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Electromagnetic wake fields and beam stability in slab-symmetric dielectric structures

A. Tremaine and J. Rosenzweig Department of Physics and Astronomy, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, California 90095

P. SchoessowDivision of High Energy Physics, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439

Received 28 March 1997

Several promising schemes for high-gradient acceleration of charged particles in slab-symmetric electro-

magnetic structures have been recently proposed. In this paper we investigate, by both computer simulation and

theoretical analysis, the longitudinal and transverse wake fields experienced by a relativistic charged particle

beam in a planar structure. We show that in the limit of an infinitely wide beam the net deflecting wake fields

vanishes. This result is verified in the limit of a large aspect ratio sheet beam by finite beam analysis based

on a Fourier decomposition of the current profile, as well as a paraxial wave analysis of the wake fields driven

by Gaussian profile beams. The Fourier analysis forms the basis of an examination of flute instability in the

sheet beam system. Practical implications of this result for beam stability and enhanced current loading in

short-wavelength advanced accelerators are discussed. S1063-651X9701611-5

PACS numbers: 41.75.Ht, 29.17.w, 29.27.a, 41.60.Bq

I. INTRODUCTION

The use of planar structures for advanced accelerator ap-

plications has been discussed recently in the context of both

metallic, disk-loaded millimeter wave structures for linearcollider applications 1 and high gradient dielectric loaded-structures excited by lasers 2. It has been noted that thistype of electromagnetic structure may have advantages overthe usual axisymmetry, in ease of external power couplingand lowered space-charge forces 2. More importantly, ithas also been speculated 1,2 that the transverse wake fields

associated with this class of structure are mitigated, thus di-minishing the beam breakup BBU instability which typi-cally limits the beam current in short-wavelength accelera-tors. This instability arises from off-axis beam currentexcitation of dipole mode wake fields which in turn steertrailing particles; a measure of the strength of this problem isthe amplitude of the transverse wake fields. In this paper weshow, by both theoretical and computational analysis, thatthe transverse wake-field amplitude is in fact diminished forasymmetric x0/2y , where indicates the rmsbeam size, and 02/c is the vacuum wavelength of theelectromagnetic wave relativistic bunched beams. Indeed, inthe limit of a structure and beam which is much larger in xthan in y , the transverse wake fields vanish, in analogy to themonopole modes of axisymmetric structures.

The organization of this paper is as follows. We begin inSec. II with some general results which constrain the pos-sible forms of the wake fields, including the relationship ofthe longitudinal to the transverse wake fields, and the addi-tional implications on the form of the wake fields arisingfrom the vacuum dispersion relation. Wake fields due tobeams in the infinitely wide limit are analyzed in Sec. III,with a frequency domain analytical theory compared to com-putational modeling in the time domain. In Sec. IV, as a firststep in the analysis of finite beam effects, we will then pro-

vide for a Fourier decomposition of the beam current in x,again employing analytical and computational tools. Theseresults are then compared in Sec. V to those obtained fromassumption of a beam current which is Gaussian in x , ananalysis which is similar in methodology to that used onparaxial photon beam modes. In Sec. VI, the transverse BBUinstability of these beams arising from the dipole modelikewake fields is analyzed in two limits: a single bunch fluteinstability, in which ripples in the beam transverse distribu-tion self-amplify, and the multibunch instability due to long-range wakes. We conclude in Sec. VII with a discussion of

the practical implications of our results for short-wavelengthaccelerators, comparing the performance of slab-symmetricand cylindrically symmetric systems from the points of viewof beam loading and transverse stability.

II. WAKE FIELDS: GENERAL CONSIDERATIONS

Before we begin an analysis of wake fields which can beexcited in slab-symmetric dielectric-loaded structures, we in-troduce some general considerations concerning the possibleforms of these fields. The first of these considerations con-cerns the relationship between the longitudinal and trans-verse components of the fields, while the second concernsthe constraints on functional form of the field componentsgiven by the Maxwell wave equations.

The possible forms of the wake fields left behind a rela-tivistic charged particle, and the net forces imparted by thesewake fields on trailing charged particles, are constrained bythe Maxwell equations, in ways that were initially describedby Panofsky and Wenzel 3. The original theorem given inRef. 3 gives a relationship between the integrated longitu-dinal and transverse momentum kicks a particle receives as ittraverses, in the constant-velocity, paraxial limit, an isolatedmedium or device with an electromagnetic excitation. Wegeneralize this theorem in two ways: 1 we include fields

