PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS, VOLUME 6, 014202 (2003)

Design and simulation of a multibeamlet injector for a high current accelerator

David P. Grote*LLNL, University of California, Livermore, California 94550

and Virtual National Laboratory for Heavy-Ion Fusion

Enrique Henestroza and Joe W. KwanLBNL, University of California, Berkeley, California 94720

and Virtual National Laboratory for Heavy-Ion Fusion(Received 9 September 2002; published 8 January 2003)

014202-1

A multibeamlet approach to a high current ion injector, whereby a large number of beamlets areaccelerated and then merged to form a single beam, offers a number of potential advantages over amonolithic single beam injector. These advantages include a smaller transverse footprint, more controlover the shaping and aiming of the beam, and more flexibility in the choice of ion sources. A potentialdrawback, however, is a larger emittance. In this paper, we seek to understand the merging of thebeamlets and how it determines the emittance. When the constraints imposed by beam propagationphysics and practical engineering issues are included, the design is reduced to a few free parameters. Wedescribe the physics design of a multibeamlet injector and produce a design for an example set ofparameters. Extensive use of 2D and 3D particle simulations was made in understanding the injector.Design tolerances and sensitivities are discussed in general and in relation to the example.

DOI: 10.1103/PhysRevSTAB.6.014202 PACS numbers: 41.85.Ar, 41.75.Ak, 29.27.Ac, 29.27.Bd

to this scaling problem is to use multiple beams,so ITotal / NJ

�3=5, where N is the number of beams.normalized free energy. Defining the occupancy factor� � 2a=�r where a is the beamlet radius and �r is the

I. INTRODUCTION

The requirement of increased particle beam luminosityhas driven the need for beam sources which simulta-neously have high current and low emittance. The prac-tical limits in single beam sources, however, limit thebrightness that can be obtained. We demonstrate here thatthe limits on brightness can be circumvented by utilizinga multibeamlet injector, showing that the scaling of mul-tibeamlet injectors allows better sources. We also showthat a multibeamlet injector can have other practicaladvantages, such as better control over the beam.

The traditional approach in a single beam ion source isa Pierce gun geometry. See, for example, [1]. The beam iscreated in a diode and accelerated to the desired energyand then must be matched into the subsequent transportchannel. The current density in an ideal parallel planediode is limited by the Child-Langmuir relation,

JCL /V3=2

d2 ; (1)

where V is the voltage drop across the diode whose lengthis given by d. The total current is then ICL � AJCL, whereA is the area of the emitting surface. Minimizing geo-metric aberrations limits the maximum value of �A=d2�.Furthermore, voltage breakdown limits the voltage to beproportional to

���d

pfor d greater than a centimeter [2,3].

These constraints result in a total current that variesinversely with the current density, ICL / J�3=5

CL —thehigher the desired current, the lower the current density,and therefore the larger the size of the emitter. A solution

CL

1098-4402=03=6(1)=014202(12)$20.00

Holding the total area of the emitter, ATotal / NA, tobe constant then leads to an increasing current densitywith increasing total current, since ITotal � NICL �NAJCL. The beams are subsequently merged to form asingle beam which propagates into the transport channel.Note that the scalings used here are independent of theparticle mass.

A major design requirement is the production of a beamwith low emittance. When the beamlets merge (hence-forth the individual beams will be referred to as ‘‘beam-lets’’ and the merged whole as the ‘‘beam’’) the emptyphase space between the beamlets can become entrainedin the beam, possibly resulting in a significantly largeemittance. The final emittance can be estimated by as-suming that all of the available free energy goes intoemittance. The free energy here is defined as the differ-ence between the self-field energy of the beamlet con-figuration and that of a single, uniform beam with thesame perveance and rms radius as the combined beam-lets. The free energy is usually normalized by the self-field energy inside the equivalent uniform beam. The finalemittance has been calculated for a configuration ofbeamlets in concentric circles assuming that the totalbeam is matched, centered, and properly aimed [4].

"2xf

"2xi

�hX2i hY2i

2hX2i

�1

QUn8hX02i

�: (2)

Here, "xf and "xi are the final and initial emittances, hX2i,hY2i, and hX02i � hY02i are the mean square beam sizesand velocities, Q is the total perveance, and Un is the

2003 The American Physical Society 014202-1

PRST-AB 6 DESIGN AND SIMULATION OF A MULTIBEAMLET . . . 014202 (2003)

center-to-center spacing of the concentric rings of beams,�0<�< 1�, an approximate expression for the normal-ized free energy was written in terms the occupancyfactor and the number of beams, N.

Un �4

N

�3

4� ln3 � ln�

3

8�2

�: (3)

These relations are expected only to give qualitativescalings since the beam conditions are outside the rangeof applicability. There is no focusing in the mergingregion for the beam to be matched to and the beamletswill be converging and accelerating.

These relations do show that the emittance growthafter the merge can be reduced by the following.

(i) Increasing the energy at the merge location, whichdecreases Q.

(ii) Decreasing the spacing between beamlets, whichincreases � and so decreases Un (note that the ln� termdominates).

(iii) Increasing the number of beamlets, which alsodecreases Un.

Use of these three criteria leads to a design wherein alarge number of small beamlets are first accelerated inseparate channels which are placed as close to each otheras engineering constraints will allow. In the next section,the details of the design of such an injector are presentedand an example given. In the last section, analytic esti-mates of the merged emittance are made using energyconservation.

