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NON-EQUILIBRIUM TRANSVERSE MOTION AND EMITTANCEGROWTH IN SPACE-CHARGE DOMINATED BEAMS

S.G. Anderson and J.B. RosenzweigUCLA Department of Physics and Astronomy, 405 Hilgard Ave, Los Angeles CA

90095

The transverse dynamics of space-charge dominated beams are investigated both analyitically and computationally, in order to understand the mechanisms for

emittance oscillations and growth due to nonlinear space-charge fields. Thiswork explores the role of space-charge dominated equilibrium and itsrelationship to phase space wave-breaking, which is responsible for theirreversible emittance growth in these systems. The physics of both coasting andaccelerating beams are examined, in order to illuminate the most effectiveapproaches to beam handling during the emittance compensation process, as wellas during subsequent beam transport.

1 Introduction

In recent years, a concerted attempt has been made to understand the space chargedominated beam dynamics of intense electron beams, mainly in the context of radio-frequency (rf) photoinjectors. The ultra-short beams in these devices undergo

transverse expansion from the photocathode in the initial cell of the rf gun, anexpansion accompanied by rapid rms emittance growth[1]. This growth has beenfound to be due in large part to correlations in between the transverse phase spaceangle described by the rms beam size and divergence , and the longitudinal

position in the beam[2]. A transverse cross-section of the beam at a givenlongitudinal position, is referred to as a beam slice , and removal of the correlation

between slice position and rms phase space angle / is a process known asemittance compensation[2,3]. As is discussed in the following section, this processis explainable in terms of linear plasma oscillations (the beam is considered to be anearly laminar, cold relativistic plasma) about equilibria dictated by the value of thecurrent at a given slice, and the applied external forces. This analysis, originally

performed by Serafini and Rosenzweig (SR)[4], lead to the identification of a newtype of space-charge dominated beam equilibrium which is found in acceleratingsystems, termed the invariant envelope . It was proposed in this analysis that theinvariant envelope is the preferred mode of beam propagation for providingoptimized emittance compensation. In fact, this point of view is not completelyconsistent, as we shall see, with the original proposed mechanism of emittancecompensation. Part of the motivation for this work is to clarify the role of theinvariant envelope in the emittance compensation process.

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Further, because the invariant envelope is a generalized equilibrium, a beamslice matched to it undergoes wave-breaking in the transverse phase space, whichcauses an irreversible emittance growth process. This emittance growth mechanismhas been studied extensively in the field of heavy-ion fusion in the context of Brillioun flow (the rigid rotor equilibrium corresponding to maximum beamdensity[5]) in coasting, solenoid-focused beams. It is well understood from theview point of microscopic phase space dynamics of coasting beams[6], andalternatively as the conversion of so-called nonlinear field energy to thermal energy,and thus emittance[7,8]. This irreversible emittance growth has been associated inOSheas analysis with the increase in the beam entropy[9]. These facts concerningwave-breaking due to nonlinear space-charge fields are also, at first glance,apparently at odds with the assertion that the invariant envelope is a preferred modeof transport in pulsed, space-charge dominated beams. This work is also intended toaddress and clarify this apparent disagreement.

This paper is concerned with the self-consistent phase space dynamics of a beam slice as it evolves under of the influence of space-charge and external forces.We analytically study the dynamics to determine the conditions under which wave-

breaking occurs, for both coasting beams, and in slab-symmetric and cylindricallysymmetric geometries. The slab-symmetric case is included mainly to allow use of exact and physically transparent results, which illustrate the mechanisms involvedin phase space wave-breaking. In practice, one is always concerned withcylindrically symmetric beams, and so we extend our discussion of this case toinclude acceleration in an rf structure. Because the dynamics of this system are nottractable after wave-breaking has occurred, we then also employ computationalsimulations to further our understanding of the cylindrically symmetric beam

physics in both the coasting and accelerating cases. The results of this analysisshow that in order to preserve (compensate) the beam emittance within a slice, inthe presence of significant nonlinearities in the space-charge field, one must avoidmatching of the beam to the generalized equilibria, and that the optimal transport of a space-charge dominated beam is typically not close to such equilibria.

2 ENVELOPE DYNAMICS AND LINEAR EMITTANCECOMPENSATION

The purpose of this section is to provide a review of the analytical theoryof emittance compensation as formulated by SR in Ref. 4. This background isneeded in order to understand the detailed nature of the problems addressed in this

paper. The invariant envelope theory begins with the writing of the cylindrically

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symmetric rms envelope equation of each beam slice in the long-beam (two-dimensional) limit. This limit is reached when the beam is highly relativisticv cb ( >>1) , and even a short pulse of particles appears elongated in thelongitudinal dimension in its rest frame. In this limit, which is assumed for theremainder of this section, the transverse defocusing due to space-charge forces isdependent only on the local value of the current I q v q cb ( )= ( ) ( ) and therms beam size at the particular slice in question r z,( ), and the envelope equationincluding acceleration is

( )+ ( ) ( )+ ( ) ( )= ( )( ) ( ) r r r e

r z

z z

z z r

z z, , ,

,8

2

3 . (1)

