serafini_1997_0209

8/14/2019 serafini_1997_0209

http://slidepdf.com/reader/full/serafini19970209 1/26

Envelope analysis of intense relativistic quasilaminar beams in rf photoinjectors:

A theory of emittance compensation

Luca Serafini Istituto Nazionale di Fisica Nucleare, Milano, Via Celoria 16, 20133 Milano, Italy

James B. Rosenzweig Department of Physics and Astronomy, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, California 90095-1547

Received 11 November 1996

In this paper we provide an analytical description for the transverse dynamics of relativistic, space-charge-

dominated beams undergoing strong acceleration, such as those typically produced by rf photoinjectors. These

beams are chiefly characterized by a fast transition, due to strong acceleration, from the nonrelativistic to the

relativistic regime in which the initially strong collective plasma effects are greatly diminished. However,

plasma oscillations in the transverse plane are still effective in significantly perturbing the evolution of the

transverse phase space distribution, introducing distortions and longitudinal-transverse correlations that cause

an increase in the rms transverse emittance of the beam as a whole. The beam envelope evolution is dominated

by such effects and not by the thermal emittance, and so the beam flow can be considered quasilaminar. The

model adopted is based on the rms envelope equation, for which we find an exact particular analytical solution

taking into account the effects of linear space-charge forces, external focusing due to applied as well as

ponderomotive RF forces, acceleration, and adiabatic damping, in the limit that the weak nonlaminarity due to

the thermal emittance may be neglected. This solution represents a special mode for beam propagation thatassures a secularly diminishing normalized rms emittance and it represents the fundamental operating condition

of a space-charge-compensated RF photoinjector. The conditions for obtaining emittance compensation in a

long, integrated photoinjector, in which the gun and linac sections are joined, as well as in the case of a short

gun followed by a drift and a booster linac, are examined. S1063-651X9710706-1

PACS numbers: 41.75.i, 41.85.p, 29.17.w, 29.25.Bx

I. INTRODUCTION

Quasilaminar, space-charge-dominated relativistic elec-

tron beams have become a subject of great interest with the

advent of short laser pulse-driven radio-frequency rf pho-

toinjectors 1,2 that are able to produce electron beams car-rying current densities well in excess of 1 kA/cm2, with the

transition from the nonrelativistic to the relativistic regime

occurring very quickly. The accelerating gradient required to

guarantee that the beam will be captured in the rf wave at

relevant wavelengths 5–25 cm ranges from 10 up to 100

MeV/m: the beam is therefore accelerated from rest at the

photocathode emissive surface, up to relativistic energy

within a fraction of a rf wavelength, which is a distance

comparable to one-half of a plasma oscillation period in the

transverse plane. The trapping condition is typically ex-

pressed as 1/2, in terms of the quantity

eE 0 /2kmc2, which represents the dimensionless amplitude

of the vector potential associated with the accelerating field,of frequency rf (k 2 rf / c) and amplitude E 0 .

Furthermore, the random, thermal component to the trans-

verse emittance is very small compared to the total rms emit-

tance, which is dominated by the dilution of the projectedtransverse phase space density due to correlations in thebeam distribution function, so that the beam is fairly laminarin both the transverse and longitudinal planes. This impliesthat in the transverse plane trajectories do not cross eachother, while in the longitudinal plane different slices of length small compared to the total bunch length do not mix

with each other. Since neighboring longitudinal slices addi-tionally do not behave in vastly different ways, precludingthe occurrence of large longitudinal density gradients in thebeam charge density, this final condition implies that thebeam may be broken up, for analysis purposes, into nearly

independent longitudinal slices that behave in the same man-ner as a continuous beam. Evidence for the validity of thismodel for photoinjector beam dynamics comes from bothmultiparticle simulations and experiments performed atBrookhaven 3.

This set of conditions, which defines the notion of aquasilaminar beam in this paper, is generally attained in rf photoinjectors, in particular when they are operated in thespace-charge-emittance compensation regime 4. This re-gime implies that the beam propagates for one transverseplasma oscillation, so that the correlations in the transversephase space that develop in the first half of the oscillation areundone in the second half by properly focusing the beam.Due to the relativistic diminishing of the space-charge forcesas the beam accelerates, one can adiabatically nearly termi-nate the plasma motion and associated emittance oscillationsas the minimum in the emittance occurs, obtaining maximumbeam brightness at the exit of the photoinjector.

In this paper we wish to provide a simple framework inwhich the beam dynamics in such a regime can be analyti-cally described and the space-charge-emittance correctiontechnique can be quantitatively explained. We begin by us-ing a heuristic model of the plasma and emittance oscilla-tions in a quasilaminar beam. This model allows the under-lying physical mechanisms involved in the complicated

PHYSICAL REVIEW E JUNE 1997VOLUME 55, NUMBER 6

551063-651X/97/556 /756526 /$10.00 7565 © 1997 The American Physical Society

8/14/2019 serafini_1997_0209


phase space dynamics of the rf photoinjector to be eluci-dated. After this discussion, we then construct the quantita-tive model for quasilaminar beam propagation. Analyticalexpressions for the beam envelope from the photocathodesurface up to the gun exit in a long, integrated rf photoinjec-tor are provided and the predictions for optimum photoinjec-tor configuration to achieve emittance correction are ex-tracted from the properties of the envelope itself. A particular

solution for the beam envelope is found that assures all thebunch slices evolve in transverse space phase with a com-mon phase space angle, which is in fact the desired final stateto achieve emittance compensation. This particular solutionis termed the invariant envelope, and is in many ways analo-gous to the equilibrium Brillouin flow of space-charge-dominated beams in constant gradient focusing channels. Al-though this study is directly applied to a description of rf photoinjectors, the concept of invariant envelope and themethod of analysis is of interest and applicable to any rela-tivistic beam that is space-charge-dominated and acceleratedin high gradient linear accelerators.

The equation we base our analysis on is Lawson’s expres-

sion for the evolution of the rms envelope in the paraxiallimit 5,

2 K r

s

3 3

n2

3 2 20, 1.1

which governs the evolution of the cylindrical symmetricrms transverse beat spot size ( z) under the effects of anexternal linear focusing channel of strength K r F r / r 2 mc2. Here the prime indicates differentiationwith respect to the independent variable z, the distance alongthe beam propagation axis, mc2 is the mean beam energy,

and b / c1 2

is the normalized mean beam ve-locity. The defocusing space charge term in Eq. 1.1 is pro-portional to the beam perveance s , and the final term rep-resents the outward pressure due to the normalized rmsemittance, which in the case of cylindrical symmetry can bewritten as

n

2r 2 r 2rr 2. 1.2

We use Eq. 1.1 under a host of assumptions, which wenow delineate. Equation 1.1 is of course only valid in a

paraxial approximation (

1) and for a narrow energyspread beam. In our envelope analysis, which is applied onlyin the region where the beam has attained relativistic veloci-ties the mean beam velocity b cc, the normalizedacceleration gradient is approximated as constant, so that ( z2) ( z1) ( z 2 z1). In the case of an unbunchedbeam the perveance takes the form s I /2 I 0 , with I 0ec / r e17 kA for electrons. Since we restrict the discus-sion to axisymmetric beams, the focusing gradient can incor-porate two different types of focusing, that applied externallyby a magnetostatic solenoidal focusing field, and the pon-deromotive rf focusing 6 produced by the nonsynchronousspatial harmonics of the accelerating rf wave, an effect that isparticularly strong in a high gradient standing wave accelera-

tor such as an rf photoinjector. As discussed in Refs. 6 and7, these two focusing sources can be cast into a singleexpression,

K r 8b 2

sin

2

, 1.3

where bcB z

/ E 0

eE 02 m

ec 2 /sin( ), k sin( )

for the particular case of a constant solenoidal magneticfield, is the particle phase with respect to the rf field wave, t kz 0 , and 0 is the rf phase of the bunch cen-troid at injection. The quantity , which is a measure of thehigher spatial harmonic amplitudes of the rf wave is definedin Sec. III and is generally quite close to unity in practical rf structures.

We have given the expression for the beam perveance inEq. 1.1 above for an unbunched beam of constant current I . In our analysis of the quasilaminar beam in Sec. III, wegeneralize this quantity to include the case of bunched beam,by incorporating a geometrical factor 8 g( ) in the per-veance that contains the longitudinal dependence of the

transverse space-charge field versus the internal bunch coor-dinate z bt .

Furthermore, Eq. 1.3 ignores possible chromatic aberra-tion effects on the transverse phase space dynamics, due toan energy-phase correlation in the bunch—this analyticalstudy is carried out assuming a monoenergetic bunch. A re-lated source of longitudinal-transverse phase space correla-tions in this system arises from the phase dependence of thetransverse rf forces, which gives rise to an emittance increase9 at the first iris of the gun cf. Fig. 3. We assume that thissource of emittance, like the chromatic effects, does not giverise to significant changes in the transverse beam dynamicsof a given slice of the bunch. This assumption is quite

good, in that these correlations are of a similar form to thethose arising from space charge, but smaller in magnitude innearly all cases of interest. In fact, because of the similarityin spatial dependence of the forces, it has been observed insimulations that the space-charge-emittance compensationprocess can also partially mitigate this source of emittance10. Although the solutions found for Eq. 1.1 can be ex-tended to any kind of charge density distribution in thebunch, the actual predictions of the rf photoinjector designedto achieve emittance compensation will be provided for adensity distribution that is Gaussian in all dimensions.

The initial model used for the photoinjector analysis as-sumes a long multicell rf structure, i.e., an integrated device

such as the AFEL advanced free electron laser injector atLos Alamos 4, and the proposed PWT plane wave trans-former injector at University of California at Los AngelesUCLA 11. The analysis is, however, sufficiently broadthat many characteristics of photoinjectors with short one ortwo cell rf guns and a postacceleration booster linac,where the space-charge compensation takes place in driftspace between the rf gun and the booster linac, can be in-ferred. In fact, the case of a short 11/2 cell rf gun fol-lowed by a drift is discussed in Sec. VI. The exact solutionfor the beam envelope is not found for this case, but theoperating conditions needed to achieve emittance compensa-tion are deduced from the general properties of the envelopeequation.

7566 55LUCA SERAFINI AND JAMES B. ROSENZWEIG

8/14/2019 serafini_1997_0209


Radial nonlinearities in the space-charge field are not ex-plicitly taken into account in this model, as they have a weak impact on the rms envelope behavior. The full influence of these effects is beyond the scope of this paper, but is impor-tant nonetheless—it is more relevant to a discussion of mini-mizing the residual emittance after compensation. Somecomments on these subjects are made in Sec. II.

In overview, the organization of the paper is as follows. In

Sec. II we provide a heuristic model to explain the basics of the emittance oscillation due to a small mismatch of a space-charge-dominated beam at injection into the focusing chan-nel. Section III is devoted to the detailed analysis of theenvelope equation and the model for a multicell photoinjec-tors; analytical solutions derived in a perturbative approxi-mation around an exact solution are presented. The conceptof the invariant envelope is introduced and illustrated in Sec.IV, and its deep relationship with the space-charge-emittancecompensation technique is discussed. Predictions relevant tophotoinjector design characteristics needed to achieve an in-variant envelope operation, i.e., emittance compensation, arepresented in Sec. V together with comparisons to numerical

simulations of existing rf photoinjector designs. Section VI isdevoted to the case of a short rf gun followed by a drift spacewhere the emittance correction takes place. Finally, the im-plications of the analysis presented in this paper are summa-rized in Sec. VII.

II. AN ILLUSTRATIVE MODEL

The simplified model that we propose for the complicatedmotion of the beam envelope and emittance evolution in highgradient linear accelerators is motivated primarily by theproblem of understanding emittance evolution in rf photoin-

jector sources. In this model we view the rms emittance asarising from the differing phase space dynamics of each lon-gitudinal slice of the beam, which is assumed to behave as anindependent, cold, laminar, space-charge-dominated plasmaevolving under the influence of linear external forces. In thiscase, even though the rms emittance of each longitudinalslice can be neglected in the analysis, the rms emittance of the ensemble can be quite large upon summation of the en-tire ensemble making up the beam.

In order to understand how this mechanism causes emit-tance growth, as well as how the emittance growth can bereversed by proper focusing of the beam, we begin by exam-ining a simplified model problem, that of an intense, cold,uniform-density beam nearly matched to an external focus-ing channel. While this model ignores the effects of accel-

eration and transverse motion due to the high gradient rf fields in the accelerator, it serves to illuminate the fundamen-tal dynamics of the emittance oscillations in these devices.When the effects of the high gradient electromagnetic fieldsare included in the subsequent analysis, analogies to thissimple model will be apparent.

