of wake fields has been found to be restricted [2]. This is because the LSE
and LSM families of modes each possess an axial electric field component Ez, and have
their symmetry axis normal to the dielectric surface. Moreover, both families are
excited by the motion of a charge bunch, so ten field components would then be
invoked in the solution, although only six of these are independent. Therefore, a
different basis has been used in the development of the new analytical theory, in which
each family of modes has all six independent components. In general, four such mode
families must be considered, corresponding to symmetric and anti-symmetric solutions
in both transverse coordinates x and y\ all four families can be excited by nonsymmetric beams, where stability issues in particular need to be examined. Orthonormalization constants have been found for all four families, following procedures
established when deriving wake fields in dielectric-lined cylindrical waveguides. The
theory provides the amplitudes of all the modes, and compact formulas have been
found for the longitudinal and transverse forces on test particles.
Some examples of the calculated longitudinal force Fz = -qEz on a test electron are
shown in Fig. 8. For these examples, one to ten 1-pC sheet bunches each with width,
height and length of 10, 150, and 1 jim respectively are sequenced periodically and
injected symmetrically into the planar dielectric structure. The bunch spacing is
chosen to be equal to the fundamental wake field period of 20 jim, so that cumulative
build-up of the field from successive bunches can occur. It is seen that the calculated
peak wake field amplitude of 40 MV/m from a single bunch is in good agreement with
the value found in the KARAT simulations. Uniform build-up of the maximum wake
field to nearly 600 MV/m from ten bunches is seen in Fig. 8c. Also to be noted is a
long-period modulation of the wake field, caused by interference with an adjacent
mode having slightly higher frequency and about 1/3 the amplitude of the dominant
mode. These results affirm the expectation that injection of a train of pC sheet bunches
into a micron-scale dielectric wake field structure can lead to accelerating fields that
approach 1 GV/m. (For example, a Ipsec macrobunch obtained from the linac can be
chopped into ~30 microbunches, and the lack of dispersion in the rectangular structure
will permit a nearly-linear superposition of the individual wake fields.) In the
companion paper [2], it is found that the transverse forces are modest, and would
permit an off-centered bunch to propagate without external guiding about 7cm before
hitting the dielectric wall. The instability is much weaker than for a bunch moving
along a cylindrical structure, for the same choice of narrow gap dimension.
368
100
200
300
400
500
600
700
800
z (microns)
FIGURE 8a. Axial force behind a single bunch, located at z = 800 (im and moving to the right.
100
200
300
400
500
iOO
700
800
z (microns)
FIGURE 8b. Same as Fig.Sa, but for two bunches spaced by 20 (im.
100
200
300
400
500
600
700
800
FIGURE 8c. Same as Fig. 8a,b, but for ten bunches spaced by 20 (im. Note expanded scale.
369
The wake field structure would be about 10 cm in length, and would be canted so
that a bunch will travel along its axis without grazing the walls. It can be pumped
along its open sides, and unlike the numerical case studied above, these would include
the top and bottom surfaces. The structure may be prepared by micro-fabrication. The
question of the dielectric material is still open: the dielectric constant we used is
compatible with a choice of NaCl or BaF2, both of which have very low losses in the IR
where the wake field modes are set up. Other materials such as silicon could prove
more versatile; indeed this would permit doping with impurities which might absorb
away any unwanted modes.
ACKNOWLEDGMENT
This work was supported by the DoE, High Energy Physics Division.
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