PROPOSAL
TO
USER’S COMMITTEE, CENTER FOR ACCELERATOR PHYSICS
BROOKHAVEN NATIONAL LABORATORY
from
Omega-P, Inc.,
Beam Physics Laboratory,
J.
L. Hirshfield, principal investigator
tel: (203) 432-5428
fax: (203) 432-6926
e-mail: jay.hirshfield@yale.edu
LASER-DRIVEN
CYCLOTRON
AUTORESONANCE
ACCELERATOR
LACARA
INTRODUCTION
Omega-P, Inc. has carried out analytical
and computational studies of a novel accelerator-driven electron acceleration
mechanism LACARA, with support in
1999-2000 under a SBIR Phase I grant from High Energy Division, Department
of Energy. This study led to a Phase
II proposal to DoE, currently pending, to support a proof-of-principal experiment
at Brookhaven National Laboratory Accelerator Test Facility. Obviously, acceptance of this proposal by the
ATF User’s Committee is also required. This
document is intended to provide background information to enable the Committee
to reach a considered judgement on this project. Included here are excerpts from the Phase II
proposal submitted to DoE.* Further
information will be available, as needed, during a verbal presentation scheduled
to be given to the Committee during its forthcoming meeting June 1-2, 2000.
*It
is expected that a decision from DoE on Phase II support for this project
could be announced prior to the June 1-2 User’s Committee meeting. If that decision is not positive, Omega-P, Inc.
will withdraw this proposal to ATF.
SIGNIFICANCE AND BACKGROUND INFORMATION,
AND TECHNICAL APPROACH
Under Topic 11a in the 1999 SBIR Program Solicitation entitled
Advanced Concepts and Technology for
High Energy Accelerators—New Concepts for Acceleration, grant applications
were sought to develop new or improved acceleration concepts to provide very
high gradient (>100 MeV/m for electrons) acceleration of intense bunches
of particles. Omega-P, Inc. submits
this proposal in response, describing Phase II of a three-phase program to
develop a laser-driven cyclotron autoresonance accelerator (LACARA).
The analysis carried out during Phase I confirms that LACARA (a) can provide an acceleration gradient in one stage of the order
of 100 MeV/m, (b) can accelerate
continuously along a 150-cm length in vacuum using an available laser, and
(c) can accelerate in a vacuum with good
uniformity all electrons within a millimeter-length bunch.* The Phase I analysis has been applied to the
experimental parameters available at Brookhaven National Laboratory Accelerator
Test Facility (BNL-ATF), where experiments to confirm the analysis using a
prototype LACARA are proposed for
Phase II.
Electron acceleration using intense lasers has engendered
significant attention within the accelerator research community. This interest stems from the enormous optical
electrical field strengths
that can be obtained with a focused
laser, i.e. of the order of
TV/m, where the intensity
is in
. Since compact terawatt focused lasers
can have
, field strengths of the order of TV/m are possible. Of course, since this field is transversely
polarized, it cannot give much net acceleration to a charged particle directly,
so an indirect means must be employed to achieve net acceleration. The basis upon which LACARA rests is cyclotron resonance, using an axial static magnetic
field. The magnetic field can be adjusted
to allow transverse deflections of electrons that move along a helical path
to be synchronous with the rotating transverse electric field of a circularly-polarized
laser beam, thereby allowing the field to do work on the electrons.
LACARA is a laser-driven
accelerator that operates in vacuum. It does not require a pre-bunched beam; nevertheless
all injected electrons can enjoy nearly the same acceleration history.
LACARA is operated without a tight laser focus, so the Rayleigh length
can be 10¢s of cm for a 10.6
laser wavelength, and continuous acceleration in vacuum over several Rayleigh
lengths can take place. Phase bunching—but
not spatial bunching—occurs in LACARA, which explains how all injected electrons can experience nearly
the same accelerating fields, since circularly-polarized laser radiation is
used. Furthermore, the effective group
velocity in LACARA exceeds the particle’s
axial velocity, so operation with strong pump depletion is possible without
causing undue energy spread for the accelerated beam. It is shown that LACARA is not limited to being a
“
-doubler” (as is its microwave counterpart
CARA), because the relativistic energy factor
____________.
*The preliminary analysis underlying
LACARA is presented in a forthcoming
publication entitled “Laser-driven cyclotron autoresonance accelerator with
production of an optically-chopped electron beam,” by J. L. Hirshfield and
Changbiao Wang, Phys. Rev. E 61, June 2000 (to be published and appended
herein.)
can be increased
by more than a factor-of-two in a single stage. This is because stalling of the electron beam
in the axial magnetic field can be avoided. (In this expression,
is the electron rest energy plus kinetic
energy, and
is the rest energy.) Another feature
of LACARA is the relatively low
level of magnetic field required for the cyclotron resonance interaction when
a
laser is employed.
For the prototype LACARA demonstration proposed here for
operation at BNL-ATF, the magnetic field required is only 6 T, a field that
can be obtained using a cryogen-free superconducting magnet system available
from a number of industrial vendors.
During Phase I, efforts were directed towards a detailed
study of LACARA, using computational
tools available to Omega-P, Inc. The
main goal is to develop a design for the prototype LACARA based on parameters of experimental facilities available at
BNL-ATF, including an rf linac to provide a 50 MeV beam to be accelerated,
and a high-power
-laser to drive the acceleration. It is necessary in Phase II for Omega-P to procure
a high-field solenoid magnet, specifications for which evolved during the
Phase I study. Some compromise in specifying
the parameters of the magnet is necessary on account of budgetary limitations,
but this is not expected to prevent confirmation of the underlying principles
of LACARA, and for quantitative
comparison between performance and theoretical predictions. A presentation by Omega-P is scheduled for
June 1-2, 2000 before the BNL-ATF Steering Committee, to request approval
for the installation and test of a prototype LACARA,
contingent upon approval by DoE of the Phase II project. A letter from Dr. Ilan Ben-Zvi, Head of ATF,
expressing strong interest in LACARA,
is enclosed in this proposal.
ANTICIPATED
BENEFITS
The
physics underlying laser-based acceleration provides a wide range of fertile
problems that continue to motivate a not-insignificant number of research
workers. Still, none of the schemes
for acceleration under study has yet produced a beam with low enough energy
spread and emittance to be considered suitable as one stage out of many in
a machine for nuclear or high energy physics experiments, even assuming that
multi-stage operation is perfected. A
single stage should be capable of uniformly accelerating a bunch containing
a significant number of electrons (1 nC, for example), with a gradient of
the order of 100 MeV/m, and producing a beam with an acceptable emittance
(<5 mm-mrad, for example). These
attributes are anticipated for LACARA. Efficiency is an oft-overlooked but critical
parameter, since the energy per pulse that will be available in a laser beam
is not unlimited. For example, for
a 1 Joule laser pulse, energy conservation sets a limit of 100 MeV that can
be gained by
electrons per pulse, corresponding
to
nC, where
is the efficiency with which laser
energy is imparted to the electrons. For
, only 10 pC can be accelerated; however, for
, as is shown below to be possible in LACARA, over 5 nC can be accelerated. This is a critical issue, since energy consumption
by an eventual high energy accelerator with acceptable luminosity dictates
that a reasonable level of efficiency for the driver is a sine qua non. Additional potential advantages of LACARA, as compared with other laser-based
accelerator schemes, include the absence of any material medium in or nearby
the accelerating region. In some vacuum
accelerator schemes, nearby mirrors with apertures or surfaces that support
surface waves are required. It has
been shown that these surfaces can suffer permanent damage within a short
time when illuminated by intense lasers. Or,
when solid dielectric loading is used to provide for wave slowing, breakdown
limits in the dielectric will limit the acceleration gradients. And in the inverse Cerenkov laser, where a low-pressure
gas fill is used to provide the wave slowing, a small degree of ionization
of the gas could be sufficient to cause a significant change in the index
of refraction of the medium; this leads to loss of synchronism between the
radiation and the accelerated electrons. This
recitation of concerns, already thoroughly discussed in the literature, is
not meant to imply that such problems cannot be overcome; rather it is to
draw attention to issues that are not inherent to a vacuum accelerator such
as LACARA.