We begin by writing the rms envelope equation for a cy-lindrically symmetric, space-charge-dominated, coasting,relativistic, charged particle beam in a focusing channel of constant strength,

K r I

2 I 0 3

n,th2

2 3. 2.1

Given our assumption of space-charge-dominated envelopemotion, we may ignore the final term on the right-hand sideof Eq. 2.1, which represents the contributions to envelopeforcing due to the emittance arising from both random, ther-malizing sources as well as the effects of nonlinear macro-scopic forces. In this analysis, we will be using Eq. 2.1 todescribe the evolution of longitudinal slices of a beammeaning infinitesimally small lengths of beam about

given values of , assuming that the motion of each slice isessentially uncorrelated to that of nearby slices, and in factdepends most strongly only the local in value of the cur-rent. This means that the normalized thermal emittance cor-responding to each beam slice is small. We define this emit-tance formally as

n,th

2r 2 r 2 rr

2, 2.2

where the subscript indicates that the average is performedonly over the distribution within a given slice.

We next generalize the expression for the space-charge

term to include an explicit dependence on the longitudinalposition by I → I g( ), where I is now defined as the maxi-mum current in the beam. The geometrical factor g( ),which is less than unity, is discussed in more detail for afinite beams below; for now let us note that in the limitwhere the beam is long ( z r ) in its rest frame g( )follows the local dependence of the current very closely.

Upon linearizing Eq. 2.1 about the equilibrium Brillouinflow condition for a slice at a given value of ,

eq„g … Ig

2 I 0 3K r

1/2

, 2.3

we obtain the equation for small amplitude motion about thispoint,

K r I g

2 I 0 3 eq2

g 0 or

2K r 0, 2.4

which gives oscillation frequencies that are dependent on theexternal focusing strength, but independent of the beam cur-rent. It is this characteristic of the space-charge-dominated,quasilaminar beam dynamics that allows emittance compen-sation.

This model can be used to illuminate the rf photoinjectorcase by assuming that the envelopes in the beam ensemblebegin at the ‘‘cathode’’ slightly mismatched to the channelwith 0 eq and 0. All envelope oscillations pro-ceed with the same frequency, given only by the externalfocusing strength, but with different amplitude and aboutequilibria that are dependent on the current if we assume theapproximation that g( ) is proportional to the current. Wethus have formally

z , eq„g … 0 eq„g …cos2K r z ,2.5

and

55 7567ENVELOPE ANALYSIS OF INTENSE RELATIVISTIC . . .

8/14/2019 serafini_1997_0209


z 2K r 0 eq„g …sin2K r z . 2.6

Since the frequency of the oscillations is independent of thevalue of current, but the amplitude is not, the rms emittanceof the beam ensemble grows, but returns periodically tominimum values.

This can be seen by noting that under our above assump-tions, the rms emittance defined by Eq. 1.2 can be calcu-lated as follows:

z 2 2 2, 2.7

where the angular bracket indicates an average weightedover the distribution of currents in the entire beam ensemble,

i.e., all of the slices. To quantify the effect of the differingtrajectories in the ensemble of beam slices, we assume thelong beam limit expands the effective distribution function incurrents to second order about the maximum current i.e.,near the peak of a symmetric beam current profile that iscontinuous though its first derivative, and obtain the emit-tance evolution

z 1

& 0 I rms

I

I I p

1

2

K r 0 eq I p I rms

I p

sin2K r z . 2.8

Figure 1 displays the emittance and envelope evolutionfor a slightly mismatched beam ensemble, beginning, as inthe case of the rf photoinjector, with a minimum beam sizeand vanishing emittance as defined by Eq. 2.7. It can beseen that there are two subsequent emittance null points, oneat the maximum in beam size, and another when it returns toits original size. These minima occur where the angles inphase space tan1( / ) are independent of the beamcurrent value. This type of behavior is in fact similar to thatobserved in rf photoinjectors, as can be seen from the mul-tiparticle simulation shown in Fig. 2, where the beam under-goes one envelope oscillation and two emittance oscillations

from the cathode to the injector end in a scaled UCLA Sat-urnus photoinjector. The emittance minimum occurring atthe maximum in the envelope is of secondary interest be-cause it occurs at large beam size and low energy inside of the primary focusing magnet of the rf photoinjector.

The qualitative similarity between the behavior predictedby simple emittance oscillation model and that found insimulation of rf photoinjectors points the way toward thefurther analysis of the photoinjector, which differs from themodel case in both acceleration and nonuniform applicationof focusing. One prediction can be gleaned from the simplemodel even before we begin, which is that one should allow

the photoinjector beam to go through only one envelope os-cillation, with further oscillations suppressed by diminishingthe space-charge forces through acceleration. This must bedone with some care, and our analysis leads eventually to aquantitative prescription for obtaining this condition in Sec.V.

This simple model has other aspects that help explain byanalogy the behavior of rf photoinjectors operated in theemittance compensation regime. The artifact of the oscilla-tion frequency about the equilibrium being dependent onlyon the applied restoring force gradient, which is what allowsthe correlated emittance developing in the beam ensemble toperiodically vanish, is not valid to all amplitudes in the beamenvelope system. To lowest significant order in the mismatch

FIG. 1. Emittance and envelope evolution for a slightly mis-

matched beam ensemble beginning with a minimum beam size and

vanishing correlated emittance, in linearized limit.

FIG. 2. a Envelope, and b emittance evolution of 1300-MHz

rf photoinjector design, from PARMELA multiparticle simulation.


8/14/2019 serafini_1997_0209


amplitude a„ I ( )… 0 eq„ I ( )… / eq„ I ( )…, the oscilla-tion frequency is

„ I …2K r 1 316 a„ I …2 . 2.9

This anharmonicity in the beam ensemble generally pre-cludes the vanishing of the correlated emittance. Thus, thereis an additional prediction that can be made on the basis of this observation: excursions in the beam size should be mini-mized to produce the best emittance compensation.

It should also be noted that, assuming laminarity there isno ‘‘wave breaking’’ in transverse phase space within a lon-gitudinal beam slice is maintained, that phase space correla-tions arising from radial nonlinearities must also behave asdo the beam slices—that is, they should also be compen-sated. Maintenance of laminarity also implies that the beamexcursions from equilibrium are limited in amplitude.

III. BEAM ENVELOPE ANALYSIS FROM CATHODE

SURFACE TO INJECTOR EXIT

In order to perform the envelope analysis of the multicellinjector presented below, we must first specify a model forthe rf photoinjector. The model we adopt for the acceleratingstructure is geometry independent, since the accelerating rf field is written in Floquet form as a sum of its spatial har-

monic amplitudes, and the rf photoinjector cavity is assumedto be a multicell structure indefinitely extending along itssymmetry axis. There is, however, some specificity in ourchoice of the model for the static longitudinal magnetic fieldproduced by the external focusing solenoids: it is assumed tohave a hard-edge longitudinal profile extending over a fewcells of the accelerating structure.

A typical multicell rf cavity employed in rf photoinjectorsis shown in Fig. 3, displaying the cross section of an axisym-metric iris-loaded structure terminated into a half cell hostingthe cathode located at z0 and operated in a TM010-

standing mode with one-half wavelength cells following thecathode cell. The general expression of the rf field compo-nents expanded linearly off axis is 6

E z E 0 n1,odd

a ncos nkz sin t 0 ,

E r kr

2E 0

n1,odd

nansin nkz sin t 0,

B ckr

2E 0

n1,odd

a ncos nkz cos t 0 , 3.1

where k 2 / / c, and a n are the spatial harmonic co-efficients that depend on the actual cavity geometry that canbe computed easily by computer codes or derived by experi-mental bead measurements. Due to the symmetry of the se-lected mode, all even a n’s vanish, a11, and E 0 becomesthe amplitude of the fundamental harmonic speed-of-lightphase velocity component of the rf wave. All higher har-monic amplitudes are therefore normalized to the value of the fundamental.

The external solenoid is assumed to be folded around thefirst 21/2 cells of the rf cavity, producing a constant mag-netic field B z B 0 from z b /8 half-way through the cath-ode cell up to z c11 /8 a quarter-way through the thirdfull cell. The beam dynamics in the photoinjector are de-scribed using a three stage procedure:

a The first one and a half cells from z0 to z z2 aretreated by using a ballistic approximation, as described inRef. 7. In this region, named region 0, the transverseplasma oscillation begins, driven by the strongly repulsivespace-charge forces. The transverse dynamics are dominatedby the defocusing effects of space charge and a transient rf kick in the region of the first iris.

b In the following cells, i.e., up to the end of the sole-noid field at z z c , the envelope equation is solved pertur-batively with a constant beam size space-charge approxima-tion. Here, the extra focusing applied by the solenoid field, inconjunction with the ponderomotive rf focusing, overcomesthe transverse space-charge force and turns the beam enve-lope from divergent to convergent. This is named region Isince it is the first region where the envelope equation isapplied and applicable.

c In the final region of the accelerator beyond z zc ,named region II, the envelope equation is solved initially asa perturbation about an approximate solution, which pro-vides a general solution to the problem of the beam dynam-ics up to the end of the photoinjector. In the case of a nearlyoptimized injector, this approximate solution can be replacedby a special exact solution called the invariant envelope. Inthis case, the normalized emittance associated with the per-turbed plasma oscillations is damped gently for a beamnearly matched to the invariant envelope, while it can beexcited to perform additional oscillations if the beam is over-focused by the solenoid, going through successive minimaand maxima.

The beam conditions 2 and 2 at the second iris, i.e., at

the end of region 0, are reported in Appendix A 213 /2 at /2. With the condition 1/2, the trap-ping threshold requirement that holds for any rf photoinjec-tor 9, the beam at this point is quite relativistic, since typi-

FIG. 3. Schematic cross section in the ( r , z) plane of a typical rf

multicell cavity of a photoinjector gun: the rf field distribution on

axis is plotted together with the electric field lines of the TM010-

mode in use for electron acceleration. The three different regions 0,

I, II used to describe analytically the beam dynamics in the photo-

injector are shown.


8/14/2019 serafini_1997_0209


cally 25 – 10. Rewriting the envelope equation assuming 1 and beam laminarity neglecting the thermal emit-tance, Eq. 1.1 becomes

8b 2

sin

2

s

3, 3.2

where the normalized beam energy is given to an excellent

approximation by 1 kz sin( ) cos( )1 z cos( ), and we now leave out the explicit indication of the dependence of on and z . The ponderomotive rf fo-cusing term displays, through the quantity , its dependenceon the higher spatial harmonic amplitudes 6,

n1

a n12a n1

22an1a n1cos a00 .

3.3

In Eq. 3.2 the perveance s( ) explicitly retains a func-tional dependence on the longitudinal position of the par-ticular slice in the bunch, so that s( ) I g( )/2 I 0 . As

shown through more detailed calculations in Appendix B, thegeometric factor g( ) is given by

g e 2 /2 z21

A2

2 1 2

z2 1

2ln A

1

for a Gaussian distribution of aspect ratio A r / z , or

g 12 A 2

2 112

L

2

80 L

4

for a uniform distribution of aspect ratio A R / L , where R isthe beam radius and L is the beam length. In this section we

will assume, for the sake of simplicity, g( )

I ( )/ I peak , sothat s( ) does not depend on , which is consistent with therelativistic approximation that the transverse space-charge-field amplitude follows the beam current distribution. As amatter of fact, since the bunch aspect ratio A is typically of the order of 1, the rest frame aspect ratio is such that A 2 / 21 in the domain where Eq. 3.2 is applied ( 2). However, the generalization of Eq. 3.2 to deal withthe analysis of bunched beam dynamics will be performed infollowing sections, together with the analysis of the emit-tance compensation mechanism.

To solve Eq. 3.2, we apply a Cauchy transformation bychanging the independent variable from z to y , defined as y ln( /

2), to obtain

d 2

d y 2 2 S

e y, 3.4

with ( y ) and S( ) I ( )/2 I 0 2 2 s( )/ 2

2 de-fined to be the Cauchy perveance. The transformation clearlyreveals, by removing the term ( / corresponding toadiabatic damping in Eq. 3.2, that the single particle beta-tron motion in the Cauchy space ( , y ) is actually, as long asthe space-charge force is neglected, the simple one of a uni-form focusing channel with constant normalized focusinggradient 2: this is basically the reason that the matrix de-rived in Ref. 6 for the transverse motion in rf linacs can be

expressed in terms of simple sinusoidal functions. TheCauchy perveance S has been introduced in order to under-line the analogy with the usually defined beam perveance,i.e., I /2 I 0

3, which turns out to be invariant only for a non-accelerated beam: before attacking the mathematical treat-ment of Eq. 3.4 it is worthwhile to anticipate that, since S isinvariant in Cauchy space, a sort of equilibrium will beachieved whenever the space-charge term on the right-hand

side RHS is exponentially damped at the same rate as thecombination of focusing and trajectory curvature on the left-hand side LHS. Since this requires a nonoscillatory behav-ior for the LHS, which actually represents the betatron mo-tion, this argument leads straight to the requirement of laminarity i.e., negligible betatron oscillations, which is,consistently, the assumption on which the whole analysis isbased.

