Focusing of Intense Laser Pulses Using Plasma
Channels
Richard F. Hubbard,1 Bahman Hafizi,2 Antonio Ting,1 Daniel F. Gordon,2
Theodore G. Jones,1 Dmitri Kaganovich,3 Joseph R. Penano,1 Phillip
Sprangle,1 and Arie Zigler2'4
Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375-5346
2
Icarus Research, Inc., P.O. Box 30780, Bethesda, MD 20824
3
LET Corp., 4431 MacArthur Boulevard., Washington, DC 20007
4
Hebrew University, Jerusalem
Abstract. Short plasma channels may provide focusing or control of laser pulses at intensities
far above the usual damage limitations of conventional optics. Analytical and simulation models
that predict the behavior of a variety of channel lens configurations are presented. A laser
wakefield accelerator (LWFA) that uses a series of plasma lenses to transport the laser pulse
over extended distances is described.
INTRODUCTION
Plasma channels have been used to guide intense laser pulses over distances of
many Rayleigh lengths [1-6]. The guiding is the result of the radial variation in the
plasma refractive index. Plasma channels are an essential component of most designs
for a future laser wakefield accelerator (LWFA) [7-9]. In these designs, laser and
plasma parameters are usually chosen so that the laser pulse propagates at its
equilibrium spot size rM. Thus, the guiding and acceleration processes are closely
coupled.
Any application of intense, short pulse lasers requires focusing and shaping of the
pulse with conventional optical elements such as lenses, mirrors, and gratings. These
optical elements must be placed at locations where the laser spot size rL is large to
avoid damage to the optics. We have recently proposed that focusing or defocusing of
laser pulses at much higher intensities should be possible using short plasma channels
[10]. Such a lens would generally be placed close to the focus where the spot size is
100 jim or less, and the intensity is well above the usual damage limits for
conventional optics. This paper reviews the plasma channel lens concept and
emphasizes potential applications to laser-driven acceleration.
The analysis presented here is based on envelope equation solutions derived from
the source dependent expansion method [11] and simulations using the LEM code
[12]. LEM is a versatile laser propagation code that employs the widely-used quasistatic and quasi-paraxial approximations and calculates the laser fields and plasma
response in a frame moving with the pulse at the speed of light. The plasma is treated
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
664
as a cold relativistic fluid. The primary quantity of interest for a lens simulation is the
spot size TL(Z), which is obtained from a Gaussian fit to the laser amplitude at an
appropriate reference point in the center of the pulse. This quantity reaches its
minimum or focused spot size r/ at a focal distance z/. Analytical calculations for
these quantities as functions of laser and plasma parameters agree well with LEM
simulations [10].
The plasma channel lens is one of several plasma-based pulse control methods that
have been analyzed. The others involve nonlinear effects that arise from self-phase
modulation or relativistic self-focusing (RSF) at high laser intensities and are
described in Ref. 13. RSF is the basis of a plasma lens proposed by Ren, et al. [14]
that shares many of the features of the plasma channel lens.
SIMPLE PLASMA CHANNEL LENSES
This Section describes the behavior of an ideal plasma channel lens in the limit of a
nonconverging beam and compares analytical focusing models with LEM simulations.
Comparison of Plasma Channel Lens with Long Guiding Channel
We consider first the behavior of an ideal Gaussian laser pulse propagating in a
long parabolic plasma density channel of the form ne(r) = no(l + (r/rcl)2). Envelope
equation models predict that the spot size rs will oscillate about its equilibrium or
'matched' value rM, given by
rM=(rc2l/m-en0)l/4.
(1)
where re is the classical electron radius, and rci is the nominal channel radius where
the plasma density is twice its on-axis value. A nonconverging pulse entering the
channel with an initial spot size TQ that is larger than rM will reach its minimum spot
size r/c at the focal location z/c = 7frM2/2^. If the length A of the channel is less than z/,
the plasma will behave as a thick lens, with a focal length z/ and focused radius r/ that
are somewhat larger than the long channel values z/c and r/c.
