Arbitrary emittance partitioning between any two dimensions for electron beams

Bruce E. Carlsten,1 Kip A. Bishofberger,1 Leanne D. Duffy,1 Steven J. Russell,1

Robert D. Ryne,2 Nikolai A. Yampolsky,1 and Alex J. Dragt3

1Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA2Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

3University of Maryland, College Park, Maryland 20742, USA(Received 7 January 2011; published 26 May 2011)

The flat-beam transform (FBT) for round symmetric beams can be extended using the concept of

eigenemittances. By tailoring the initial beam conditions at the cathode, including adding arbitrary

correlations between any two dimensions, this extension can be used to provide greater freedom in

controlling the beam’s final emittances. In principle, this technique can be used to generate extraordinarily

transversely bright electron beams. Examples are provided where an equivalent FBT is established

between the horizontal and the longitudinal beam dimensions.

DOI: 10.1103/PhysRevSTAB.14.050706 PACS numbers: 41.60.�m, 29.27.�a, 41.85.Lc

I. INTRODUCTION

Currently, free-electron lasers (FELs) are the only de-vices which can be scaled to generate coherent radiationfrom microwaves to x rays. The technological limit ongenerating hard x-ray radiation is mainly determined bythe availability of transversely bright electron beams. Inorder for an FEL to lase optimally, the normalized trans-verse beam emittance must be small, "n � ���x-ray=4�,

to ensure overlap between the electron and x-ray phasespaces, where � and � are the beam’s velocity normalizedto the speed of light c and its relativistic factor, respec-tively, and �x-ray is the x-ray wavelength. This condition is

easier to satisfy at higher beam energies and this approachwas used to design Linac Coherent Light Source [1] whichgenerates 8 keV photons using a 15 GeV electron beam.

The next generation hard x-ray FELs (XFELs), produc-ing photons well above 10 keV, cannot use the sameapproach due to increased beam energy spread caused bythe single-particle synchrotron radiation at high energies[2]. At the end of an undulator, this spread is equal to

�Erms

E¼

ffiffiffiffiffiffiffiffiffiffiffiffi55

48ffiffiffi3p

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@e6B3

4�"0m5c6

s�L1=2; (1)

where B is the rms undulator field strength, L is theundulator length, and e, m, @, and "0 are the electroncharge, electron mass, normalized Planck’s constant, andfree-space permittivity. FEL performance is degradedwhen the total beam energy spread is greater than thegain parameter. Estimates for the proposed XFEL for LosAlamos’ Matter and Radiation in Extremes (MaRIE)

facility [3] generating 50-keV photons (with a 100-meterlong undulator with a 2.4-cm period and a 1-T rms field,driven by a 3.4-kA beam) show that the beam energy islimited to about 20 GeV. At this energy both the inducedenergy spread and the FEL gain parameter are close to0.015% and the FEL efficiency rapidly drops at higherenergies. This limitation on the beam energy puts a con-straint on the normalized transverse beam emittance.Numerical simulations for this case indicate that emittan-ces as high as 0:15 �m are marginally acceptable (thesetransverse emittances are about a factor of 2 more relaxedthan the optimal constraint, with some corresponding lossof efficiency), and with performance degrading signifi-cantly as the emittance is increased above that. In contrast,the longitudinal emittance for a 150-fs bunch and 0.01%energy spread can be as high as 180 �m.Using the scaling law for transverse emittances in current

high-brightness photoinjectors, "n � 1 �m ðq=nCÞ1=2, onecan find that the required emittance can be achieved forbunch charges q on the order of 20 pC. A peak current of3.4 kA implies that such a bunch should be about 6 fs(2 �m) long and this ultrashort bunch may be severelyimpacted by three-dimensional effects of coherent synchro-tron radiation (CSR) in the compressing chicane.Alternatively, a 500-pC, 3.4-kA bunch would be 150 fslong which is consistent with demonstrated chicane per-formance and could produce over an order of magnitudemore photons.A 500-pC electron bunch produced with currently avail-

able high-brightness photoinjectors would have normalizedbeam emittances of "x;n="y;n="z;n of 0:7=0:7=1:4 �m, with

a total phase-space volume of 0:7 �m3. At the same time,the electron bunch required for a 50 keV XFEL must havenormalized beam emittances not exceeding "x;n="y;n="z;nof 0:15=0:15=180, with a total phase-space volume of4 �m3. Therefore, currently available photoinjectors cangenerate bunches with sufficiently small phase-space