PHYSICAL REVIEW E DECEMBER 1997VOLUME 56, NUMBER 6

561063-651X/97/566/720413 /$10.00 7204 1997 The American Physical Society

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arising possibly from free charges, assuming that fields andpotentials vanish at infinity; and 2 we give an alternateform of the theorem applicable in the infinite medium orstructure limit, which is of use when describing a very longinteraction between charged particles and their environment.The assumptions leading to these generalized forms of thePanofsky-Wenzel deflection theorem still include, explicitly,that the particle receiving the kick travels parallel to the z

axis, with a velocity which is constant.The net Lorentz force on a charged particle consists of the

sum of electric and magnetic components. In general, the

electric field E may be derived from a scalar potential

and a vector potential (A),

E1

c

A

t. 2.1

Inserting Eq. 2.1 into the Lorentz force equation, with the

beam particle velocity b

b

c and charge q , we have

FqFbBqW , 2.2

where, for a particle traveling parallel to the z axis, we maywrite

bBbz

Ab Az Az . 2.3

We can obtain from Eqs. 2.12.3 an expression for W

,the net force per unit charge q , as

W1

c

A

tbAz Az . 2.4

Using the assumed condition that b is constant an excellentapproximation for relativistic beams, b1 we can expressthe partial derivatives in Eq. 2.4 in terms of the convec-tive full derivative in z,

1

c A

tb

A

z 1

c

dA

dtb

dA

dz. 2.5

The full momentum transfer, through a region R , due to thenet Lorentz force, can now be rewritten as

pq

bc

Rb dAdz bAzdz . 2.6

For an isolated system where E and B are nonvanishing only

in the region R , A0, when evaluated at the end points ofintegration. This condition makes the first term inside theintegrand in Eq. 2.6 vanish, and we obtain

p qbc RbAzdz . 2.7

Since the integrated momentum kick field p can be derivedfrom a potential, its curl must vanish,

p0. 2.8

This expression is more usefully written to display the rela-tionship between the longitudinal and transverse kicks,

pzp

z. 2.9

It should be noted that this form of the deflection theorem ismuch more general than that due to Panofsky and Wenzel, inthat can be applied to interactions in which free charges andcurrents come in contact with the beam, such as plasma wakefields 4 and the beam-beam interaction 5. It states that, fora deflection kick to occur, there must be transverse variationof the longitudinal kick imparted by the fields in the isolatedsystem under consideration.

The equivalent expression of the deflection theorem in thecase of continuously applied wave fields can also be derived.Here we make the wave ansatz, that the fields longitudinaland temporal dependence can be expressed solely in terms of

zt. In this case we ignore all transients in the prob-lem, which are in any event covered well by the form of thetheorem given in Eq. 2.8. Explicitly examining the curl ofthe net force on a particle traveling parallel to the z axis withspeed b , we have

WEbBEb

BE

b

B

t. 2.10

Thus, for b, which is always assumed true for wake

fields in the relativistic beam limit a wake field satisfyingthe wave ansatz must satisfy this condition, we have that thewake field is conservative,

WE1

c

B

t0, or Wz

W

.

2.11

This form of the theorem basically states that when the wavephase velocity is equal to the beam velocity, a transformationto the rest frame gives a purely electrostatic net force, whichis always conservative. Since we are concerned here with a

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mode, as was done in previous analyses of this two-

dimensional problem 9. By assumption, the longitudinal

and temporal dependence of the fields of the n th mode ex-

cited by the beam is of the form expikn, where knn /c. As the excited modes in this limit are purely TM,we need only solve for Ez in this analysis. We shall verify inthe following sections that this behavior is obtained in thelimit of a very wide, yet finite-sized, beam. For the TM case

we need only additionally consider the vertical dependenceof the longitudinal electric field to determine the mode fieldscompletely. Inside of the gap ( y a), we have

Ez,nE0exp i kn , 3.1

with no x or y dependence, where E0 is an arbitrary ampli-tude, while in the dielectric ( ayb) we must have

Ez,nA nE0exp iknsinsy ,nyb , 3.2

with sy ,nkn1. Application of the boundary conditionsat ya continuity of Ez and Dy , which is trivially derived

FIG. 1. Color Schematic of a

slab-symmetric, dielectric-loaded

structure, with a vacuum gap half-

height a , dielectric layers of per-

mittivity , and a thickness ba ,

with metal boundaries at y

b.

FIG. 2. Color False color contour map of Wz for a slab-symmetric structure with a vacuum gap half-height a2.5 m, a dielectric of

permittivity 4 in the regions ay b between the gap and the conducting boundaries at y b5 m, from time-domain electro-magnetic field solver. The ultrarelativistic beam distribution is infinite in x , with a line charge density dq/dx2/z , infinitesimal in

y at y1 m and Gaussian in z with standard deviation z0.5 m, centered at about z27.5 m. The color map is linear and in

spectral order, with red most negative, violet most positive, and blue-green the zero-field strength.


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from Ez from the relation D0 allows a determination

of the eigenvalue of each mode through the transcendentalrelation

cotkn1bakna1

, 3.3

and the amplitude of the longitudinal electric field within thedielectric,

A ncsckn1ba . 3.4

Note that, for higher wave-number modes, Eq. 3.3 impliesthat the phase variation in the dielectric must be approxi-mately (n 12 ), and thus A n becomes large, as the field isconcentrated in the dielectric.