This behavior is illustrated in Fig. 1, which plots r^(z) from the LEM simulation
model for a laser pulse with TQ = 35 jim, /I = 1 jim, and PO = 1.5 TW, propagating into
a channel with no = 4xl018 cm"3 and rci = 57.7 jim. The dotted curve is for
propagation in a long plasma channel, while the solid curve is from a LEM simulation
for a plasma channel lens with on-axis density equal to HO and nominal thickness A =
0.0375 cm. The analytical matched radius rM = 17.5 jim, corresponding to a nominal
focal length z/c = 0.151 cm. The long-channel simulation gives z/c = 0.152 cm, and
reaches a minimum spot size r/c of 7.48 jim.
As expected, the plasma lens simulation (solid curve) gives a slightly larger focal
length (z/ = 0.195 cm) and focused spot size (r/ = 18.8 jim) than the long channel
simulation. The peak laser intensity on the front side of the plasma lens is SxlO1
W/cm2, which is orders of magnitude above the damage limit for conventional optics.
665
Analytical Models for Plasma Channel Lenses
Reference 10 describes an analytical model that calculates r/ and z/ as functions of
the laser and plasma parameters. The model is an application of the Hafizi, et al.
generalized ponderomotive channeling model [15]. For a nonconverging (collimated)
beam with drjdz = 0 at z = 0, the envelope equation inside the lens may be integrated
to give
2r 2
cos
= 1-
2A(z-z 0 )
(2)
where /lp = 2nc/a>p is the plasma wavelength based on the on-axis plasma frequency
(Op, Pr is the critical power for relativistic focusing [14-17], and N = (4P/7tPr)2(hp/rC2)2.
If the plasma density is assumed to be zero for z - ZQ > A, then the focal spot size and
focal length may be obtained by matching rs and drjdz at the end of the lens with the
well-known vacuum solution expressions. This procedure is described in the
Appendix of [10].
60
6O
40
\ n0 (lens)
0.0
0.2
0.4
0.6
z (cm)
FIGURE 1. Laser spot size rL(z) from a LEM simulations in a long plasma channel (dotted line) and a
short plasma channel lens (solid line). The dashed line gives the on-axis plasma density for the lens.
Laser and plasma parameters are given in the text.
It is convenient to characterize the laser and plasma parameters by a normalized
injection spot size po = (n/^prci)ll2r0 and normalized thickness 8 = A/Zr0 = (h/7zr02)A.
If PIPr « 1, and A is sufficiently small so that the cosine term in Eq. (2) can be
expanded, the focal length and spot size can be expressed as [10]
666
(3)
and
l-(p 0 4 -l)<5 2
1/2
(4)
l + (p 0 4 -l) 2 5 2
These simple expressions are remarkably accurate even if the thin lens
approximation is breaking down or PIPr approaches unity.
Comparison of Analytical Models with Simulation
The solid line in the left frame of Fig. 2 plots r/ as a function of the lens thickness A
using Eq. (4). The other laser parameters (ro = 35 jim, A = 1 jim, and PQ = 1.5 TW)
and plasma parameters (no = 4xl018 cm"3 and rcl = 57.7 jim.) are the same as in Fig. 1.
The normalized injection spot size po is 2.00, and ZRQ = 0.385 cm in all cases. The
focused spot size r/ always decreases with increasing A. The corresponding focusing
distance z/ from Eq. (3) is shown in the right frame. The dashed lines in the two
figures are calculated from the thick lens model described in the Appendix of Ref. 10,
and the asterisks are LEM simulation results
0.25
40
0.20
30
8o °-15
20
7 0.10
10
0.05
0.00
0.00 0.02 0.04 0.06 0.08 0.10
A (cm)
0.00 0.02 0.04 0.06 0.08 0.10
A (cm)
FIGURE 2. Laser focusing model results for (a) focal spot size r/ and (b) focal length z/, as functions
of the lens thickness A. Other laser and plasma parameters are the same as in Fig. 1. The solid lines are
from the thin lens model, the dashed lines are the more accurate thick lens model, and the asterisks are
from LEM simulations.
The agreement between the simulations and the thin lens analytical model for both
r/ and z/ is very good for A < 0.04 cm but becomes progressively worse as the lens
thickness is increased. This is primarily due to a breakdown in the thin lens
approximation; the cosine argument 2hA/hprci in Eq. (2) actually exceeds unity for A >
0.05 cm. However, the agreement between the thick lens analytical model (dashed
line) and the simulations is excellent. The thickest lens simulation, with A = 0.0625
667
cm,
cm, has
has r/rf == 13.1
13.1 jim,
µm, corresponding
corresponding to
to aa relatively
relatively modest
modest spot
spot size
size reduction
reduction of
of aa
factor
factorof
of2.6
2.6from
fromthe
the35
35 jim
µm value
value at
at injection.
injection.
