Wake field in dielectric acceleration structures
L. Scha
¨
chter,
1
R. L. Byer,
2
and R. H. Siemann
3
1
Department of Electrical Engineering, Technion-IIT, Haifa 32000, Israel
2
Department of Applied Physics, Stanford University, Stanford, California 94305-4085, USA
3
SLAC, Stanford University, Stanford, California 94305-4085, USA
Received 20 November 2002; revised manuscript received 5 March 2003; published 5 September 2003
In this study we present a general approach for the analysis of the wake field of a point charge moving in a
vacuum tunnel bored in dielectric material that is uniform in the direction parallel to the motion of the bunch.
In the transverse direction the structure surrounding the dielectric may have arbitrary geometry. A quasianalytic
expression that relates the decelerating force with the first dielectric layer, the radius of the vacuum tunnel
where the charge moves, and the reflection characteristics of the structure has been developed. Simulation
results for a simple structure indicate that, if the effective location where the reflection occurs in the dielectric
is sufficiently apart from the edge of the vacuum tunnel, it has no effect on the point charge. In fact, the
decelerating field converges exponentially as this distance increases, to the asymptotic value determined by the
first dielectric layer. An estimate of the trailing wake when the structure supports a specific mode is also
provided.
DOI: 10.1103/PhysRevE.68.036502 PACS numbers: 41.75.Lx, 41.20.Jb, 41.60.Bq
I. INTRODUCTION
One of the appealing paradigms for future particle accel-
erators relies on dielectric slow-wave structures confining a
laser field. Conceptually, this is quite similar to today’s linear
accelerators driven by microwave sources. Efforts are under
way 1 for a proof of principle at the level of the interaction
of electrons with a laser field in a single cell, but eventually
any practical accelerator will consist of a series of extended
slow-wave structures that need to satisfy several conditions.
Beyond slowing down the phase velocity to the speed of
light, it needs to ensure a maximum longitudinal electric
field at the location of the electrons for a given laser power,
minimize dissipation loss, and provide good heat transfer
characteristics. Moreover, in order to avoid breakdown it is
important to ensure minimum electric field at the vacuum
interface as well as in the dielectric, entailing a need for
some trade-off between the latter and the need for maximum
power imposed by the maximum gradient condition. At the
high intensities involved, the laser field may affect the di-
electric coefficient of the structure optical Kerr effect thus
altering the wave’s phase relative to the accelerated bunch.
Finally, when a bunch is injected into a dielectric accelera-
tion structure, its deceleration ought to be as small as pos-
sible. It is the wake field that is responsible for this deceler-
ating field, and it is its analysis that is the focus of the current
study.
Throughout the years extended studies of wake fields
have been conducted, many of which have been summarized
in reviews by Heifets and Kheifets 2 and Chao 3. How-
ever, the large majority of these studies address wakes in
azimuthally symmetric metallic disk-loaded structures or
metallic cavities used in microwave accelerators which have
only limited relevance to optical dielectric structures. In the
study that follows, a general approach is being developed for
estimation of the impact of a dielectric structure that is uni-
form in the direction parallel to the moving charge and has
virtually arbitrary characteristics in the transverse directions.
We present a quasianalytic expression that relates the decel-
erating force to the first dielectric layer of the structure, the
radius of the vacuum tunnel where the charge moves, and the
reflection characteristics of the structure.
II. PRIMARY FIELD
Whatever structure surrounds the vacuum tunnel where
the electron bunch propagates, it can be represented math-
ematically by a matrix that relates the outgoing waves with
the incoming ones. The top frame of Fig. 1 illustrates the
conceptual configuration of a vacuum tunnel surrounded by a
dielectric medium, and the reflecting structure is schemati-
cally represented by a ‘‘reflecting wall.’’ Two examples of
reflecting structures are illustrated at the bottom. The first is
a nonsymmetric structure bottom left frame consisting of an
array of vacuum cylinders surrounding the central one; this
structure is better known as a photonic band gap PBG
structure see Ref. 4. Another example is an azimuthally
symmetric Bragg structure bottom right frame consisting of
a series of concentric dielectric layers. Both structures and
other variants have been investigated in the context of optics
applications e.g., 5,6; therefore, we shall skip here the
analysis of propagation of homogeneous waves and limit the
discussion to the effect of nonhomogeneous waves linked to
the motion of a point charge in the central bore.
For this purpose we introduce a cylindrical coordinate
system whose z axis coincides with the axis of the central
vacuum tunnel. With this coordinate system it is possible to
attribute to a charge q moving at a constant velocity v par-
allel to the z axis and located in the transverse plane at r
=r
0
and =
0
a current density J
z
( r , z , , t ) =
-q v (1/r ) ( r -r
0
) ( -
0
) ( z -v t ). In the absence of
the dielectric structure this current density generates a pri-
mary field which satisfies the nonhomogeneous wave equa-
tion
PHYSICAL REVIEW E 68, 036502 2003
1063-651X/2003/683/03650210/$20.00 ©2003 The American Physical Society 68 036502-1