Once the fields have been determined, the response of thestructure to the passage of an ultrarelativistic, horizontallyoriented line charge of constant density e(yy 0)() within the vacuum gap (y 0a) can be calcu-lated by energy balance. It can be shown that, for a linearwake field 10 the net decelerating field on a line chargeassociated with a wake amplitude of E0 is EdecE0 /2. We

can equate this energy loss per unit length with the fieldenergy per unit length left behind the ultrarelativistic linearcharge density as

Edec u emSz d y , 3.5

where uem12 (Ez

2Ey

2)Hx2 is the electromagnetic en-

ergy density, Sz(4)1EyHx is the longitudinal Poynting

flux, and HxEy . The longitudinal wake field behind thecharge obtained from this expression is thus simply

Wz,nEz,n4

aAn2ba

coskn, 3.6

where is the Heaviside function which explicitly showsthe causal nature of the wake fields. It should be noted fromthis expression that the longitudinal wake field is in generallargest for the lowest frequency mode. For the slab-symmetric, laser-pumped accelerator proposed in Ref. 2,however, the examples given have the device operating on ahigher-frequency mode of the structure. This could pose abeam-loading problem for this and other overmoded laseracceleration schemes, as the beam gains energy from a mode

FIG. 3. Color False color contour map of Wy for the structure and beam described in Fig. 1. The transverse wake field essentially

vanishes in the vacuum gap for an infinitely wide in x beam.

7208 56A. TREMAINE, J. ROSENZWEIG, AND P. SCHOESSOW

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which it is poorly coupled to, in comparison to the modeswhich it loses energy to in the form of wake fields.

To obtain the full wake field response from a beam withan arbitrary longitudinal current profile e(yy 0)f() , we must perform a longitudinal convolutionover the point response,

Wzn

fWz,nd. 3.7

The predictions of Eq. 3.7 have been verified by use ofnumerical simulation of the wake fields in a planar structure,performed using a custom two-dimensional finite-differencetime domain electromagnetic simulation code. The beam isassumed to be a rigid current distribution, infinitesimally thinin the vertical (y) direction and with a fixed offset from thesymmetry plane y0. The beam is also taken to be ultra-relativistic and traveling in the z direction, with a Gaussianlongitudinal of standard deviation z . The line charge den-

sity of the beam in x is normalized to 2/z statcoul/m,with z in m. The fields are advanced using the standardleapfrog time integration algorithm. Figure 2 shows a false-color contour map of Wz for a case similar to the infraredwavelength examples given in Ref. 2. One can clearly seeboth the uniform speed-of-light phase fronts in the vacuum,and the Cerenkov nature of the wake field in the dielectric,which displays the expected propagation angle. It should alsobe noted that, even though the beam current is asymmetricwith respect to the y0 plane, the excited longitudinal wakeis nearly symmetric after propagation away from the simu-lation boundary, indicating symmetric coupling of the beamfields to the dielectric and the accompanying suppression of

the transverse variations of Wz which lead to a transversewake field.

Figure 3 shows the net vertical wake field WyEyBxexcited by the beam in this case; one can see that Wy essen-tially vanishes inside of the vacuum gap. The lack of fullcancellation ofEyBx is due mainly to the electric and mag-netic field centers in the calculation being one-half of both aspatial and time step apart. Figure 4 displays a comparisonbetween the simulation and analytical results for Wz in thiscase. The results are in good agreement, with some discrep-ancies due to the transient fields found in the time-domainsimulation which are not present in the Fourier-based ana-lytical treatment.

IV. WAKE FIELDS IN FINITE BEAMS:

FOURIER ANALYSIS

With the method of determining the wake-field couplingestablished in Sec. III, we now turn our attention to wakefields in finite beams. We begin by generalizing the aboveanalysis to a beam of finite horizontal extent by examiningthe wake fields due to beams of a harmonic in x chargeprofile

e,kxkxcoskxx yy 0, 4.1

where is now the peak line charge density. This profile canbe viewed as a Fourier component of a finite beam, i.e.,e(x)e,kxcos(kxx), where we implicitly assume that the

waveguide now has conducting sidewalls with separation inLx allowed wave numbers kxm/Lx , m1,3,5 . . . , andthe beam distribution in x is centered and symmetric withinthese walls, as is shown in Fig. 5. Partial wake fields ob-

tained from this harmonic analysis can therefore be summedto find the complete wake fields.

The longitudinal electric field associated with the nthmode of the wake fields that the harmonic beam can coupleto has the following form in the vacuum region:

Ez,nE0exp ikn coshkxy sinhkxy coskxx . 4.2

The cosh(kxy) dependence indicates the monopolelike, or ac-celerating, component independent of y in first order forsmall vertical offsets and the sinh(kxy) is the dipolelike, ordeflecting component, which couples to the longitudinal fieldwith strength approximately linear in y and produces deflect-ing forces nearly independent of y for small vertical offsets.Note that these modes have explicitly been described asmonopolelike and dipolelikethey are exact modes whichdisplay a specified multipole characteristic only in lowestorder in y .