ALTERNATIVE
ALTERNATIVE PLASMA
PLASMA CHANNEL
CHANNEL LENS
LENS CONFIGURATIONS
CONFIGURATIONS
The
The simple
simple case
case of
of aa single
single lens
lens and
and aa collimated
collimated or
or nonconverging
nonconverging beam
beam isis of
of
limited
limitedpractical
practical interest
interest because
because intense
intense laser
laser pulses
pulses are
are already
already routinely
routinely focused
focused to
to
much
much smaller
smaller spot
spot sizes
sizes that
that the
the 35
35 jim
µm value
value used
used in
in the
the previous
previous section.
section. In
In this
this
Section,
Section, we
we discuss
discuss several
several alternative
alternative configurations.
configurations. Figure
Figure 33 shows
shows four
four such
such
configurations.
configurations.
(a)
(a)
(b)
(c)
| Converging
Converging pulse
pulse
focusing
lens
focusing lens
(d)
Converging
Converging pulse
pulse
defocusing
defocusing lens
lens
Multi-lens
Overmoded
Overmoded lens
lens
FIGURE
FIGURE3.3. Examples
Examplesof
ofother
otherplasma
plasma channel
channel lens
lens configurations.
configurations. These
These include
include (a)
(a) aa focusing
focusing lens
lens
with
with aa converging
convergingpulse,
pulse, (b)
(b) aa defocusing
defocusing lens
lens with
with aa converging
converging pulse,
pulse, (c)
(c) aa multiple
multiple lens
lens transport
transport
system,
system,and
and(d)
(d)aathick,
thick,overmoded
overmodedlens.
lens.
Focusing
Focusing and
and Defocusing
Defocusing of
of aa Converging
Converging Optical
Optical Pulse
Pulse
AAconverging
convergingor
orprefocused
prefocused pulse
pulse has
has drjdz
drL/dz << 00 and
and an
an injection
injection spot
spot size
size TQr0 at
at zz ==
0.0. For
For an
an ideal
ideal Gaussian
Gaussian pulse
pulse propagating
propagating in
in vacuum,
vacuum, the
the pulse
pulse would
would focus
focus to
to the
the
diffraction-limited
diffraction-limited spot
spot size
size rjo
rf0 atat the
the focal
focal distance
distance z/0.
zf0. IfIf aa plasma
plasma channel
channel focusing
focusing
lens
lensisisplaced
placedin
inthe
thepath
pathof
ofthe
theconverging
convergingpulse,
pulse, both
both r/rf and
and z/zf will
will be
be reduced.
reduced. This
This
will
will increase
increase the
the intensity
intensity and
and fluence
fluence of
of the
the laser
laser pulse
pulse at
at the
the focus.
focus. The
The plasma
plasma
channel
channel thus
thus enhances
enhances the
the focusing
focusing of
of an
an optical
optical beam
beam in
in aa manner
manner similar
similar to
to that
that
exhibited
exhibited by
by plasma
plasma lenses
lenses for
for electron
electron beam
beam [18].
[18]. Simulations
Simulations that
that exhibit
exhibit this
this
enhanced
enhanced focusing
focusing are
are reported
reported in
in Refs
Refs 10
10 and
and 15.
15. Because
Because the
the focusing
focusing effect
effect is
is
668
approximately linear, the increase in intensity compared with the vacuum limit scales
roughly as (r/o/r/)2. Reductions in spot size of more than a factor of 3 have been
produced, with an order of magnitude increase in focused intensity [15].
Reference 15 also contains a generalization of the analytical model to the
converging beam case where drLldz < 0 at the front of the lens. Again, the agreement
between the model and LEM simulations is excellent.
The most likely application of this configuration is for laser target experiments
where maximum intensity or fluence is desired. This is usually not the case for laser
accelerator applications since extremely small spot size is usually not necessary. In
addition to a straightforward enhancement of fluence on target, a plasma channel
offers the option of using a higher f-number focusing element and placing it farther
from the target. Also, the plasma channel lens is tunable, making it possible to move
the location of the focus without physically moving any optical element.
