Abstract It is demonstrated that a planar Bragg reflection waveguide consisting of a series of dielectric layers may form an acceleration structure. It is shown that an interaction impedance per wavelength of over 100Ω is feasible with existing materials, Silica (ε = 2.1) and Zirconia (ε = 4), and if materials of high dielectric coefficient become available in the future, they may facilitate an interaction impedance per wavelength closer to 500Ω

A Mizrahi

L Schächter

SLAC, MS- R H Siemann

CiteSeerX

DIELECTRIC BRAGG ACCELERATOR
A. Mizrahi, L. Scḧachter, Electrical Engineering Department, Technion, Haifa 32000, Israel
R. L. Byer, Department of Applied Physics, Stanford University – MC4085, Stanford, CA 94305, USA
R. H. Siemann, SLAC, MS-7, Stanford University, P.O. Box 4349, Stanford, CA 94309, USA
Abstract
It is demonstrated that a planar Bragg reflection waveg-
uide consisting of a series of dielectric layers may form
an acceleration structure. It is shown that an interaction
impedance per wavelength of over100Ω is feasible with
existing materials, Silica (ε = 2.1) and Zirconia (ε =
4), and if materials of high dielectric coefficient become
available in the future, they may facilitate an interaction
impedance per wavelength closer to500Ω.
1 INTRODUCTION
Indications that solid-state lasers will reach wall-plug
to light efficiencies of 30% or more make a laser-driven
vacuum-accelerator increasingly appealing. Since at the
wavelength of relevant lasers, dielectrics may sustain a sig-
nificantly higher electric field and transmit power with re-
duced loss comparing to metals, the basic assumption is
that laser accelerator structures will be made of dielectrics.
Closed optical structures and near-field accelerators with
dimensions comparable to the wavelength are both being
considered. Examples of these two are: a) the LEAP [1]
crossed laser beam accelerator where the interaction be-
tween the crossed laser beams and the particles is limited
by slits to satisfy the Lawson-Woodward theorem [2, 3],
and b) thephotonic band-gap (PBG) concept where a laser
pulse is guided in a dielectric structure with a vacuum tun-
nel bored in its center [4]. Lithography, which would result
in planar structures, and optical fiber drawing are manu-
facturing techniques that seem well suited for laser driven
structures that have typical dimensions of a few microns.
Motivated by the low-loss Bragg dielectricplanar mir-
rors used in high-power lasers, it is suggested to harness
this concept in order to confine the laser-field in an opti-
cal acceleration structure. Its essence is to form a hollow
dielectric waveguide consisting of an almost perfect reflec-
tor made of a planar array of quarter-wavelength dielectric
layers. In the transverse direction the geometry is similar
to that of a dielectric mirror, however, its characteristics are
slightly different since the wave number has a significant
component (k = ω/c) parallel to the dielectric surfaces,
whereas in the case of a high-power laser mirror, the wave
impingesperpendicularly.
2 DESCRIPTION OF THE SYSTEM
Consider a planar symmetric dielectric waveguide
(∂/∂y = 0), as illustrated in Fig. 1, which has a vac-
uum inner layer of width2Dint and surrounding periodic
layers made of two lossless materialsεI, εII, the first layer
Figure 1: Planar dielectric waveguide.
having a relative dielectric coefficientεI. Each layer has
a thickness∆ν and a dielectric coefficientεν . Assuming
a steady-state regime (jωt) we focus on a mode having a
phase velocity equal to the speed of light, or explicitly in
the vacuum layer the electromagnetic field reads,
Ez = E0e−j
ω
c z
Ex = j
ω
c
xE0e
−j ωc z
Hy =
j
η0
ω
c
xE0e
−j ωc z
(1)
and the field components within some dielectric layerν a e
Ez =
[
Aνe
−jkνx + Bνe+jkνx
]
e−j
ω
c z
Ex =
−1√
εν − 1
[
Aνe
−jkνx − Bνe+jkνx
]
e−j
ω
c z
Hy =
−1
Zν
[
Aνe
−jkνx − Bνe+jkνx
]
e−j
ω
c z
(2)
wherein
Zν  η0
√
εν − 1/εν (3)
is the transverse wave impedance and
kν 
ω
c
√
εν − 1 (4)
is the transverse wavenumber. Imposing the continuity of
Ez andHy at the interfaces between the dielectric layers, a
plane-wave matrix formulation is obtained. Given the ge-
ometry and the dielectric coefficients of a waveguide that
0-7803-7739-9 ©2003 IEEE 728
Proceedings of the 2003 Particle Accelerator Conference
supports the mode (1), the amplitudes in the second layer
are dependent only on the vacuum fields. Imposing the
continuity of the fields at the second boundary, we get that
the amplitudes in the second layer depend only on the first
