Omega-P, Inc., Suite 100, 345 Whitney Avenue, New Haven CT 06511; and

Beam Physics Laboratory, Yale University, New Haven CT 06520


J. L. Hirshfield, principal investigator

tel:  (203) 432-5428

fax:  (203) 432-6926








May 15, 2000






            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.





            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.





            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.







            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




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.







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









Task b









Task c









Task d









Task e









Task f









Task g









Task h









Task i









Task j









Task k









Task l









Task m










Schedule of tasks. 

*This task is expected to be completed prior to the start of Phase II.









            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.







            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 Yale University facilities, including libraries, on account of their official Research Affiliate status.  But for the project proposed here, experimental work will be carried out Brookhaven National Laboratory Accelerator Test Facility.  Facilities at BNL-ATF are described on the BNL-ATF web-site, at


            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 Bethlehem, PA, as discussed above.  Other minor items, such as the beam pipe and associated mirror mounting and alignment fittings, will be fabricated by Omega-P in shops available to it, or will be procured from local vendors.






            The research institution collaborating directly with Omega-P on this project is Yale University.  Yale is particularly well-suited as a collaborator with Omega-P on development of LACARA because of its long working relationship with Omega-P, and because of the close proximity between personnel and facilities of the two organizations.   But, at least as important, is the availability for this research program of Dr. Changbiao Wang, Associate Research Scientist in the Yale Physics Department, who will be Principal Investigator for the Yale effort.  Dr. Wang has had extensive experience with LACARA, as one of the originators of the idea (see reference on p. 4).  Dr Wang carried out the simulation studies that led to the theoretical predictions and designs given in this proposal.  Examples of some of his publications can be found in the list given in the Phase I proposal.  Dr. Wang graduated with a Ph.D. in electrophysics from University of Electronic Science and Technology of China, Chengdu, Sichuan in 1987, and worked there as a postdoctoral researcher for two years.  From August 1991 to December 1993, he worked as a visiting researcher in the Center for Beam Physics at Lawrence Berkeley Laboratory.  Since January 1994 he has been at Yale University, as Associate Research Scientist in the Department of Physics.  His research interests are in microwave electronics, relativistic electron beam physics, two-beam acceleration and cyclotron autoresonance accelerators.  He is expected to devote approximately 40% of his time to this research program, working on evaluation of LACARA over a wide range of parameters, on analysis to support the experiments, and on the future multi-stage version of LACARA  The proposed sub-contract to Yale includes provision for Dr. Wang, and an allowance for travel to scientific meetings, as shown in the following Yale budget.


                                                                                    9/00 – 8/01                  9/01-8/02


            Dr. Changbiao Wang, PI (40%, 41.6%)             

            Fringe benefits/hospitalization               


               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.







            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, Yale University, New Haven, CT 06520-8120

2Omega-P, Inc., Suite 100, 345 Whitney Avenue, New Haven, CT 06511










            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)






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



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.





[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).


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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