We obtain solutions to Eq. 3.4 employing two differenttechniques appropriate to two distinct domains of propaga-tion. As anticipated, we are proceeding from region 0 to Iand II by connecting output conditions of each region ininput to the following. In region I, defined by z 2 z z c (0

y y c) with y cln1(5/21/4) / 2 , the beam isexposed to external solenoidal focusing. In this domain wehave 2

( /8b 2)/sin2( ), where bcB 0 / E 0 . Region II,defined by z z c , is solenoid free and hence 0

( / 8)sin( ). In region I the beam size varies slightlywith respect to 2 , allowing the approximation 2 in thenonlinear term on the right-hand side of Eq. 3.4. The gen-eral solution I of the resulting linearized equation is

I 2 cos y ˙ 2

sin y

S

e ycos y

sin y

212, 3.5

where ˙ d / dy and ˙ 2 2 2 / . Setting c I( y c)

and ˙c ˙ I( y c), we can perturbatively solve Eq. 3.4 in

region II, assuming that the nonlinear term on the right-handside may be represented by a particular solution of the formgiven in Eq. 3.5.

The perturbative solution in the second region II thenbecomes

II c S e yc

c cos0 y y c

S e y y c y ˙

c / c

c

˙c

S e yc1 ˙c / c

csin0 y y c / 0 ,

3.6

where 02(1 ˙

c / c)2. The combination of Eqs.3.5 and 3.6, together with Eqs. A4 and A5, allows thedescription of the beam envelope from the initial conditionsat the photocathode surface up to the photoinjector exit.While this treatment of the behavior of II is quite general, it


8/14/2019 serafini_1997_0209


will ultimately prove less useful than one based on the in-variant envelope given in the next section.

The whole system, i.e., the beam and the external RF andmagnetic field, can be specified by means of ten operationalquantities: the main quantities are the laser pulse character-istics spot size at the cathode r , pulse length zc t , theextracted bunch charge Q b , the rf field quantities field am-plitude E 0 and rf frequency rf , the magnetic field amplitudeof the solenoid B 0 , and the initial and final positions for thesolenoid field distribution, namely, z b and zc . The somewhatancillary parameters associated with description of the rf field are and defined in Appendix A, which depend onthe set of spatial harmonic coefficients a

n

. In the followingwe will consider the special case of 1, i.e., a purefirst harmonic rf field, because it greatly simplifies the analy-sis without significant loss of generality. In the following wetake also /2, which corresponds to the phase of maxi-mum acceleration.

The beam envelopes resulting from this analysis appliedto a multicell photoinjector are shown in Figs. 4 and 5, withseveral different values of the bunch charge Q b and solenoidfield amplitude B 0 for a typical set of photoinjector param-eters. In the upper diagram of Fig. 4 the bunch aspect ratio is A1.25, with r 1.5 mm, corresponding to a peak current

I 100 A at Q b1 nC ( I Q bc / 2 z) and 400 A at Q b

4 nC z279 mm, 28.7, 1.64; in the lower dia-

gram A0.83 with r 2.0 mm, giving I 200 A at 4 nC

z2174 mm,

28.6, 1.62 no solenoid field is

present in the data reported in Fig. 4. The simulations wereperformed with the codes ATRAP 12 for the S-band gunFig. 4, upper diagram and ITACA 13 for the L-band caseFig. 4, lower diagram. A similar comparison is shown inFig. 5, where the extra focusing due to the magnetic field of the solenoid is clearly displayed. It should be noted that byswitching off the space charge term in Eqs. 3.5 and 3.6one obtains the dotted curve plotted in Fig. 5, for the case of B00.5 kG, which is clearly mismatched with respect of thesimulation curve, indicating the relevance of the nonlinearspace-charge term in our analysis.

It is useful to note, as explained in Ref. 6, that a tran-sient angular kick /2 corresponding to ˙

/2 must be added to the secular beam envelope at thegun exit in order to transform it back into the actual enve-

lope. What is meant by the distinction secular in describingthe envelope is the following: the secular envelope repre-sents the actual envelope averaged over the cell-to-cell oscil-lations caused by the alternating gradient focusing effect as-sociated with the backward component in the rf standingwave, as discussed in Ref. 6. The good agreement betweenthe analytically predicted envelopes dashed lines and thenumerical simulation data solid lines gives a significantconfirmation of the capability of the present model to predictcorrectly, within the domain of interest, the beam envelopecharacteristics.

It is interesting to note that the first two terms on the

right-hand side of Eq. 3.5, which scale linearly with theinitial conditions 2 and ˙ 2 , correspond exactly, as far asb0 is set no superimposed solenoid field, to the lineartransport matrix elements derived in Ref. 6 for the evolu-tion of the secular envelope in rf linacs. Therefore, Eq. 3.5represents the extension of the analysis performed in Ref. 6to the case of an external magnetic focusing added to the rf ponderomotive focusing, as well as the contribution from thespace-charge field, which is given by the third term on theright-hand side of Eq. 3.5.

IV. THE CONCEPT OF INVARIANT ENVELOPE

In view of the excellent agreement between the analyticaland numerical solutions to the envelope equation for space-

FIG. 4. Beam envelopes through two different 101/2 cell rf

guns operated without external solenoid focusing, i.e., B00

rf 2.856 GHz E 0100 MV/m upper diagram, rf 1.3 GHz E 045 MV/m lower diagram. Dashed lines give the secular orbits

analytically predicted, while solid lines are numerical simulation

results. Various bunch charges have been used, as indicated.

FIG. 5. Beam envelopes through a 101/2 cell L-band rf gun

E 045 MV/m, I 200 A, Qb4 nC at different amplitudes B0

in kG of the solenoid magnetic field. Dashed lines give the secular

orbits analytically predicted, while solid lines are numerical simu-

lation results.


8/14/2019 serafini_1997_0209


charge-dominated, strongly accelerating beams in the pre-ceding section, we now extend our analysis to finding a par-ticular beam propagation mode. This mode will be shown tobe analogous to the Brillouin flow for space-charge-dominated beams in focusing channels discussed in Sec. II.

First of all, we begin by transforming the envelope de-scription in Eq. 3.4 from the Cauchy space ( , y ) into adimensionless Cauchy ( , y) space, which displays the fun-

damental parametric dependence that governs the beam sizeevolution. By defining the dimensionless quantity / S

we are now, for the sake of compactness, leaving implicitthe dependence of S on , the envelope equation in thedimensionless Cauchy space ( , y ) reads

d 2

dy 22 e y

. 4.1

The scaling of the beam size with the square root of theperveance in this analysis naturally agrees with the scalinglaws set down in Ref. 17, in that the beam plasma fre-quency is the same for any envelope of the same .

We are interested in the third region ( z zc), where, tak-ing the case of 1 and /2, which imply that

01/ 8, the envelope equation reads

d 2

d y 2

8

e y

, 4.2

which is a universal scaled equation, independent of any ex-ternal parameter.

Since the quantity S which has units of a lengthcan be related to the transverse plasma frequency

p4 n ee2 / 3m e(c / ) I /2 I 0 3 by p(c / )

S 2 / , it is interesting to note thatthe function can be expressed as e y /2 / k

p( / )/ k p 2 / with k p p / c. In this form it isclearly shown that scales as the ratio between the localplasma wavelength p2 / k p , which sets the defocusinglength of the beam, and the local incremental energy gainlength L g / , which sets both the beam adiabatic damp-ing and rf focusing lengths.

Equation 4.2 has, like Eq. 3.4, a general perturbativesolution,

c e y c

ccos0 y y c

e y y c y ˙c / c

c

˙ c

e y c 1 ˙c / c

c sin0 y y c / 0 ,

4.3

with 181 ˙ c / c2 and 01/ 8.

Within this family of solutions there is a notable particu-lar solution,

ˆ 8/3e y /2, 4.4

corresponding to ˆ c8/3e y c /2, ˆ

c2/3e y c /2, and

38. This solution is characterized by having a plasma fre-

quency k ˆ p3/2 / , which is proportional to the pondero-

motive rf focusing frequency and imaginary adiabatic

damping frequency up to a fixed constant. Another way of

viewing this is to note that the ratio between the two funda-mental scale lengths, Lg and p , is in this case exactly con-

stant, i.e., p / L g2 2/3. This is achieved because the

scaling of the plasma frequency as n b 3/2 is exactly

matched to the energy gain, including the reduction of thebeam size ˆ with energy, which scales as 1/2, namely, ˆ

(2/ ) I /3 I 0 . This constant relationship between L

gand

p also explicitly indicates that the invariant envelope is in-deed an equilibrium solution in the Cauchy space. Further, itobviously displays the equilibriumlike characteristic thatthere are no periodic oscillations associated with it, butnearby orbits will oscillate about it; these oscillations will bestudied in Sec. V.

The solution ˆ given by Eq. 4.4 has also the extremelyrelevant property that it is the only solution displaying aconstant phase space angle ( / ), independent of ini-tial conditions c and ˙

c in all of the three spaces Cauchydimensionless ( , y ), Cauchy ( , y), configuration space

( , z). In fact, ˆ ˆ / ˆ ˆ / ˆ 1/2, so that in both Cauchyspaces the phase space angle is a universal constant, while in

configuration space the phase space angle ˆ c ˆ / ˆ

/2 is a constant the trace space angle is ˆ / ˆ

/2 . The negative sign of ˆ , implying a convergentbeam, is a clear signature of the adiabatic damping due toacceleration: on the other hand, the same quantity is vanish-ing in the case of Brillouin flow, where see Eq. 2.3 Br

eq / eq0.

Further, the most important attribute of ˆ on the invariantenvelope is that it does not depend on the beam current,which is embedded in the perveance scaled variables c and ˙

c . For this reason the solution ˆ will be called the invariant

envelope; its invariance in phase space angle with respect to

current is exactly the basic condition to obtain a vanishinglinear correlated emittance as the final state of the beam. Infact, it is well known that the emittance growth from linearspace-charge effects is due to the angular spread in the phasespace distribution of different bunch slices, which receivedifferent kicks from the space-charge field. In analogy to thediscussion of the emittance oscillations in the beam mis-matched to the solenoid in Sec. II, these different beam slicesmay be represented by different current amplitudes in Eq.3.4, with the full beam represented by the ensemble of beam slices. Since Eq. 2.8 predicts emittance oscillations of amplitude scaling like the spread in phase space angles, it isnatural to anticipate that this property of the invariant enve-lope will be crucial to achieve emittance correction, as dis-cussed in detail in Sec. V.

It is interesting to observe that, under the invariant enve-lope conditions, the space-charge term in Eq. 4.1, e y / ˆ

3/8e y /2, is dominant over the focusing term, which isonly one-third of the magnitude of the space-charge term,

ˆ /8(1/3)3/8e y /2. Adiabatic damping of the angular di-vergence due to acceleration provides an additional damp-ing term that counteracts the space-charge defocusing in theenvelope equation, but it should be noted that the secondderivative of the invariant envelope is always positive, thusclassifying this trajectory as unstable. This in fact must bethe case, since a stable trajectory would imply oscillatory, ornonlaminar, behavior: one of the main consequences of such


8/14/2019 serafini_1997_0209


a characteristic of the invariant envelope is the simultaneousdamping of the beam spot size and the beam transversemomentum p ˆ ˆ /2 as 1/2.

Since we solve the envelope equation under the assump-tion of laminarity, the range of validity of such a hypothesisshould be investigated. Rewriting Eq. 4.1 by taking intoaccount also the thermal emittance term defined by Eq. 2.2we find

d 2

d y 22 e y

2 n,th 2 I 0

I

2 1

3. 4.5

When the beam is on the invariant envelope ˆ , thesecond term on the right-hand side of Eq. 4.5 growsas e3 y /2, while the space-charge term decreases as e y /2.In order to preserve the condition of quasilaminarity, sothat the beam can be considered space-charge dominated,the space-charge term must be larger than the emit-tance term. This condition holds up to a position y l

ln(8/3) I /2 I 0 n,th 2 , beyond which the beam entersthe region where it becomes emittance dominated. This po-sition corresponds to an energy l given by

l8/3 I

2 I 0 n,th . 4.6

Since the thermal emittance n,th is typically of the order of 1mm mrad, and taking the relatively high accelerating gradi-ent found in the plane-wave transformer PWT linac atUCLA, which is 30 m1 ( E 030.6 MV/m), we have l1.6 I A . This energy is quite a bit larger than that ob-tained at the UCLA PWT 16 MeV, which, like all existingstanding wave photoinjectors, has a peak energy less than 25MeV, but with peak beam currents in excess of 50 A con-sidered typical.

Another relevant assumption made above was that of lon-gitudinal laminarity, which means that different slices do notmix with each other. This assumption is not violated in gen-eral since, as previously discussed, the longitudinal plasmaperiod is much longer than the typical time scale of emit-tance compensation i.e., of one plasma oscillation in thetransverse plane. Since the longitudinal plasma frequency,which is suppressed in comparison to the transverse fre-

quency by a geometrical factor p pg( z)11,

where for large beam rest frame aspect ratios g1, the num-ber of plasma oscillations in the longitudinal plane is typi-cally much smaller than 1. The major result of this longitu-

dinal plasma motion, which, unlike the transverse motion,has little restoring force, is to lengthen the pulse in a laminarfashion; there is relative motion of the beam slices, but theydo not overtake each other.