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 14, 050706 (2011)

1098-4402=11=14(5)=050706(16) 050706-1 Published by the American Physical Society

volumes, but the partitioning of this phase space into lon-gitudinal and transverse emittances is not correct. Thisconcept is shown notionally in Fig. 1, where we plot thenumber of photons which can be generated in an FEL usingan electron bunch from a currently available photoinjector[black line, using the constraints "n � ���x-ray=2� and

"n � 1 �mðq=nCÞ1=2 to determine the bunch charge, cap-ping the maximum bunch charge at 100 pC] and if theoptimal partitioning of the bunch phase-space volume be-tween the longitudinal and the transverse emittances couldbe achieved (red line). For the red line, a bunch charge of500 pC is used up to about 120 keV photon energies,because the excess transverse emittance can be reparti-tioned into the longitudinal dimension, after which themaximum allowable bunch charge also scales inversely asthe square of the photon energy. For both lines, an electron-beam energy of 20 GeV and an FEL efficiency (from totalelectron energy to total photon energy) of �3� 10�4 areassumed, for ease of comparison.

A novel technique for partitioning the phase-space vol-ume suitable for linear colliders and XFELs was recentlyproposed [4]. This technique relies on the fact that once thebeam is generated, three quantities, known as eigenemit-tances [5], are conserved through the beam line. This fact isa consequence of the symplectic properties of Hamiltoniansystems; preservation of these quantities is only limited bynonlinear effects. A large asymmetry in eigenemittancescan be set up by introducing cross-dimensional correla-tions in the beam as it is formed at the cathode surface [4].

These cross correlations can be removed downstream athigh energy, and the eigenemittances can be recovered asthe three beam rms emittances resulting in a transverselyvery bright beam.The flat-beam transform (FBT) [6–9] has been described

in this context [4]. The FBT employs x-y0 and y-x0 corre-lations in a symmetric beam to tailor the beam eigenemit-tances which are recovered as the transverse beamemittances by using a specific optics algorithm that re-moves the correlations [6,9]. One of the resulting emittan-ces is decreased and another is increased compared to theuncorrelated beam. The total transverse phase-space vol-ume does not change when the correlations are introducedin a FBT, which makes it a flexible option for significantreduction of one eigenemittance. The FBT scheme wasdemonstrated experimentally [10], achieving one emit-tance even less than the initial beam thermal emittance.Another example of the eigenemittance concept, nowwhere there are no initial cross correlations, is the emit-tance exchanger (EEX) [11,12], which swaps the emittancefrom one transverse dimension with the axial emittance(and also experimentally demonstrated [13]).Kim proposed an idea of using a temporally very short

electron bunch (and thus very low longitudinal emittance),followed by an FBT and an EEX in succession to achievethe low transverse emittances required for an XFEL [14].For example, consider the case a 200-fs laser generates abeam with 2:1=2:1=0:15 �m intrinsic emittances. TheFBT adjusts these numbers to 0:15=30=0:15 �m, and theEEX swaps "y;n and "z;n, to finally yield 0:15=0:15=30 �m

emittances, reasonably suitable for an XFEL. However,photoinjector design does not scale linearly enough toproduce the needed initial emittances, particularly forsuch a short axial beam size.Earlier studies of FBTs were limited to the case of an

initially symmetric beam. In this paper we generalize theFBT concept which will allow eigenemittance partitioningof more general and asymmetric initial beams, with threeimportant results. First, we present a simple algorithm tocalculate eigenemittances from any set of physically realiz-able correlations between any two dimensions. Second, weshow that the optics elements employed for the symmetricFBT case can also recover the eigenemittances for anyarbitrary cross-diagonal beam matrix, with only simpleretuning of the optics. Third, we present the beam lineoptics for recovering eigenemittances as rms beam emit-tances for a generalized FBTwhich introduces correlationsbetween either x-y or x-z phase-space planes, for any physi-cally realizable beam. In particular, we study a flat-beamtransform with x-z correlations (XZFBT) and demonstratethat, among other configurations, an XZFBTwith an ellip-tical cathode can produce transversely bright electron beam.Section II briefly discusses the choice of canonical

variables and beam units used in our analysis to ensurethe following development is symplecticly consistent.