Here = z ct is the internal longitudinal coordinate of a fixed positionwithin the beam (and thus labels a slice), z is the distance along the beam

propagation direction, and we have suppressed the thermal emittance term, whichmeans we are assuming a space-charge dominated beam. Also, the parameter isa measure of the second-order focusing, e.g. nonsynchronous rf wave[10,11] and/or solenoid focusing[4], applied to the beam as it accelerates with normalized, average(over an rf period) spatial rate = q E m c z / 0 2 . For a standing wave accelerator 1 , while for a disk-loaded travelling wave accelerator it is an order of magnitude smaller[11]. If solenoid focusing is also applied,

+2 2b , where

b B E z z= / .When the beam is focused by a solenoid, but not accelerating, = 0 , we

recover the familiar rms envelope equation

( )+ ( )= ( )( )

r r e

r

z k zr

z, ,

,2

3 , (2)

where k qB m c qB m c z z = / / 0 2 0 2 is the spatial betatron frequency[12], whichin this case is identical to the Larmor frequency of the particle. Eq. ? is a nonlinear differential equation with no general analytical solution, but does have a particular,equilibrium solution

eqe

k

r

( )=( )13

. (3)

This steady state envelope corresponds to a rigid rotor equilibrium known asBrillouin flow, in which the beams canonical angular momentum is zero. The

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typical way of dealing with Eq. 3 is to expand it to first order about itsequilibrium, in the parameter r r eq eq=

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vanishes. This definition of emittance is identical to that of the standard radial rms

emittance rms r r rr = 2 22 if each slice of the beam is a line in r r , ( )

trace space, which connects the origin to the edge of the slice distribution throughthe value r r , ( ). This case, which is physically realized when the beamsdensity distribution is uniform inside of the radius r r = 2 and vanishing outsideof this radius, was the subject of the envelope dynamics analyzed in Ref. 4.

r

r

1 2 3

eq1 eq2 eq3

'

2 1<

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trace space area described by the three slices at k z p =0 2 3 2 2, / , / , . It can beseen that the trajectories fan out to produce a large summed (or projected ) emittanceat k z p = / , / 2 3 2 , while to lowest order the emittance vanishes at k z p =0 2, and also at k z p = (not shown). These emittance oscillations repeat twice every

plasma oscillation, but eventually decohere due to small, higher order differences inthe nonlinear plasma frequency in each slice[13]. The proper execution of such anemittance oscillation due to differential slice motion is termed emittancecompensation in the context of high current, space-charge dominated beams in rf

photoinjectors. This simple picture is complicated somewhat by acceleration, asdiscussed below, but essentially illustrates the relevant physics of compensation

process.

r

r '

k z=3 /2 p

k z=0,2 p

1

2

3

1< .

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While the picture in Fig. 4 gives a similar schematic view of emittanceoscillations as Fig. 3, it has two notable differences with the non-accelerating case.The first is simply that the emittance one needs to be concerned with when the

beam accelerates is the normalized emittance r n r r r r , = 2 22

r r r r 2 22

, which is a measure of the transverse phase space area,

and is thus conserved under linear transport and acceleration. The adiabaticdamping of the trace space area is emphasized in Fig. 4 by rescaling of the verticalaxis by (we have set =1 in this analysis) to account for this effect. The seconddifference is that all mismatch oscillations have end-points attached to the the line

= r r / / 2instead of

=

r 0 . As the invariant envelope associated with the

slices becomes smaller with increasing energy as 1 2 / (the ensemble of ellipsesshown slides up the line = r r / / 2towards the origin), the area associatedwith the emittance not only oscillates, but secularly damps as 1 2 / . Note that theoffset phase space area described by the mismatch oscillations (the ellipses in Fig.4.) is actually conserved, as can be seen by Eqs. 11 and 12. This means that anensemble of slices initially placed all at the same initial condition, but withdifferent ( ), the set of points which make up the section of the phase space

boundary not attached to the origin form a line with varying length, but no area.This ensemble line stretches and rotates about the invariant envelope of the matchedslice. If the invariant envelope slice is actually present in the beam, the ensembleline passes through the invariant envelope line = r r / / 2, rotates about theintersection point of these two lines. Thus the matched invariant envelope is a

generalized fixed-point in the envelope phase space. This is an importantobservation having implications for particle motion within a slice.

3 LAMINAR AND NONLAMINAR MOTION IN COASTING SLABBEAMS

As can be seen by the analysis above, the self-consistent collective motion of particle beam in cylindrical symmetry is complicated somewhat by the need toapproximate the solutions to the differential equations which are encountered.Because of this it is most instructive to begin our analysis using a Cartesian, or slab-symmetric (sheet) beam, following the general methods introduced by O.Anderson in Ref. 6.

We start this discussion by examining a freely expanding (unfocused) laminar beam, with initial ( z=0) density profile, infinite in the infinite in the y and zdimensions, and propagating in the + z direction

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n xa

f x n f x f bb

b00

0 0 0 0 1( )= ( )= ( ) ( )= , ,with (13)

where b is the beam charge per unit (slab) area, and a nb b0 0= / is the effectiveinitial beam width. The case of free-expansion can be considered to represent themost non-equilibrium situation it is possible to encounter. It can also be thought of as forming one portion of propagation under periodic application of thin lensesseparated by drifts, or free-expansion regions.