These
advantages for LACARA may thus provide
a basis for electron and positron accelerators using powerful lasers, to be
designed and built to take advantage of the high electric fields lasers provide,
to generate an accelerated beam with a small energy spread and low emittance,
and to transfer laser pulse energy to the beam with high efficiency.
Laboratory proof of these virtues in the SBIR Phase II project proposed
here by Omega-P could open the door towards realizing a high-gradient electron/positron
accelerator free of many irksome features of other laser-based schemes. The potential market for the large number of
magnets and optical stages of LACARA needed to provide a beam of interest
to the high energy physics community is very large indeed, and represents
a highly attractive future business opportunity.
DEGREE
TO WHICH PHASE I HAS DEMONSTRATED
TECHNICAL
FEASIBILITY
The overall goals of the Phase I program include these general objectives:
·
refinement of the theory and computations that underlie LACARA;
·
analysis of LACARA performance
for a range of experimental parameters;
·
determination, through consultations with Brookhaven ATF personnel, of
a range of parameters that could be available for a proof-of-principle test
of LACARA;
·
consultations with vendors, to define specifications and cost for a cryomagnet
designed to meet the needs of the proof-of-principle LACARA, and to fit within the ATF experimental
hall;
·
design of the LACARA proof-of-principle
apparatus to be built during Phase II; and
·
obtaining approval from the ATF Steering Committee for a Phase II proof-of-principal
test of LACARA, contingent upon
approval by DoE of the SBIR Phase II program.
Detailed discussion of tasks undertaken to pursue each of
these goals is given below. But before
elaborating on these tasks, performance for the prototype LACARA that is proposed for construction and evaluation during Phase
II is first summarized. A sketch of
LACARA is shown in Fig. 1. Copper mirrors direct a laser beam to pass along
the axis of a 6 T solenoidal magnetic field set up by the surrounding cryomagnet.
For this prototype, the
10.6 mm CO2 laser power
is taken to be 2 TW,* the minimum laser spot radius is taken to be
1.0 mm, with a Rayleigh length
29.6 cm. The electron beam and the
laser radiation interact over a length of
178 cm, but the uniform portion of the magnetic field only extends for about
150 cm. The (nominal) 1 A, 50 MeV beam
injected at z = 0 has a normalized
emittance of 2.0 mm-mrad.** Compromises
were made in selecting these parameters to be the basis for design of the
prototype LACARA, mainly on account
of the high cost of a magnet with a more suitable field profile.
The mirror spacing is 225 cm, and the 8-cm i.d. coil length of 180
cm provides a nearly uniform field region of 150 cm in length.
Electron orbits are computed from one mirror to the other, all through
the fringing fields at the ends of the coil.
Fig. 1. Sketch of LACARA
prototype, not to scale. Accelerating charge bunch is shown at center.
Fig. 2 shows, with the solid lines, the magnetic field profile
and the average relativistic energy
factor
as they vary along the axis of LACARA.
Using dashed lines, the same quantities are shown
for the ideal resonance magnetic field profile. Table I compares results for the actual and
ideal magnetic field profiles.
_____________.
*The design output power for
the ATF CO2 laser is 3 TW. (I.
Ben-Zvi, private communication).
**I. Ben-Zvi, private communication.
Fig. 2. Energy gain and magnetic field profile for the
prototype LACARA (solid lines).
Dashed lines show these parameters for the idealized resonant magnetic
field profile.
| |
one-coil B-field profile |
ideal B-field profile |
| final beam energy |
120.9 MeV |
135.5 MeV |
| average accel. gradient |
47.2 MeV/m |
57.0 MeV/m |
| maximum accel. gradient |
75.7 MeV/m |
100.6 MeV/ |
Table I. Comparison of LACARA performance for the proposed affordable one-coil magnet system
with that for the ideal resonant magnetic field profile.
These results for a LACARA
prototype employing an affordable magnet are seen to extract some sacrifice
in achievable acceleration, but not so severe a sacrifice to prevent a careful
comparison to be made between prototype performance and predictions of the
theory. It might even be argued that,
until full confidence is established in the viability of LACARA based on laboratory results, investment in a more sophisticated
magnet system might not even be warranted. Moreover, the (nominal) 6 T, 150-cm long uniform
region, 36-mm room-temperature bore, non-cryogen, superconducting magnet that
Omega-P proposes to acquire for the prototype LACARA is a versatile laboratory instrument that can find other applications
in the future. This might not be so
for a magnet with a more specialized field profile.
Tasks for achieving the Phase I goals listed above are described
in the Phase I proposal, which is enclosed herein. The title of each task is given in bold italics
below, together with details of the results obtained during Phase I.
Task A: Include finite emittance and energy spread
for the injected beam in computations.
Since examples given in the Phase I proposal were for an initially
cold beam, it is important to judge how a finite initial transverse beam emittance
can affect LACARA performance. In carrying out this task, acceleration was
computed for finite initial transverse emittance, and for several initial
beam energies to simulate the influence of a finite energy spread. The normalized transverse emittance for the
BNL-ATF beam that is to be used with the prototype LACARA experiment is
= 2.0 mm-mrad [mm].
At 50 MeV, this implies a rms emittance of
0.0202 mm-mrad. For the simulations described below, it was
more convenient to specify initial beam coordinates and momenta using the
phase-space transverse “98% emittances”
and
, which are here defined by the area in
and
phase space within which
98.2% of the particles are found.
For a beam having Gaussian distributions of coordinates and momenta,
and with
0.0202 mm-mrad, it was found that
=
= 0.125p mm-mrad. In the examples shown in this section, laser
parameters were as in the example of Fig. 2.
For some of the examples shown in this section, the magnetic field
profile shown in Fig. 2 (the “one-coil profile”) was employed.
For other examples a profile that is closer to the ideal resonant profile
(the “two-coil profile”) was employed; the two-coil profile is shown in Fig.
3. From this figure, it is seen that
final beam energy, and average and maximum acceleration gradients achievable
with the two-coil profile are 128.0 MeV, 52.0 MeV/m, and 87.7 MeV/m.
Fig. 3. Energy gain and magnetic field profile for a
LACARA using the two-coil magnet (solid
lines). Dashed lines show these parameters
for the resonant magnetic field profile.
It is important to note a significant
difference in the model used for the one- and two-coil examples, and that
for the resonance profile. It is taken
in the computations that the beam for the one- and two-coil cases enters and
exits the laser interaction region between the mirrors by moving along the
full fringing magnetic fields at the edges of the magnet coils. However, in computations for the resonance profile,
the interaction only occurs between 50 and 200 cm, as shown by the dashed
lines in Fig. 3. In practice, this
could be arranged in the laboratory by positioning two mirrors in the magnet
bore to deflect the laser beam in and out of the beam path once the coil field
reached the resonance value. To accomplish
this in practice would require a cryomagnet of larger room temperature bore
diameter than the 36-mm for the one-coil affordable system to be used in the
LACARA prototype.
Fig. 4 shows, for the two-coil profile, and for fixed value
of
= 0.125p mm-mrad, the relativistic energy
factor reached during acceleration, for various values of initial electron
beam radius
. Smaller
implies a larger range of initial transverse
momentum that leads to loss of phase synchronism during acceleration, with
a concomitant decrease in final achievable energy. Larger
allows peripheral electron orbits to
move out of the intense core of the laser spot, and thereby to achieve a lower
net acceleration. The optimum value
of initial beam radius found is
= 0.3 mm, for which the final average
beam energy is 128.0 MeV.
Fig. 4. Achievable final average beam energy factors
using the two-coil profile, for three values of initial beam radius
, with fixed initial emittance
= 0.125p mm-mrad.