II. INJECTOR DESIGN

The basic layout of the injector consists of two sec-tions, the preaccelerator and the merging region; see Fig. 1.In the merging region, the beamlets are both merged andaimed so that the combined beam enters the transportchannel. The merging region can also be used to furtheraccelerate the beam up to the design value of the transportchannel. The ability to aim the beamlets can be exploitedby arranging for the combined beam to enter the channel

FIG. 1. (Color) Layout of the multibeamlet inindependently and then merged. The total beam i

014202-2

exactly matched to it. In an alternating gradient, quadru-pole focusing channel, for example, if the first quadru-pole is half-length, then the combined beam is matched ifit has zero convergence and the correct transverse dimen-sions. This offers a substantial advantage by eliminatingthe traditional matching section and its associated prob-lems. The matching section typically must transverselycompress a large beam to fit into the smaller transportchannel. The beam manipulations to do the compressionoften have been found to cause emittance growth.Eliminating the matching section also can reduce thetransverse footprint since, in the traditional case, match-ing a round beam to an elliptical beam in the alternatinggradient transport channel can require large excursions inthe beam envelope [5].

For the preaccelerator, a novel Einzel lens design isused, a series of alternating gradient Einzel lenses with anet accelerating voltage. The voltage differences betweenaperture plates alternate in sign and there is a net voltagedrop along the column. The beamlets are accelerated inparallel channels. The Einzel lenses confine the beamletstransversely against space-charge repulsion and thermalexpansion. The focusing strength of such lenses dimin-ishes with increasing beam energy however, so the lengthof the column is limited. This limit must be balancedwith the desirability of merging at higher energy. Thebeamlet size, for a given beamlet current, is constrainedby the focusing strength of the Einzel lenses, whichdepends on the voltage differences and distance betweenplates, the aperture radius, and the beam energy.

To minimize emittance growth from the merging, thebeamlet channels are placed as close to each other aspossible. Some a minimal amount of material is neededbetween the apertures for mechanical stability, prevent-ing excessive warping from mechanical and thermalstresses. Also, the beamlets must be far enough awayfrom each other, compared to the aperture plate separa-tion, so that the electric forces between the beamlets areshielded and are small. Transverse offsets of the apertures(due to construction errors and misalignments) give the

jector. Each of the beamlets is accelerateds matched to the subsequent transport lattice.

014202-2

TABLE I. Parameters required at the entrance to the exampletransport lattice.

Ion K

Generalized perveance 0.001�0, � 65�, 7�

Energy 1.6 MeVCurrent 0.57 AEmittance 1�mmmradBeam major and minor radii 2.0, 1.3 cm

PRST-AB 6 DAVID P. GROTE, ENRIQUE HENESTROZA, AND JOE W. KWAN 014202 (2003)

beamlets transverse kicks, moving them off axis.The apertures must be large enough so that, with thebeamlets off axis due to the errors, they do not touchthe apertures.

The end conditions of the merging section are given bythe matched beam parameters in the transport channel—the current, energy, and beam size and divergence. Withfixed rms beam size at the start of the merge (averagingover all of the beamlets), and fixed beam size and diver-gence at the start of the transport, the transverse rmsenvelope solution for the propagation between the twopoints is unique. This sets unique values for the length ofthe merging region and the beam rms convergence at thestart of the region. Note that the unique solution dependson any acceleration in the merging region. The uniquesolution can be found by following the beam backwardfrom the start of the transport channel—the transverserms envelope expands freely to match to the size of thebeamlet arrangement of the final plate of the acceleratingcolumn. Note that the length of the merging region in-creases with the merging energy. This puts a lower limiton the merging energy—with decreasing merging energy,the increasing acceleration needed to get to the designenergy of the transport channel must happen over adecreasing merging length and is limited by voltagebreakdown constraints.

Given the constraints, there are two primary free pa-rameters in the design: the number of beamlets and theenergy at which the beamlets are merged. The number ofbeamlets, N, along with the transport channel current,gives the current per beamlet. The focusing strength ofthe Einzel lenses, as well as the maximal current densityobtainable from emitters, constrains the beamlet radiusfor that current. The beamlet radius, the clearance spacebetween the beamlet and aperture, and the amount ofmetal between apertures all then determine the transversesize of the configuration of N beamlets. The uniquesolution to the envelope equation in the merging regionthen provides the length of the region, as well as thebeamlet aiming angle at the final aperture plate. Thisnearly completes the design.

There is flexibility with some parameters, allowingoptimization of the final emittance of the merged beam.For example, the transverse convergence of the individualbeamlets as they exit the accelerating column can bevaried. Ideally, if the individual convergences of thebeamlets are the same as the aiming convergence at thepoint where the beamlets just touch each other, the emit-tance would be minimized. In the transverse phase space,this can be visualized as closing the blinds on a window,where the slats lay against each other to form a nearly flatsurface with no holes. However, because of the dynamics,including space-charge effects, the optimal convergencefor a given configuration is different from that value andis difficult to estimate analytically. It is straightforward,though, to compute specific cases via simulation.

014202-3

The emittance of the individual beamlets at the sourceis also not constrained. The emittance depends mostly onthe type and temperature of ion source used. Two poten-tial sources are surface ionization and plasma sources [1],which have different ion temperatures. As will be shownin the example, for small to modest source temperatures,the final emittance is only weakly dependent on thesource emittance. This insensitivity can allow muchmore flexibility in the choice of ion source. We assumehere that there will be negligible emittance growth as thebeamlets propagate through the accelerating column,which is borne out by simulation.

III. INJECTOR EXAMPLE

As an example, a multibeamlet injector suitable for anintermediate term facility for heavy-ion fusion is de-signed. The principal parameters desired at the beginningof the transport are given in Table I. The most stringentdesign parameter is the low emittance. The goal of thedesign will be keeping the emittance as low as possiblegiven the other constraints on the design. The acceleratorwill be a multibeam accelerator with the beams acceler-ated side by side. Note that cost optimization requires thebeams to be packed closely together. In an electric trans-port lattice, the beams share electrodes with their neigh-bors. Ideally, the beam lines would follow a straight linepath from the injector into the transport line, thus greatlyreducing the complexity (no bends) and the cost. Thetransverse footprint of the injector would then have tobe less than or equal to the transverse size of the transportline. In the example case here, this will be roughly 7 cm inradius.