The equations of motion for the electron position for the free-expansionscenario are, under laminar flow conditions,

( )= ( ) ( )= ( ) = x z k F x F x f x dx p x

02

0 00

0

, constant, (14)

where the local value of the initial (spatial) plasma frequency in the plane of symmetry has been defined as

k r n

pc b

02 0

2 34=

. (15)

If laminarity is obeyed, the integral F x0( ) is constant and these equations of havesolutions dependent only on initial conditions,

x x z xk z

F x p

0 0

2

02,( )= +( ) ( ). (16)

The density distribution is also a simple function of its initial state, as conservationof probability gives f x x z dx f x dx0 0 0( )( ) = ( ), or

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f x x z f x

dx x

dx

f x

k z f x

p

00

0

0

02

01 2

( )( )= ( )( ) =( )

+( ) ( ), . (17)

In the freely expanding case, the density distribution becomes more uniform as itexpands over many plasma radians ( k z p >>1 ),

f x x f x

k z f x k z p p0

02

0

2

12

2( )( )= ( )

+( ) ( )

( ). (18)

This observation is critical, as it implies that the transport is more linear, sincethe space-charge defocusing for a uniform beam becomes approximately linearlydependent on offset,

( )= ( ) x z k F x x z p02 0 22 / (19)

This will in turn imply that the phase space wave-breaking effects which lead toirreversible emittance growth are mitigated, as the angle a particle makes in phasespace becomes more linearly correlated with position,

= ( )+ ( ) x

x

k zF x

x k z F x z p

p

02

0

012 0

2 20

2. (20)

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0.0

0.20

0.40

0.60

0.80

1 .0

1 .2

0.0 0.20 0.40 0.60 0.80 1 .0 1.2

initialfinal

f / f ( 0 )

x/xmax

Figure 5. Initially parabolic slab beam distribution (solid line), mapped to moreuniform (normalized) distribution (dashed line) after a drift length k z p =4 .Distribution shown as a function of relative offset position, x x / max .

As an example of this increased distribution uniformity is shown in Fig.5, where a beam with initial parabolic profile

f xxa0

2

1( )= (21)

has freely expanded for a distance k z p =4 . The profile has become noticeablyflattened during this expansion.

It is instructive at this point to calculate the emittance evolution associatedwith this freely expanding beam. In order to do so, we consider a number of

possible forms of the distribution, gaussian, parabolic, and uniform (flat-top). ).The single particle equations of motion and the condition of laminar flow allow thecalculation of the second moments of the distribution and consequently the RMSemittance. Laminar flow implies

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n x z dx n x dxb b o,( ) = ( )0 (22)

Thus, the second moments of the distribution in trace space can be simplycalculated by integrating with respect to the initial particle positions. For example 2 is:

x x x z n x dxb2 2

0 0 0= ( ) ( )

, (23)

Through this method the second moments are calculated and the emittance is found.The emittance evolution of the drifting laminar beam can be written in thefollowing general form,

= k z p02 0 , (24)

where 0 is the initial rms spread in the distribution, and is a form factor dependent on the initial beam distribution type. The values of are summarized inTable 1; for a uniform initial distribution, there are no nonlinear forces, and thus noemittance growth. Note that in the case of free-expansion that the emittance growslinearly with distance from the launching point, but has no dependence on initial

beam size, as k p b02

0 . While this linear growth is a worrisome phenomenon,it turns out not to be valid for cylindrically symmetric beams in this case thegrowth is reversed after a time during expansion, and after application of a thinlens, nearly a perfect oscillation of this nonlinear space-charge force-inducedemittance can be made to occur. This compensation of the nonlinearity-derivedemittance, which is the central phenomenon under study in this paper, will bediscussed in following sections.

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PROFILE

Gaussian 3

3

Parabolic 23675

Flat-top 0

Table 1. Values of the form factor for various initial slab-symmetric distribution types.

Wave-breaking occurs in phase space when the value of x z( )somewhere in thedistribution becomes independent of x0 , and the transverse momentum distribution

becomes a multiple valued function of transverse offset. According to Eq. 14, thiscondition ( dx dx / 0 =0) also implies that the density would become singular at

these points. Note that there is no wave-breaking for the free-expansion slab-symmetric case, as

dxdx

k z f x

p

0

2

01 21 0= +( ) ( )> > . (25)

This will change when we introduce focusing, but one conclusion remains from thisanalysis: one must allow the beam to stay far from equilibrium in order to avoidthe most serious consequences of wave-breaking.

There are two ways to proceed from this point. One is to introduce thinlenses to produce a periodic transport system with an rms matched (in the sense thatthe envelope has the same periodicity and symmetry as the applied focusing forces)

beam. The other is to introduce a uniform-gradient focusing channel (akin to thesolenoid commonly used in cylindrically symmetric systems), but to allow amismatch between the beam and the channel. In the interest of simplicity, we willfirst follow the latter course first.