and so forth, so that the amplitudes are determined from
inside out.
3 FIELD CONFINEMENT
Analysis of the transverse propagation in thex direction
is performed similarly to [5]. Assuming an infinite pe-
riodic structure, Floquet theorem leads to an eigen-value
problem for the possible modes, and maximum attenua-
tion per unit cell is seen to be achieved for the case of
quarter-wavelength width layers. According to (4), each
layer width should beλ/(4
√
ε − 1), and an exponential
decay is obtained. The attenuation in one period is given
by the ratio of the lower to higher wave impedances of the
two dielectric layers. In case of a plane wave impinging
perpendicularly upon a planar interface, the impedance is
Z = η0/
√
ε, whereas if the wave has a phase velocity
in the z direction then the wave impedance is (3), which
clearly has a maximum forε = 2.
The above description of the confinement process, is of
an infinite and ideal Bragg reflector. In case of afinite
structure imposed by the vacuum tunnel, the design con-
straints on the first layer are different. The first layer, whose
amplitudes are completely determined byDint/λ andεI,
should be of such width that at the interface with the next
layer, the perfect reflection condition is met. It may be con-
ceived as a matching layer between the vacuum region to
the subsequent periodic structure, as it rotates the ampli-
tude vector dictated by the vacuum mode, to overlap the
eigen-vector of the periodic structure. Confinement entails
vanishingreal part of the transverse component of the com-
plex Poynting vector, meaning that in each dielectric layer
there is a standing wave. TakingE0 to be real without loss
of generality, we getAν = B∗ν . It is therefore evident that
for the structure to truly support the mode, there must be
an infinite number of layers, otherwise energy would ”es-
cape” and there would be no confinement. In a practical
structure, the number of layers should be sufficient so that
the outward power flow is negligible.
Fig. 2 illustrates a typical spatial distribution of the lon-
gitudinal electric field as well as the total electric field for a
structure made of Silica (εI = 2.1) and Zirconia (εII = 4),
and Dint = 0.3λ. It shows thatEz is uniform in the
vacuum layer while oscillating and decaying exponentially
in the Bragg layers. Another feature is thatEz vanishes
and achieves a maximum alternately at the discontinuities,
which can be proved to be mathematically equivalent to
the quarter- wavelength condition. Accordingly, the trans-
verse electric fieldEx, derived fromEz with respect tox, is
maximal and discontinuous wheneverEz is zero, and zero
wheneverEz peaks. The total electric field in turn under-
goes a discontinuity every second interface.
Since our goal is to keep the energy as confined as pos-
sible in the vacuum inner layer, i.e., maximal interaction
impedance, we may choose one of the dielectric materials
to haveε = 2, and the other dielectric as large as pos-
ible. This will indeed create maximum attenuation per
unit cell, however the interaction impedance depends on
the total flowing power outside the vacuum layer, which
tends to grow when low dielectric coefficient materials are
used and the layers become thicker. Consequently, there
exists a tradeoff between creating high contrast between
the two materials, and using low dielectric coefficient ma-
terials. For instance, choosing one dielectric material to
haveε = 2 and the the other as small as possible would re-
sult in high attenuation per period, but very low interaction
impedance.
0 1 2 3 4 5 6 7 8
−2
0
2
x/λ
E
z/
E
0
0 1 2 3 4 5 6 7 8
−2
0
2
x/λ
|E
to
ta
l/E
0|
Figure 2: A typical distribution of the longitudinal electric
field and the total electric field(Dint = 0.3λ, εI = 2.1,
εII = 4).
4 ACCELERATOR PARAMETERS
4.1 Interaction Impedance
The interaction impedance is a measure of the accel-
erating gradient experienced by the electrons for a given
amount of power injected into the system. Denoting by
P the flowing power per unit length ofy, the interac-
tion impedance per unit length is defined byZint[Ωm] 
|λE0|2 /P . Assuming that the materials’ characteristics are
known (εI = 2.1, εII = 4) and so is the laser wavelength
(λ), the only free parameter left is the width of the internal
vacuum layer2Dint. Based on simulations, it was found
that for0.3 ≤ Dint/λ ≤ 0.8, the best fit for the interaction
impedance is given by
Zint
η0λ
(εI = 2.1, εII = 4)  1.124 − 3.561Dint
λ
+ 4.258
(
Dint
λ
)2
− 1.823
(
Dint
λ
)3
(5)
729
Proceedings of the 2003 Particle Accelerator Conference