FIG. 1. Rough scaling of photons per pulse versus photonenergy for conventional technology (black) and from usingeigenemittance partitioning (red). The scaling for conventionaltechnology is dashed beyond 10 keV, due to uncertainty in theCSR model’s validity for ultrashort bunches in the compressors.

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In Sec. III, theoretical definitions are presented for initialbeam correlations, and Sec. IVexpands the eigenemittanceconcept beyond previous work [4] to include all possiblecorrelations and a simple set of formulas to calculate theeigenemittances. After a review of x-y FBTs in Sec. V,explicit configurations for XZFBTs, including the neces-sary correlation-removing optics, is offered in Sec. VI. Afinal section discusses practical limitations due to photo-injector nonlinearities and compares correlations couplingall three dimensions, and includes a numerical example forthe proposed MaRIE XFEL injector, another for the JLAB500-kV DC gun, and one using an energy-attenuating foilat higher energy.

II. DEFINITION OF CONSISTENTBEAM VARIABLES

A key concept in this paper is that each eigenemittanceis not tied to a specific dimension; while conserved throughlinear symplectic transformations, they can be ultimatelyexpressed in any dimension depending on the specificbeam optics. The invariance under linear symplectic trans-formations demands that our approach is traceable to ca-nonical variables. In this section, we show that the beamcoordinates we use in the following sections are directlytraceable to canonical coordinates, and identify that theonly approximation used for the beam coordinates, whichwe will alternatively call the beam trajectory, is the para-xial approximation.

In our analysis we use the arclength s (or the longitudi-nal coordinate, z, in rectilinear systems) as the independentvariable. In Cartesian coordinates, the 6-vector of canoni-cal coordinates and momenta is given by &Tcan ¼ðx; px; y; py; t; ptÞ, where x and y denote transverse

Cartesian coordinates, px and py denote transverse canoni-

cal momenta, ~p ¼ ~pmech þ q ~A (where ~pmech is the

mechanical momenta, q is the charge, and ~A is the vectorpotential), t is the arrival time at a location s in the beamline, and pt is the negative of the total energy, ��mc2.Beam dynamics codes often use variables that are dimen-sionless deviations from a reference trajectory. To avoidnotational clutter, we will write the dimensionless devia-tions as ðx; px; y; py; t; ptÞ, derived from the dimensional

variables, the design trajectory &Tcan;0, and the scaling quan-

tities l, �, and!, as will be described in a future paper [15]:

x ðx� x0Þ=l; px ðpx � px0Þ=�;y ðy� y0Þ=l; py ðpy � py0Þ=�;t !ðt� t0Þ; pt ðpt � pt0Þ=ð!l�Þ:

(2)

Because we will use unnormalized quantities we set � ¼�0�0mc. Additionally, we use l ¼ 1 m and !l=c ¼ �1.With these choices, and making use of the definitions of px,py, and pt, the 6-vector of canonical coordinates

and momenta is therefore &Tcan ¼ ½x; ð��x=�0�0Þ; y;ð��y=�0�0Þ; c�t; ð��=�0�0Þ� in drift regions where the

vector potential vanishes, and where now we have identi-fied �t as a time variable which is positive for the head ofthe bunch and negative for the tail.For the purposes of this paper, we transform these

canonical variables into the more traditional (noncanoni-cal) formulation used in linac design. Let x0 ¼ dx=dz andy0 ¼ dy=dz. It follows that the unnormalized beam vectorcan be written in the paraxial approximation as &T ¼½x; x0; y; y0; c�t;�ð��Þ=�0�, where we have used �� ¼��ð��Þ ¼ �� �ð��Þ