In a system with uniform gradient focusing, Eq. 14 becomes

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( )+ ( )= ( ) x z k x z k F x p 2 02 0 , , (26)

where we have introduced the betatron wave-number k associated with free

oscillations under the influence of the focusing gradient The equilibrium solutionfor a given initial particle position is simply

x xk

k F x

eq

p

0

02

2 0( )= ( )

, . (27)

It can be seen that this equilibrium can be made self-consistent, in the sense that no particles will move after the distribution is launched, if F x0 1( )= , and k k p02 2= . If any other distribution other than a uniform one is employed, there will besubsequent motion, and associated rearrangement of the distribution. In this moregeneral case, we may write the solution to Eq. 22 as

x x z x x x x x k zeq eq0 0 0 0, cos .( )= ( )+ ( )[ ] ( ) (28)

The wave-breaking condition associated with this motion is

x

x

k

k f x

k

k f x k z p p

0

02

2 00

2

2 01 0= ( )+ ( ) ( )=cos , or (29)

f xk

k

k z

k z p0

2

02 22 2

( )= ( )( )

cos

sin / . (30)

It can be seen that wave-breaking always occurs for a sufficiently small value of f x0( ), i.e. portions of the beam found in a long continuous tail, assuming a

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monotonically decreasing function f x0( ). Quantitatively, Eq. 30 states that wave- breaking eventually occurs for all f x k k p0

20

22( )< , with the most interior value of x0 undergoing wave-breaking at k z = (for distributions which smoothlyapproach zero, wave-breaking begins in these tails at k z = / 2). It is also apparentthat wave-breaking can be avoided by a combination of removal of the distributiontails, so that f x0( )discontinuously goes to zero at a hard-edge beam boundary, and

by making the ratio k k p 2

02 become small. When this ratio is near unity, the beam

is closely matched to the external focusing, and when the ratio is much smaller than unity the beam is mismatched, with the focusing being too weak to control the

beam distribution at its launch size. Another way of understanding wave-breaking isthat the equilibrium beam size xeq associated with the initial wave-breaking in tracespace is in fact a fixed point of the oscillation. On the other hand, we know thatthe origin in trace space is also a fixed point, with an opposing sense of rotation.The existence of two such fixed points guarantees that the trace space will filamentafter wave-breaking, and the emittance will grow irreversibly. The trace space

picture of this system is shown in Fig. 6.

-0.20

-0.15

-0.10

-0.050

0.0

0.050

0.10

0.15

0.20

0.0 0.20 0.40 0.60 0.80 1.0 1.2

x '

x

Fixed point

Trace spacerotation

Distribution

Figure 6. Trace space picture of slab symmetric beam a t wave-breaking onset ( k z = /2), for caseof k k p

20

2 =2/3, showing two fixed points with opposing direction of rotation.

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Thus we deduce that a mismatched beam is more likely to preserve its laminar flow, under mismatched conditions, which is an extension and deepening of whatwe have learned from the case of free expansion. To emphasize this point, in Fig. 7,we show a plot of normalized beam density at the maximal wave-breaking pointk z = for a cut-off (at the 25% intensity level) parabolic distribution in nearlymatched ( k k p

20

2 4 3 / / = ) and highly mismatched ( k k p 2 02 1 3 / / = ) cases. The nearlymatched case barely evades wave-breaking, and displays a very large density spikeat the beam edge, while the highly mismatched beam easily maintains laminarity,giving a smaller density spike.

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.20 0.40 0.60 0.80 1.0 1.2

mismatchedmatched

f / f ( 0 )

x /xm a x

Figure 7. Normalized beam density f f / 0( )for a beam with initially parabolic slab beam distribution(cut-off at 0.25 normalized density) at k z = , for distribution in nearly matched ( k k p 2 02 4 3 / / = )and highly mismatched ( k k p

20

2 1 3 / / = ) cases. Offset x is normalized to its maximum value in thedistribution.

In order to calculate the emittance evolution in the case of the slab beam in afocusing channel, we follow the same procedure as in the drifting beam case to upto the point of wave-breaking, where strictly laminar flow ends and this analysis

breaks down. Assuming a cold beam initially at a waist ( = x0 0 ) the emittance isfound to be

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= ( )k k k z p2

0 sin ,

(31)

where again is a constant depending of the form of the initial distribution. Wenote from this that the predicted maximum emittance occurs at k z = /2, as withthe correlated inter-slice emittance studied in Ref. 4. It should also be emphasizedthat this is the same longitudinal position as that the initial wave-breaking occursin for a distribution with a continuous tail.

4 LAMINAR AND NONLAMINAR MOTION IN COASTINGCYLINDRICAL BEAMS

The density of a continuous beam in an axisymmetric system can be described bythe expression

n r z f r zb b, ,( )= ( ) , (32)

where b I qv= / is the beams axial charge density. The electromagnetic force on a particle in such a distribution is

F r zqr

n r z r dr q r

r r br

, , .( )= ( ) ( )2 2 220

02

0

(33)

The force has been written in terms of the enclosed current at an initial point r z0 0( ),

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r n r z rdr br

00

020

( )= ( ), , (34)

which for laminar flow is a constant of the motion.The equation of motion for a particle with no canonical angular momentum

experiencing both a solenoidal restoring force and the repulsive space charge forcecorresponding to Eq. 33 is

( )+ ( )= ( )r z k r z r r r

e

2 02 3

2, (35)

which is a nonlinear equation not amenable to exact solution in general. We can begin an approximate analysis, however, by defining an equilibrium radiuscorresponding to each value of r 0 ,

r r r r

k r

k r

k eqe p

00

2 2 3 002

2( )= ( )

( )

. (36)

Here we have introduced an average beam plasma frequency

k r r n r r r

r pc b c2

00

2 30

02 2 3

4 2( )= ( )= ( )

. (37)

which corresponds to the mean enclosed initial density at r 0 .We now proceed to linearize Eq. 35 about the equilibria given in Eq. ? to

obtain

+ =r k r 2 02 , (38)