When the beam leaves the accelerating structure one mustadd a positive defocusing kick /2 , as previ-ously mentioned, to obtain the correct connection betweenthe secular envelope in the gun and the actual envelope out-side. Since the corresponding kick in the Cauchy space is ˙ /2, and in Cauchy dimensionless ˙ /2, it canbe clearly seen that a beam propagating through the structure

on the invariant envelope, for which ˆ ˆ /2, will exit therf structure as a parallel beam, i.e., with ˙ ˙ 0 and

f 2/ I /3 I 0 f , 4.7

where f is the exit beam energy. This condition is a usefulexperimental diagnostic of emittance compensation in prac-tice.

The parallel exit condition on the beam envelope pointsout the analogy between the invariant envelope and the Bril-

louin flow. In fact the two flows can be matched at the exit of a standing wave linac; equating Eq. 4.7 with Eq. 2.3, one

can find that a focusing gradient K r (3/8 / f )2 pro-

duced by a solenoid of field amplitude B z3/2m ec / ecan achieve this match, preserving the beam’s mean angle inphase space to be vanishing after the linac.

The converse of the exit condition just discussed is thefollowing entrance condition: a beam entering a standingwave linac must have initial beam size given by

i 2/ I /3 I 0 i, 4.8

with vanishing divergence. In other words, the beam mustalso enter on a parallel trajectory. The implications of thiscondition for operation of a split photoinjector, consisting of a short rf gun followed by a drift space and a booster linac,are discussed in Sec. VI.

V. EMITTANCE COMPENSATION

External control of the beam spot size and emittance evo-lution in a long rf photoinjector is accomplished through thevariation of the solenoid field strength, which allows one tolaunch, at z zc , a beam envelope that may be optimized forachieving low emittance performance. It is obvious from the

previous section’s discussion that this particular envelope so-lution is of interest from the point of view of emittance con-trol, and so we now concern ourselves with the examinationof two issues. The first is how to achieve this ‘‘matching’’ of the beam to the invariant envelope at the end of the solenoid,while the second is the investigation of the subsequent phasespace dynamics of a real beam ensemble with a spread intrajectories. Both of these issues are critical in understandingthe phenomenon of emittance compensation. We have ar-gued that operation on the invariant envelope is the conditionfor optimum emittance compensation, in the sense that thebeam the ensemble of all beam slices fully matched to theinvariant envelope displays no further emittance oscillations.It will be shown below that this is only part of the story;beam slices that are not directly on the invariant envelopeperform stable oscillations around the invariant envelope,leading to secular damping of the normalized emittance of the full beam ensemble.

At this point, we wish to find proper gun operating con-ditions, in terms of the six free parameters spot size at thecathode r , pulse length z , bunch charge Q b , rf field am-plitude E 0 , frequency rf , and magnetic field amplitude B0, able to achieve a beam matched at z zc to the invariantenvelope. In order to reduce the number of free parameterswe need to specify the matching conditions in the Cauchydimensionless space, we turn to a set of four free parameters, , A , , and b, defined by


8/14/2019 serafini_1997_0209


eE 0

2mc 2k , A

r

z,

I

r 2 , b

cB 0

E 0.

5.1

These quantities are physically described as follows: is thedimensionless amplitude of the rf vector potential, A thebunch aspect ratio, the Cauchy current density, and b themagnetic-to-rf focusing ratio. These tuning parameters are

linked to the six previous free parameters by k 2 rf / c,

eE 0 / mc2, I Qc / 2 z , while the Cauchy currentdensity is given 14 in terms of the current density J by2 J / 2 and is linked to the Cauchy perveance S by

2 I 0 2S / r 2. The merit of the set of four parameters

given in Eq. 5.1 consists in the possibility of expressing thebeam conditions 2 and ˙ 2 at the exit of the second cell ( z

z2) entirely in terms of these four, as reported in AppendixA, instead of the previously used six parameters.

The matching conditions at z zc can be expressed as

8/3e y c /2 2cos y c ˙ 2sin y c / e yccos y csin y c /

2 12 5.2a

2/3e y c /2 2sin y c ˙ 2cos y c e y c sin y ccos y c

212 , 5.2b

where 2 2( , A , ,b) and ˙ 2 ˙ 2( , A , ,b) are given by Eq. A7, while y cln1(5/21/4) /(13 /2) and 2(1/8b 2). The first two parameters are restricted by practical considerations to a limited range, that is, 0.7 3 and

1/4 A2. Therefore, we solve the two expressions in Eqs. 5.2a and 5.2b by expressing their roots as ss( , A) and

b sb s( , A), which can be well approximated by the expressions

skA235266 283 A188 A80 A 25.3 A 3

15.6413.2 A

1.330.94 A

2

5.3

and

bs102 5.7 28.577.84.8 A

1/4 0.15s , A s , A 5.7

1/40.840.28 A 7.1 A . 5.4

These quantities are plotted in Figs. 6 and 7, respectively,

as functions of at different values for A , i.e., A2, 1, 1/2,and 1/4. In general, the Cauchy current density s increaseswith up to a maximum max beyond which s is no longerdefined. The behavior of the values of bs are plotted in Fig.7, for the same values of A, up to each corresponding max .

The upper part of the operating diagram in Fig. 6, theregion above the dotted line, is in fact forbidden, because inthis region the bunch charge is in excess of the maximum,which can be extracted from the cathode surface. This fun-damental limitation, as predicted theoretically 15 and ob-served experimentally 16, sets the maximum achievablecurrent density J max , according to the nonrelativistic Child-

Langmuir law, in the form J maxA/cm2

300 / bps,valid for short bunches ( bps1180/ ). The conditioncan be cast in terms of max as

maxkA324

°rf . 5.5

The dotted line plotted in Fig. 6 represents the limitationmax for a typical bunch length of 2°. Clearly, the op-timum operating points should be relatively far from themax line, because of the severe energy spread induced bythe longitudinal space-charge field at extraction from thecathode surface if operating on the line, the photoelectrons

in the bunch tail would actually see a vanishing accelerating

field at the cathode surface due to the canceling of rf field bythe space-charge one. Such a correlated energy spread, i.e.,the dependence of energy on the phase or slice position inthe bunch, produces chromatic aberrations in the transportthrough the solenoid field and rf focusing channel, which canprevent the emittance correction process from proceedingcorrectly. An obvious cure is the use of off-crest accelerationto compensate the space-charge-induced energy spread withan opposite effect from the rf field. In practice, this mayimply operation far off crest if the Cauchy current densityapproaches the limit given by Eq. 5.5.

It is interesting, for the sake of illustration, to plot thecurrent density J and cathode spot size r corresponding tothe line

s

( , A1) , drawn in the Cauchy operating dia-gram of Fig. 6, once the rf frequency rf 2 has beenfixed to some representative values, namely, 650 MHz, 1.3GHz, 2.856 GHz, and 6 GHz, as shown in Figs. 8 and 9. Asanticipated from the Cauchy operating diagram, each fre-quency has a definite window in the rf field amplitude inwhich operation of the injector in the space-charge-compensation regime is possible: the dashed lines set themaximum current density limit corresponding to a bunchlength of 5 ps for the upper line and 10 ps for the lower line.The cathode spot size r , plotted in Fig. 9, corresponds to abunch charge of 1 nC; it is given by the relation r

3(c / 2 )Q b A / 2, showing the expected scaling as

Q b1/3 as anticipated in Ref. 17.


8/14/2019 serafini_1997_0209


In order to better illustrate the predictions of the operatingdiagram in Fig. 6, we choose a point on the diagram, specifi-cally one on the A1 line at 1.3, which corresponds tos144 kA. Choosing the rf frequency to be 1300 MHz L band we obtain E 036 MV/m for the peak cathode fieldand r 0.87 mm for the cathode spot size and r 1.36° rf for the bunch length, once we choose the bunch charge Q b tobe 1 nC. The peak current comes out to be I 137 A whileb s0.85 implies a magnetic field B 01.02 kG. The threerepresentative currents, corresponding to three slices, are inthis case cf. Appendix B, Eqs. B9 and B10 I 137, I

163, and I 97. The numerical integration of the enve-

lope equation is shown in Fig. 10a solid lines, where thecase for B 00 no solenoid focusing is also plotted dashedlines. The corresponding three slice emittance is shown inFig. 10b: as predicted, the normalized emittance is actuallycorrected only for the case of a beam following the invariantenvelope.

We are now in a position to discuss in more detail howthe emittance correction process works when the injector isoperated under the invariant envelope mode; i.e., it is set at

the prescribed s and bs for a chosen A . Let us rewrite Eq.3.4 in the third region ( y y c) by explicitly showing the

dependence of the Cauchy perveance S and the ponderomo-tive focusing frequency 0 on the slice position in thebunch:

d 2

dy 2

1

8 sin2 k y , S

y , e y ,

5.6

where the Cauchy perveance now becomes S( ) Ig( )/ 2 I 0 c( ) ( )2, the average accelerating gradient ( ) k sin( k ) with defined as the bunch av-erage phase, and the initial normalized energy c( )1(3 /2)sin( k ) cos( k ). As indicated by the

subscript in the independent variable y , Eq. 5.6 repre-sents actually a family of equations, one for each slice lo-cated at a distance from the bunch central slice, in thevariable

y ln 2

ln 1 kz sin k cos k

2 .

FIG. 6. Operating diagram in the ( , A) plane for an indefinitely

long multicell photoinjector.

FIG. 7. Parameter bs , plotted as a function of , at some values

of the bunch aspect ratio A, for an indefinitely long multicell pho-

toinjector.

FIG. 8. Current density J plotted solid lines vs the cathode

peak field E 0 MV/m, at different rf frequencies indicated in

MHz, for a multicell photoinjector operated in the emittance cor-

rection regime. The dashed lines show the limit of maximum J for

two different bunch lengths.

FIG. 9. Cathode spot size r , plotted vs the cathode peak field

E 0 MV/m , at different rf frequencies, for a multicell photoinjec-

tor operated in the emittance correction regime.


8/14/2019 serafini_1997_0209


This family of equations can be transformed, in analogy toEq. 4.2, to read

d 2

dy 2

1

8 sin2 k y , e y

y , , 5.7

where ( y , ) ( y , )/ S( ). The invariant envelopethen reads

ˆ 2e y /2

11/2 sin2 k , 5.8

which in configuration space is simply

z , 2

Ig

2 I 011/2 sin2 k

1/2

.

5.9

As already discussed in Sec. IV, we know that the neces-sary condition for vanishing correlated emittance growth isthat the phase space angles of different slices are equal. Inthis respect, any effect that induces a correlation, i.e., a dependence, will produce an emittance increase through thespread of phase space angles of the different slices. The rf effects are basically chromatic, and their phase dependenceis typically quite a bit weaker than the dependence of theeffective perveance in Ig( ) in Eq. 5.6, since z . Wethus neglect the chromatic contributions to the dynamics, andconcentrate on the charge dependent effects in which thelongitudinal correlation of the transverse space-charge field

gives the Cauchy perveance S a slice dependence through thegeometrical factor g( ). This condition makes the transfor-mation from Cauchy space ( , y) to the dimensionlessCauchy space ( , y) dependent on , an effect that is absentin continuous beams, as previously discussed in Sec. II andanalyzed further in Ref. 7. In this spirit we also set theaverage phase of the bunch to /2, which corresponds tomaximum acceleration. We therefore write Eq. 5.9, under

these approximations, as ( z, )(2/ ) Ig( )/3 I 0. It ob-viously is straightforward to generalize the following analy-sis to include the arbitrary accelerating phase.

Under the assumption of a monoenergetic bunched beam,the Cauchy transformation from z to y is again indepen-dent, and we now write SS( )( I /2 I 0 2

2)g( ). Wehave already shown that the invariant envelope is character-ized by a phase space angle independent of the Cauchy per-veance S , and hence on the current. We now demonstratethat this condition corresponds to a vanishing correlated

emittance growth. Since we are dealing with transverseforces that are linear in the radial coordinate, the transversetrace space distribution of the quasilaminar beam ( r ,r ) isrepresented by an ensemble of straight segments, one foreach slice in the bunch, as depicted schematically in Fig. 11.In this figure only two of these line segments are drawn, onefor the central slice located at 0, having spot size and

divergence , which is subject to the peak space-charge

field and another for a slice located at z with trace

space variables and , where the space-charge field issmaller for typically encountered current distributions, andthis reduction is represented by the geometrical factorg( z)1. The normalized rms transverse emittance, defined

by the relation n( z) ( z) ( z), with ( z) given inEq. 2.7, is explicitly evaluated as

n z

2

2

2 2

2 2

2 2

2 .

5.10

As can be seen from Eq. 5.10, the rms emittance in thistwo-slice case is identical to the common geometrical defi-

FIG. 10. a Envelope and b emittance evolution as predicted

by the operating diagram in Fig. 6, obtained by numerical integra-

tion of the envelope equation, with and without solenoid focusing

applied solid and dashed lines, respectively.