As a reference case, we choose 91 beamlets (five ringsplus the center) merging at 1.2 MeV. See Fig. 2(a) for thelayout of the beamlets. For a system of beamlets asassumed in deriving Eq. (3), 100 beamlets is just beyondthe ‘‘knee’’ in the curve where more beams give littlefurther decrease in the merged emittance.When using theconfiguration of concentric rings of beamlets plus the oneat the center, the number of beamlets is given by N �1 3M�M 1�, where M is the number of rings aroundthe center beamlet. With M � 5, there are 91 beamlets.The beamlet radius, r, can be estimated by balancing thefocusing force of the Einzel lens with the space-charge

014202-3

0.0 0.5 1.0 0

1

2

3

4

−4 −2 0 2 4−4

−2

0

2

4

Z (m)

X, Y (cm)

Merge Region

Transport

Half Length

0.4 MV drop

Lattice

Quad

Envelope Solution

X (cm)

Y (cm)

a) b)

FIG. 2. The configuration of apertures at the last accelerating plate. Each aperture has aradius of 2 mm. The elliptical arrangement allows the combined beam to be exactly matchedinto the quadrupole transport channel.

PRST-AB 6 DESIGN AND SIMULATION OF A MULTIBEAMLET . . . 014202 (2003)

force [6]:

r2 �9

8�"0I

����������m2qV

r �L

�V

�2: (4)

Here I is the current per beamlet, V is the beamlet energyin volts, �V is the voltage difference between plates, L isthe plate separation, andm and q are the mass and chargeof the particles. With the values for this example at theend of the accelerating column, where V � 1:2 MeV, anda voltage gradient, �V=L, of 100 kV=cm, gives r�1 mm. With a clearance of 1 mm between the beamletand aperture, the aperture radius is 2 mm. Another pa-rameter is the distance between beamlets, which is con-strained by the need for mechanical rigidity of theaperture plates as well as the need for electrostatic shield-ing between the beamlets. It was felt that 2 mm was areasonable value, giving enough material to support thearray of aperture holes. This gives a total of 6 mm center-to-center ring separation. The design of the preacceleratorcolumn is described in the next section. The design of themerging section, discussed immediately below, assumesbeamlets with those parameters.

A. Merging region

The transverse envelope equation solver in WARP [7]was used to find the unique solution in the merging region.The parameters fixed by the design are the minimum rmsradius (set by the minimum beamlet-to-beamlet spacing),the acceleration (up to the desired total injection energy),and matching into the transport lattice. The beam sizeis held fixed in one direction to the minimum rms radius.In the other, it is allowed to increase, giving an ellipticalarrangement. The free parameters are then the initialconvergence angle in the two planes, the major radii ofthe ellipse, and the length of the merging region. Forthe case with 91 beamlets, 6 mm beamlet-to-beamlet

014202-4

spacing, and 1.2 MeV merging energy, the envelope solu-tion gives major and minor radii of 39.7 and 32.7 mm,convergence angles of 51.4 and 53.1 mrad, and a length ofthe merging region of 0.7378 m. The configuration ofbeamlets is shown in Fig. 2(a). Figure 2(b) shows theenvelope solution in the merging region and into the firstfew quadrupoles. The beamlets along each transverse axisconverge toward a point. However, with the ellipticallayout, the focus is astigmatic, the focal plane for thetwo axes are different. The convergence angles also givethe aiming angles of the accelerating columns. Because ofthe aiming, each aperture plate is curved, with radii ofcurvature (different in the two transverse axes) equalto the outer semiaxes of the beamlets divided by theconvergence angle, which are approximately 0.77 and0.62 m, respectively, for the last plate of the preacceler-ator column.

To characterize the design, transverse slice simulationsof the merging region and following transport were car-ried out using the WARPxy code [7]. For most of theparameters, there is a monotonic change in emittance asthe parameter is varied. Other constraints, such as voltagebreakdown, place limits on the parameters—the goal isto minimize the emittance within the constraints. Theindividual beamlet convergence angle at the start of themerge region, however, has, for a given set of the otherparameters, an optimal value minimizing the emittance.For this reason, for each set of parameters studied, arange of convergences was examined to find the mini-mum. Some of the data presented here are graphs ofemittance versus convergence angle. Note that for thesesimulations, it was assumed that an accelerating columncould be designed which produced beamlets with thecorrect parameters.

Some care was needed in setting the resolution ofthe slice simulations. It was found that fairly high reso-lution was needed to reach numerical convergence. The

014202-4

z=4.1 m

z=0.5 m

z=1.9 m

X’

X’

X’

Y

Y

Y

X X

FIG. 4. (Color) Density plots of (x-y) and (x-x0) spaces. In the(x-x0) plots, the gray scale is brightened. All densities greaterthan one-quarter of the maximum are drawn in white. Fromthere, the gray scale linearly goes down to zero. The z is thedistance from the end of the preaccelerator column.

0 0.5 1.0 0 20 40 600

0.5

1.0

Z (m)

0

0.5

1.0

Z (m)

Nor

mal

ized

em

ittan

ce (

m

m m

rad)

Nor

mal

ized

em

ittan

ce (

m

m m

rad)

a) b)

End of merge

Y

X

End of merge

120 Lattice periods

π π

FIG. 3. Long term emittance behavior after merging. (a) Detail of the emittance over thefirst meter. (b) History over 120 lattice periods.

PRST-AB 6 DAVID P. GROTE, ENRIQUE HENESTROZA, AND JOE W. KWAN 014202 (2003)

individual beamlets needed to be well resolved. In thesimulations discussed here, the mesh size was 512� 512,giving about ten grid cells across the radius of eachbeamlet. To reduce particle noise, 20 000 particles wereused for each beamlet. Higher resolution and particlenumber does not significantly alter the result. Thebulk of the simulations were carried out on the IBM SPat NERSC and typically required 15 min each on 16processors.