Since the modes under consideration are not pure TM, butare hybrid modes, we must find the longitudinal magneticfield to specify all the fields. In the gap region, this field hasthe form

Bz,nE0exp ikn

sinhkxy

coshkxy sinkxx . 4.3

We again have obviously taken the ultrarelativistic limit,and in this case one must be very careful in finding the trans-verse components of the fields in the vacuum region. Theyare

Ex,niknE0

2kxexp ikn coshkxy sinhkxy sinkxx ,

Ey ,niknE0

2kxexpi kn sinhkxy coshkxy coskxx ,

Bx,niE0 kn2kxkx

kn exp i kn sinhkxy coshkxy

coskxx ,

By,ni E0 kn2kxkx

kn exp ikn coshkxy sinhkxy

sinkxx . 4.4

It should be noted that one cannot obtain the case of theuniform beam by taking the limit kx0 in Eqs. 4.4, be-


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cause these expressions were obtained by assuming that kxkn /, where (1

2)1/2 is the Lorentz factor of thebeam. One also obtains immediately from Eqs. 4.4 thegratifying result that the transverse forces on a relativisticparticle of charge q, due to the modes described by Eqs.4.2 and 4.3,

Fx,nqWx,nqEx,nBy ,ni kx

knE0exp ikn

coshkxy sinhkxy sinkxx ,

Fy ,nqWy ,nqEy ,nBx,nikx

knE0exp ikn

sinhkxy coshkxy coskxx , 4.5

vanish in the limit that kx0, as we had found in the uni-

form line charge ( kx0) beam case. The form of this resultcould have been directly deduced from the generalizedPanofsky-Wenzel theorem given by Eq. 2.11. Equation4.5 reinforces the primary point of the present analysis.Simply stated, for highly asymmetric for rms beam sizes

xy , with associated kxx1 beams in slab-symmetric

structures, transverse wake fields are strongly suppressed. Infact, since all wake fields are proportional to the linearcharge density , the transverse wake fields scale at constant

charge per bunch as x2. This result mitigates one of the

major objections to use of high-frequency accelerating struc-tures, that the transverse wake fields scale prohibitively withfrequency. This objection holds for cylindrically symmetricstructures, but can be greatly eased by use of slab structures.

The fields in the dielectric, unlike those in the gap, can befound by standard wave-guide analysis; for brevity theirderivation is omitted. Following the same prescription used

to obtain Eqs. 3.13.3, we obtain a transcendental expres-sion for the eigenvalues of the symmetric modes,

cot1 kn2kx

2bacothkxa1 kn /kx

21

1

2 kn

kx

2

10. 4.6

The eigenvalues of the antisymmetric modes are simply ob-tained by substitution of tanh(kxa) for coth(kxa) in Eq. 2.11.The longitudinal wake fields associated with the symmetricmodes are

Wz,nkx4 coshkxy 0coshkxy coskxx coskn

sinh2kxa

2kx kx

kn

2

1 cosh2kxa

sin2snba

sinh2kxa

cos2s nba kx

s n

2

2 ba2

sin2snba

4sn sinh

2kxa

cos2snba

cosh2kxa

sin2snba kx

sn

2

4kxsn

kn21

coshkxasinhkxa

sins nbacossnba ;

4.7

those of the antisymmetric modes are obtained by substitu-

tion of sinh(kxy) for cosh(kxy ), and vice versa, where ytakes

on the values of a , y , and y 0 in Eq. 4.7.

A discrete sum and convolution integral similar to Eq.3.7 which sums over both symmetric and antisymmetricmodes, as well as the beam charge distribution, must be per-formed to obtain the full wake fields for a beam of finiteextent in configuration space. The resultant expressions havealso been compared to simulations of this periodic in xsystem, with Wy and Wz obtained by both methods shown in

Fig. 6. The algorithm in the numerical simulations used in

this case is based on discretizing the Maxwell equations after

Fourier transforming with respect to x . The discrepancies in

the two approaches due to transient effects are slightly morepronounced in this case, but again the agreement is quitegood. Parametric studies performed with these simulationshave also verified the suppression of transverse wake fieldsfor wide beams.

The Fourier analysis of the beam current is most appro-priate for quantifying the behavior of the wake fields in, e.g.,

FIG. 4. Comparison of the values of Wz as a function of z at

y1 m given by a time-domain electromagnetic field solver and

the predictions of Eqs. 3.6 and 3.7, for cases of Figs. 1 and 2.


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millimeter-wave structures 1, which have sidewalls not too

distant from the beam. It is also, as will be seen below, most

useful for analyzing the flute, or filamentation instability of

asymmetric beams in slab-symmetric structures. This work,

however, is motivated by the investigation of ultrashort

wavelength infrared to optical accelerator structures suchas those discussed in Ref. 2 in which the sidewalls are very

distant from the beam. In order to handle this case, in which

the beam current is rather isolated horizontally, we now ex-

amine the wake fields generated by a beam with a Gaussian

profile. It should be noted that adoption of this physical

model has the additional motivation that the Gaussian beam

is commonly encountered in practice, as it is produced in

systems in thermal equilibrium.

V. FINITE-BEAM EFFECTS: GAUSSIAN

BEAM ANALYSIS

We now turn to the analysis of the wakes produced by thecommonly encountered Gaussian horizontal beam profile asit traverses the very wide dielectric structure shown sche-matically in Fig. 1. The structure is assumed to be wideenough such that the deviation in the wakes due to the finitewidth effects can be ignored. In this limit, the solutions to thewave equation can be found in the paraxial approximation byutilizing methods developed for analysis of Gaussian photonbeams in lasers and optical transport systems.