Fig. 5 shows the acceleration history near the end of LACARA for different initial beam energies
that are 1% above and below the design energy of 50.0 MeV; for all cases
= 0.125p mm-mrad. As
is seen, the final beam energy variation is less than ±1%. This
strongly suggests that, for a beam with initial energy spread within ±1%, that the acceleration will
be essentially the same as for a mono-energetic beam.
Fig.
5. Average beam energy for values of
initial beam energy that are 1% above and below the design energy of 50.0
MeV.
The effect of varying the initial transverse emittance was computed,
with results shown in Fig. 6. Here,
for the two-coil magnetic field profile, and for
= 0.30 mm, results are shown for
=
= 0.125p mm-mrad (case 1), 0.175p mm-mrad (case 2), and 0.225p mm-mrad (case 3). As is evident, substantial decrease in achievable
final beam energy ensues as the initial
Fig. 6. Acceleration in LACARA for three different initial beam emittances.
beam
emittance increases. It is likely that
a somewhat greater final beam energy could be realized in the higher emittance
cases by optimization of the initial beam radius in each case. Nevertheless, the lesson taught in this example
is the necessity for employing a beam of reasonably small initial emittance
in order to realize the full potential of LACARA.
The evolution of transverse emittance
during acceleration in LACARA has
also been examined. This is an unusual
situation to evaluate, since the beam executes an orderly gyration about which
random variations in coordinates and momenta occur, the latter arising from
random variations in the initial values. The orderly variations can be described by “geometric”
emittances, defined as
, and
, where
are orderly deviations in the respective
variables, quantities that are normally zero for solid laminar beam.
Geometric emittances in LACARA presumably can be made to approach
zero using a sequence of quadrupole lenses. (This point is discussed in sub-section E below.) Figs. 7 and 8 illustrate both the orderly and
random nature of the beam, wherein plots in
and
phase space are shown for an ensemble
of 1889 particles that are injected at four equally-spaced intervals during
one optical cycle (35 fs). In this
exercise, the resonance magnetic field profile was chosen.
Fig. 7. Phase space
(a) at z = 0, (b) at z = L/2,
and (c) at z = L.
Fig. 8. Phase space
(a) at z
= 0, (b) at z = L/2, and (c) at z = L.
The
phase-space plots are shown at the point of injection (z = 0), at a point mid-way along LACARA (z =
= 74.1 cm = L/2), and at the end of LACARA (z =
= 148.2 cm = L). Variations
in arrival times
and
at z = L/2
and z = L for groups of particles that departed at the same time are seen,
due to slightly differing histories of axial velocity. These variations are of the order of 4-5 fsec
out of
fs (at z = L/2) and 7-8 fs out of
fs (at z = L). The variations are fractionally tiny, but still
amount to a significant fraction of the optical period (35 fs). From the phase space plots, one can estimate
that
(0.4)(5)p
= 2p mm-mrad, and similarly
for
. However, the random variations, evaluated
at fixed arrival time at z = L, give maximum values
(0.3)(0.5)p
mm-mrad, only slightly larger than the initial transverse areal emittances.
In any case, the method of estimation isn’t accurate enough to distinguish
the initial and final values. Thus,
to the extent that geometrical emittance can be manipulated and reduced using
quadrupole lenses, it appears that emittance growth—as it is usually described—is
not in principle a serious issue in LACARA,
when the resonance magnetic field profile is used.
Task B: Examine effects of errors in magnetic field
profile on LACARA performance.
Results shown in the Phase I proposal, and in the discussion
above relating to Task A, are for
the magnetic profile of a single coil, of two coils, or for the ideal resonance
magnetic profile. While use of the
ideal resonance profile is preferable, two issues arise in implementing it
in the laboratory. The first issue
is the need for a complex coil system to reproduce the required profile, while
the second issue is the need for limiting the beam interaction with laser
radiation to be only where the magnetic field conforms to the resonance profile,
and not in the fringing fields at the entrance and exit of the cryostat. Both issues can be satisfied by careful design
of the cryomagnet: a complex coil system
can reproduce the resonance profile with good accuracy, and a large room-temperature
bore diameter in the cryostat can allow space for mirrors to deflect the laser
beam in and out of the beam path only where the field conforms to the resonance
profile. However, consultations with
cryomagnet vendors indicated that implementation of these remedies would raise
the cost of the magnet system to a level well above what can be accommodated
in a Phase II SBIR grant budget.
Performance with simple one-coil system was examined, to determine
if the severe differences between its magnetic field profile and the resonance
profile would be unduly detrimental to demonstration of the principle of LACARA. The single-coil was taken to have a rectangular
cross section and uniform current density, with inner and outer diameters
of 8 cm and 16 cm, and a length of 180 cm. Fig. 2 shows the magnetic field profile and
the energy gain, for the one-coil system and for the ideal resonance magnetic
field profile, both for the same beam and laser parameters. For the one-coil magnet with an optimized peak
field of 60.6 kG, the final average beam energy is 120.9 MeV (
), and the average and peak acceleration gradients are 47 and 76 MeV/m.
Comparisons, discussed above, between the acceleration achievable with
the three different field profiles indicate that the LACARA
mechanism is not at all defeated by magnetic field profile variations away
from the ideal profile, although a clear penalty is extracted when the profile
deviates significantly from the resonance profile.
One measure of sensitivity to magnetic field is shown by evaluating
acceleration for a beam of fixed parameters, with a laser of fixed parameters,
with a single magnet coil of fixed geometry, but with a coil current that
is varied. Fig. 9 shows the acceleration
along LACARA for three different
peak magnetic field values, the optimum value of 60.6 kG and two other values
58.5 kG and 62.7 kG that are 3.5% higher and lower.
This degree of mistuning of the magnetic field strength is seen to
degrade the final beam energy from 121 MeV to 113 MeV, a 7% decrease in energy
gain. This calculation shows that it
is important to set the magnetic field strength to the desired value within
a fraction of a percent, so as not to cause a diminution in energy gain. This requirement is not particularly demanding.
Fig. 9. Acceleration in a one-coil magnetic profile
LACARA for three values of peak magnetic
field. The loss in energy gain is 7%
for a 3.5% change in peak magnetic field.
Another practical issue is
the question of alignment between the magnetic axis and the laser beam axis.
The effect of misalignment has been computed, for the one-coil example
as shown in Fig. 2, except that the magnetic field is taken to be tilted about
the center point of the
Fig. 10. Effect upon acceleration of a misalignment between the magnetic axis and the laser beam axis. For cases 1-4, the misalignment angles are 0, 0.675, 1.35, and 2.7 mr.
system by an angle
. Fig. 10 shows the mean energy gain
profiles, for three different tilt angles. For comparison, the result with perfect alignment
is also shown (case 1).
Cases 2, 3 and 4 are for tilt angles
= 0.675 mr, 1.35 mr, and 2.70 mr, respectively.
These correspond to transverse displacements
of
respectively. Not surprisingly, when
misalignment causes the two axes to be displaced by more than about two laser
waist radii, the acceleration diminishes significantly.
This is shown clearly in Fig. 10. The
lesson taught by this exercise is that provision is required to align the
magnetic and laser beam axes to within about 1 mr.
This is a degree of precision required in many laser-acceleration schemes.
Task C: Include
beam loading.
All
the examples of LACARA performance
discussed so far are for a 1 A beam. If
the average energy gain per electron is 71 MeV (as for the one-coil magnetic
profile), then the power added to the beam is 71 MW, far less than the 2 TW
in the laser beam. Fig. 11a shows the
power profiles of the beam and laser. Here,
the efficiency
for transfer of laser power to electron
beam power is
%, and beam loading is negligible.
Thus the laser power is seen to not diminish along the acceleration
path. To examine performance of LACARA
when beam loading is not negligible, higher beam currents were introduced,
namely 10, 20, and 40 kA. Power profiles
are shown for these currents in Figs. 11b, 11c, and 11d, respectively.