As the beamlets merge, the overall emittance changesrapidly, reaching its final value just as the beam enters thefirst quadrupole, showing a rapid mixing of the beamlets.Figure 3 shows the emittance history of that case withmerging at 1.2 MeV, beamlet radii of 1.5 mm, 0.5 mmclearance, and convergence of �3:5 mrad. Simulationscovering several hundred additional lattice periods showno further emittance growth. The x and y emittancesinitially rise to different values because of the ellipticalarrangement of beamlets, differing by as much as 50% insome cases. As the beam propagates further, the emittan-ces equilibrate to approximately the average value in afew undepressed betatron periods. Others have shown thatin some cases when a small number of beamlets aremerged, the initial pattern reappears, the location de-pending on the depressed phase advance [8]. Such reap-pearance has not been observed in the simulations carriedout here.

The merging of the beamlets creates complex space-charge waves that propagate across the beam and mix.Figure 4 shows contours of density at several locations inconfiguration space (x-y) and in phase space (x-x0). Theyshow that the individuality of the beamlets is quickly lost,leaving a nearly uniform beam with short wavelengthperturbations. Those perturbations diminish after severaldepressed betatron periods and do not cause any furtherincrease in emittance. The merging process does producea small halo population. In the simulation of the typicalcase, after propagating about one depressed betatron

014202-5

period beyond the merge, 10 m in this case, about 1% ofthe beam is in the halo.

The merged emittance is examined, varying the twoprincipal parameters: the number of beamlets and the

014202-5

A

A

A

B

B

C

C

DD

D

EE

E

A 1.4 MeV (1.6 MeV final)B 1.2 MeV (1.6 MeV final)C 0.8 MeV (1.6 MeV final)D 0.6 MeV (1.6 MeV final)E 0.5 MeV (0.5 MeV final)

−8 −6 −4 −2 00.0

0.5

1.0

1.5

Initial beamlet convergence (mrad)

Nor

mal

ized

em

ittan

ce (

m

m m

rad)

π

FIG. 5. (Color) Results from 2D slice simulations; emittance isshown versus the initial beamlet convergence angle. Comparescases with differing merging energies, for 91 beamlets andclearance rc � 0:5 mm.

PRST-AB 6 DESIGN AND SIMULATION OF A MULTIBEAMLET . . . 014202 (2003)

energy at which the beams are merged. Figure 5 shows theresults of several series of simulations with varying merg-ing energy. Keeping the configuration of beamlets emerg-ing from the accelerating channel unchanged, the finalemittance of the beam is seen to decrease with increasingmerging energy as expected. The beamlet configuration isas determined above, but with a 0.5 mm clearance be-tween the apertures of the last plate and the beamletradius— the beamlet radii are 1.5 mm. The length of themerging region decreased with decreasing merging en-ergy. The case with merging at 0.5 MeV is included as apossible design that might be used as a proof-of-principleexperiment.

A

AA

B

B

B

C

C

C

C

C

C

DD

D D

DE E

E

E

E

A 8 rings (169 beamlets)B 7 rings (127 beamlets)C 6 rings (91 beamlets)D 5 rings (61 beamlets)E 4 rings (37 beamlets)

−8 −6 −4 −2 0Initial beamlet convergence (mrad)

Nor

mal

ized

em

ittan

ce (

m

m m

rad)

0.0

0.5

1.0

1.5a) b

r = 0.5 mm

r = 1.0 mmc

cπ

FIG. 6. (Color) Results from simulations

014202-6

Figure 6 shows results with a varying number of beam-lets. In Fig. 6(a), the number of beamlets is varied,holding fixed the initial charge density of the beamletsand rc; the clearance between the beamlets and the aper-tures (the difference in radii of the two). As the number ofbeamlets increases, the current per beamlet decreases,and therefore the beamlet size decreases. Holding theclearance fixed, the overall size of the configuration in-creases with an increasing number of beamlets. Note thatcases with more than 91 beamlets have a larger footprintthan the 7 cm limit. The emittance is seen to decreasewith an increasing number of beamlets, as expected.Withthe charge density held fixed, the Einzel lens scaling,Eq. (4), gives a constant focusing strength. The initialtemperature of the beamlets was taken to be 0.17 eV.

In Fig. 6(b), the number of beamlets was varied hold-ing fixed the center-to-center beamlet spacing. To in-crease the number of beamlets, a new ring of beamletswas added to the outside of the configuration, increasingthe overall size. Note that cases with fewer than 91beamlets were not examined since the increased currentper beamlet cannot be confined by the Einzel lenses.

These results show that there is quite a bit of flexibilityin the design of the system without significant impact onthe resulting emittance. The parameter with the mostsignificant impact is the beamlet size. Increasing thenumber of beamlets gives only a small decrease in emit-tance as expected. Note also that in both cases, the widthof the minimum becomes smaller with more beamlets.So, depending on how well the preaccelerator can ac-tually be designed to give beamlet parameters near theminimum, the emittance advantage of more beamletsmay further decrease.

Figure 7 shows the results from a series of simulationswith varying beamlet size but with constant clearance,and with 91 beamlets merging at 1.2 MeV. The overall

A

A

A

A

A

A

B

B

B

B

B

B

C

C

CC

C

C

A 6 rings (91 beamlets)B 7 rings (127 beamlets)C 8 rings (169 beamlets)

−8 −6 −4 −2 00.0

0.5

1.0

1.5

Initial beamlet convergence (mrad)

Nor

mal

ized

em

ittan

ce (

m

m m

rad)

r = 1.5 mm

r = 1 mm

)

π

with a differing number of beamlets.

014202-6

A

AAB

BC C

A beamlet radius = 2.0 mmB beamlet radius = 1.5 mmC beamlet radius = 1.0 mm

−8 −6 −4 −2 00.0

0.5

1.0

1.5

Initial beamlet convergence (mrad)

Nor

mal

ized

em

ittan

ce (

m

m m

rad)

π

FIG. 7. (Color) The beamlet radii are varied while the clear-ance is held to a constant 0.5 mm.