FIG. 11. Description of a bunched beam via two representative

slices in the trace space (r ,r ) .


8/14/2019 serafini_1997_0209


nition of emittance; it is simply the area of the triangle given

by the origin and the two rms phase space points correspond-

ing to the slices.

It should be emphasized at this point that each slice is

represented for simplicity by a straight segment in phase

space, which is a zero emittance distribution, because we are

neglecting the thermal emittance n,th , according to our as-

sumption of quasilaminarity. In practice, this emittance,

which is added in squares with spatially correlated sources of

emittance, can be estimated to be n,th r kT / m ec 2, wherethe effective rest frame temperature T of the beam elec-trons is determined by the photoemission process, which formetal photocathodes is less than 1 eV, and semiconductorcathodes are expected to be one order of magnitude smaller.With these thermal effects, each beam slice’s phase spacewould be a bi-Gaussian distribution whose rms ellipse has anarea the slice emittance proportional to the thermal emit-tance n,th . The emittance n( z) defined by Eq. 5.10 rep-resents a reversible emittance growth that can be correctedby proper beam manipulation, as we are discussing, whilethe thermal emittance n,th does not arise from reversible

transformations and is, in this sense, a true Liouvillian in-variant, as discussed in Ref. 15. It should also be recalledthat we are neglecting the emittance due to nonlinear space-charge fields in this discussion as well.

Assuming for the sake of discussion that thetwo representative slices follow their own invari-

ant envelopes, we have ˆ ˆ

ˆ 8/3e y /2,

which implies (2/ )( I /3 I 0 )g( 0 ) and (2/ )( I /3 I 0 )g( z). For the invariant envelope, we

have ˆ / ˆ 1/2, and thus ( /2 ) and

( /2 ) . It is readily verified that under these condi-tions where the normalized emittance defined by Eq. 5.10vanishes, the invariant envelope is the propagation modewhere all the bunch slices are aligned in the transverse phasespace.

Clearly, to achieve this ideal beam propagation mode ev-ery slice in the bunch must be matched at the invariant en-

velope, that is, c( )/ S( ) ˆ c8/3e y c /2

8 2 /3 c . This is an impossible condition to fulfilland, in practice, only a small section of beam can be exactlymatched. In this regard, the matching discussed above, deal-ing with the conditions s and b s necessary to operate thephotoinjector on the invariant envelope, is clearly a rmsmatching, because the beam conditions 2 and ˙ 2 Eqs. A7and A8 are given in terms of the rf and space-charge kicksaveraged in the rms sense over the Gaussian charge distri-bution. Because of this, only a beam slice equivalent to rmsbeam is matched; the other beam slices can be considered ingeneral to be mismatched from their invariant envelopes.The dynamics of these mismatched envelopes can be ana-lyzed by perturbation of Eq. 5.6 or its equivalent about theinvariant envelope of the matched slice. We thus assume inthe following a rms matching and take the equivalent rms

beam slice to be ˆ ˆ S , with S S rmsS( z), matched to the invariant envelope ˆ , while ( ˆ

)S 0 with S0( 0) is slightly mismatched by aquantity from its invariant envelope.

Substituting and in Eq. 5.10, and recalling that

( / ) ˙S , we obtain

n z

2

S ˆ

2 2 ˙ . 5.11

Expressing n( z) in terms of physical quantities associ-

ated with the invariant envelope, we find that

n z 1

Ig z

3 I 0

1/2

2

ˆ

2 2 , 5.12

where we set g(0)1. We can see that the normalized emit-tance is proportional to the beam size, which is monotoni-cally decreasing on the invariant envelope; we shall nowshow that the term inside of the absolute value sign is in factbounded, and so the emittance also displays a generally

monotonically decreasing behavior.We first study the behavior of deviations from the invari-ant envelope in Cauchy space, by linearizing Eq. 5.6around the solution represented by Eq. 5.8 to obtain, for thesmall-amplitude motion about the invariant envelope

¨ 2 e y

ˆ 2 ¨ 22

1

4 0, 5.13

showing an oscillatory behavior

ccos y y c ˙

c

sin y y c ,

5.14

˙ c

sin y y c ˙ ccos y y c.

for around the invariant envelope with frequency

221/4, with the constants of integration derived from

conditions at y y c . Since 2(1/8)sin2 1/8, the mo-

tion around the invariant envelope is stable, so that any beaminjected slightly mismatched to the invariant envelope willfollow a trajectory oscillating about it.

This stable motion has, like the small amplitude oscilla-tions discussed in Sec. II, frequency independent of thespace-charge strength. This is in fact a general property of the superposition of a linear focusing force with associatedfrequency and a repulsive inverse power law force power , which has a particular equilibriumlike solution, in thiscase the invariant envelope. The small amplitude oscillationsabout this particular solution then have frequency1 , which depends on the power exponent of the re-pulsive force, but not its strength, and is always proportionalto the linear focusing force strength.

It can be seen in this case that the total potential mustexhibit a local parabolic well at the intersection point of theattractive and repulsive force terms. The potential terms thatgive rise to these forces are shown by the Hamiltonian asso-ciated with Eq. 5.7,


8/14/2019 serafini_1997_0209


H y , p

2

2

4

2

e y ln , p ˙ , 5.15

in the conjugate variables ( , p ). The Hamiltonian H is nota constant of the motion, as indicated by the explicit depen-dence on the independent variable y . The perturbed Hamil-tonian, however, is a constant for small amplitude motionabout the invariant envelope. The resultant simple-harmonicsmall amplitude motion can be seen to be manifestly Liou-villian, and the ( , p ) phase space area an emittance, whichwe discuss further below is a constant of the motion as well.This fact guarantees that the normalized emittance mustdamp as 1/2, as illustrated by the two-point emittancegiven by the second form of Eq. 5.11.

To further illustrate these points, it is perhaps more in-structive to view the oscillations around the invariant enve-lope in physical variables at this point. The physical spaceanalogue to Eq. 5.13, which describes oscillations about theinvariant envelope can be written as

1

2

2

0. 5.16

This equation has the general solution

ccos & c

csin , with

5.17

1

& csin c c cos ,

where (1/ &)ln( / c), c c(2/ ) I /3 I 0 c , and

c c I /3 I 0 c3/2 for the mismatched core envelope. It

can be seen that the determinant of the matrix of the

( , ) transformation is simply c / , which is expected

from adiabatic damping of the transverse oscillations. Thus

we see that the normalized offset emittance associated with

the phase space of the perturbed oscillations centered on the

invariant envelope is conserved. This is to be expected fromthe Liouvillian nature of the perturbed envelope system.Before discussing a general distribution, we first examine

the behavior of the two-slice case introduced in Eqs. 5.12.In this case the emittance is given by

n z 1

Ig z

3 I 0

1/2

c 2 c ccos

c c c & sin . 5.18

Equation 5.18 shows the expected 1/2 damping of thenormalized emittance, with anharmonic oscillations of peri-odicity 2 times shorter than the period of the perturbationsabout the invariant envelope.

For the case of a general n-slice distribution, with a sym-metric spread in mismatch amplitude about the invariant en-velope, it can be shown by extending the above argumentsthat the normalized emittance that is projected by this distri-bution of phase space orbits offset from the origin in phasespace has the form

n z 2

2

2

ˆ 2

2

2 2

2

1/2

off 2 ˆ 2 2

2

2

2 , 1/2 5.19

where we have defined the normalized offset emittance of the distribution,

off 2 2 2, 5.20

and the indicated averages are over the n-slice distri-bution. The normalized offset emittance is a constant of the motion; it can be evaluated, for example, at the beginn-ing of invariant envelope propagation as off

c2 c

2 c c2. It is also clear from Eqs.

5.17 that the term inside the square brackets in Eq. 5.19 isbounded and oscillatory. Therefore we can write the generalform of emittance evolution as

n z off 2 ˆ 2 ab cos2 c, 5.21

where a , b, and c are constants describing the orientationof the offset distribution, and is as defined below Eq.5.17.

FIG. 12. Schematic drawing of the phase space of a photoinjec-

tor beam rms matched to an invariant envelope. The offset phase

space area is a Liouvillian invariant.


8/14/2019 serafini_1997_0209


A schematic picture of the phase space of a beam that is

matched in the rms sense to the invariant envelope is shown

in Fig. 12. For the sake of illustration, the familiar form of an

ellipse is used to indicate the offset phase space distribution

boundary. This ellipse has an invariant area off , and ro-

tates with the same frequency as the envelope oscillations,

4 p / ). Figure 12 illuminates, by a phase space dia-gram, the physical mechanism of emittance compensation;reassuringly, there is a Liouvillian space of orbits about theinvariant envelope with conserved phase space area—thephase-space-centered rms emittance of the beam damps asthe offset from the origin ( ˆ , ˆ ) of the distribution ap-proaches the origin in phase space. The normalized offsetemittance can be therefore thought of as a strict lower boundon the phase-space-centered normalized emittance of the dis-tribution. One cannot actually extrapolate the damping of theemittance to this level, however, as this would violate theassumption of quasilaminarity, which requires that the offsetbe larger than the spread of beam sizes in the distribution.This argument implies the normalized emittance must beseveral times larger than off , when the emittance compen-

sation process is halted by nonlaminar crossover beam trajectories.

As a final note on the dynamics of emittance compensa-tion in a long accelerating structure, we point out that as thebeam exits the focusing solenoid, it has just passed its maxi-mum in beam size and local minimum in emittance. Theemittance would, in the absence of acceleration, tend to riseagain, but this rise is held in check by the 1/2 dampingeffects predicted by Eqs. 5.18–5.21. The emittance tendsnot to damp much for one-quarter of a perturbed beamplasma oscillation after this point, because the orientationof the offset ellipse major axis moves towards normal to thenominal phase space angle of the invariant envelope. Only

after the ellipse major axis begins rotating back towardsalignment with the average phase space angle does the emit-tance damping become more apparent, as has been deducedfrom multiparticle simulations. In fact, this rotation tends tooccur after the beam exits the accelerating structure, becausethe phase advance of the plasma oscillation is slowed by theacceleration process. This can be quantified as follows: thephase advance between where the beam is focused onto theinvariant envelope the exit of the solenoid and where theemittance minimum occurs must be between one-quarterand one-half of a wavelength, or /2(1/ &)ln( / c) recalling that the phase advance is defined as

(1/ &)ln( / c). This gives a multiplication of the energy

beyond the solenoid of between 9.2 and 85. Even the smallerof these numbers implies a structure over 25 cells long,which is longer than any integrated photoinjector yet built bya factor of 2. Thus the final compensation must occur withinthe drift space after the exit of the accelerating structure,where the beam is small and nearly parallel, as is discussedin Sec. VII, and its plasma frequency nearly constant, in-stead of diminishing as p 1 z1 in the acceleratingcase.

These effects are illustrated well by a PARMELA simu-lation of a new 111/2 cell, 2856-MHz rf photoinjector cur-rently under design at UCLA. In this case, a solenoid field,as calculated by the magnetostatic simulation code POISSON,is localized in the initial 2.5 cells of the structure to closely

approximate the analytical model we have used. The beamcharge is 1 nC and the rms bunch length is 0.78 mm, with anassociated current of 111 A, and the peak acceleration gra-dient on axis in this structure is 60 MeV/m ( 0.98). Theevolution of the rms emittance, the rms beam envelope, andthe invariant envelope obtained from this simulation areshown in Fig. 13. It can be seen that the beam envelope isclose to the invariant envelope in the photoinjector, butslightly larger and more convergent. After the photoinjectorexit the beam envelope converges to below the invariant en-

velope, and compensation of the emittance takes an addi-tional 2 m drift. The convergence of the beam to a denserfocus in the drift also serves to increase the plasma frequencyand decrease the length to compensation.

This example illustrates that one must be careful in inter-preting the 1/2 dependence of the emittance evolutionfound in the above analysis. It may be more illustrative tocast the assertion in a different light, by stating that the emit-tance scales as , and that the invariant envelope propagationis the generalized equilibrium mode that must be followed toachieve this scaling with bunch size.

VI. PHOTOINJECTORS WITH COMPACT GUNS

AND BOOSTER LINACS

While the long rf photoinjector analyzed thus far is en-countered in practice, with the noted examples of the LANLLos Alamos National Laboratory photoinjectors APEXadvanced photoinjector experiment and AFEL, it is muchmore common experimentally to employ a compact rf gun N 1/2 cells, N 3 followed by a drift space and a boosterlinac. In this configuration, the beam is focused near the endof the gun by a strong solenoidal field. The beam then driftsafter focusing, undergoing a diminishing phase of an emit-tance oscillation as the beam becomes smaller, eventuallyminimizing as a space-charge-dominated beam waist isreached. The booster linac entrance is placed at this point to

FIG. 13. PARMELA simulation of 111/2 cell, 2856-MHz rf

photoinjector currently under design at UCLA. The solenoid field is

localized in the initial 2.5 cells of the structure, Q1 nC and z0.78 mm ( I 111 A), and E 060 MeV/m. Resulting evolution

of rms emittance, and beam envelope, as well as the invariant en-

velope are plotted. Compensation occurs far after the photoinjectorexit.