In the present case, the beam current profile is assumed tohave the form

e,wx,nwx,ne

x2/wx ,02

yy 0, 5.1

where wx,0&x,b is, following the notation of Siegman11 for photon beam analysis, the horizontal particle beamsize, which is identical to the horizontal width of the wakefield extent in x directly behind the exciting line charge (0). This current will drive wake fields which are inmany ways similar to Gaussian photon beams, but with somenotable differences. In photon beam propagation, the fieldsare derivable from the paraxial wave equation, and areGaussian in both transverse coordinates. For the structureinduced wake fields, the fields will have a Gaussian depen-dence in only the x direction, where the paraxial approxima-tion in its familiar form ( knwx,01) can still be employed.Since the solution to the paraxial wave equation is separablein the x and y coordinates with similar types of solutionsHermite-Gaussian for each coordinate, one might supposethat the vertical dependence of the fields also be Gaussian. Inthe limit that wxa, however, the vertical dependence of the

FIG. 5. Color Dielectric waveguide with conducting sidewalls

of separation in Lx for Fourier beam analysis, with allowed wave

numbers kxm/Lx (m1,3,5 . . . ).

FIG. 6. Comparison of the values of a Wz and b Wy as a

function of z at y1 m given by the time-domain electromag-

netic field solver and the predictions of Eqs. 6 and 12, for the

identical structure and beam of Figs. 13, but with the beam charge

distribution modulated with kx0.4 m1.

FIG. 7. Color Diagram of the wake-field diffraction pattern in

the x-z plane, for Gaussian beam and wake-field model calcula-

tions.


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fields is expected to be negligible of second order in the

small parameter a/wx,0 and, as in the infinite beam and

structure case, the field has no significant vertical depen-

dence in the vacuum gap.

As mentioned above, the horizontal dependence of the

longitudinal electric field directly behind the infinitesimally

short beam takes the form of the current, Ezexp(x2/w

x,0

2 ). We note that this wake field can be viewed

as similar to the wake for the Fourier analysis, Eq. 4.2, inthat it has a local maximum in x. In fact, for small kxy ,

cos(kxx), and exp(x2/wx,n

2 ) are locally equivalent if we sub-

stitute wx,nkx /2. Once we have found this local equiva-lency, in fact, the Gaussian wakes can be found by substitu-tion, exploiting the Fourier wakes as a model. The resultingfields can be verified as being solutions to Maxwell equa-tions in the paraxial limit.

It should also be noted that the above comparison can alsobe done for the antisymmetric wake field in Eq. 4.2. Thedependence of the dipolelike Fourier wake becomes linear iny for small kxy . A linearly dependent wake can be seen to bean acceptable solution to the paraxial wave equation if it is

noted that for small y , the first odd-symmetry Hermite-Gaussian solution is also linear. Thus the antisymmetricGaussian wake can be expressed as Ezy exp(x2/wx,n

2 ), which is equivalent to the dipole modedescribed in Eq. 4.2.

The wake fields created by the beam are to be initially ata waist, then diffract out behind the beam, as is shown sche-matically in Fig. 7. The waist, as well as the entire fieldpattern, moves forward in z at the speed of the beam zc. It is again interesting to contrast this form with that of alaser beam. A focused laser beam envelope is minimized at aspatially stationary waist, after which it diffracts away trans-versely. While the electromagnetic waist of the wake fields is

stationary in the beam rest frame, not the laboratory frame,the laser and wake fields hold in common the expected be-havior that if one travels with a phase front, diffraction of thefields is observed as time advances.

The full longitudinal symmetric wake field in the struc-ture can be found from paraxial wave solutions to be

Ez,nE0,nex2/wx,n

2ikx2/Rnexpi knnz

5.2

where n() tan1(/zR,n) is the Guoy phase shift of a given

mode, zR,nwx,n2 (0)/nknx

2 is its Raleigh diffraction

length, R n()1(zR,n /)2 is the local radius of curva-

ture of the phase fronts, and wx,n2 ()wx,0

21(/zR,n)

2 isthe horizontal spot size. It should be emphasized that at thepoint of creation directly behind the beam the diffractingwake field is at a waist. Here the beam size is a minimum,and the Guoy phase shift is most rapidly changing. In light ofthe dispersion relation Eq. 2.12, this phase shift can beviewed as equivalent to a longitudinal wave-number shift inthe region of largest horizontal wave-number kx .

Using the Fourier wake analysis as a guide, we found theremaining fields in the gap:

Bz,nE0,n2xy

wx,n2

ex2/w

x,n2

ikx2/Rnexp ikn

n ,

Ex,niE0,n

2

knx ex2/wx,n

2ikx2/Rnexp ikn

nz ,

Ey ,niE0,n

2kny e

x2/wx,n2

ikx2/Rnexpi kn

nz ,

Bx,niE0,n

2 1 2

knwx,n

2

kny e

x

2

/wx

2

ikx

2

/Rnexp i knnz ,

By ,niE0,n

2 1 2

knwx,n

2

knx e

x2/wx,n2

ikx2/Rnexp iknnz .