Convergence (mrad)

C

C

B

B

AA

C

C

C

A

CB

BA

AB

BA

−8 −6 −4 −2 0 0

1

2

C 1.062 AmpsB 0.572 AmpsA 0.286 Amps

Nor

mal

ized

em

ittan

ce (

m

m m

rad)

π

FIG. 9. (Color) Emittance versus convergence angle for variousvalues of current, for the same number of beamlets, merging at1.2 MeV. The upper curve in each pair is with beamlet radii of1 mm, and the lower is with 1.5 mm.

PRST-AB 6 DAVID P. GROTE, ENRIQUE HENESTROZA, AND JOE W. KWAN 014202 (2003)

configuration size increases with increasing beamletradii. This further shows the significance of the beamletsize on the emittance.

Also examined is the initial temperature (individualemittance) of the beamlets. Using the configuration asdetermined above (91 beamlets merging at 1.2 MeV), aseries of simulations was done, varying the initial tem-perature of the beamlets. The final emittance, as shown inFig. 8, is quite insensitive to the initial emittance of thebeamlets. This allows much flexibility in the type ofsource used. For example, plasma sources, which typi-cally have a higher operating temperature than othersources (�1:2 eV), but still within the range examined,could be used.

A r = 1.5 mm, 3.5 mradB r = 1.0 mm, 3.5 mradC r = 1.0 mm, 1.0 mrad

0.0

0.5

1.0

1.5

Source temperature (eV)0.0 2.0 4.0 6.0 8.0

A

A

C

A

B

C B

Nor

mal

ized

em

ittan

ce (

m

m m

rad)

π

FIG. 8. (Color) Variation of final emittance with a range ofinitial source temperatures.

014202-7

Alternative designs of the accelerator may require ahigher or lower current per beam. Figure 9 shows thescaling of the emittance versus convergence angle fordifferent currents. Here, the total number of beamletswas held fixed (to 169, eight rings) and the current perbeamlet was adjusted appropriately. The merged emit-tance is roughly proportional to the current. The bright-ness (current divided by emittance squared) then variesinversely proportional to the total current.

In order to examine sensitivities and tolerances in thedesign, transverse slice simulations of the merging regionwere carried out with errors in the initial position, aimingangle, size, and convergence of the beamlets. The pri-mary measure of interest is the resulting beam emittanceafter the merge. In all cases, the emittance was found tobe insensitive to the errors. With up to 0.7 mm errors inbeam position and size, 0.4 mrad in aiming angle, and0.7 mrad in beam convergence, the largest increase inemittance over the case with no errors was of the orderof 20%. It is expected that the errors in the acceleratingcolumn are all within that range. The effect of nonuni-form density distribution in the beamlets was also exam-ined. Distributions examined range from severely hollow(zero density on axis) to moderately peaked (50% higherdensity on axis than edge). As with the other errors, theemittance was insensitive, varying only by 5%–10%.Curiously, the emittance drops monotonically with in-creasing hollowness. The effect is small though, droppingonly 10% for the most severely hollow beam.

B. Preaccelerator

In designing the preaccelerator, the same base case willbe used as above, i.e., with 91 beamlets. Present-day

014202-7

0.0 0.1 0.2 0.30

1

2

1

2

0.1 0.2 0.3 0.40.00

5375041100 954 846 264 488 226306626 2801000 969 609 462 0

4783967311100 732 265 461

Z (m)

r (mm)

b)a)

Z (m)

r (mm)

211400600225563 372

0187350321

507419

591679

697

700

700

FIG. 10. (Color) Example layout of the aperture plates with the beam envelope. (a) Merging at1.2 MeV for a case with 91 beamlets. (b) Merging at 0.8 MeV for a case with 169 beamlets. Bothcases are well optimized. The voltages of the plates are given in kV.

PRST-AB 6 DESIGN AND SIMULATION OF A MULTIBEAMLET . . . 014202 (2003)

sources can emit reasonably well on the order of100 mA=cm2 [9]. To produce the specified current perbeamlet at that current density, sources of radius1.414 mm are required. See Fig. 10 for example layoutsof the aperture plates for the 1.2 MeV and 0.8 MeV cases.

In designing the preaccelerator, there is a balance be-tween getting a large beamlet size and getting the con-vergence angle close to the optimum (minimizing theemittance). The design becomes more constrained whenincluding clearance between the beamlet and aperture,which is needed to allow for beamlet movement off axisdue to construction errors. In the design shown inFig. 10(a), the beamlet size at the exit of the column is1.22 mm and the convergence is �1:19 mrad (i.e., con-verging), which is near the optimal value for this case.The merged emittance is estimated to be 1:11�mmmrad.Given the limited focusing strength of the Einzel lenses,it appears that the convergence cannot be further opti-mized without decreasing the beamlet size, and viceversa. Going to higher merge energy may improve thedesign, since the optimal convergence angle decreaseswith merge energy (see Fig. 5), but the focusing getsweaker at the end of the column and so the achievableconvergence angle decreases.

To optimize the preaccelerator design, the multidimen-sional functional minimization algorithm SimultaneousPerturbation Stochastic Approximation was used [10].The distances between the plates and the voltages on theplates were allowed to vary (except for the first threeplates which were held fixed). This gave either 28 or40 parameters (for 14 or 20 plates varied) for the twocases with the merging energies of 0.8 and 1.2 MeV. Thecalculation was done using an envelope solver and in-cludes the focusing from the Einzel lenses and the accel-eration. The field of the first three plates is obtained froma 3D Laplace solution. For the rest of the plates, the radialfield given by the paraxial approximation is used.