Fig. 11a. Laser and beam power profiles along LACARA for a 1 A beam, where beam loading
is negligible. Final beam energy is
120.86 MeV.
%.
Fig. 11b. Laser and beam power profiles along LACARA for a 10 kA beam, where beam loading
is significant. Final beam energy is
114.05 MeV.
%.
Fig. 11c. Laser and beam power profiles along LACARA for a 20 kA beam, where beam loading
is more significant. Final beam energy
is 107.24 MeV.
%.
Fig. 11d. Laser and beam power profiles along LACARA for a 40 kA beam, where beam loading
is even more significant. Final beam
energy is 94.2 MeV.
%.
Several
messages are conveyed by the results shown in Figs. 11b-11d. First,
LACARA is a laser-based accelerator that can have very high efficiency
for transfer of laser power to electron beam power.
Second, the efficiency is
evidently not critically dependent upon the choice of magnetic field profile,
since the results shown here are for a profile with significant deviations
from the ideal resonance magnetic field profile.
Third, operation with higher
beam loading and higher efficiency carries a penalty of lower net acceleration;
this is a feature that is natural and is common with other traveling-wave
accelerators, where the accelerating field is reduced as the particle beam
soaks up power from the radiation beam. The
high efficiency values that are shown here serve to distinguish LACARA from other laser-based accelerators
where efficiency is so small as to be hardly ever discussed.
Task D: Examine finite bunch-length effects and slippage
for a finite length radiation pulse.
All
of the computations presented so far in this proposal assume that the laser
field is at a steady-state level during passage of the electrons. Invariably, however, both the laser pulse and
the electron beam bunch length are finite in duration and comparable, typically
in the psec range. Operation of LACARA then requires precise synchronism
in timing between injection of the electron bunch and the laser pulse. In this case, it is possible for there to be
an excess energy spread that arises during acceleration from two sources:
(i) laser amplitude variations if the laser
pulse is not wider than the electron bunch, even with good synchronization;
and (ii) slippage of the laser pulse
over the electron bunch, since the two do not move at exactly the same speeds.
In the experiments proposed at BNL-ATF, it is intended to operate LACARA with an electron bunch width of less than 1 ps, if possible,
and with a laser pulse of width of greater than 1 psec. Thus, if the slip between the pulse and the
bunch is rather less than 1 psec, slippage should not cause significant non-uniformity
in the electron acceleration. To estimate
slippage, the transit times for electron bunches
and for laser pulses
were computed, using the one-coil magnetic
field profile with all other parameters identical to those taken above.
The transit times were defined as
,
and
,
where normalized group and phase velocities for the
laser pulse are related by
, and
with
given on p. 7 of the Phase I proposal,
enclosed herein. The group velocity
is used here since it is motion of a wave packet (laser pulse) that influences
the field amplitudes that act on the electrons; phase variations between the
laser fields and electron orbits are already included in the dynamics as analyzed
for a continuous laser beam. The slip time is given by
,
since
. Fig. 12 is a plot of
along LACARA.
Numerical integration of this curve from z = 25 cm to z = 255 cm
gives
= 0.27 ps. This suggests, for example, that a 1 ps electron
bunch can remain within a 1.3 psec laser pulse without slipping out of the
accelerating field, when the laser pulse and electron bunch start off in perfect
synchronism. The
Fig. 12. Slip between laser pulse and electron bunch
along LACARA.
relatively small margin for slip without a strong effect
on acceleration indicated by this estimate suggests that care must be taken
in the experiments. Moreoever, it appears
that actual time-
dependent simulations of acceleration history should be carried out in Phase
II for a finite laser pulse width and a finite electron beam width, in order
to better model this important phenomenon.
Task E: Examine
effects of orbit gyration.
Issues to explore in this task included (i) estimation of the limits to LACARA performance from synchrotron radiation
due to the particle gyrations; and (ii) exploration of transverse geometrical emittance in LACARA, defined above under Task A as
, and
, where
are orderly deviations in the variables.
The total synchrotron radiation power for an electron moving
on a helical orbit in a constant uniform magnetic field has been derived by
Sokolov and Ternov.* Their result is
watts.
An estimate for the radiated
power will be given, based on behavior of LACARA
analyzed so far. Two features act to
minimize synchrotron radiation in this case, as compared with a conventional
cyclotron where particles move on circular orbits. First
is the factor
, which in the examples considered in this proposal never exceeds about
; it becomes even lower as
increases. (A plot of typical variation of
vs z is given in Fig. 2 on p. 8 of the Phase I proposal.) Second
is the product
, which in LACARA can be approximated
as being nearly constant. [This follows
from the autoresonance condition
, when
.] For l = 10.6 mm,
. The numerical value of synchrotron
radiation power under these conditions is
watts/electron. For the highest current
example considered in sub-section C, namely 40 kA, the number of electrons in a 1 psec (0.3 mm) bunch
is
. It is a good approximation to assume
that these electrons radiate incoherently, since most of the synchrotron spectrum
is at wavelengths below the bunch length. Then the net synchrotron radiation power loss
is
watt. This is clearly quite negligible
compared with the beam power of ~1 TW. From these considerations, it is concluded that—for
the range of parameters where LACARA
is likely to operate—synchrotron radiation should not impose any limitation.
The transverse geometrical emittance that is characteristic
of a beam accelerated in LACARA arises from the orderly gyrations
of the beam about the magnetic axis. While
individual electrons only make a few gyrations in their transit through LACARA,
electrons follow
__________.
*A. A. Sokolov and I. M. Ternov,
Radiation from Relativistic Electrons,
(Am. Inst. Phys., New York, 1986) p. 80.
one another on orbits that
are displaced in phase at the optical frequency. Thus a complete cycle
in the transverse plane for
particle and momentum coordinates is completed in 35 fs.
The image of this cycle for a cold beam, after exiting LACARA and progressing down the magnetic
field taper, is shown in Fig. 13. To
simplify the interpretation, the injected beam taken for this example has
a beam radius
0.10 mm, and zero emittance; this translates into initial trajectories that
are all parallel to the system axis. For this beam with a 2 TW laser having other
parameters as above, the optimum single-coil magnetic field peak strength
is 5.78 T, and the final beam energy is 138.7 MeV.
At the location of the second mirror z
= 255 cm, the particle phase space
is shown in Fig. 13.
A similar plot exists for the
phase space.
Fig.
13. Phase space
for an 50 MeV initially cold beam
after acceleration to 138 MeV in LACARA, and after passage down the magnetic field taper.
This
phase plot shows particles out of the magnetic field after navigating the
fringing field of the magnet, as they move on straight line trajectories that
lie on the surface of a cone in x-y-z. Successive trajectories in time are rotated
in angle around the cone, with phase variations at the optical frequency. The cone angle is about 2.5 mr, and the beam
orbit radius at z = 255 cm is about
0.9 mm. It is reasonable to expect
that these straight line orbits can be brought to a focus using a quadrupole
doublet, but analysis of this situation is beyond the scope of the Phase I
project; it is a topic for further study during Phase II. Focusing and matching are expected to be critical
issues for a multi-stage LACARA,
but this too is a topic for further study in Phase II.
Task F: Optimize choice of cryomagnet for proof-of-principle
LACARA experiment.
Discussions with BNL-ATF management, and close inspection of
the ATF experimental hall revealed several general limitations on the installation
of a large cryomagnet, as required for the prototype LACARA. First,
is the impossibility—due to low ceiling height—for utilization of customary
overhead direct transfer of liquid He and N2 into the cryostat.
This implies that a transfer system be installed to allow introduction
of cryogens from outside of the experimental hall.
Second, is the need for a BNL safety review
to determine if the ventilation system in the experimental hall is adequate
to handle the gas evolution during a quench. Doubt was expressed on this count, indicating
that a new ventilation system would be needed in order to proceed.
These two obstacles caused a decision to be made by Omega-P to seek
vendors that can supply a cryogen-free magnet, i.e. a superconducting coil
system that is cooled by circulation of He gas using a cryocooler refrigerator.