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where r r r eq= . This equation yields a familiar form of solution, for adistribution beginning with no radial momentum (or angular momentum in the

beams Larmor frame)

r r z r r r r r k zeq eq0 0 0 0 2, cos( )= ( )+ ( )[ ] ( ) (39)

The wave-breaking condition is again given by

r r

r

r

r

r k zeq eq

0 0 01 2 0= + ( )=cos , or

r

r

k z

k z

eq

02

2

2 2= ( )

( )cos

sin / . (40)

The quantity on the left-hand side of Eq. 40 can be written as

r

r

r

r r

k r

k r eq p p

p0 0 0

20

02 2=

( )= ( )

( ), (41)

where we have employed the local measure of the initial beam plasma frequency,

k r r n r r r

r pc b c2

00

2 30

2 302

4 2( )= ( )= ( )

. (42)

As an illustrative example, let us examine the wave-breaking condition for thecase of an initially Gaussian beam, where

n r n r b b r 0 0 02 22( )= ( )exp / . (43)

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In this case

r n r r dr

n r

n n r

b r

r

b r r

r b b

0 02 2

0

02

02 2

20 0

2 2

2 1 2

2

0

( )= ( )

( )[ ] ( )[ ]

exp /

exp /

,

=

=

(44)

and the wave-breaking condition can be written as

k

k r r

r

k z

k z

p

r

r

r

0 0 02 2

02 2 2

2

1 2

2

2

exp /

exp /

cos

sin / ,

( ) ( )

= ( )( )

with k k r n

p pc b

02 2 0

2 304= ( )=

.(45)

For wave-breaking to be avoided, we have that the left-hand side of Eq. 45must be greater than unity,

k

k r r

r

k

k g r p

r

r

r

p0 0 02 2

02 2

00

2

1 21

exp /

exp / .

( ) ( )

( )> (46)

The function g r 0( )is shown in Fig. 8 with f r 0( ) also displayed for comparison. Itcan be seen that g r 0( )approximately follows the density, and thus the threshold for wave-breaking is estimated as

k

k f r p0 0 1

( ). (47)

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This is in contrast to the equivalent condition found in the slab beam case,

k

k f y p

02

2 021

( ), (48)

which has a much stronger dependence on the mismatch parameter k k p0 / .

0 .0

0 .20

0 .40

0 .60

0 .80

1 .0

1 .2

0 .0 0 .50 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5

f g

f / f ( 0 ) , g

/ g ( 0 )

r/ r

Figure 8. A comparison of the function g r 0( )with Gaussian f r 0( ).

As the linear dynamics of the axisymmetirc beam have been seen to beformally quite similar to those of the slab beam, it is not surprising that theemittance evolution is similar as well. Given the same initial conditions asassumed in the slab case, we find the emittance to be of the same form as well.

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= ( )02 0 2k k z p sin . (49)

Here is again a form factor, defined as in the previous section. The numericalvalues of found for the cylindrically symmetric case are shown in Table 2. Wewill see that Eq. 49 provides a very accurate description of the emittance evolutionup until wave-breaking. Note that the emittance in Eq. 49 in fact linearly dependenton 0 , as k p0 0

1

.

PROFILE

Gaussian 0.141

Parabolic 0.065

Flat-top 0

Table 2. Values of the form factor for various initial cylindrical beam slice distribution types.

5 SIMULATION OF COASTING CYLINDRICAL BEAMS

The analytical treatments of intra-slice transverse space charge detailed above arelimited to the laminar flow regime, and in the case of cylindrical beams are onlyapproximate. They do however, predict where wave-breaking will occur and that i tcan be minimized or avoided by mismatching the beam. In order to test these

predictions and examine the behavior of a beam slice after wave-breaking, we useself-consistent simulations that follow the evolution of the beam using the spacecharge force of Eq. 33.

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We found in the case of the slab beam expanding under its space charge forcethat there was no wave-breaking for any type of distribution. Equation 33 tells usthat this is not the case for a freely expanding cylindrical beam if the initialdistribution function falls off, so that the integral of the charge density does notincrease proportionally with r. In this case we expect wave breaking, and look tosimulations for understanding the beam behavior after wave-breaking.

The emittance evolution of the freely expanding beam shows the effects of wave-breaking. As in the focusing channel, the emittance increases to a maximumat p0/4 where wave-breaking occurs. As the beam continues to expand the particlesin the vicinity of the initial wave-breaking point (where the maximum outward forceis found) effectively rotate, as the entire distribution expands around this point.This rotation causes the tail particles to tuck under in phase space in a distance a

bit longer than the initial plasma half-wavelength (the plasma frequency is notconstant, but decreases as the beam expands), as would be expected, and theemittance decreases during this initial rotation. The emittance growth is not

perfectly compensated by this nonlinear effect however and the emittance reaches alocal minimum. After that is becomes simply proportional to as the beamcontinues to expand. Examination of the beam phase space evolution, shown inFigs. 10, illustrates this process.

Figure 9. Results of a simulation of an the free-expansion of an initially Gaussian beam. The beam size(solid line) increases monotonically while the emittance (dashed line) has a local maximum andminimum.