Er � 0:5r�E�z � E

z �=l: (5)

014202-8

Here, r is the beamlet radius, E�z is the axial field to the

left of the plate, Ez is the field to the right, and l is an

arbitrary length of the lens, taken in the calculation hereto be the width of the plates, 1 mm. The accelerationbetween plates is taken to be uniform. The axial fieldsare calculated knowing the voltage drop between platesand the plate separation. The envelope solution is initiatedat the first plate using parameters obtained from steady-state particle simulations. Note that in the particle simu-lations, the emitter radius needed to be adjusted to obtainthe desired current in each case since the injected currentdensity was affected by the emitter radius. For the enve-lope calculation, the first three plates are held fixed so thatthe starting values from the particle simulation wouldbe fixed.

Transverse slice (2D) particle simulations were used toobtain the final merged emittance as a function of thebeamlet radii and convergence angles, and polynomial fitswere made of the data. Using the fits, the merged emit-tance then could be estimated given the final beamlet sizeand convergence from the envelope solution of the pre-accelerator. The optimization minimized this estimatedemittance, while maintaining a clearance of 0.5 mmbetween beamlet and aperture and limiting the voltagegradients. (The reasons for the choice of 0.5 mm for theclearance are discussed in the next section.) Also, whilethe locations of the plates were allowed to vary forflexibility, the optimizer minimized the rms average ofthe plate separation, leading to nearly uniform platespacing. The results from several optimized designs areshown in Table II. Note that the designs are not neces-sarily completely optimized, but are close. For example,the plate locations are not exactly evenly spaced.

The results show that, given the constraints, the merg-ing beamlet configuration can achieve a normalizedemittance near or below 1�mmmrad. There is little dif-ference in the emittances with the differing mergingenergies. Even though the emittance will be lower withhigher merging energy given the same beamlet size and

014202-8

TABLE II. Optimized aperture columns. Two cases were examined, with differing limits on the voltage gradients. In the first case(a), the gradient is independent of the plate separation. In the second case (b), the gradient depends on the plate separation, d, givenhere in centimeters.

(a) max Ez � 100 kV=cm (b) max Ez � 100 kV=cm � d2=3

Source radius Merge energy a a0 "n a a0 "n

# Beamlets (mm) (MeV) (mm) (mrad) (�mmmrad) (mm) (mrad) (�mmmrad)

91 1.454 1.2 1.22 �1:19 1.11 1.35 1.00 1.410.8 1.16 �2:33 1.17 1.20 0.04 1.39

127 1.125 1.2 1.18 �1:70 1.01 1.14 �0:79 1.150.8 1.18 �3:07 1.04 1.13 �1:57 1.19

169 1.099 1.2 1.23 �2:02 0.89 1.18 �0:76 1.100.8 1.18 �2:80 0.99 1.10 �1:86 1.12

PRST-AB 6 DAVID P. GROTE, ENRIQUE HENESTROZA, AND JOE W. KWAN 014202 (2003)

convergence, the column design is able to get closer to theoptimum for the lower energy case since the focusing isstronger at the end of the column. The cases with addi-tional rings were included to examine what advantagecould be gained by increasing the configuration sizebeyond the footprint limit. Note that the center-to-centerspacing was held fixed, so the outer radius increases withthe square root of the number of beams.

C. Preaccelerator errors

Various errors were applied to the preaccelerator, suchas voltage errors, transverse and longitudinal apertureoffsets, and aperture radius errors. For each type of error,a large ensemble of calculations with differing randomseeds was done for a range of magnitudes of maximumerrors. Each individual error was chosen from a uniformrandom distribution within the range of plus/minus themaximum error. The beamlet was found to be fairlyinsensitive to all but the transverse aperture offsets.With a maximum error of 700 m in the longitudinalplate location and aperture radius, and a maximum

0.0 0.1 0.2 0.30.0

0.5

1.0a)

Z (m)

0.05 mm

0.10 mm

0.15 mmMax offs

2 R

MS

Bea

mle

t cen

troi

d (m

m)

FIG. 11. Simulation results of a single beamletTwice the rms average of each ensemble versus z.root of the number of elements. (b) Twice theaperture offset.

014202-9

error of 7 kV in the plate voltages, the beamlet size,convergence, and emittance did not vary by more than15%, and in no cases were particles lost to the apertureplates.

With aperture offset errors, the beamlet is kicked offaxis, leading in some cases to partial or complete beamloss. Figure 11(a) shows twice the rms beamlet offsettaken over each ensemble for three different maximumoffsets. (Assuming that the beamlet offsets will follow aGaussian distribution, the rms offset is equivalent to �,the Gaussian width. So, the probability that the offset willfall within twice the rms value, 2�, is 95.4%.) The finalvalues of the beamlet offsets are show in Fig. 11(b). Thesedata were generated using an envelope and centroid equa-tion solver. This gives good agreement with the particlecode and is much faster, allowing a very large number ofsample cases (with differing random seeds). This as-sumes that the offsets are small enough to scale linearlyand that the beam does not hit an aperture. The field iscalculated for each aperture plate with a small offset, andthe dipole component is extracted and saved. Then, foreach calculation, the dipoles are scaled by a random

0.00 0.05 0.10 0.15 0.200.0

0.5

1.0b)

Maximum aperture offet (mm)

et

2 R

MS

Bea

mle

t cen

troi

d (m

m)

channel with transverse aperture offsets. (a)The offsets increase roughly with the squarefinal rms beamlet offset versus maximum

014202-9

PRST-AB 6 DESIGN AND SIMULATION OF A MULTIBEAMLET . . . 014202 (2003)

offset. The focusing and acceleration components areassumed to be unchanged by aperture offsets.