� and kept both the transverse and lon-

gitudinal momentum deviations to first order in the smallquantities. When these conditions (lowest order paraxialexpansion in a drift space) are satisfied, we can constructthe following unnormalized beam matrix,

� ¼

hx2i hxx0i hxyi hxy0i hxðc�tÞi�x �ð��Þ

�0

�

hxx0i hx02i hx0yi hx0y0i hx0ðc�tÞi�x0 �ð��Þ�0

�

hxyi hx0yi hy2i hyy0i hyðc�tÞi�y �ð��Þ

�0

�

hxy0i hx0y0i hyy0i hy02i hy0ðc�tÞi�y0 �ð��Þ�0

�

hxðc�tÞi hx0ðc�tÞi hyðc�tÞi hy0ðc�tÞi hðc�tÞ2i�ðc�tÞ �ð��Þ�0

��x �ð��Þ

�0

� �x0 �ð��Þ�0

� �y �ð��Þ

�0

� �y0 �ð��Þ�0

� �ðc�tÞ �ð��Þ�0

� ���ð��Þ�0

�2�

0BBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCA

; (3)

where the brackets indicate ensemble averages over theentire electron bunch distribution. In the absence of cou-pling among the phase planes, the rms unnormalizedgeometrical emittances in this limit follow immediately

by inspection of the 2� 2 determinants on the diagonal[where we use the subscript ‘‘z’’ to denote quantitiesassociated with the longitudinal (temporal) phasespace]:

ARBITRARY EMITTANCE PARTITIONING BETWEEN ANY . . . Phys. Rev. ST Accel. Beams 14, 050706 (2011)

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"x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihx2ihx02i � hxx0i2

q"y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihy2ihy02i � hyy0i2

q

"z ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihðc�tÞ2i

���ð��Þ�0

�2��

�ðc�tÞ�ð��Þ

�0

�2

s: (4)

In summary, we have used the definition of canonicalvariables to motivate the unnormalized beam matrix intraditional variables, where cð�tÞ is the appropriate longi-tudinal coordinate and �ð��Þ=�0 is the appropriatelongitudinal momentum deviation term. [Equivalently,we could have used ��=ð�0�0Þ but chose this alternativeform to simplify the coordinate when the beam is at verylow energies near the cathode.]

It is worth recalling that the beam trajectory parametersare not canonical because they do not include the vectorpotential, so these expressions (in terms of identifyingeigenemittances) are only valid in drift regions where thevector potential vanishes. Thus, this formalism is accuratewhen one is considering the transfer matrix of an opticselement from the field-free region before the element toanother after the element, as we do in the following analy-sis. Likewise, they can be used to describe correlations inthe beam from magnetic field present at the cathode onlyafter the beam has entered a field-free region.

III. BEAM CORRELATIONS AND TRANSFORMS

In general, the beam matrix transforms through anylinear optics from position 1 to position 2 by

�2 ¼ R�1RT; (5)

where the transform matrix R transforms an unnormalizedparticle vector ~&2 ¼ R~&1 and ~&T ¼ ½x; x0; y; y0; ðc�tÞ;�ð��Þ=�0� as shown above. All electrodynamic motionof charged particles in the electromagnetic field is sym-plectic, so all transfer matrices R obey

J6 ¼ RTJ6R; (6)

where now

J6 ¼

0 1 0 0 0 0

�1 0 0 0 0 0

0 0 0 1 0 0

0 0 �1 0 0 0

0 0 0 0 0 1

0 0 0 0 �1 0

0BBBBBBBBBBB@

1CCCCCCCCCCCA: (7)

Following Kim [8], we use the form of nonsymplectictransformations on an initially diagonal beam matrix torepresent initial beam correlations, specifically once thebeam is in a field-free region. We start with general corre-lations, followed by specific symplectic transformationswhich establish x-y and x-z correlations. In theAppendix, we provide specific x-y transformations andx-z transformations that are needed in subsequent sections.