Two such vendors were identified, and
quotations received from both. Such a cryomagnet requires no liquid cryogens,
even for initial cool-down; the coil is brought to 4 deg K operating temperature
after a few days of cryocooler operation. Use of HTS current leads allows operation with
connection to the current supply, a necessary requirement during LACARA experiments. One vendor was unwilling to quote on the two-coil
magnet (see Fig. 3), but the second vendor’s quote for the two-coil magnet
was $350,000—too high to be accommodated within a Phase II budget. Both vendors provided quotes for the one-coil
6 Tesla magnet (see Fig. 2) that were in the $250,000 range, but one of the
vendors (Everson Electric Co.) provided the full system within its quote,
including the power supply, 8-channel temperature monitor, 4K cryocooler,
and a two-day installation visit. Moreover,
Everson provided design curves for the magnetic field profile that were the
basis for simulations provided in this proposal.
In view of these considerations, of Everson’s strong reputation for
delivering high-quality magnets for accelerator applications, and of their
willingness to enter into a Phase III commitment for this project, Omega-P’s
intention is to award its purchase order for the magnet system to Everson.
Task G: Design optical layout for LACARA proof-of-principle
experiment.
BNL-ATF management has indicated that space will be available
on beam-line #2 in the ATF experimental hall, since the HGHG experiment is
expected to vacate this space before the LACARA prototype is ready for installation. BNL-ATF management has also assured Omega-P
that it will assume responsibility for design and implementation of the transport
system to convey CO2 laser pulses to the LACARA prototype, and for the laser beam
dump. The laser beam is to be in a
near-Gaussian form with focus to a 1-mm waist radius. Omega-P is to supply two flat copper mirrors,
with central holes for beam passage, as shown in Fig. 1. The flat mirror mounts are to be designed to
allow precision alignment of the laser beam, while the cryomagnet and associated
¾² beam pipe are to be designed
to allow their precision alignment as well. Installation of the cyromagnet and optical system
are to be closely coordinated between Omega-P and BNL-ATF.
Task H: Electron energy spectrometer.
It has been determined, through discussion with BNL-ATF management,
that a suitable energy analyzer is available for this experiment.
Its energy resolution can be arranged to be 100 keV or less, precise
enough for determining the energy spectrum of the beam after interaction in
the LACARA prototype. BNL-ATF is also responsible for the electron
beam dump. Radiation shielding in the
ATF experimental hall is approved for electron beam energies up to 200 MeV,
so that the predicted acceleration to 121 MeV in the LACARA prototype (see Fig. 2) will be well within this approved range.
Task I: Seek approval from ATF Steering Committee for
LACARA proof-of-principle experiment.
Omega-P is presenting an application to BNL-ATF for an allocation
of time to install and run LACARA
during 2001-2002, and a verbal presentation is scheduled before the ATF Steering
Committee on June 1-2, 2000. The application
to ATF will be contingent upon Omega-P receiving this Phase II SBIR grant
from DoE. The disposition of the contingent
application to ATF will be made known to the DoE program manager as soon as
it is known to Omega-P.
Summary of Results of Phase I Study
Conceptual
design has been completed of a prototype LACARA
(Laser Cyclotron Autoresonance
Accelerator), a sketch of which is shown
in Fig. 1. LACARA is a laser-driven vacuum accelerator capable of a ~100 MeV/m acceleration gradient acting on all
electrons in a mm-scale bunch, using an available 2-3 TW CO2 laser.
Parameters for the prototype LACARA accelerator that is proposed for
installation at Brookhaven National Laboratory Accelerator Test Facility are
given in Table II.
injected
beam energy
50 MeV
injected
normalized beam emittance
2 mm-mrad
injected
beam 98% emittance
0.125p mm-mrad
injected
charge/bunch
~1 nC
laser
wavelength
10.6 mm
laser
power
2 TW
minimum
laser spot radius
1.0 mm
laser
Rayleigh length
29.64 cm
mirror
spacing
230 cm
magnet
coil length (single coil) 180
cm
peak
magnetic field
6.06 T
cryostat
warm bore diameter
36 mm
final
beam energy
120.86 MeV
maximum
accelerating gradient 75.65
MeV/m
final
beam 98% emittance
~0.15p mm-mrad
final
beam geometric emittance ~2p mm-mrad
Table II. Parameters of prototype LACARA.
The design chosen for this prototype LACARA is a compromise based on use of
an affordable magnet, but with a maximum accelerating gradient that is not
too much less than the ideal value of 100.62 MeV/m that would obtain for the
ideal resonance magnetic field profile. In
the course of developing the conceptual design, issues that were examined
included the following:
·
sensitivity of acceleration to magnetic field profile (Fig. 3);
·
sensitivity of acceleration to magnetic field amplitude (Fig. 9);
·
sensitivity of acceleration to magnetic field alignment (Fig. 10);
·
sensitivity of acceleration to initial beam radius (Fig. 4);
·
sensitivity of acceleration to initial beam energy (Fig. 5);
·
sensitivity of acceleration to initial beam emittance (Fig. 6);
·
evolution of emittance during acceleration (Figs. 8, 13);
·
acceleration efficiency and beam loading effects (Figs. 11a-d);
·
slip between laser pulse and electron bunch during acceleration (Fig.
12).
The analysis during Phase I
formed the basis for preliminary discussions with BNL-ATF management, and
will form the basis for a proposal to the BNL-ATF steering committee for approval
to install and operate the prototype LACARA
using ATF facilities and support staff. It
is anticipated that the prototype LACARA
could be installed and operating by Fall 2001.
THE
PHASE II PROJECT
Technical Objectives
The
principle technical objectives for Phase II are to build, install and evaluate
the prototype LACARA, parameters
for which are summarized on p. 19 of this document.
The prototype LACARA is designed
to be installed on beamline #2 at BNL-ATF, where a 50 MeV injected beam is
available from a rf linac, and where 2-3 TW laser pulses are to be available
from a CO2 laser. The major
new component to be procured for LACARA is a 6 T solenoid cryomagnet with a 3.6 cm diameter warm bore
and a 150 cm uniform field length. In addition to design and construction
of this cryomagnet, optics and beamline designs are needed, and continued
analysis is planned to enable conceptual design of a future staged version
of LACARA for achievement of higher beam energy.
In
addition to close collaboration in Phase II between Omega-P, Inc. and BNL-ATF
staff, work during Phase II is also to be within the ongoing collaboration
with beam physics personnel from Columbia University, including Professor
Thomas C. Marshall (an Omega-P consultant) and Dr. J.-Y. Fang, a Columbia
staff research scientist. Dr. Changbiao
Wang, Associate in Research in Physics at Yale University, will also continue
(under a sub-contract to Yale) in his central role in providing supporting
analysis for the project.
Phase II Work Plan
Tasks to achieve the aforementioned objectives
are as follows:
Task a: Present proposal to BNL-ATF Steering Committee
at June 1-2, 2000 meeting, based on material in this document, for installation
and beam time for operation and evaluation of prototype LACARA.
Task b: Review available space on beamline #2 in BNL-ATF
experimental hall to confirm maximum dimensions for cryomagnet system and
layout of optics.
Task c: Review design of cryomagnet with vendor engineeering
staff, to confirm final operating parameters, and to obtain parameters needed
for design of precision support structure for the cryomagnet and beam pipe.
Place order for cryomagnet system.
Task d: Design and fabricate precision support for cryomagnet
and beam pipe.
Task e: Design and fabricate beam pipe for LACARA, including housings and mounts for
copper mirrors and bellows adjustments for precision alignment.
Task f: Review laser and accelerator synchronism and
energy analysis instrumentation and controls with BNL-ATF staff, and organize
protocols for experimental trials of prototype LACARA.
Task g: Conduct acceptance test of cryomagnet system
at vendor’s plant.
Task h: Install cryomagnet and beam pipe in BNL-ATF
experimental hall.