8/14/2019 serafini_1997_0209


begin acceleration, extending further the process of emit-tance compensation. This waist should be chosen to bothgive a small emittance at the waist point and to match ontothe invariant envelope associated with the beam current and

energy, as well as the linac accelerating gradient. An illus-trative example of such a system is shown in Fig. 14a,which displays the rms envelope and emittance evolution of the beam in the TTF-FEL Tesla Test Facility–free electronlaser photoinjector as obtained from a particle-in-cell simu-lation performed with ITACA 13. The analytical predictionof the correct invariant envelope in this case is shown forcomparison in Fig. 14b. It is very close to this optimizationfound by performing many such simulations, thus validatingthe approach to photoinjector design we have deduced fromthis analysis.

The number N of full cells in the gun is a variable in thisanalysis, but the validity of the approach followed here isconfined to a few cells. The most commonly encountered

case in practice, of course, is one full cell. We will alsoconsider, as a particular case, the possibility to slightly varythe length of the first half cell, as it is known from experi-ence that a slightly longer typically 0.625 instead of 0.5first cell gives better performances in terms of emittance cor-rection. This generalization, while a departure from themodel employed in the previous sections, is necessary for anaccurate comparison of the theory to actual rf photoinjector

configurations.The basic strategy of the analysis presented in this sec-

tion, in which we must specify the optimum envelope behav-ior in the drift space, is not a search for an invariant en-velopelike solution, but a matching of the beam envelopefrom the drift space to the invariant envelope of the boosterlinac. The model in this case is slightly changed with respectto that shown in Fig. 3: first, the point zc is now located at avariable position given by zc(1d )/2 N ( /2), whichbecomes the end of the rf gun cavity structure, and the be-ginning of the drift section. Here the quantity d accounts fora change in the first half cell length, i.e., d 0.25 indicates a25% lengthening 0.625 cell. We will also show that, as

expected, the optimum field profile for the solenoid magnetis different from the previous long gun case, where themagnetic field begins at z B1 /8 and ends at z B2(5/41/8) . For the compact injector the field start position isshifted downward at z B1 /2, while the end position isshifted downward at z B2(7/4). As discussed in the fol-lowing, the longer magnetic field profile is needed to providemore focusing from the solenoid in a case where the pon-deromotive rf focusing in the following cells is not onlymissing, but the exit defocusing transient kick at the end of the gun must be overcome.

The beam energy c at z z c now becomes c 2( N 1) , with

2 13

2 1

5

12d 5

2

2

3 d 2

24

again the energy at the second iris position as derived inAppendix A. Since the drift space downstream of zc is freefrom any accelerating and/or focusing force, the rms enve-lope equation becomes, in this case,

P

n,th2

3 c20, 6.1

where d 2 / dz2 and P I /2 I 0 c3 is now defined as the

beam perveance the assumption c1 is understood. Ac-

cording to the assumption of quasilaminarity, we neglect theemittance term and cast Eq. 6.1 into the space ( , z) as

1/ 0, 6.2

where ( z) ( z)/ P. Typical values for the perveance are,in case of a 100-A beam at c15 /29, P4106,so that for a 1-mm beam spot size the quantity is of theorder of 1, as is when is a few mrad. Equation 6.2

can be derived from a Hamiltonian H p 2 /2ln with p

, where H is now a constant of the motion, so that

c22 ln / c, 6.3

FIG. 14. a Envelope and emittance evolution of the beam in

the TTF-FEL photoinjector as obtained from particle-in-cell simu-

lation. The analytical prediction of the correct invariant envelope in

this case is shown for comparison in b.


8/14/2019 serafini_1997_0209


which gives the trajectory solution for Eq. 6.2 in integralform:

1

/ c dx

c22 ln x

z z c

c. 6.4

This equation actually represents a universally scaledbeam spreading curve, representing a universally scaledspace-charge beam blowup.

The integral in Eq. 6.4 is not analytically solvable unlessthe approximation 1 / c1 is assumed, which is in facttypical of a rf gun operated in the emittance compensationregime, as the beam size oscillations must be kept small bothto prevent nonlaminar trajectories and to keep the oscillationfrequency nearly independent of the perveance. Indeed, inpractice, the beam exits the gun with a small negative diver-

gence c , so that it is transported up to a space-charge-dominated waist with a spot size usually slightly smaller than c . In this case the approximate solution is

z c 2

c c z

c c z 1 c

2

1 c c z / c c z 21/2

, 6.5

where z z z c . It is interesting noticing that Eq. 6.5actually represents a generalization of a previous result de-

rived by Reiser 18. The initial conditions ( c , c) aregiven, in terms of the beam conditions at the gun exit( c , ˙ c), by

c c

c

3

2

1/2

, c ˙ c c

2 c 2

1/2

, 6.6

recalling that ( c , c) correspond to actual envelope vari-

ables, which Eq. 6.6 connects to ( c , ˙c), which are secular

envelope variables.In order to match to the invariant envelope at the entrance

of the booster linac, we need to find the conditions underwhich the phase space angle corresponding to the solution toEq. 6.2 is invariant with respect to the beam current I , or,equivalently, to the perveance P . This is also equivalent torequiring

d /

d P

d /

d P

1

2 d

d P

d

d P0. 6.7

Since d / d P(1/ )d / d P see Eq. 6.3, we haved ( / )/ d P(1/ 2)(1 2)d / d P. Therefore, d ( / )/ d P0 if either 2

1 for all z which is not possible tofulfill, because it is not a solution of Eq. 6.2 or d / d P

0. Since depends on P through the initial conditions( c , c), the condition d / d P0 is equivalent to

d c

d P

d c

d P0, 6.8a

which in turn is equivalent to the condition

d c

d 0,

d ˙ c

d 0, 6.8b

where the Cauchy current density is given by I / ( r )

2.It should be noted, however, that the invariance of the

phase space angle at the end of the drift given by Eq. 6.7 isachieved only through the invariance of the initial conditionsversus the current. The reason for this is that the phase spaceangle / associated with Eq. 6.5 is not intrinsically in-variant, unlike the case of the invariant envelope, where the

phase space angle is a constant ˆ / ˆ 1/2. In this respect,Eq. 6.2 does not display any invariant envelope solution,i.e., any solution for which / const equivalent to theBrillouin flow condition given in Eq. 2.2, where the phase

space angle is again a constant eq / eq0.

To better clarify this point let us examine the equation forsmall deviations around an equilibrium solution 0 of Eq.6.2. Assuming / 01, we find

02 0, 6.9

giving stable oscillations with frequency 1/ 0 around the

equilibrium solution 0 . As far as the beam envelope can berepresented by the approximate solution Eq. 6.5 in thedrift space, implying that the beam size varies slightly be-tween the initial condition c and the beam spot at the waist w , we may identify 0 roughly with the expression of Eq.6.5, so that the drift space up to the waist and slightlyfurther away is comparable to a quasi-Brillouin flow condi-tion with a local stability condition similar to the one de-scribed in Sec. II. Since the beam size, in the absence of anyfocusing, grows indefinitely after the waist, the frequency of oscillation 1/ 0 around the equilibrium solution is decreasingand the nonlinearities see Eq. 2.9 in the oscillations pre-clude any further vanishing point in the correlated emittance,as clearly illustrated in examples shown below.

It is interesting to notice that Eqs. 6.8 are equivalent to avanishing correlated emittance at the waist position; this re-quires that the waist position z w zw zc and the waistbeam spot size w are independent of the perveance,

d z w

d P0,

d w

d P0,

with


8/14/2019 serafini_1997_0209


z w c f c, f ce c

2 /2

1 dx

c22 ln x

, w ce c2 /2. 6.10

Note the elegant expression for w as a function of c and

c , which physically indicates that a beam focused too

strongly i.e., c2 will not come to a laminar space-charge-dominated waist but will likely come to an emittance-dominated nonlaminar waist.

The function f ( c) is plotted versus c in Fig. 15. For the

purpose of further analysis, we note that f ( c) can be ap-

proximated within a 5% error by the function g( c

)1.09 c /(1.69 c

2)0.423 ce0.296 c

2

, in the range c6. This range easily covers all rf photoinjectors of interest,

as a larger value of c implies a strongly convergent beamthat will be susceptible to nonlaminar behavior near thewaist. The conditions in Eq. 6.10 are now written explicitlyin terms of the initial conditions at the start of the drift as

f cd c

d P c

d f c

d c

d c

d P0,

d c

d P c cd c

d P 0. 6.11

This system of equations allows solutions different from

d c / d Pd c / d P0 if the determinant of the coefficient

matrix, det M c c f ( c)df ( c)/ d c, is vanishing. Byapplying the Leibniz formula for the derivation of definite

integrals, we find d f ( c)/ d c1 c f ( c), so that det M

c , which can never be vanishing, implying that the con-

ditions d c / d Pd c / d P0 and d zw / d P0; d w / d P

0 are in fact equivalent.In order to derive the solution to Eq. 6.8 in terms of the

beam conditions ( 2 , ˙ 2) at the second iris position ( z

z2), as reported in Appendix A, we approximate the enve-lope equation in the region of drift with applied solenoidfield i.e., z2 z z B2 as

K r 1/ 2 , 6.12

where 2 2 / I /(2 I 0 23), and K r (b / 2) 2, and so the

space-charge term is taken as a constant, its value assumed atthe gun exit. This is valid at the present point only for a 11/2 cell gun, for which the beam energy at the exit, 2 , isconstant all over the drift space: however, the treatment canbe easily generalized to the case of a N 1/2(1d ) cell gun.In the N 1 the drift space is divided into two parts. In thefirst one, from z z 2(3/4) up to z z c z B2(7/4), thebeam is subject to a focusing solenoid field, while for z

zc the drift is in free space. Under this approximation Eq.6.12 can be easily solved to find

c 2cos 1cos

2K r

2sin

K r

,

6.13

c 2K r sin sin

2K r

2 cos ,

where K r ( z z2), and hence

d c

d P

d 2

d P cos 1cos

22K r

d 2

d P

sin

K r

,

6.14

d c

d P

d 2

d P K r sin sin

22K r

d 2

d Pcos .

FIG. 15. Plot of the function

f ( c

) e c2 /2

1

dx / vc

2

2 ln xdots: the solid line gives a fit of

the function, namely, g( c)

1.09 c /1.69 c2

0.423 ce0.296 c2

.


8/14/2019 serafini_1997_0209


The determinant of the coefficient matrix is derived to bedet M 21(1cos )/ 22K r , which clearly implies det M 2

1, indicating that the only solution is d 2 / d Pd 2 / d P

0, which we know to be equivalent to

d 2

d 0,

d ˙ 2

d 0, 6.15

where 2 and ˙ 2 are specified in Eqs. A7 and A8 as func-tions of ( , A , ,b). Following the same procedure as inSec. V, we solve these two equations by expressing theirroots as ( , A) and bb( , A). The system is highlynonlinear, so that we start by expressing the first part of Eqs.6.15 as

d 2

d

1

2 2

I 0

1/ 2 A

4 1

ln 2

21 2 /20,

6.16

which can be solved for the variable

b& 1 1 A

4 I 0 1

ln 2

21

1/2 ln 2 b

.

Substituting back into the second part of Eqs. 6.15, weobtain an equation in , , and A . By a fitting procedure weobtain the following solutions:

optkA57.312.4 2.63 226.2 A1.78 A

1.86 A 2 6.17

and

b opt1.49

1.67

2.07

1/4 . 6.18

The Cauchy perveance opt is plotted in Fig. 16 solidlines as a function of for some usual values of A . It isinteresting to note that opt is nearly independent of for 1.5, while it decreases almost linearly with for

1.5. Its dependence on A is fairly linear in most of thediagram, so that the following scaling laws hold:

opt A if 1.5,

A3 if 1.5,6.19

which can be cast in terms of the bunch charge Q b , cathodespot size r , accelerating gradient , and rf wave numberk ,

Q b r 3 2 if 1.5

r 3 23 / k if 1.5,

6.20

which resemble the scaling laws reported in Ref. 17.As in Fig. 6, the Cauchy perveance max see Eq. 5.5

compatible with the maximum charge limit is plotted in Fig.16 dotted lines for a bunch length 2° the higher lineand 4° the lower line. The parameter bopt, which rep-

resents the ratio between magnetic and rf focusing, turns outto be nearly independent of the aspect ratio A, as shown inFig. 17, where b opt is plotted as a function of for differentvalues of A it should be noted that Eq. 6.18 displays asimplified form for b opt, which has already removed the veryweak dependence on A. It should be also noted that bopt ismuch higher than the analogous parameter b s see Fig. 7required in the case of the indefinitely long photoinjector.This is due to the facts that, not only is the additional rf focusing applied in the long photoinjector missing here, butthe transient defocusing at the end of the gun occurs at alarge beam spot size; these effects create the need for en-hanced focusing from the external solenoid.