which meansZint/λ decreases monotonically from about
147Ω to about26Ω.
Holding the vacuum region width to a constant value
(Dint = 0.3λ), Fig. 3 shows the contours of constant
interaction impedance as a function of (εI, εII). Since
no confinement is expected when the medium is uniform
(εI = εII), the interaction impedance is virtually zero on
the diagonal. Similarly, when either one of the dielec-
tric coefficients is close to unity, the thickness of the layer
being proportional to1/
√
ε − 1 implies large confinement
space and therefore, low interaction impedance. In between
these three minima regions there are twoasymmetric re-
gions of maximum interaction impedance. The asymme-
try is dictated by the choice of the dielectric consisting the
first layer (εI). For the vacuum layer width chosen here,
a larger impedance is obtained when the first layer is of
lower value, but this trend may change for a different value
of Dint. Clearly, ahigh dielectric coefficient can signif-
icantly increase the interaction impedance. For example,
taking Silica for the first layer and a material withε = 25
for the second layer, leads to an interaction impedance per
wavelength of471Ω.
0
100
200
300
400
500
600
700
1 5 10 15 20 25 30
1
5 
10
15
20
25
30
εI
εII
Figure 3: Contours of constant interaction impedance
Zint/λ[Ω] as a function of (εI, εII), with Dint = 0.3λ.
4.2 Energy Velocity
Denoting the energy density bywEN, we define the av-
erage energy per unit length asW 
∫ ∞
−∞ dx wEN(x) and
the energy velocity is defined byvEN/c  P/(cW ). Ac-
cording to our simulations, for0.3 ≤ Dint/λ ≤ 0.8, the
energy velocity increases monotonically from about0.42c
to about0.53c. As before, the best fit of the simulation
results is given by
vEN
c
(εI = 2.1, εII = 4)  0.342 + 0.290
(
Dint
λ
)
− 0.061
(
Dint
λ
)2
(6)
4.3 Maximum Electric Field
The last parameter of interest is the maximum electric
field sustained by the structure before breakdown. Ac-
cording to (1) the magnitude of the electric field vector in-
creases from a local minimum on axis, to a larger value
Emax
E0
=
√
1 +
(
2πDint
λ
)2
(7)
at the vacuum-dielectric discontinuity; for most practical
purposes this may also be considered the maximum electric
field – fact revealed also by the bottom frame of Fig. 2. For
avoiding breakdown it is assumed that the fluence threshold
of the material limits the maximum field to about 2GV/m.
Therefore, bearing in mind that the gradient of interest is
of the order 1GV/m, then Eq. (7) entails the inner layer
half-width should be
Emax
E0
= 2 ⇒ Dint  0.28λ (8)
5 CONCLUSION
In the present study we have designed and analyzed
an accelerator based on a Bragg reflection waveguide,
where the layers have a width of a quarter of the trans-
verse propagation wavelengthλ/(4
√
ε − 1). An interac-
tion impedance per wavelength of over100Ω is feasible
with existing materials. Materials of high dielectric coef-
ficient can significantly improve the interaction impedance
per wavelength to hundreds of Ohms.
The planar structure provides a good analogy, especially
asymptotically, to a hollowcylindrical Bragg fiber acceler-
ator. A cylindrical structure would have a higher interac-
tion impedance than a planar structure for an internal ra-
dius Rint = Dint for two reasons. For a given gradient
the maximum field would be smaller as the radial electric
field Er = j2
ω
c rE0e
−j ωc z would be smaller by a factor of
2. The second reason is that in addition to the radial expo-
nential decay, in the cylindrical case there would be a1/r
decay due to the cylindrical wave functions.
6 REFERENCES
[1] Y. C. Huang, D. Zheng, W. M. Tulloch, and R. L. Byer, “Pro-
posed structure for a crossed-laser beam, GeV per meter gra-
dient, vacuum electron linear accelerator,”Applied Physics
Letters, vol. 68, pp. 753–755, February 1996.
[2] J. D. Lawson, “Lasers and accelerators,”IEEE Transactions
on Nuclear Science, vol. 26, pp. 4217–19, June 1979.
[3] P. M. Woodward,J. IEE, vol. 93, p. 1554, 1947.
[4] X. E. Lin, “Photonic band gap fiber accelerator,”Physical
Review Special Topics – Accelerators and Beams, vol. 4,
pp. 051301 1–7, May 2001.
[5] P. Yeh and A. Yariv, “Bragg reflection waveguides,”Optics
Communications, vol. 19, pp. 427–430, December 1976.
730
Proceedings of the 2003 Particle Accelerator Conference


DIELECTRIC BRAGG ACCELERATOR

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=3C545AE93958869352046B5129A179B1?doi=10.1.1.1076.1115&rep=rep1&type=pdf

DIELECTRIC BRAGG ACCELERATOR

Abstract

Similar works

Full text

Available Versions

CiteSeerX