A small number of cases were run with the three-dimensional particle code for validation and to includeemittance growth and particle scraping. For the casessimulated which had a clearance of 1 mm between thebeamlet and the apertures, particle loss occurred withmaximum offsets greater than 0.2 mm. That result is lesspessimistic than the envelope calculation which givesenvelope excursions (centroid � envelope) greater than2 mm for maximum offsets greater than about 0.15 mm.In the cases with no particle loss, there was little effect onthe emittance, no more than a 20% increase. Note alsothat the merged emittance was very insensitive to thebeamlet emittance. Similarly, the beamlet size and con-vergence show little change, less than 10% variation.Figure 12 shows one sample case from the particle sim-ulations, with 0.5 mm clearance and 0.2 mm maximumerrors.

Design tolerances of the order of 100 m (0.1 mm) arerealistically achievable. With this tolerance, the order of

0 1 2 3

−2

2

0

X (mm)

Z (m)

FIG. 12. Sample particle simulation showing the beamletoffset caused by aperture offsets.

014202-10

0.5 to 0.6 mm clearance will be required between thebeamlets and the apertures. In the Einzel lenses used inthese error studies, the beam was somewhat overfocusedin the first half of the column as compared to the optimaldesigns in Sec. IIIB. Decreasing the focusing will relaxthe tolerances somewhat. So, the value of 0.5 mm wasused for the clearance in the column optimization.

IV. ESTIMATE OF MERGED EMITTANCE

An attempt was made to estimate the merged emittanceusing energy conservation. The initial energy of the ar-rangement of beamlets at the exit of the acceleratingcolumn (the point at which the beamlets interact) canbe calculated either analytically or via the simulation as asum of the kinetic energy and the field energy due to thespace charge. An analytic expression for the space-chargefield energy, Usc, of N nonoverlapping beamlets in thecylindrical pipe with radius R can be derived [11].

Usc�1

4�"0

�XNi�1

$2i

�lnaT

ai ln

�1�%2

i =R2

1�%2=R2

���XNj�1

Xi<j

$i$j

�1

2 ln

�j%i�%jj

2

a2T

�ln

�1�

%2

R2

�2�ln

�1�

2%i �%jR2

%2i %

2j

R4

��

PNi�1$

2i

4

�14ln

RaT

�: (6)

Here, $i, %i, and ai, are, respectively, the line charge,centroid locations, and radii of the beamlets. The centerof mass is

% �

�XNi�1

$i%i

�=$; (7)

and the total beam size is

a2T �

�XNi�1

$i�a2i � 2j%2j 2j%2

i j�

�=$; (8)

where $ is the total line charge. For the cases describedhere, the center of mass % is always zero and ai; and a0i(the beamlet convergence) are the same for all beamlets(written below without the subscript i). The kinetic en-ergy of a single beamlet can be calculated by integrating12mv

2f over the beamlet, where f is the distributionfunction, f � 1=�a2 inside the beamlet and zero outside,and v2 � v2

z�X02 Y02�. The transverse velocities, X0 andY0, each have two components, the convergence of thebeamlet and the centroid motion, X0 � x a

0

a %ixA0A and

Y0 � y a0

a %iyB0

B , where x and y are the coordinates rela-tive to the beam center, and A, B, A0, and B0, are theoverall beam size and convergence given from the enve-lope solution. Note that for a round beamlet configura-tion, A � B � aT. The result of integration is summedover the N beamlets to yield the total kinetic energy, Uke.

Uke �$i2qmv2

z

�1

2Na02

XNi�1

��%ixA0

A

�2

�%iyB0

B

�2�:

(9)

The normal procedure is to equate the initial and finalenergy, where for the final energy, the space-charge

014202-10

PRST-AB 6 DAVID P. GROTE, ENRIQUE HENESTROZA, AND JOE W. KWAN 014202 (2003)

energy term is for a uniform beam and all of the kineticpart is assumed to be in the emittance [4]. There is adifficulty since the procedure relies on knowing what thematched beam radius is. The merged beam size producedby the simulation was typically smaller than the envelopesolution by a few percent. The small difference had a largeeffect since the emittance is calculated by subtractingenergies of similar magnitude. Using the larger beamsize from the envelope gave an overestimate of emittanceof a factor of 2 or more. Furthermore, the acceleration inthe merging region complicates the procedure.

The following heuristic was found to produce a rea-sonably good estimate, even in the presence of accelera-tion. Assume that the ‘‘coherent’’ energy stays coherent,and the ‘‘incoherent’’ incoherent. The initial coherentenergy is the sum of the space-charge energy from auniform beam with equivalent rms size, Ucoherent

sc , andthe kinetic energy is from the overall convergence,Ucoherent

ke . The space-charge energy of a uniform beam ina round pipe can be written as

Ucoherentsc �

$2

16�"0

�1 2 ln

2R2

A2 B2 2 ln2�A2 B2�

�A B�2

�:

(10)

The coherent kinetic energy term can be written as

Ucoherentke �

$2qmv2

z

�hXX0i2

hX2i

hYY0i2

hY2i

�: (11)

Here, X and Y are the coordinates, which can be writtenfor each beamlet as X � %ix x and Y � %iy y. Theaverages for each beamlet are

hXX0ii �1

4aa0 %2

ixA0

A; (12)

hX2ii �1

4a2 %2

ix: (13)

BC

A

BCA

B

C

A

B

C

A

B

C

AB

C

A

−6 −4 −2 00.0

0.5

1.0

1.5

Initial beamlet convergence (mrad)

Nor

mal

ized

em

ittan

ce (

m

m m

rad)

A 5 rings (91 beamlets)B 6 rings (127 beamlets)C 7 rings (169 beamlets)

π

FIG. 13. (Color) Comparison of the estimated emThe solid curves are from the simulations, the d

014202-11

Averaging over the beamlets gives

hXX0i �1

4aa0 j%2

ixjA0

A; (14)

hX2i �1

4a2 j%2

ixj �1

4A2: (15)

The coherent kinetic energy then evaluates to

Ucoherentke �

2$qmv2

z

�1

A2

�aa0

4 j%2

ixjA0

A

�2

1

A2

�aa0

4 j%2

iyjB0

B

�2�: (16)