5.3

The transverse forces associated with these fields,

Fx,nqWx,n

qEx,n

By ,n

iE02x

knwx,n2

ex2/2

x2

expi knz ,

Fy ,nqWy ,nqEy ,nBx,ni E02y

knwx,n2

ey2/2

x2

exp iknz , 5.4

again will diminish for very wide beams. This result is ingood agreement with the analysis of the Fourier beam case,

as well as the limiting infinite beam case.Employing the same techniques as used to derive Eqs.

3.3 and 4.6, we find the transcendental eigenvalue expres-sion for the symmetric Gaussian mode to be

x

acot 1kn2 1x2

1/2

ba1 knx21

1

2knx

210. 5.5

The longitudinal wake-field amplitude is similarly given by


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Wn40

a1 12kn2x2a3

3x2 1

kn2x

21

2

sin2snba

1

2kn2x

21

sin2snba

4sna

sin2snba

sn2a2

cos2snbab2

a2

2x2

cos2snba

1

kn21

sn2

sin2snba

4sna

x2 sin2snba

a2

x2 cos2snba

ba2 sinnsba

4sn,

and was found using energy balance. The wake fields coming off the beam initially have all energy flowing in the longitudinal dir

flow is transferred to the transverse directions; thus it was necessary to calculate the wakes when at a waist.

As mentioned above, the Gaussian modes for an antisymmetric profile in y can also be found. The form of the longitudinal e

EzE0&y

x,nex

2/wx,n2iknx

2/Rnexpiknn.

Following the prescription outlined above, the transverse wakes

FxieE02&xy

knx,n3

ex2/wx,n

2iknx

2/Rn expiknn,

FyieE0&

knx,nex

2/wx,n2iknx

2/Rnexpiknn

have an associated amplitude of

Wn40

a12 1kn2x2 a3

3x2 12kn2x21 a

2

x2

sin2snba 1

2kn21 a

2

x4 sin2snba

4sna

x2 sin2snba

sn

2

cos2snba

12 cos2snba1

kn21

sn2a2

x2 sin2snba

4sna

x2 sin2snba

1

x2 cos2snba

ba2 sinsnba

4sn,

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and we again observe that for wide beams the dipolelike

mode transverse wake fields are considerably reduced.

This Gaussian wake analysis can easily be extended to

include arbitrary beam profiles by using Hermite-Gaussian

functions 11 as a basis set to represent the beam. This

allows the analysis of beams which are modulated in x, as in

the Fourier case, but with finite extent and no reference to

existence of sidewalls. A useful result of this analysis is that

a beam with N horizontal modulations has an effective Ray-

leigh length which is shortened by a factor of N. Thushigher frequency-modulations on a beam create wakes which

diffract out more quickly, and the long range wake field is

eventually dominated by the lowest spatial frequency simple

Gaussian component.

VI. TRANSVERSE INSTABILITIES

IN ASYMMETRIC BEAMS

With the analysis of the wake-field excitation complete,

the wake-field-induced transverse instabilities in very asym-

metric beams can now be analyzed. These instabilities fall

into two categories: the short-range, single-bunch beam

breakup BBU instability which tends to ripple or flute

the charge distribution in x with a fairly large wave number

(kxa1), which couple most strongly to the structure but

diffract away quickly, and the long-range, multibunch BBU

instability, in which the Gaussian wakes of the bunches as a

whole, rather than horizontal Fourier structure of the

bunches, are dominant. These instabilities are analyzed fol-

lowing the formalisms developed for the flute instability by

Whittum 12, using our results on the Fourier components

of the wakes, and for the multibunch BBU by Thompson and

Ruth 13 using the Gaussian wake analysis of Sec. V.We begin by examining the flute instability, employing

the physical and mathematical model of the beam dynamics

developed in Ref. 12 to analyze this type of instability inthe dense beam-plasma interaction. In this model, the beamis viewed in lowest order as a uniform slab lying symmetri-cally about the symmetry plane of the device. A more de-tailed analysis, in which the beam is not taken to be uniformin x , with a harmonic perturbation, but instead is modeled asa Gaussian with periodic perturbations may be performed byuse of Hermite-Gaussian functions. It should be emphasizedthat we do not mean to imply that we are discussing thephysically uninteresting case of a uniform sheet beam with a

sharp cutoff horizontal edge, which would have significantFourier components which would drive BBU instability.This type of beam would not in general be found in, e.g.,high-energy linear collider beams derived from dampingrings, which are approximately thermally equilibrated in allphase planes, giving Gaussian beam profiles. Keeping themodels applicability in mind, a vertical perturbation (z,)of a small amplitude and harmonic with wavenumber kxdependence in x is assumed, which can be shown 12 to beequivalent to a harmonic charge-density perturbation. Thewake fields arising from this perturbation can then be calcu-lated using the results of Sec. IV.