0 . 0

5. 0

1 0

1 5

2 0

0

2 . 5

5

7 . 5

1 0

0 . 0 0 . 2 5 0.50 0.75 1 . 0

/

0

E mi t

t an c e

[ A r b i t r ar y

U ni t s

]

k p

z/ 2

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We note from Fig. 10(b) that this tuck under effect on the emittance occurs onlyafter the RMS beam size has grown substantially (recall that k z p0 > , and the

beam has had a large distance in which to expand), as the emittance minimumoccurs when /0 8.5.

While the drifting beam is instructive, we are interested in beam transportinvolving focusing elements. We proceed therefore, by examining two cases:

periodic thin lenses separated by drifts, and a focusing channel. In the case of thinlens focusing we can directly apply the result of the drifting beam. We find that for a given transport length fewer lenses and larger beam size oscillations will producea better emittance at the end of the transport line provided that the beam makes an

integer number of oscillations. Figures 11 and 12 below show two simulations of a beam with the same initial conditions and transported through the same length of drift. In the first there is one thin lens applied when /0 = 8.5. In the second, inorder to approximate a beam which is more closely matched to a uniform focusingchannel, a lens is applied each time the beam size doubles its initial value. It isclear from the graphs that when the beam is allowed to expand enough to takeadvantage of the tuck under effect observed in the drifting beam above, much of the emittance growth can be reversed when the beam is focused back down. In thecase where the beam size oscillations are kept smaller we see that the emittanceoscillates around its peak value but never drops to as low a level as in the first case.

0 . 0

0.020

0.040

0.060

0.080

0 . 1 0

0 . 0 1.0 2 .0 3.0 4.0 5 .0

r ' [ r a

d i a n s

]

r [mm]( a )

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0 . 0

0.030

0.060

0.090

0 . 1 2

0.15

0 . 0 5.0 1 0 1 5

r ' [ r a

d i a n s

]

r [mm]( b )

Figure 10. Trace space plots of a freely expanding, initially Gaussian beam at the initial emittance (a )maximum and (b) minimum.

The striking performance of the scheme shown in Fig. 11 for minimizingthe emittance at the envelope minimum in other words, compensation of thenonlinear field-derived emittance, is understandable in a number of different ways.If the dynamics being described were only the linear slice dynamics, Figs. 2 and 3illustrate that the emittance performance would be qualitatively the same in Figs.11 and 12. They are not, however, and this is because of the strong wave-breakinginduced in the intra-slice dynamics by the beam being too close to equilibrium. Inother words, the existence of the off-origin fixed point in trace space gives rise towave-breaking, trace space filamentation, and associated irreversible emittancegrowth. OShea has identified irreversible emittance growth of this type with anincrease in the entropy which, we note, is also equivalent to loss of order or information in the system. In the case of Fig. 11, the emittance increase due to fieldnonlinearities is reversed (compensated) and the information about the beamsinitial state is preserved. An excellent illustration of this phenomenon is shown inFigs. 13, which illustrate the beam distribution in r at three points in the

propagation shown in Fig. 11 the initial and final states, as well as the thin lens position. It can be seen that by this judicious choice of focusing that the final beam distribution reproduces the initial distribution remarkably well, consideringhow distorted it becomes in intermediate points in the propagation.

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0 . 0

2. 5

5. 0

7. 5

1 0

0

1 . 5

3

4 . 5

6

0 . 0 0 0.29 0.58 0.87 1 .16 1 .45

/

0

E mi t t an

c e

[ A r b i t r ar y

U ni t s

]

/

Figure 11. Evolution of beam size and emittance in simulation, with thin focusing lens applied at the point of initial emittance minimum. Lens strength chosen to reverse the envelope angle.

0 . 0

0 . 5 0

1 . 0

1. 5

2. 0

0

1 . 5

3

4 . 5

6

0 . 0 0.3 0.6 0.9 1.2 1.5

/

0

E mi t t an

c e

[ A r b i t r ar y

U ni t s

]

/

Figure 12. Evolution of beam size and emittance in simulation, with thin focusing lens applied at the points of beam envelope doubling, and lens strength chosen to reverse the envelope angle. Thesimulation is followed for the same number of plasma periods as in Fig. 10.

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0

1 0 0

20 0

30 0

40 0

50 0

60 0

70 0

0.0 0 . 20 0.40 0.60 0.80 1 . 0 1.2

D e n s

i t y

[ M a c r o p a r t

i c l e s

/ m m

2 ]

r [mm](a)

0

2

4

6

8

10

12

14

16

0 . 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

D e n s

i t y

[ M a c r o p a r t

i c l e s

/ m m

2 ]

r [mm](b )

0

1 0 0

20 0

30 0

40 0

50 0

60 0

70 0

0.0 0 . 2 0 0.40 0.60 0.80 1 . 0 1.2

D e n s

i t y

[ M a c r o p a r t

i c l e s

/ m m

2 ]

r [mm](c )

Figure 13. Evolution of beam distribution during simulation shown in Fig. 10, at the (a) beginning, (b)focusing lens (midpoint) and (c) the endpoint (emittance minimum).

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It is natural to consider the limit suggested by the second case above, in whichthe beam size does not vary the case of a beam matched in the rms sense to auniform solenoidal focusing channel. We can also compare these simulations withthe prediction of Eq. 49, at least until the onset of wave-breaking. The emittanceevolution found by simulation of an initially parabolic beam RMS matched to afocusing channel along with the emittance predicted by Eq. 49 is shown in Fig. 14.