Task i: Carry out experimental tests of prototype LACARA, including measurements of beam
energy spectrum without laser and magnetic field, with laser but without magnetic
field, and with both laser and magnetic field. Optimize acceleration by adjustment of synchronism
between laser and electron bunch, magnetic field strength, beam energy, alignment
between laser and magnetic field, and bunch charge.
Task j: Provide theoretical and computed predictions
for actual conditions of the experiments carried out in Task i, to allow comparisons between the two.
Task k: Extend Phase I analysis of evolution of beam
emittance during acceleration in LACARA,
including possible reduction in geometrical emittance using a quadrupole pair
or triplet after the fringing magnetic field at the exit of LACARA.
Task l: Examine performance of a two-stage LACARA, with straight-ahead injection into
stage #2 from stage #1, and with refocusing of original laser beam.
Task m: Reports at six-month intervals during Phase
II will be prepared and submitted to detail achievements during each period
of the project. It is also anticipated
that periodic status reports on the project will be presented at the forthcoming
Particle Accelerator Conference, and at Advanced Accelerator Concepts and
RF Technology for Accelerators Workshops, as appropriate.
Performance Schedule
The tasks described above will
be carried out according to the following approximate schedule, where the
time intervals are divided into quarterly segments.
| |
qtr I |
qtr II |
qtr III |
qtr IV |
qtr V |
qtr VI |
qtr VII |
qtr VIII |
Task a |
x* |
|
|
|
|
|
|
|
| Task b |
x |
|
|
|
|
|
|
|
| Task c |
x |
|
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| Task d |
|
x |
x |
|
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|
|
| Task e |
|
x |
x |
|
|
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| Task f |
|
|
x |
x |
|
|
|
|
| Task g |
|
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|
x |
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|
| Task h |
|
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|
x |
|
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|
| Task i |
|
|
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|
x |
x |
x |
x |
| Task j |
|
|
|
x |
x |
x |
x |
x |
| Task k |
x |
x |
x |
|
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Task l |
|
|
x |
x |
x |
x |
x |
|
| Task m |
|
x |
|
x |
|
x |
|
x |
Schedule of tasks.
*This task is expected to be
completed prior to the start of Phase II.
PRINCIPAL INVESTIGATOR AND OTHER KEY PERSONNEL
Principal Investigator for the Phase II program is to be Dr. Jay L. Hirshfield, who functioned in that role for the Phase I program. His resumé and list of recent publications are included in the enclosed Phase I proposal. Dr. Hirshfield is committed to devote at least 1040 hours to the Phase II program, corresponding to a minimum of 10 hours/week, averaged over the 24-month span of Phase II. Other Omega-P employees who will participate in the project include Dr. M. A. Lapointe, experimental physicist; Mr. S. Finkelshteyn, technician; and an as yet unidentified Ph.D. level physicist.
FACILITIES AND EQUIPMENT
The work proposed for Phase I involves computational, engineering design, and experimental tasks. The proposed computational and design work will be carried out at facilities maintained by Omega-P in a suite of offices at 345 Whitney Avenue, New Haven, CT, and at the Yale Beam Physics Laboratory at 272 Whitney Avenue, just across the street. The Omega-P offices contain several high-capacity computer terminals that operate a number of microwave design codes and relativistic particle simulation studies on a stand-alone basis, or by interconnection with NERSC. KARAT is installed on two computer work stations at Yale. These facilities have been used successfully for similar simulation studies and design projects, such as the vacuum beat wave accelerator project; 10 MW and 36 MW CARA’s; and 1.3 GHz, 11.4 GHz and 34.3 GHz magnicons.
Omega-P
has available to it facilities for beam physics research at the Yale Beam
Physics Laboratory, under a collaborative arrangement that has been functioning
since 1991. This is made possible since
Dr. J. L. Hirshfield of Omega-P is Adjunct Professor of Physics at Yale, and
has—in collaboration with other Omega-P staff—built up the facilities. Omega-P
personnel have access to
The facilities at ATF of main significance include a rf linac, with photocathode rf gun injector, that is capable of accelerating ~1 nC electron bunches to 70 MeV. A new CO2 laser, now in the commissioning phase, is designed to generate 3 TW of power at 10.6 mm wavelength. Support for the notion of carrying out evaluation of the prototype LACARA is attested to in the enclosed letter, signed by Dr. Ilan Ben-Zvi, Head of the Accelerator Test Facility at Brookhaven.
Only one item of new capital equipment
is needed for the Phase II program, namely the cryomagnet system described
on pp. 13-15. The magnet system will
be procured from Everson Electric Co., of
SUB-CONTRACTOR
The
research institution collaborating directly with Omega-P on this project is
9/00 – 8/01 9/01-8/02
Dr. Changbiao Wang, PI (40%, 41.6%)
Fringe benefits/hospitalization
Travel
Total, subject to indirect costs
Indirect
costs, 63.5%
Yearly totals
Two-year total
An enclosed letter, signed by Dr. Suzanne K. Polmar, Director of the Yale Office of Grant and Contract Administration, gives official approval for Yale participation as a sub-contractor in the Phase II program, with the approved budget figure.
APPENDIX
Attached
hereto is a preprint of the forthcoming article on LACARA, to be published in the June 2000 issue of Physical Review E.
Laser-driven
electron cyclotron autoresonance accelerator
with
production of an optically-chopped electron beam
J.
L. Hirshfield1,2 and Changbiao Wang1
1Department
of Physics,
2Omega-P, Inc.,
Abstract
Gyroresonant acceleration of electrons
in vacuum using a focused laser is analyzed. Continuous and equal acceleration is shown for
electrons injected at all optical phases over an interaction length of 10’s
of cm. Beam stalling is avoidable as
beam energy increases. Acceleration
from 50 to 178 MeV is predicted, for a 4 TW, 10.6 mm laser focused to a waist radius
of 1.0 mm; these parameters correspond to a planned experiment. A beam stop with an off-axis hole after acceleration
is shown to create a train of optically-chopped bunches with 3 fs bunch lengths
and 35 fs period.
PACS numbers: 41.75.Jv, 41.75.Lx,
96.50.Pw.
Electron acceleration using
intense lasers has engendered wide-spread attention within the accelerator
research community, stimulated mainly by the enormous optical electrical field
strengths
that can be obtained with a focused laser in vacuum, i.e. of the order of
TV/m, where the intensity
is in
[1]. Since compact terawatt focused
lasers can have
, field strengths of the order of TV/m are possible. Of course, since this field is transversely
polarized, it cannot give much net energy gain to a charged particle directly,
so a number of indirect means have been devised to achieve cumulative acceleration.
For example, in the laser wake field accelerator [2] an intense laser
pulse is used to create a strong longitudinally-polarized plasma wake field
for acceleration. In the vacuum beat-wave accelerator [3], two
laser pulses of different frequencies are combined to create a slow optical
ponderomotive beat wave that can exert a strong force for acceleration. Electron acceleration to over 100 MeV has been
observed in laser wake field accelerator experiments, corresponding to an
acceleration gradient of the order of 30 GeV/m [4]. The energy spread of electrons that are accelerated
in this manner is usually not small, since particles are acted upon throughout
the non-uniform plasma wake. Moreover,
the acceleration length is limited to a few Rayleigh lengths, usually less
than a few millimeters for tightly-focused optical radiation. These facts have led to experiments in which
an optically pre-bunched beam is created, so that injected electrons in an
inverse Cerenkov accelerator might all enjoy nearly the same acceleration
[5,6]; or by exploitation of an injection mechanism (LIPA), wherein an energetic
highly directed bunched beam is born within the optical focus [7].
Channeling has been suggested as a means to allow acceleration over
many Rayleigh lengths [8]. Progress
in the wide field of laser-based accelerators is summarized in a recent review
[9].
This Brief Report describes a laser-driven
acceleration mechanism in vacuum that does not require a pre-bunched beam;
nevertheless all injected electrons can enjoy nearly the same acceleration
history, regardless of their initial optical phase. A tight laser focus is not required, so the
Rayleigh length can be 10’s of cm’s for a 10.6
laser wavelength, and continuous acceleration over meter-length paths is predicted.