A converging pulse with a defocusing plasma channel lens is also shown in Fig. 3.
A plasma column with an on-axis density maximum will defocus an optical beam and
thus can act as a diverging or negative lens. If such a short "inverse" plasma channel
is placed near the vacuum focal point of an optical beam, both r/ and z/ will increase.
Simulations that exhibit this behavior were reported in Ref. 10.
Negative or inverse plasma channel lenses offer additional flexibility in designing
intense laser pulse systems. The most obvious applications involve situations for
which a larger spot size is desired than that produced by the vacuum optical system,
and higher f-number optics are unavailable or impractical because of space
constraints. An example would be matched injection of a laser pulse into a long
plasma channel for laser wakefield accelerator or x-ray laser applications. The
matched spot size in such channels is typically 20-50 jim.
A LWFA with Period Focusing Lenses
Figure 3 also shows an example of a multi-lens transport system with a series of
weak, thin plasma channel lenses. This configuration has also been described
previously [10]. If a collimated pulse enters a single thin lens of thickness A at z = 0,
the laser pulse will first focus and then expand to its injected spot size TQ at z ~ 2z/. A
second lens with identical plasma parameters and thickness 2A, with its center located
near 2z/will slow the laser pulse expansion rate to near zero near the center of the lens,
and refocus the beam to a similar focal length and spot size as the first lens. In
principle, the laser pulse may be transported over arbitrarily long distances using this
approach.
The use of a series of thin lenses in a periodic focusing system offers the
possibility of decoupling the guiding requirements from the conditions in the
interaction. This could be particularly useful for a LWFA because of the difficulties in
producing a single long, low-density plasma channel. For a 100 fsec (FWHM) laser
pulse, the desired resonant density is 2-3xl017 cm"3, and the desired channel length
exceeds 10 cm.
Figure 4 shows the evolution of the spot size rL(z) and on-axis density HO(Z) from a
LEM simulation of a LWFA with period focusing. The laser has YQ = 40 jim, PQ = 26
TW, /I = 0.8 jim, and TL = 100 fsec. The background plasma has a uniform density of
669
2.5xl017 cm"3. The higher density channel lenses have no = 1018, rci = 40 |im, a
downstream thickness of 0.06 cm, and a spacing of 0.6 cm. The filling factor for the
lenses is therefore 10%. The spot size exhibits the expected refocusing effects at each
downstream lens, and the variation of spot size between the lenses is modest. No
attempt was made to fine tune the transport; with modest changes in lens parameters
or location, the variations in the minimum and maximum spot sizes could have been
reduced.
50
40
30
20
10
0
0.0
I"! n0 (xlO 17 )
0.5
1.0
z (cm)
1.5
2.0
FIGURE 4. Evolution of the laser spot size rL (solid line) and on-axis plasma density n0 (dashed line)
in a period-focused LWFA. The plasma is assumed to have a uniform density in the region between the
lenses. Parameters are given in the text.
Figure 5 plots the axial (Ez) and radial (Er) electric fields near the axis versus
position £ = z -ct within the pulse. The pulse location is at z = 1.5 cm, which lies in
the uniform, low plasma density region after the third lens. The original head of the
laser pulse is at £ = 0. Both the axial and radial fields are well-behaved and regular.
Maximum acceleration occurs at £ = -70 jim, where Ez = -12 GV/m. If additional
lenses were added to extend the transport to the full dephasing length (-20 cm), the
LWFA performance would approach that of a similar LWFA simulation in a long
axially-uniform channel that we reported previously [8]. That simulation produced
energy gain that exceeded 1 GeV.
Thick Plasma Channel Lenses
An example of a thick lens configuration is the overmoded channel lens shown in
Fig. 3. In an overmoded lens, the laser spot size rL(z) undergoes one or more complete
oscillation before the pulse exits the lens. For an ideal, nonconverging pulse in a
parabolic channel, the period of spot size oscillations in the channel in the channel is
he = rfrulk. Thus, a lens with thickness AN = Nhe + AQ should have similar focusing
properties to one with thickness AQ. Overmoded lenses have been examined in [10].