Note that the emittance again follows the same pattern shown above in that itincreases rapidly in a quarter of a plasma oscillation to a maximum[6]. Since wave-

breaking does not occur until this maximum is reached, the excellent agreement between theory and to match the simulation up to that is not surprising. We willencounter a similar type of emittance behavior in accelerating systems in thefollowing sections.

0 . 0

0 . 2 0

0.40

0.60

0.80

1 . 0

1. 2

0.0 5.0 10 2 1.0 10 3 1.5 10 3 2.0 10 3 2.5 10 3

SimulationTheory N

o r m .

E m

i t .

[ m m

m r a

d ]

Z [mm]

Figure 14. Evolution of emittance for beam rms matched to a uniform focusing channel, fromsimulation and analytical prediction (Eq. 49).

6 LAMINAR AND NONLAMINAR MOTION IN ACCELERATINGCYLINDRICAL BEAMS

In the case of a beam accelerating under the influence of radio-frequency fields, the

paraxial equation of motion for a particle in a laminar flow conditions now containsterms arising from adiabatic damping and ponderomotive (alternating transversegradient) forces,

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( )+ ( )

( )+ ( )

( )=

( )( )r z z r z z r z

r r

z r e

82

20

3 ,

(50)

which is again a nonlinear equation without an analytical solution. In this system,there is also an equilibrium-like particular solution to Eq. 50, which is analogousto the invariant envelope discussed above, corresponding to each value of r 0 ,

As in previous sections we proceed by finding an analytical formula for the

emittance of a matched beam. In the case of acceleration by matched we meanthat the RMS size of the beam follows the invariant envelope. This situation isslightly different to that of coasting beams because we are required to reference to the non-stationary particular solution

r r zr r

z

r k r

k z

pe

p

00

00 0

42

12

,

.

( )= ( )

+( ) ( )

( ) + ( )

. (51)

In Eq. 51 we have identified k = 8 , and can see that the particular solution isagain proportional to the initial ratio of k r k p 0( ) / . We can again proceed tolinearize Eq. 51 about these particular solutions, to obtain

+ +

+

=

r r r 1

40

2

. (52)

where r r r p= . This equation has a general form of solution Therefore, we solveEq. 50 to find the single particle motion as

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r r z r r z r r r r r r p p p0 0 0 0 00

0 0 001

21

1

12

, , cos ln sin ln( )= ( )+ ( )[ ] + +

+ ( )[ ] +

(53)

where the integration constants are chosen so that

= r r 0 01

2

. (54)

The wave-breaking condition is now given by

r

r p

0

0

2 0

12

21

8

=

+

+

cos ln

sin ln

. (55)

The quantity on the right-hand-side of Eq. 55 can be recast to give

r

r r n r

r z

k r

k p b c p

0

0

0

20

0

2 0

42 2 2

12

21

8

= +( ) ( ) ( ) =( )

+ =

+

+

cos ln

sin ln

(56)

and we see that wave-breaking is again averted by cutting the tails off of thedistribution

To proceed in the analysis, we again use the laminarity condition to integrateover the initial beam distribution and determine the second moments of the

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distribution and the emittance. We find the (geometric) emittance evolution for a beam rms-matched to the invariant envelope is

geom

e br = +( )

+

4

1

12

03

0sin ln . (57)

Here again is a unitless constant depending on the initial beam distribution, withvalues listed in Table 3

DistributionType

Gaussian 0.1704Parabolic 0.0561Flat top 0

Table 3. Values of the form factor for various initial cylindrical beam slice distribution types,accelerating case.

The expression for the emittance evolution given in Eq. 54 is valid (in the linear

approximation r r p

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0

0 . 2

0.4

0.6

0.8

1

1 . 2

1.4

0 1000 2000 3000 4000 5000 6000

TheorySimulation

E m

i t t a n c e

[ m m

m r a

d ]

Z [mm]

Figure 15. Emittance evolution of an initially parabolic beam matched to the invariant envelope with a

60 MV/m peak accelerating field gradient (These beam and accel erator parameters a re the same asthose in the booster linac at the Neptune Advanced Accelerator Laboratory). The dashed line is thesimulation result and the solid is produced by Eq. 54.

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-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0 0.1 0.2 0.3 0.4 0.5 0.6

r ' [ r a

d i a n s ]

r [mm]

Figure 16. Trace space of an initially parabolic beam slice at the maximum emittance point inaccelerating beam simulation. Wave breaking has just occurred.

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7 SIMULATION OF ACCELERATING CYLINDRICAL BEAMS

In this section we study the behavior of an initially parabolic profile accelerating beam matched to in the rms sense the invariant envelope, and compare simulationto analytical results. Figure 15 shows the normalized emittance evolution in asimulation of such a case along with emittance predicted by Eq. 54. Again we seethat the emittance ( n geom geom= ) rapidly increases to a local maximum.We also see from the figure that the analytical formula for the emittance agrees wellwith the simulation up to the emittance maximum. However because Eq. 52 is thelinearized equation of motion, and r has constant amplitude while

r z p 1 2 1 2 / / decreases, the agreement between theory and simulation is not asstriking as with the coasting beam. Also we see that theory and simulation do notagree after the emittance maximum. This is in keeping with the coasting beam asthe beam undergoes wave breaking near the emittance maximum and theassumption of laminarity used in Eqs. 50-54 is no longer true. This wave breakingis easily seen in the beam trace space at the peak emittance shown in Fig. 16.