Furthermore, since the accelerated beam gyrates in a transverse plane
at the laser-frequency, an interposed beam stop with a judiciously-placed
off-axis hole can be employed to produce a transmitted beam comprising a train
of optically-chopped bunches with bunch lengths below 1
(3 fs). An example is presented of
acceleration of a beam from 50 to 178 MeV using a 4 TW, 10.6 mm wavelength laser in a non-uniform
magnetic field peaking at 81 kG; a rf linac and a
laser with these parameters are to
be available for a proposed experiment at Brookhaven National Laboratory [10].
The underlying mechanism, cyclotron
autoresonance acceleration (CARA),
has been studied heretofore mainly as a microwave interaction, where theory
predicts and experiments show efficiencies exceeding 95% for transforming
microwave energy into directed beam energy [11]. Phase bunching—but not spatial bunching—occurs
in CARA, so that all injected electrons
can be arranged to experience nearly the same magnitude of accelerating fields.
However, a microwave CARA
is in practice only a “
-doubler,” in that the relativistic energy factor
cannot in practice be increased much
beyond a factor-of-two in a single stage, due to stalling of the electron
beam in the required up-tapered guide magnetic field. In this expression,
is the electron rest energy
plus kinetic energy.
It will be shown below that the CARA stalling limit can be circumvented
when a focused optical field is used in place of a guided microwave field
since, in the optical case, the axial magnetic need not necessarily be continuously
up-tapered. Another feature of the
laser-driven CARA, hereafter dubbed
LACARA, is the relatively low level
of magnetic field required for the cyclotron resonance interaction: for high
beam energies, the magnetic field scales roughly as
, where
is the optical wavelength.
As a result, state-of-the-art superconducting solenoid magnets are
suitable for a 100 MeV demonstration LACARA operating at
.
Laser acceleration based on cyclotron
resonance was first analyzed by Sprangle, Vlahos and Tang [12], using fields
approximating a focused Gaussian. These
authors identified the need for a non-uniform guide magnetic field to preserve
gyroresonance, and gave an example with an acceleration gradient of 31 MeV/m
for a
, I =
laser. More recently, other authors
have analyzed acceleration that is based on cyclotron resonance [13, 14, 15].
However, none of this prior analysis considered acceleration into and
beyond a laser focus, and thus failed to show that the magnetic field will
fall in magnitude after the focus, thereby avoiding stalling so that no upper
limit to acceleration is imposed.
The analysis presented below is for a traveling Gaussian
laser beam focused by a parabolic mirror, as shown in Fig. 1. The second mirror is solely to direct the spent
laser beam away from the beam axis. The
electron beam is taken to be injected and extracted through holes in each
mirror. An axisymmetric non-uniform
magnetic field is imposed on the system, as provided by a system of surrounding
coils not shown in the figure. The
form of this magnetic field is determined self-consistently by requiring that
gyroresonance be maintained along the particle orbits. The electromagnetic fields in cylindrical coordinates
for the lowest-order Gaussian mode
in such a configuration excited with circular polarization are given by [3]
;
(1)
; (2)
; (3)
and
(4)
where the waist
radius
in the focal plane
and the Rayleigh length
are related by
, with
the radiation wavelength.
The radius of the radiation pattern for
is
, the phase is
, and the radius of curvature of the ray normal surfaces is given by
. The coordinates
on surfaces of constant phase, i.e.
where
, leads to
, with axial and radial phase velocities
and
. The effective axial and radial wavenumbers,
found from
and
, are
and
. (5)
Laser power
is related to the electric field amplitude
by
. Some prior analyses [13, 14] of gyroresonant
acceleration consider uniform optical fields, thereby neglecting both diffraction
and axial components.
The condition
to be met for gyroresonance between electrons the electromagnetic wave is
given by the relation
, where
is the axial component of the electron
velocity,
, and where the gyrofrequency on axis is
with
the axially-symmetric static magnetic
field. The on-axis field can be used
in the resonance condition in this case since radial excursions are much smaller
than both the Rayleigh length and the scale length
. Clearly, resonance can be maintained
along the orbit for an accelerated electron if the magnetic field is tailored
in space to track the variations in
. An equivalent way to write the resonance
condition is
, where
and where
are the normalized axial velocity
and effective index of refraction. When ensembles of particles with narrow variances
are considered, the resonance condition can be approximated as
, where the angle brackets indicate ensemble average values.
The above
field equations, Eqs. 1-4, together with the relativistic equation of motion
for the electrons
(6)
allow solutions
to be found for single-particle orbits. In the results of computations to be shown below,
iterative solutions for the position, velocity and energy of the particles
are found at each computational stage by specifying the change in guide magnetic
field value necessary to maintain resonance. This insures that the solutions are internally
consistent.
An example of predicted LACARA performance is shown in Fig. 2 for
an incident 10.6
laser power of 4 TW, an initial electron
beam energy of 50 MeV, a current of 1 A, and an initial normalized beam rms-emittance
mm-mrad [10]. In the computation,
total of 904 computational particles were injected, uniformly distributed
in optical phase, and in transverse phase space within emittance ellipses
having major and minor axes
and
, with the beam radius
= 0.1 mm and
. The waist radius
was chosen to be 1.0 mm, or approximately
100 optical wavelengths. This leads
to a Rayleigh length
= 29.64 cm. A mirror separation of
= 148.2 cm was chosen, with mirror radius of 85.95 cm. Fig. 2a shows the computed mean axial and transverse
normalized velocities 1-
and
as functions of distance along the
axis, showing that the transverse momentum never exceeds 0.63% of the axial
momentum; this insures that the motion remains well within the 2-mm diameter
optical waist. Furthermore,
throughout the interaction.
The electrons execute about five gyrations in traversing the 148-cm
inter-mirror distance, reflecting the strong Doppler down-shifted laser frequency
experienced by the electrons, due to the small value of
. Fig. 2b shows the mean relativistic
energy factor
and the magnetic field strength
both versus axial distance. It is seen that the mean beam energy is predicted
to rise monotonically from 50 MeV to 178 MeV in a distance of 148 cm, corresponding
to a maximum acceleration gradient at z = 75 cm of 147 MeV/m and an average acceleration gradient of 86.6
MeV/m. The resonance magnetic field
required rises from 52 to about 81 kG near the laser focus, and then falls
back to about 60 kG. It is the fall
in magnetic field beyond the focus that allows continuous acceleration without
stalling; this fall in magnetic field can be traced to the fall in
beyond the focus. This example demonstrates that LACARA is not limited to be a
-doubler. In fact, indefinite acceleration
beyond the focus is possible—albeit with ever-diminishing acceleration gradient.
In general,
computations for a range of laser power levels, waist radii and acceleration
lengths show that energy gain increases as laser power increases, but more
slowly than linearly; that energy gain falls with increasing initial beam
energy; that energy gain is approximately independent of waist radius when
the overall interaction length is held constant; and that average acceleration
gradient increases as waist radius decreases, provided the overall acceleration
length contains a constant number of Rayleigh lengths. For example, with
= 4.0 TW,
= 0.60 mm,
= 53.4 cm, and initial beam energy
of 50 MeV; one finds an energy gain of 65.6 MeV and an average acceleration
gradient of 123 MeV/m. For
= 1.0 TW, and other parameters unchanged, the average acceleration gradient
falls to 46.9 MeV/m. For
= 4.0 TW, but initial beam energy increased
to 80 MeV and other parameters unchanged, the average acceleration gradient
falls to 42.5 MeV/m. In yet another
example, acceleration from 0.50 GeV to 1.50 GeV is predicted for a 4.0 PW,
10.6 mm laser with a waist of 0.30 cm, over a distance of
= 16 m, for an average acceleration
gradient of 64.4 MeV/m. The magnetic
field for this example varies from 6 kG, up to about 24 kG, then down to 13
kG.