The current density J corresponding to the s

lines isplotted solid lines in Fig. 18, for some selected rf frequen-cies 0.65, 1.3, 3, and 6 GHz, at aspect ratio A1. Themaximum limits for the current density are also reported, inthe case of l5 psec higher dashed line and t

10 psec lower dashed line. The required cathode spotsize r for the case of a 1-nC bunch charge are plotted inFig. 19, for the same set of frequencies and at differentbunch aspect ratios.

It should be noted that the operating point to achieve theemittance compensation shown in Figs. 14a and 14b hasbeen derived by the operating diagram in Fig. 16, selectingthe point 1.8, A1/2, 56 kA, giving at the L band E 050 MV/m, r 0.76 mm, and I 78 A, I 97 A,

FIG. 16. Operating diagram in the ( , A) plane for a short 1

1/2 cell photoinjector.

FIG. 17. Parameter bopt, plotted as a function of , at some

values of the bunch aspect ratio A .


8/14/2019 serafini_1997_0209


I 53 A for a 1-nC bunch charge. The magnetic field pa-rameter is b opt

0.94, corresponding to B 01.57 kG.We have already solved exactly for the beam waist size

w ce c

2 /2, but have not explicitly given an expression

for the position of the space-charge-dominated waist whenthe gun is operated under the conditions specified by Eqs.6.17 and 6.18, as required to achieve emittance correc-tion. We restrict the discussion here to the case of N 1, butthe results are easily extendible to N 2. At the gun exit ( z

z2) we have cf. Eq. 6.6 2opt( 2

opt / ) 2 , 2opt

( ˙2opt 2

opt /2), where the superscript opt indicates that wehave substituted Eqs. 6.17 and 6.18 into A7 and A8,

so that 2opt , ˙

2opt , 2

opt are now functions of only and A ,

while 2opt can be represented by a function of and A di-

vided by , e.g., 2opt f ( , A)/ . Substituting 2

opt ,

2opt for 2 , 2 in Eq. 6.13, we find

copt

1 f cos opt

1cos opt

f b opt2 22 2

opt sin opt

bopt 2 ,

(6.21)

copt f sin opt / 2

sin opt

f bopt 2 2

optcos opt,

where, again, copt is a function of only and A , and c

opt

can be represented by a function of and A divided by

, since the term inside square brackets is a function of only and A opt

b opt( c 2)/ 2 .Now employing Eq. 6.10 we finally find the waist posi-

tion z wopt and spot size w

opt under optimum operating condi-

tions,

z wopt

74

1

10.8 1.48 21.18 A1.07 A3 ,

6.22

woptmm

I A

m13.761.56 1.58 20.26 3

0.56 A0.914 A0.11 2 A0.15 A 2 .

6.23

As an example, for an L-band injector operated at rf

1.3 GHz, with E 056 MV/m 55 m1, and 2the optimum at A1 will be opt

68 kA, while b opt

0.92, so that the solenoid field will be B01.7 kG.Assuming a bunch charge Q b1 nC, the laser spot sizeat the cathode r will be, recalling that r

3 (c / 2 )Q b A / 2, r 0.84 mm. The beam current

will be given by I Q bc / 2 zQ bcA / 2 r 142 A,

so that the beam spot size at the waist will be wopt

0.94 mm, while at the gun exit 21.9 mm. The waist

FIG. 18. Current density J plotted as a func-

tion of the cathode peak field E 0 MV/m , at

different rf frequencies, for a short photoinjectoroperated in the emittance correction regime.

FIG. 19. Cathode spot size r , plotted vs the

cathode peak field E 0 MV/m , at different rf

frequencies, for a short photoinjector operated in

the emittance correction regime.


8/14/2019 serafini_1997_0209


will be located at zwopt0.793 m. It is remarkable to note that

the waist position scales with explicit dependence only onthe rf field and wavelength, not the bunch charge and/orcurrent which are derived quantities, in agreement withwhat is observed in Ref. 17.

For the case where the first half cell is lengthened by20%, as is done in many new rf gun designs, we have recal-culated opt and b opt to obtain

optkA58.3311.83 2.52 227.34 A1.816 A

1.88 A 2 6.24a

and

b opt1.38

1.52

1.86

1/4 , 6.24b

i.e., behavior very close to that of the standard half cell. Thepredicted position and spot of the waist now become

z wopt

74

1

9.94 2.59 20.18 A2.65 A2.96,

6.25

woptmm

I A

m1 5.22.14 1.77 20.27 3

0.19 A0.95 A0.058 2 A0.19 A2 .

6.26

Taking the same example as before, i.e., for an injectoroperated at the L band ( rf 1.3 GHz) and E 055.7 MV/m hence 54.6 m1 and 2 the optimum, at A1, is now opt

70.4 kA, while bopt0.89, so that

the solenoid field will be B01.64 kG. The relaxation of themagnetic field required reflects the additional rf focusing thatthe elongated cell provides. With a bunch charge Q b

1 nC, the laser spot size at the cathode r is now r

0.83 mm, and the beam current becomes I 144 A, so thatthe main parameters are almost unchanged, as is the waist

position, which is zwopt0.804 m. On the other hand, since

the beam does not expand as much before solenoid focusing,

the beam spot size at the waist is now larger, being wopt

1.3 mm.The behavior of the emittance evolution after the injection

of the beam into the booster linac is closely related to thatdiscussed at the end of Sec. V concerning the case of a longinjector. In the present case, however, the beam has beenalready compensated at the entrance of the linac, in the sensethat the beam has undergone a full envelope oscillation, and

has achieved a local minimum in both emittance and enve-lope. Therefore the emittance will again tend to rise, but thedamping effects of operation near the invariant envelopehold this rise in check. A further diminishing of the emit-tance is expected after an additional perturbed beam enve-lope plasma period, as is illustrated in Fig. 14a. This‘‘double’’ compensation shows the power of the split gun–booster linac configuration.

VII. CONCLUSIONS

We have discussed, in some detail, the properties of the

invariant envelope, which is a particular beam propagationmode characterized by a phase space angle that is a globalconstant. Under the hypothesis of quasilaminarity, which isequivalent to the assumption that the beam is space-chargedominated and the number of plasma oscillations consideredare small in order to avoid transverse and longitudinal mix-ing, we have shown that the invariant envelope is a propa-gation mode that damps the correlated emittance—providesemittance compensation—so that the possible emittance di-lution of a beam, due to longitudinal-transverse correlationscaused by either space charge or any other source rf, etc.,can be corrected by transporting the beam under an invariantenvelope mode. While we have concentrated here on stand-ing wave linacs, the invariant envelope exists in other types

TABLE I. Properties of invariant envelope flow, where possible, in various types of beam transport and

acceleration.

Beam transport Invariant envelope Space Emittance damping

Standing wave linac Secular

2e y /2

/2 sin2 1

Cauchy

dimensionless

Yes

Standing wave linacplus solenoid

Secular

e y /2

/8b 2 /sin2 14

Cauchydimensionless

Yes

Traveling wave

linac

Actual

2e y /2

Cauchy

dimensionless

Yes

Traveling wave

linac plus solenoid

Actual

e y /2

b 2 /sin2 14

Cauchy

dimensionless

Yes

Drift No Real dimensionless No

Drift plus solenoid Brillouin Flow

eqP / K r

Real dimensionless No


8/14/2019 serafini_1997_0209


of structures, with and without externally applied focusing

forces, as summarized in Table I.

In general, rf linacs allow acceleration under the invariant

envelope both in standing and traveling wave operation

where the correlated emittance oscillations are damped due

to acceleration, as on the invariant envelope the beam spot,

and eventual compensated emittance diminishes as 1/2.For completeness, we have shown in the table the invariantenvelope associated with the cases of both traveling andstanding wave linacs with additional uniform solenoidal fo-cusing. Traveling wave linacs operated with an extra mag-netic focusing may be in principle equivalent to standingwave linacs, where the focusing is provided by the rf pon-deromotive effect, as long as the magnetic focusing ratio b

cB 0 / E 0 is chosen to have the value b /2.Drift spaces, on the other hand can be operated in the

invariant envelope only with an external focusing to set up atrue Brillouin flow condition: the motion in the drift spaceafter a compact photoinjector is, in this respect, only an ap-proximation of the invariant envelope. In any event, space-charge-driven emittance oscillations are not damped in drifts,

so that one must quickly accelerate the beam after the drift,starting from the first emittance minimum, in order to avoidthe onset of wave-breaking processes i.e., nonlaminar ef-fects due to nonlinearities, which transform the reversibleemittance oscillation into an irreversible, thermal-like emit-tance growth as discussed in Ref. 19.

A transport line made by different sections that are alloperated under their own invariant envelope mode is of course a globally invariant envelope beam propagation. Thefinal design of a photoinjector that is be operated in the idealemittance correction regime will be therefore made up by aninterlocked sequence of accelerating and drift sections prop-erly matched and operated under invariant envelopes. Care

must be taken that this array includes the final transport fromthe booster linac to the application, so as to avoid emittancegrowth after initial compensation: this is of course true as faras the beam is still space-charge dominated in the sense dis-cussed above in this paper. In general this means, for trans-port that is longer than one-quarter of a plasma wavelength,that the beam be focused often enough typically by quadru-poles to approximate Brillouin flow after the photoinjectorlinac, with the beam controlled so as to not make large ex-cursions in spot size. For discrete focusing elements such asquadrupoles, this means that the elements must be placedwithin one-quarter of a plasma wavelength of each other.

In matching different sections one should, however, be

careful about what kind of orbit the invariant envelope isexpressed as: secular or actual. In standing wave linacs theenvelope is given in terms of a secular orbit, i.e., an orbitaveraged over the cell-to-cell oscillations, so that at the en-trance of the structure one must subtract a focusing kick of strength /2 to the beam envelope conditionsof the previous section in order to match the secular enve-lope. In the case the previous section is the drift space be-tween the short rf gun and a booster linac, one should posi-tion the space-charge-dominated waist directly at theentrance of the linac, as discussed in Sec. VI: the initialdivergence of the secular envelope in the booster will be inthis way /2 , which is exactly the first conditionon the invariant envelope. The second condition, i.e.,

(2/ ) I /3 I 0 , can be easily achieved by simply tuning

the accelerating gradient for a given energy, current, and

spot size of the beam at the booster entrance. It is remarkable

to note that this prescription on the matching condition has

been observed in several simulations of rf photoinjectors20.

If the booster linac is as well approximated as the puretraveling wave structure the matching conditions may seemdifferent because the invariant envelope is expressed in terms

of the actual physical orbit (2/ ) I /2 I 0 . It should benoted, however, that this orbit still has the same associatedphase space angle /2 , and that physical transientkick applied at the booster entrance is identical to that ap-plied in the standing wave case. Thus a parallel beam shouldstill be injected into a traveling wave booster in order tomatch the invariant envelope conditions, as is again in agree-ment with the results of multiparticle simulations 21. It cantherefore be seen that the natural matching of a parallel beamfrom a drift to the invariant envelope in an accelerating sec-tion is in fact due to the fortuitous relationship between adia-batic damping in trace space and the transient kick the par-

ticles feel as they enter the accelerator. This relationship canbe explained by a Hamiltonian approach to the dynamics; theradial entrance kick is set by the need to conserve canonicalmomentum. As the particle enters the accelerator, it mustpick up a radial mechanical momentum opposite to the radialfield momentum, which is proportional to , a conditionthat guarantees that a generalized Brillouin equilibrium in-variant envelope can be obtained.

As a final related example, in the case of a long multicellrf photoinjector structure we recall that the beam, if trans-ported under the invariant envelope, must also leave the pho-toinjector cavity with zero divergence. Due to the typicalhigh energy of the beam, as in the case of the 101/2 cell

AFEL photoinjector 22, the beam envelope is assumed tostay parallel for a long drift after leaving the photoinjector23. Therefore, a parallel matched beam emerging from along multicell photoinjector is therefore a sign of proper op-eration in the emittance compensation regime, as experimen-tally observed 24. To restate the conclusions of the previ-ous paragraphs, parallel beams, which are the analogue of the invariant envelope in a drift, match invariant envelopesupon both entrance and exit to accelerating sections.

We have pointed out that most emittance compensatedphotoinjectors are operated near the optimum conditions pre-dicted by the theory presented in this paper. These condi-tions, in which the beam is nearly matched to generalized

equilibria, have been discussed in detail in this work, andprescriptions for obtaining them have been given for manycases of interest. These prescriptions, and the general physi-cal reasoning behind their generation, can therefore be con-sidered to be guides for optimized performance of space-charge-dominated beams undergoing acceleration, focusing,and drifts.