We see from the simulation that the emittance does not change significantlyshortly after the emittance maximum. Since the transverse plasma frequency of the

beam decreases as 3 2 / , the acceleration process essentialy stops the plasmaoscillations and the beam becomes emttance dominated. The initial emittancegrowth caused by space-chagre field non-uniformities then is frozen in and the

beam has a finite irreversible emittance. We can use Eq. 57 to estimate the final

emittance of the beam and therefore its size in the emittance dominated limit. Todo this we start by finding the position of the emittance maximum by

= = +( )

+

+

+ +

+

n e b

zr 2 2

02

20 0 00

16

1

1

2

1

21

1

2sin ln sin ln cos ln

(58)

or

tan ln1

210

+

= +

. (59)

Therefore, the position of the emittance maximum is

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z

e

max tan

= +( )0

2 1

2

111 (60)

and the maximum emittance is simply

n

e br ,max /

exp tansin tan=

+( ) +( )( )[ ]

+( )[ ]4

1

1

2 11

0 11 2

1 .

(61)

The final beam size in the simulations is estimated by ignoring the spacecharge term in the envelope equation and assuming a steady state solution based ona constant normalized emittance equal to the maximum as predicted by Eq. 61,

min,max=

814 n . (62)

A comparison between the final rms beam size achieved in simulation and the prediction of Eq. 62 for the simulation case of Fig. 16 is shown in Fig. 17. Theagreement is quite good in the asymptotic region, where the simulated beam sizeapproaches a constant value very close to that predicted from the above analysis.Thus one can determine, simply by knowing the degree of nonuniformity of thedistribution (which is parameterized by ) at the beginning of acceleration withtransverse matching to the invariant envelope.

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1000 2000 3000 4000 5000

SimulationInvariant EnvelopeEmittance Limit

R M S B e a m

S i z e

[ m m

]

Z [mm]

Figure 17. The beam envelope evolution for the same simulation as Fig. 16. Here the beam sizefollows the invariant envelope initially, but levels off as it approches the limit predicted by Eq. 62.

As an example of the potential final emittance, we take the nominal LCLS photoinjector design parameters, in which a 100 A beam is emitted in a highgradient rf gun, accelerated to 0 12 , and then focused into a matched invariantenvelope at the beginning of a high gradient linac. For a standard SLAC S-bandtravelling wave ( .3 ) linac (average accelerating gradient of 17 MeV/m), one

obtains an asymptotic emittance of n,max .=6 5 mm-mrad. Even though aroughly uniform beam is planned to be launched at the cathode, i t will benonuniform at the injection to the linac due to nonlinearities in the space-chargeforces at very low velocities, as well as imperfections in the drive laser spatio-temporal profile. To see the potential effects of such nonlinearities, if we assume 0 1. (between a gaussian and a parabolic profile), then the emittance due tononlinearites alone is n,max .=0 65 mm-mrad, which is nearly equal to the fullallowed design emittance in the LCLS. An alternative design, which is discussed

below, uses the high gradient (30 MeV/m) standing wave ( 1 ) PWT linacdeveloped at UCLA for the acceleration after the gun. In this case, we have n,max .=2 75 , which produces a more tolerable margin for emittance due tononlinearities and wave-breaking.

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8 CONCLUSIONS

In this paper, we have explored the consequences wave-breaking and assoctiateemittance growth of the choice of beam envelope trajectory, i.e. the degree towhich a beam is matched to a generalized equilibrium. In cases where thenonlinearity of the field is tolerable, running the beam essentially on the invariantenvelope in a booster linac works well, as predicted by the analysis of SR.

On the other hand, when a moderately non-uniform beam is propagated througha transport section, it was found that mismatching the general equilibriumminimizes the initial wave breaking and allows the emittance oscillation to come toa smaller minimum. The minimum emittance associated with matching theinvariant envelope is given by Eq. 61, which serves a useful guide to estimation of the best performance possible for a given injector configuration.

In conclusion, we have in this work attempted to unify the microscopicconcepts of linear emittance compensation and nonlinear wave-breaking, showingtheir relationship to one another in the context of high brightness photoinjectors.This understanding aids in the classification of global characteristics of beamdistributions, such as nonlinear field energy and entropy, which have beenoriginally introduced in the field of intense ion beams. It is hoped practioners from

both fields will make use of these results.

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References

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[3] X. Qiu et al. , Phys. Rev. Lett. 76 , 3723 (1996 ).

[4] Luca Serafini and J.B. Rosenzweig Physical Review E 55 ,7565 (1997).

[5] L. Brillouin, Phys. Rev ., 67 , 260 (1945)

[6] O. A. Anderson, Part. Accel. 21 , 197 (1987 )

[7] T.P. Wangler et al ., IEEE Trans. Nucl. Sci . 32 , 2196 (1985).

[8] I. Hoffman and J. Struckmeier, Part. Accel ., 21 , 69 (1987).

[9] Patrick G. O'Shea , Physical Review E 57 , 1081 (1998).

[10] S.C. Hartman and J.B. Rosenzweig, Phys. Rev. E 47 , 2031 (1993).

[11] J.B. Rosenzweig and L. Serafini, Physical Review E 49 , 1499 (1994).

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[13] Bruce E. Carlsten, Physical Review E 60 , 2280 (1999)

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