To calculate the evolution of the flute amplitude (z ,),we write the beam breakup equation 12

z

zkyz ,

0

dWyz ,,

6.1

where WyyWy is the wake function produced by a har-

monic beam perturbation with wave number kx and is pro-portional to the Fy,n /y given by Eq. 4.5, and ky is thebetatron wave number associated with the applied verticalfocusing. Since we are considering beams with small phaseextent in the fundamental accelerating wave, Wysin(kn)kn. Following Whittum, the asymptotic flute amplitude fora coasting beam (constant) due to the nth antisymmetricmode is found by a saddle-point analysis of Eq. 6.1 to be,in the limit of strong focusing instability growth length L gsatisfying kyL g1,

z,texp3)

4 Y,nz

2

ky

1/3

, 6.2

where

Y,n re

42Nzx

c2kn Fy,n

y; 6.3

re is the classical electron radius, and N is the number ofelectrons per bunch. As an example, relevant to experimen-tation in a 10.6-m wavelength accelerator similar in designto that considered in Ref. 2, we choose a structure andbeam with a5 m, b6.05 m, x100 m, ky5 cm1 equilibrium beta function y ,eqky

12 mm, N

105, and 100. Using the mode with highest coupling tothe dipolelike mode near kxa

1 we find a growth lengthof L g8 cm, which justifies the strong focusing approxima-tion.

It should be possible to stabilize this instability by use ofhorizontal focusing of sufficient strength to mix the horizon-tal positions of the beam particles in a time shorter that thegrowth time. This is quantified by condition kxL g1,which implies that the horizontal betatron wave number mustscaled to the vertical by the ratio kyL g , which in our ex-ample yields kxky/40.

This ratio may appear to be arbitrarily chosen, but it infact may be constrained by other considerationsprimarilythat the beam be in thermal equilibrium. Because the tran-verse wakes in this case couple the x and y motion in anonlinear fashion, one may expect that if the temperature isdifferent in the two dimensions that it would soon equalize.In terms of standard beam characteristics, this proposed re-quirement TxTy is equivalent to x /eq,xy /eq,y , orxy(ky /kx)y(kyL g). This in turn implies a con-

straint on the beam sizes, as x /xkxy /kyxkyL g , which is approximately 40 in our case. This is aborderline problem for our example, in that yx/402.5 m, which only one-fourth of the total vertical gap inthe structure. This potential problem, which demands rigor-ous analysis in future work, certainly emphasizes the need tohave very strong vertical focusing in the device.

The problem of instabilities due to long range wake fieldscan be studied using the multibunch BBU formalism previ-ously developed by Thompson and Ruth 13. We employ astrongly damped wake approximation, the daisy-chain


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model, where a bunch is only affected by the wake fieldsfrom the previous bunch. For optical accelerators, the hori-zontal diffraction of the wakes make this model promising;on the other hand, since each bucket will undoubtedly befilled in such a short wavelength device, the wakes must bevery heavily damped or have large frequency spreads in therelevant modes for the model to be accurate. We additionallyassume that the high-frequency wakes responsible for the

single-bunch flute instability are ignorable from the multi-bunch point of view, again because the enhanced diffractionof these components of the wake fields quickly dampen theireffect on the bunch train.

In order to include the effects of acceleration and adia-batic damping, it is convenient to define an effective distance

z effky1(0)0

zky(z)dz, where the betatron wave number

now is considered as a function a distance z down the struc-

ture. Assuming k(z)(0)/(z)k(0) , with the approxi-mation that the energy is much greater at the end of thestructure than at the beginning, the effective distance be-

comes z eff20z/, where eEacc/m ec2 is the nor-

malized accelerating gradient. The equations of motion in the

effective length approximation become 13

2y 1

zeff2 ky

2 y 10,

2y n

z eff2 ky

2y n

Nre

1

2x

Fy

yy n1 , 6.4

where the first bunch is explicitly unaffected by transversewake fields. Assuming solutions of the form nA ne

ik z, it isfound the deflecting amplitude A n grows with effec-tive growth length is approximately Lgeff

4x(/Nre)k(Fy/y)

1

. We again use the parameters x100 m, N105, and (0)100, which assuming everyoptical accelerating bucket is filled, and that the acceler-ating gradient is eEacc1 GeV/m, implies we have a beamloaded Q103, similar to the expected unloaded Q of theoptical structure itself 2. For ky5 cm

1, the growthlength of the multibunch BBU instability is 15 cm. This isrelatively gentle growth, which can be controlled by a vari-ety of methods 13, including detuning of the dipole modefrequencies, and tuning the strongest frequencies near half-integer harmonics of the accelerating frequency, thus placingthe bunches near zero crossings of the dipole-mode wakes.

VII. DISCUSSION

The results we have obtained above allow much greaterfreedom in imagining linear accelerator designs at muchshorter wavelengths, as they mitigate the scaling 14 of thetransverse wake coupling strength which limits the current in

these devices, from Wk

3k03 k is the transverse wave

number of the mode, analogous to our present kx in the

cylindrical case, to Wkx

3x3 at a constant beam

charge, independent of accelerating wavelength in the slabgeometry for the case of multibunch BBU. The scaling of the

flute instability is is a bit less dramatic, Wkx

3x2k0 , and

indeed we see that the growth of the flute instability is a bitstronger than for multibunch BBU.