Furthermore, the aa0 terms are dropped since they resultfrom the convergence of the individual beamlets:

Ucoherentke �

2$qmv2

z

�1

A2

�j%2ixjA0

A

�2

1

A2

�j%2iyjB0

B

�2�:

(17)

After merging, all of the coherent energy goes into thespace-charge energy. Assuming a round, uniform mergedbeam, Eq. (10) can be used to give the final space-chargeenergy. Setting A and B equal to afinal, the final beamradius can then be calculated.

afinal � R exp

��

1

4

�Ucoherent

init

16�"0

$2 � 1

��: (18)

The remaining energy is incoherent and is all in theemittance, which can be found by writing the incoherentpart of the kinetic energy (with the assumption that the xand y emittances are the same) as

Uincoherentke �

$4qmv2

z"2x

a2final

� Utotalinit � �Ucoherent

sc Ucoherentke �: (19)

Here, "x is the unnormalized emittance.

−6 −4 −2 00.0

0.5

1.0

1.5

Initial beamlet convergence (mrad)

Nor

mal

ized

em

ittan

ce (

m

m m

rad) A

A

A A

AA

B

B

BB

B

B

C

C

CC

C

C

DD

D

D

D

D

A beamlet radius = 0.8 mmB beamlet radius = 1.0 mmC beamlet radius = 1.2 mmD beamlet radius = 1.5 mm

π

ittance and the results from the simulations.ashed curves the estimates.

014202-11

PRST-AB 6 DESIGN AND SIMULATION OF A MULTIBEAMLET . . . 014202 (2003)

Results from this estimate are shown in Fig. 13. Theestimate is consistently low and gives a minimum at adifferent location, but is close enough to be useful forrough scaling.

V. CONCLUSION

Given the results of these scaling studies, it appearsthat a multibeamlet injector can be built with a normal-ized emittance less than 1�mmmrad. For the parametersrequired in ‘‘an intermediate term facility for heavy-ionfusion’’ system, the case with 91 beamlets merging at0.8 MeV appears close to optimal. This design gives anemittance close to the value of 1�mmmrad. Increasingthe merging energy does not give a significant decrease inthe emittance due to the weakening focusing of theEinzel lenses. With other things being equal, the0.8 MeV design is preferable over the 1.2 MeV simplybecause it uses fewer plates, simplifying construction.Merging at significantly less than 0.8 MeV is less desirablesince the voltage gradient in the subsequent acceleratingregion exceeds voltage breakdown limits.

For this design to work, the construction requires tighterror tolerances, less than 0.1 mm error in the aperturelocation. If smaller errors are achievable, some reductionin emittance could be obtained by increasing the beamletsize, increasing the beamlet convergence angle, and/ordecreasing the center-to-center spacing.

There are several additional areas that need furtherresearch. Little study has been done of the time-dependentbehavior of the beam in the system. The rise time ofthe voltage on the column greatly affects the shape ofthe beamlet head. It will be important to resolve how thebeamlet head can be made to travel through the channelwith minimal particle loss to the apertures. Another issueis the strength of the interbeamlet forces in the column,including shielding by the aperture plates. Initial simu-lations (not reported here) show that this effect is small,but this needs closer examination. One other issue is therole of stray electrons, especially if any ions hit anaperture. It is expected that electrons will be swept outof the beamlets since the beamlet potentials are small.This can only be fully resolved via experiment.

014202-12

ACKNOWLEDGMENTS

This work was performed under the auspices ofthe U.S. Department of Energy by the Universityof California, Lawrence Livermore, and LawrenceBerkeley National Laboratories under Contracts No.W-7405-Eng-48 and No. DE-AC03-76SF00098.

*Electronic address: DPGrote@lbl.gov[1] J.W. Kwan, F. M. Bieniosek, E. Henestroza, L. Prost, and

P. Seidl, ‘‘A 1.8 MeV K Injector for the High CurrentBeam Transport Experiment,’’ in Proceedings of the14th International Symposium on Heavy-Ion InertialFusion, Moscow, Russia, 2002 [Laser Part. Beams (tobe published)].

[2] High Voltage Technology, edited by L. L. Alston (OxfordUniversity Press, New York, 1968), p. 65.

[3] L. Cranberg, J. Appl. Phys. 23, 518 (1952).[4] O. A. Anderson, Fusion Eng. Des. 32–33, 209–217

(1996).[5] J.W. Kwan, O. A. Anderson, D. N. Beck, F. M. Bieniosek,

C. F. Chan, A. Faltens, E. Henestroza, S. A. MacLaren,P. A. Seidl, L. Ahle, D. P. Grote, E. Halaxa, and C.T.Sangster, in Proceedings of the 1999 ParticleAccelerator Conference, New York (IEEE, Piscataway,NJ, 1999), p. 1937.

[6] J.W. Kwan, L. Ahle, D. N. Beck, F. M. Bieniosek,A. Faltens, D. P. Grote, E. Halaxa, E. Henestroza, W. B.Herrmannsfeldt, V. Karpenko, and T. C. Sangster, Nucl.Instrum. Methods Phys. Res., Sect. A 464, 379–387(2001).

[7] D. P. Grote, A. Friedman, I. Haber, W. Fawley, and J. L.Vay, Nucl. Instrum. Methods Phys. Res., Sect. A 415,428–432 (1998).

[8] I. Haber, D. Kehne, M. Reiser, and H. Rudd, Phys. Rev. A44, 5194 (1991).

[9] L. Ahle, R. Hall, A. Molvik, E. Chacon-Golcher, J.W.Kwan, K. N. Leung, and J. Reijonen, in Proceedings ofthe Particle Accelerator Conference, Chicago, IL, 2001(IEEE, Piscataway, NJ, 2001), p. 726.

[10] J. C. Spall, IEEE Trans. Autom. Control 37, 332 (1992).[11] E. P. Lee (private communication).

014202-12