In the future, this work will be extended to allow an esti-mate of the residual emittance of a beam after compensation.The most readily identifiable sources of residual emittanceare thermal cathode effects, radial nonlinearities, amplitudedependence of beam oscillation frequencies, and beam phasespace bifurcations. The effects of radial nonlinearitieswithin a slice under laminar conditions are not problematic,


8/14/2019 serafini_1997_0209


since the radial electric field will depend only on the beamsize and the enclosed current. Under these conditions, theenvelope approach can be easily extended to show that thissource of emittance behaves exactly as the linear longitudi-nally correlated emittance discussed so far in this paper.Likewise, amplitude-dependent frequency effects, anharmo-nicity of different slice motion, which do give rise to a re-sidual correlated emittance due to dynamical nonlinearity,

are treatable within the context of a laminar beam analysis.On the other hand, thermal cathode effects and phase

space bifurcations are beyond the reach of laminar beamanalysis, by definition. Because cathode effects are truly sto-chastic and not dependent on beam dynamics, they thereforecan be considered to be an uncorrelated source of emittance,which can neither grow nor diminish. This source of residualemittance must be determined by experimental study of thephotoelectric emission process. The subject of phase spacebifurcations is perhaps the most challenging from the ana-lytical point of view. The bifurcation in this case is due tobeam slices that have ‘‘crossover’’ waists, in which the elec-trons cross the axis, and the minimum beam size is deter-

mined by the slice emittance. This type of event is termedwave breaking in phase space, and represents a completelocal violation of the laminarity condition. When bifurca-tions are typically encountered, one enters a hybrid regimewhere the beam tails are emittance dominated while thebeam core is still in laminar space-charge flow. While it isnot clear how to deal with this case within the context of thepresent approach, it is clear that this effect is a major con-tributor to emittance growth in rf photoinjector beams, andmust therefore be seriously explored.

In addition to these extensions to cylindrically symmetricbeam analysis, the generalization of this analysis to highlyasymmetric beam ( x y) and emittance x y photoin-

jectors 20,25 is straightforward. This subject which is of high interest for linear collider electron source applications,is planned to be explored in a forthcoming investigation.

ACKNOWLEDGMENTS

The authors would like to thank B. Carlsten, E. Colby,K.J. Kim, P. O’Shea, D. Palmer, and R. Sheffield for usefuldiscussions and comments. This work was supported in partby U.S. Department of Energy Grants DE-FG03-93ER40796and DE-FG03-92ER40693, the Alfred P. Sloan FoundationGrant BR-3225, and the Instituto Nazionale di Fisica Nucle-are.

APPENDIX A

The expressions for the beam exit conditions 2 and 2 at

the second iris location z z 2 are reported in this Appendix.The expressions given have been derived for a Gaussiancharge distribution in the bunch, of dimensions r and z ,with the range of validity specified by 1/2 and Q bnC E 0MV/m /10, as extensively discussed elsewhere 7.The formulas reported here correspond to the particular case /2 examined in Ref. 7, augmented with the focusingeffects of the solenoidal magnetic field.

Let us define the beam energy 1 and 2 at the first andsecond iris locations, namely, 11 /2 and 21

3 /2. The solenoid field starts at z b /8 and extends to zc(1/85/4) , and the transverse forces imparted to beamelectrons during acceleration are represented by a defocusingrf term p rf , a defocusing space-charge term SC , and afocusing term B produced by the magnetic field of the so-lenoid. Expressed in terms of two auxiliary quantities,

1

a n 1 for an ideal first harmonic field for which

a11, a 3a5•••0 and SC I ( A)/4 I 0 2 r

2, they

take the form

p rf 1

ln 1

11 „1 /4!2…ln 2

8

1 ln 1

3 2

2 11 , A1

SCSC 1ln 2

21 , A2

Bb2ln2 2 / b /2. A3

In practice the rf term, which is a function only of , isnearly constant with a slight oscillation around the value1.06 all over the range 1/2 3, as shown in Ref. 7, forthe case of 1. In the following we will therefore take p rf

1.06.

Finally, the beam exit conditions 2 and 2 are

2 r 1SC B, A4

2 orb

2 2 2 , A5

where 2 is the secular envelope divergence, while orb isthe actual orbit divergence:

orb

2 r 1.06SC2 B 1SC B /ln 2 / b .

A6

The space-charge impulse factor SC contains a geometricform factor ( A) see Appendix B, which depends on thebunch aspect ratio A approximately as ( A)1/(2.451.82 A 5/4

0.55 A 3/2).According to the normalization applied in Sec. IV to

transform the transverse beam size into the dimensionless

quantity , defined as / S with S I /2 I 0 2 2 , we

give here the corresponding quantities 2 2 / S and ˙2

2 2 / S . Using the quantity defined in Sec. V,

I / 2 r 2, the dimensionless beam conditions 2 , ˙ 2 , and

2orb become:


8/14/2019 serafini_1997_0209


2 2 I 0 2

1/2

1 A

4 I 0 1

ln 2

21b 2ln2 2 / b /2 , A7

˙ 2 2 I 0 2

1/2

0.561

2 1

ln 2

21 A

8 I 0 1

ln 2

21b 2 ln2 2

b 4b 2 ln 2

b 1 A

4 I 0

1

ln 2

2

1 b2ln

2

b 2

, A8a

˙2orb 2 I 0 2

1/2

1.06 A

4 I 0b 2ln 2

b 1 A

4 I 0 1

ln 2

21b 2ln2 2

b 2 , A8b

which are functions of only four parameters: , A , , and b recalling that 1 , 2 , and b are functions only of .Let us assume now that the first half cell may be different, in length, from an exact quarter of rf wavelength, so that the first

iris is located at z1(1d ) /4 and the second one at z2 z 1 /2. Following the calculations by Serafini 26 we can expressthe beam energies 1 and 2 in the form

11

2 1

5

4d 5

16

2

24 d 2 , 21

3

2 1

5

12d 5

2

2

3 d 2

24 , A9

while the term p rf and SC become

p rf 1

ln 1

11 „1 /4!2…ln 2

8 1

ln 13 2

2 11 1

2

3d

d 2

2 , A10

SC 10.2475d SC 1ln 2

21 . A11

The actual orbit divergence at the second iris, orb , is found to be

orb

2 r 1.06 1

2

3d

d 2

2SC2 B 1SC B /ln 2 / b . A12

Finally, the dimensionless beam exit conditions 2 and ˙2orb at the second iris are

2 2 I 0 2

1/2

110.2475d A

4 I 0 1

ln 2

21b 2ln2 2 / b /2 , A13

and

˙2orb 2 I 0 2

1/2

1.06 12

3d

d 2

2 A

4 I 0b 2ln 2

b 110.2475d

A

4 I 0 1

ln 2

21

b2ln2 2 b

2 . A14

APPENDIX B

The transverse rms kick due to the space-charge field is represented by the factor SC in Eq. A2 for a Gaussiandistribution of transverse size r and longitudinal z , with total charge Q b . As extensively reported elsewhere 7, SC is

calculated by averaging in the rms sense the transverse electric field component of the bunch E r SC at rest in the laboratory

frame,

E r sc r ,

Q

2 0 2 3/2 r 2 z

r 0

dxexp 1/2 r 2 / r

21 x 2 / z

2 1 A 2 x

1 x 21 A 2 x B1

over the charge density distribution

r , Q b

2 3/2 r 2 z

exp r 2

2 r 2

2

2 z2


8/14/2019 serafini_1997_0209


to get

SC 2

2 E 0 r 1

2Q

V

r , E r 2 r , r d r d d

1/2

, B2

which can be cast in the form SC I ( A)/4 I 0 2 r

2, with

A

0

dx 1

0

dx 2

1 A 2 x 11 A 2 x2 2 A 2 x1 x 21/2

1 x 11 x 22 x1 x 22

1/2

. B3

( A) can be represented within 1% error in the range 0 A6 by the function

A 1

2.451.82 A 5/40.55 A 3/2 . B4

Since SC is actually the global rms space-charge kick on the bunch, we are interested in evaluating the kick SC applied

on the central bunch slice, located at 0, and the one SC applied to the slice located at z . These are given by

SC

2

2 E 0 r

2

z

Q

0

2

d 0

r ,0 E r 2 r ,0r dr

1/2

B5

and

SC

2

2 E 0 r 2

1/2 z

Q

0

2

d 0

r , z E r 2

r , zr dr 1/2

, B6

which can be cast in the form SC I ( A)/4 I 0

2 r 2 and SC

I ( A)/4 I 0

2 r 2, with

A 0

dx10

dx2

1 A 2 x1 1 A 2 x2 1/2

1 x1 1 x2 2 x1 x 221/2

B7

and

A

0

dx10

dx2

exp11/ 1 A 2 x 11/ 1 A 2 x 2 /2

1 x1

1 x2

2 x1 x

221 A 2 x

11 A 2 x

21/2

1/2

. B8

It is convenient to redefine the kicks SC and SC

in terms of rescaled currents I and I : SC

I ( A) ( A)/4 I 0 2 r

2 and SC I ( A) ( A)/4 I 0

2 r 2, where

I A I 2.451.82 A5/4

0.55 A 3/2

1.841.95 A5/40.65 A 3/2 B9

and

I A I 2.451.82 A5/4

0.55 A 3/2

3.841.74 A5/40.34 A 3/2 B10

are valid approximations for I and I in the range 0 A6. At A1 we have I 1.19 I and I 0.71 I .

In order to calculate the geometrical form factor g( ) we consider here only the linear component E lin( ) of the spacecharge field in Eq. B1, i.e.,

g E lin 0

dxe 1/2 2 /

z2

1 A¯ 2 x

1 x 21 A¯ 2 x ,

which is the source of the perveance term s( ) in the envelope equation; we want to study its dependence on the slice

position for low aspect ratios A¯ A / in the rest reference frame. For highly relativistic beams, i.e., 1 the transversespace-charge-field dependence versus the longitudinal position resembles the behavior of the charge density distribution: infact, we have

g —— →

A¯ →0

exp 2 /2 z2

,


8/14/2019 serafini_1997_0209


as expected. For small A¯ we approximate g( ) with its Taylor expansion up to second order in A¯ around A¯ 0, to obtain

g e 2 /2 z2 1 A¯ 2 1

2

z2 1

2ln A¯ 1 O A¯ 4 . B11

1 J. S. Fraser et al., IEEE Trans. Nucl. Sci. 32, 1791 1985.

2 C. Travier, Part. Accel. 36, 33 1991.

3 X. Qiu et al., Phys. Rev. Lett. 76, 3723 1996.

4 B. E. Carlsten, Nucl. Instrum. Methods A 285, 313 1989.

5 J. D. Lawson, The Physics of Charged Particle Beams, 2nd ed.

Oxford University Press, New York, 1988.

6 J. B. Rosenzweig and L. Serafini, Phys. Rev. E 49, 1599

1994; S. C. Hartman and J. B. Rosenzweig, ibid . 47, 2031

1993.

7 L. Serafini, Part. Accel. 49, 253 1995.

8 P. Lapostolle, IEEE Trans. Nucl. Sci. NS-18, 1101 1971.

9 K. J. Kim, Nucl. Instrum. Methods A 275, 201 1989.

10 J. Coacolo private communication.11 L. Serafini, R. Zhang, and C. Pellegrini, Nucl. Instrum. Meth-

ods A 387, 305 1997.

12 J. M. Dolique and J. C. Coacolo, Nucl. Instrum. Methods A

340, 231 1994.

13 L. Serafini and C. Pagani, Proceedings of the European Par-

ticle Accelerator Conference World Scientific, Singapore,

1988, p. 866.

14 The current density J is not the real beam current density along

acceleration: it is just a reference value calculated with the

beam spot size at the cathode r .

15 C. Travier, Ph.D. thesis, Universite De Paris Sud, Orsay-LAL,

1995 unpublished.

16 J. B. Rosenzweig et al., Nucl. Instrum. Methods A 341, 379

1994.

17 J. B. Rosenzweig and E. Colby, in Advanced Accelerator Con-

cepts, edited by P. Schoessow, AIP Conf. Proc. No. 335 AIP,

New York, 1995, p. 724.

18 M. Reiser, Theory and Design of Charged Particle Beams

John Wiley & Sons, New York, 1994, p. 201.

19 O. A. Anderson, Part. Accel. 21, 197 1987.

20 E. Colby, J.-F. Ostiguy, and J. B. Rosenzweig, in Advanced

Accelerator Concepts Ref. 17, p. 708.

21 D. T. Palmer et al., in Proceedings of the 1995 IEEE Particle

Accelerator Conference, IEEE Cat. No. 95CH35843, 1996, p.2432.

22 D. C. Nguyen et al., Nucl. Instrum. Methods A 341, 29 1994.

23 Taking indeed a 100 A beam at 19 MeV, the beam spot at the

photoinjector exit, as predicted by the invariant envelope

mode, is 0.9 mm, so that Eq. 6.1 predicts a spot size

increase of less than 125 m over 1-m drift.

24 R. Sheffield private communication.

25 J. B. Rosenzweig et al., in Advanced Accelerator Concepts

Ref. 17, p. 684.

26 L. Serafini, in Advanced Accelerator Concepts, edited by J. S.

Wurtele, AIP Conf. Proc. No. 279 AIP, New York, 1993,

p. 645.


serafini_1997_0209

Documents