An accelerated
beam emerges from LACARA with electrons
on helical orbits. The number of gyrations
executed by a single electron during the interaction is few, but the gyration
phases for orbits of successive electrons advance rapidly, i.e. at the laser
frequency. Thus, if a beam stop with
an off-axis hole is interposed after acceleration, the transmitted beam will
be chopped at a frequency equal to the laser frequency.
A phase plot that illustrates this possibility is shown in Fig. 3a,
where two cycles of the
coordinates are plotted for an ensemble
of 18,080 electrons after acceleration through an interaction length of
cm , with other parameters the same
as in the example shown in Fig. 2. The
correlation between coordinate and time is evident, so that if an aperture
limits beam transmission in x, then
a temporally-chopped beam will emerge. Similar
considerations apply in y. A computed example of such an optically-chopped
beam is shown in Fig. 3b which shows the two cycles of beam current after
transmission through a 0.08 mm radius hole in a 2-cm thick tungsten beam stop.
In this example, the beam is accelerated from 50 to 110 MeV (corresponding
to an average acceleration gradient of 100 MeV/m) while the resonance magnetic
field varies between 54 and 67 kG. The
entrance aperture in the beam stop is centered at x
= -0.32 mm, y = 0; and the exit
is centered at x = -0.305 mm, y = -0.11 mm; so the beam channel is inclined
at an angle of 0.32 deg with respect to the z-axis. These limits in x are shown in Fig. 3. As is seen in Fig. 3b, the peak transmitted
current is 0.6 A (out of an incident 1.0 A) in a train of 3 fs (FWHM) bunches
with a period of 35 fs, or equivalently 0.9 mm bunches spaced by 10.6 mm. Such an optically-chopped
beam could find application as an injector for other laser-based accelerators,
in a harmonic generator of optical radiation, for generating fs x-ray pulses,
or in studies of excitation and lifetimes in electron-induced nuclear reactions.
This Brief
Report has presented computed predictions for the acceleration of electrons
in a circularly-polarized, focussed, CO2 Gaussian laser beam, under
conditions where gyroresonance is maintained along the electron trajectory.
For currently-achievable CO2 laser power levels in the multi-TW
range, acceleration gradients of over 100 MeV/m and overall acceleration in
150 cm of a 50 MeV beam to 178 MeV have been shown to be achievable for a
mild laser focus with a beam waist of 1.0 mm. Parameters selected for the examples presented
correspond to those soon to be available for a proposed experiment [10].
This laser-based accelerator treats all electrons in a bunch nearly
identically, providing the laser pulse width exceeds the bunch width.
The acceleration occurs in vacuum, without any proximate material medium—except
for a mirror to focus the laser. Experience
with the microwave CARA, where high
efficiency (>95%) for transfer of rf power to beam power has been observed
[11], suggests that the laser-driven version LACARA could be similarly efficient. It has also been shown that if a beam stop with
a small slightly-inclined beam tunnel is interspersed after acceleration with
LACARA, then an optically-chopped
beam can be produced that consists of a train of fs-bunches spaced by the
laser period. This simple idea is,
to the author’s knowledge, the only mechanism yet proposed for production
of a beam fully-chopped on an optical time-scale.
Acknowledgment: Constructive discussions with B. Hafizi, T. C. Marshall, M.
A. LaPointe and V. L. Bratman are acknowledged.
This research was supported by the US Department of Energy.
REFERENCES
[1]
P. Sprangle, E. Esarey, B. Hafizi, R. Hubbard, J. Krall, and A. Ting,
in Advanced Accelerator Concepts, edited by
S. Chattopadhyay, J. McCullough, and P. Dahl, AIP Conf. Proc. 398 (AIP, New York, 1996), p. 96.
[2]
K. Nakajima, et al, Phys. Rev. Lett. 74, 4428
(1995).
[3]
E. Esarey, P. Sprangle, and J. Krall, Phys. Rev. E 52, 5443 (1995); P. Sprangle, E. Esarey, J. Krall, and A. Ting, Opt. Commun. 124, 69 (1996); B. Hafizi, A. Ting, E. Esarey, P. Sprangle, and J.
Krall, Phys. Rev. E 55, 5924 (1997).
[4]
K. Nakajima, et al, in Advanced Accelerator Concepts, edited by
S. Chattopadhyay, J. McCullough, and P. Dahl, AIP Conf. Proc. 398 (AIP, New York, 1996), p. 83; D. Gordon,
et al, Phys. Rev. Lett. 80, 2133 (1998).
[5]
W. D. Kimura, et al, in Advanced Accelerator Concepts, edited by
S. Chattopadhyay, J. McCullough, and P. Dahl, AIP Conf. Proc. 398 (AIP, New York, 1996), p. 608.
[6]
W. D. Kimura, et al, in Advanced Accelerator Concepts, edited by
W. Lawson, C. Bellamy, and D. F. Brosius, AIP Conf. Proc. 472 (AIP, New York, 1998), p. 563.
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[8]
P. Sprangle, B. Hafizi, and P. Serafin, Phys.
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C. B. Schroeder, D. H. Whittum, and J. S. Wurtele, Phys. Rev. Lett. 82, 1177
(1999); F. Dorchies, et al, Phys. Rev. Lett. 82, 4655 (1999).
[9]
E. Esarey, P. Sprangle, and A. Ting, IEEE
Trans. Plasma Sci. PS-24, 252
(1996).
[10] Experiments based on the analysis presented in this Brief Report
have been proposed to Brookhaven National Laboratory Accelerator Test Facility.
(I. Ben-Zvi, private communication).
[11] J. L. Hirshfield, M. A. LaPointe, A. K. Ganguly, R. B. Yoder, and
C. Wang, Phys. Plasmas 3, 2163 (1996); M. A. LaPointe, R. B.
Yoder, C. Wang, A. K. Ganguly, and J. L. Hirshfield, Phys. Rev. Lett. 76, 2718
(1996).
[12] P. Sprangle, L. Vlahos, and C. M. Tang, IEEE Trans. Nucl. Sci. NS-30,
3177 (1983).
[13] A. Loeb and L. Friedland, Phys.
Rev. A 33, 1828 (1986).
[14] C. Chen, Phys. Rev. A 46, 6654 (1986).
[15] C.
Wang and J. L. Hirshfield, Phys. Rev.
E 51, 2456 (1995); Phys. Rev. E 57, 7184 (1998).
FIGURE CAPTIONS
Fig. 1.
Schematic diagram of LACARA.
The incoming Gaussian CO2 laser beam is focused by the left
mirror, travels with the accelerating electron beam, and is deflected out
of the beam path by the right mirror. Gyrations
of the beam orbit are too small to be seen on the scale of this diagram.
Fig 2. (a) Average normalized transverse velocity
and normalized axial velocity, plotted as
; and (b) average relativistic energy factor
and axial magnetic field
; as functions of axial coordinate z in LACARA. For this example the initial beam energy is
50 MeV, beam current is 1 A, Rayleigh length
cm, laser beam waist
= 1.0 mm, interaction length
cm, and laser power
TW. The acceleration gradient at z = 75 cm is 147 MeV/m.
Fig. 3.
(a) Phase plot for two optical
cycles in the
plane at
cm for an ensemble of beam particles with initial normalized rms-emittance
= 2.0 mm-mrad, after acceleration from 50 to 110 MeV. Horizontal dashed lines show limits in x of the aperture in the beam stop. A similar plot, shifted one quarter-cycle in
phase, depicts the
phase space. (b) Current transmitted through the 0.08 mm
radius tunnel in a 2-cm beam stop. Incident
current is 1.0 A. Microbunches in
this example have FWHM widths of 3 fs, with an interbunch spacing of 35 fs.
See text for details.
Fig. 1. Hirshfield and Wang
Fig. 2
Hirshfield and Wang
Fig. 3a is not reproduced in this version.
Fig. 3
Hirshfield and Wang