Alternatively, if we allow the beam charge to be varied,

one can see that, by spreading the beam out in one dimension

while keeping efficient coupling of the accelerating wave to

the structure in the narrow dimension, much more charge per

bunch can be accelerated. This comes at a price, of course,

which is that the electromagnetic stored energy is much

larger in the case of a slab-symmetric device as opposed to a

cylindrically symmetric accelerator of equivalent beam hole

dimensionthe shunt impedance of the slab-symmetric ac-

celerator is very low. At optical or infrared wavelengths,

however, the stored energy is not a problem, as laser powers

large enough to drive ultrahigh accelerating fields in slab-

symmetric structure are easily obtainable, and the useful field

amplitudes are limited by structure breakdown consider-

ations 15. In the slab-symmetric case the low shunt imped-

ance can be understood as being due to a large number of

equivalent cylindrically symmetric accelerators operating inparallel, yielding a small impedance.

This multiple-channel, parallel accelerator is a useful

analogy for helping understand the transverse wake fields aswell. It is true that the transverse impedance that the beamsees at the highest transverse coupling to the structure(kxa

1) is very close to that of the cylindrical structure ofthe same vacuum radius a , and so the beam breakup issimilarthe flute instability is nearly identical for the por-tion of the beam within x2a as for single-bunch BBU forthe same charge beam in the equivalent cylindrically sym-metric structure. The advantage of the slab-symmetric struc-ture is twofold, however; it allows much more beam chargeto be accelerated for the equivalent BBU problem, and theflute BBU can be stabilized by a mechanism, horizontal mix-ing, that is unavailable in the cylindrically symmetric struc-ture. In addition, if the flute instability is stabilized, then themultibunch BBU is much more stable, for equivalent beamloading, than the cylindrically symmetric accelerator.

To illustrate this point, we give a list of parameters de-scribing two equivalent designs with slab and cylindrical ge-ometry, respectively, in Table I. In both designs, we obtain alinear accelerator with an average accelerating gradient of 1GeV/m, wavelength 02/k010.6 m, the beam half-gap a5 m, and beam loading quality factor Q1000. Itcan be seen that in the slab case, with a beam width of x100 m, much higher average current can be accelerated

TABLE I. Comparison multibunch BBU of a cylindrical and

slab-symmetric linear accelerator with an average accelerating gra-

dient of 1 GeV/m, fundamental wavelength 02/k010.6 m, a2.5 m, and beam loading quality factor Q

1000; only the lowest frequency dipolelike mode is considered,

with x100 m in the slab case. Comparison parameters: average

current eN c/0 , transverse wake strength W/eN, and BBU

growth length Lg .

Slab case Cylindrical case

Average current 490 mA 16 mA

Transverse wake

dominant dipole

30 V/mm2 fC 105 V/mm2 fC

Multibunch BBU

growth length

15 cm 1.4 cm


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at this beam loading level, and it can be propagated stablylonger.

VIII. CONCLUSIONS

In conclusion, we have theoretically and computationallyanalyzed the transverse wake fields in a slab-symmetricdielectric-loaded structure. We have found and quantified the

suppression of the transverse wake fields for wide beams inthese structures, using infinite beam, harmonic beam, andGaussian beam models, and made an analysis of the beambreakup instability in this type of structure. The additionaladvantages of using wide beams in a slab symmetric struc-ture for accelerating larger beam currents was noted as abyproduct of this analysis. While we have concentrated onlaser-driven optical or infrared accelerators, as they are likelyto use dielectrics, the conclusions reached are quite general,and should be easily applied to submillimeter wave metalliccavity structures as well.

The results we have presented allow more serious consid-eration of short-wavelength advanced accelerator schemes,which have potential application to linear colliders as well asradiation-producing accelerators free-electron lasers, Comp-ton scattering x-ray sources, etc.. These advantages, we be-lieve, present significant motivation for further theoreticaland experimental work in this field; we are presently work-ing on a test of these results using asymmetric, high charge

beams, which produce cm wavelength wake fields, as well asan improved analysis of instabilities in slab-symmetric accel-erators.

ACKNOWLEDGMENTS

The authors have benefitted from discussions with A.Chao, W. Gai, H. Henke, and L. Serafini. This work wassupported by the U.S. Department of Energy under GrantNos. DE-FG03-93ER40796 and W-31-109-ENG-38, and theAlfred P. Sloan Foundation under Grant No. BR-3225.

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Schoessow AIP, New York, 1995, Vol. 335, p. 800.

2 J. Rosenzweig et al., Phys. Rev. Lett. 74, 2467 1995.

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the pure slab symmetric limit are by A. Tremaine et al. and A.

W. Chao et al., in Proceedings of the 1996 DPB/DPF Summer

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University Publications, Palo Alto, CA, 1997, p. 399 and p.

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