Making short plasma channels with capillary discharges has proven difficult in
670
practice, so operating in the overmoded regime may be an attractive alternative. An
analysis in Ref. 10 of a previously exported capillary discharge guiding experiment by
Ehrlich, et al. [5] suggests that overmoded plasma channel lens focusing has already
been demonstrated experimentally.
CT (cm) =
1.504
15
10
?
\
>
5
2-JH
o
^
—5
-10
-15
-140
-120 -100 -80
-60
-40
-20
FIGURE 5. Electric fields Ez (solid line) and Er (dashed line) near the axis versus f = z -c£ at a location
(z = 1.5 cm) beyond the third lens. Results are from the periodic-focused LWFA simulation shown in
Fig. 4.
Another possible application of thick plasma channel lenses is to use them as
spatial filters. The radial profile of the plasma density ne(r) normally peaks at some
location Rch and falls off beyond that point. Portions of the laser pulse that lie inside
RCh will be guided by the channel, while those on the outside will be expelled or
clipped. The channel would thus function in much the same way as a conventional
spatial filter, except that the pulse intensity could be much higher for the plasma
spatial filter.
Another potential configuration would be to bend a thick plasma lens into an arc,
thus bending the optical pulse. This has been demonstrated experimentally using
longer plasma channels. The ability to introduce a slight bend in the optical pulse
could be useful for a staged LWFA that has multiple pulsed drive lasers.
CHANNEL CREATION TECHNIQUES
The same techniques used to create long plasma channel may in principle be
applied to short plasma channels. The general mechanism involved in most channel
creation techniques requires the creation of a hotter plasma near the channel axis. The
plasma then undergoes hydrodynamic expansion and reaches a pressure balance where
the density is reduced near the channel axis. For laser-generated channels, the heating
is supplied by a separate laser pulse. It may not be necessary to employ a line focus or
axicon lens since the energy will need to be deposited out to a large radius, and the
671
Rayleigh length associated with the heating laser should be much smaller than the lens
thickness. A gas jet would presumably be used to determine the lens thickness.
A similar process is involved in capillary discharge channels except that ohmic
heating of electrons by the discharge current provides the primary heating mechanism.
In the typical plasma channel experiment, the channel length is two orders of
magnitude larger than the channel radius or laser spot size, so radial plasma
hydrodynamics dominates except near the entrance and exit of the channel. This
aspect ratio is much smaller for thin plasma lenses, so the hydrodynamic behavior may
be different, and edge effects are more important. In addition, there are problems with
producing short discharge channels since the electrodes will be so close together.
Another possible technique is to use a laser-ablated capillary in place of the
discharge. Laser guiding experiments on this technique with longer channels have
been carried out at NRL [19] but have not demonstrated effective guiding. It may
actually be easier to employ this approach for lensing since the short capillary length
is probably an advantage.
Another possible approach is to tailor the neutral density rather than relying on
localized heating to produce the desired plasma density profile. This could involve
complicated gas jet nozzles or spinning capillaries. These options have not yet been
seriously examined.
Plasma channel lenses have a number of potentially attractive features in addition
their tolerance of high laser intensities. The lenses can in principle be tuned to
different densities, thus altering their focusing properties without moving the lens.
There is the potential for very good beam quality due to the "linear" focusing
properties of an ideal plasma lens. Finally, this is a versatile technology that could add
significant flexibility to future high power optical systems.
However, there are a number of key issues that must be resolved. There is
considerable complexity involved in producing appropriate plasma channels. A
practical focusing system must be highly reproducible and robust. Aberrations from
relativistic effects may significantly degrade performance at high powers. The
sensitivity to nonideal laser and channel profiles has not been examined yet. Finally,
incomplete preionization could degrade performance, especially if the channel
contains partially stripped high-Z material.
SUMMARY
A short plasma channel will focus a laser pulse in a manner similar to a
conventional solid lens, provided the laser spot size at the entrance to the channel
exceeds the equilibrium spot size in the plasma channel. However, a plasma channel
lens will be able to tolerate laser intensities far above the usual damage limits for
conventional optics. This offers the possibility of manipulating intense laser pulses
when the spot size is tens of microns and placing the focusing or defocusing optics
much closer to the target or interaction region.
The focusing properties of the plasma channel lens have been analyzed for the ideal
case of a fundamental Gaussian laser pulse and a parabolic plasma density profile.
Analytical calculations of the focal length and focused spot size agree well with LEM
672
simulation results, particular when a more accurate thick lens analytical model is used.
A variety of channel lens configurations have been examined, including a laser
wakefield accelerator that uses a series of plasma lenses to transport the laser pulse
over extended distances. Techniques used to generate long plasma channels for
continuous optical guiding can probably be adapted to produce shorter channel lenses.
ACKNOWLEDGMENTS
Conversations with C. Moore, T. M. Antonsen, Jr., and J. Grun are gratefully
acknowledged. This work was supported by the Department of Energy under
interagency agreement DE-AI02-93ER40797 and by the Office of Naval Research.
REFERENCES
1.
C.G. Durfee and H.M. Milchberg, Phys. Rev. E 71, 2142 (1993); C.G. Durfee, J. Lynch and H.M.
Milchberg, Phys. Rev. E 51, 2368 (1995).
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
Y. Ehrlich, C. Cohen, A. Zigler, J. Krall, P. Sprangle and E. Esarey, Phys. Rev. Lett. 77, 4186
(1996).
E. W. Gaul, S. P. Le Blanc, A. R. Rundquist, R. Zgadzaj, H. Langhoff and M. C. Downer, Appl.
Phys. Lett. 77, 4112 (2000).
P. Volfbeyn, E. Esarey, and W. P. Leemans, Phys. Plasmas 6, 2269 (1999).
Y. Ehrlich, C. Cohen, D. Kaganovich, A. Zigler, R. F. Hubbard, P. Sprangle, and E. Esarey, J. Opt.
Soc. Am. B 15, 2416 (1998).
D. Kaganovich, A. Zigler, R. F. Hubbard, P. Sprangle, and A. Ting, Appl. Phys. Lett. 78, 3175
(2001).
W. P. Leemans, C. W. Siders, E. Esarey, N. E. Andreev, G. Shvets, and W. B. Mori, IEEE Trans.
Plasma Sci 24, 331(1996).
R. F. Hubbard, D. Kaganovich, B. Hafizi, C. I. Moore, P. Sprangle, A. Ting and A. Zigler, Phys.
Rev. E 63, 036502 (2001).
P. Sprangle, B. Hafizi, J. R. Penano, R. F. Hubbard, A. Ting, C. I. Moore, D. F. Gordon, A. Zigler,
D. Kaganovich and T. M. Antonsen, Phys. Rev. E 63, 056405 (2001).
R. F. Hubbard, B. Hafizi, A. Ting, D. Kaganovich, P. Sprangle, and A. Zigler, Phys. Plasmas 9,
1431 (2002).
P. Sprangle, A. Ting and C.M. Tang, Phys. Rev. Lett. 59, 202 (1987).
J. Krall, A. Ting, E. Esarey and P. Sprangle, Phys. Rev. E 48, 2157 (1993).
R. F. Hubbard, P. Sprangle, T. G. Jones, C.I. Moore, A. Ting, B. Hafizi, D. Kaganovich, J. R.
Penano, D. Gordon, T. M. Antonsen, A. Zigler, and P. Mora, in Proceedings of the 2001 Particle
Accelerator Conference, edited by P. Lucas, IEEE, New York, 2001, pp. 3990-3992.
C. Ren, B. J. Duda, R. G. Hemker, W. B. Mori, T. Katsouleas, T. M. Antonsen and P. Mora, Phys.
Rev. E 63, 026411(2001).
B. Hafizi, A. Ting, R. F. Hubbard, P. Sprangle, and J. R. Penano, "Intense laser Beam Propagation
in a Tapered Plasma Channel," to be published.
F. S. Felber, Phys. Fluids 23, 1410 (1980).
P. Sprangle, C. M. Tang and E. Esarey, IEEE Trans. Plasma Sci. PS-15, 145 (1987).
P. Chen, Part. Accel. 20, 171 (1987).
T. G. Jones, C. I. Moore, A. Ting, P. Sprangle, D. Kaganovich, and K. Krushelnick, in Proceedings
of the 2001 Particle Accelerator Conference, edited by P. Lucas, IEEE, New York, 2001, pp. 3993-
3995.
673