A. Initial beam correlations

The purpose of this section is to define initial beamcorrelations in a convenient way, and to establish themost general kind of beam matrices we need to considerfor general FBTs. The uncorrelated diagonal beam matrix,�0, defined by Eq. (3), is

�0 ¼

�2x 0 0 0 0 0

0 �2x0 0 0 0 0

0 0 �2y 0 0 0

0 0 0 �2y0 0 0

0 0 0 0 �2z 0

0 0 0 0 0 �2z0

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA; (8)

where we have introduced the notation �2x ¼ hx2i, �2

y ¼hy2i, �2

z ¼ hðc�tÞ2i, �2x0 ¼ hx02i, �2

y0 ¼ hy02i, and �2z0 ¼

hð�ð��Þ�0Þ2i. Following Kim [8], suppose that the correlated

beam matrix can be established by the following transform[analogous to Eq. (5)]:

�corr ¼ ðIþ CÞ�0ðI þ CÞT; (9)

where C is any specifically chosen correlation matrix anddetðIþ CÞ ¼ 1. In this paper, we focus on two-dimensional couplings, so at this point we will mostlyrestrict our analysis to 4� 4 beam and transfer matrices,where we consider alternatively x-y or x-z phase spaces.Entries in the correlation matrix establish beam correla-

tions, but the addition of single correlations does notchange the determinant of the beam matrix. They alsodefine which beam element depends on which other one.For example, consider the two x-y correlation matrices

C1 ¼

0 0 0 c14

0 0 0 0

0 0 0 0

0 0 0 0

0BBBBB@

1CCCCCA (10)

and

C2 ¼

0 0 0 0

0 0 0 0

0 0 0 0

c14 0 0 0

0BBBBB@

1CCCCCA: (11)

These correlations lead to these correlated beam matri-ces:

�C1 ¼

�2x þ c214�

2y0 0 0 c14�

2y0

0 �2x0 0 0

0 0 �2y 0

c14�2y0 0 0 �2

y0

0BBBBBB@

1CCCCCCA (12)

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and

�C2 ¼

�2x 0 0 c41�

2x

0 �2x0 0 0

0 0 �2y 0

c41�2x 0 0 �2

y0 þ c241�2x

0BBBBBB@

1CCCCCCA: (13)

In �C1, the horizontal beam size �2x is increased through

a �2y0 contribution, while in �C2, the �

2y0 term is increased

through a �2x contribution. It is easy to show, however, that

both beam matrices have determinants equal to the uncor-related beam matrix.

Correlations can be stacked in a multiplicative manner,so that the final correlation matrix can be given as a productof n individual correlation matrices:

I þ Ctotal ¼Yni¼1ðIþ CiÞ: (14)

If the correlations are decoupled, the correlation coef-ficients from the Ci matrices reappear in Ctotal. For ex-ample, an axial magnetic field on the cathode is the firststep of an x-y FBT, which creates x-y0 and x0-y correlationsonce the beam enters a field-free region. The total corre-lation matrix (within the two dimensions) becomes

IþCtotal¼ Iþ

0 0 0 0

0 0 0 0

0 0 0 0

�a 0 0 0

0BBBBB@

1CCCCCA

2666664

3777775 Iþ

0 0 0 0

0 0 a 0

0 0 0 0

0 0 0 0

0BBBBB@

1CCCCCA

2666664

3777775

¼ Iþ

0 0 0 0

0 0 a 0

0 0 0 0

�a 0 0 0

0BBBBB@

1CCCCCA

2666664

3777775: (15)

The beammatrix, at a waist outside of the axial field, canthen be written:

�axial field ¼

�2x 0 0 �a�2

x

0 �2x0 þ a2�2

y a�2y 0

0 a�2y �2

y 0

�a�2x 0 0 �2

y0 þ a2�2x

0BBBBBB@

1CCCCCCA;

(16)

where now a ¼ e2��mc BcathðRcath=RbeamÞ2 and Rcath=Rbeam

is the ratio of the beam size at the cathode to the beam sizeat the waist. These correlations are antisymmetric (oppo-site signs), and the angular momentum can be representedby L ¼ jhxy0 � yx0ij=2 ¼ jajð�2

x þ �2yÞ=2. Also, the cor-

relations in Eq. (15) were written as if y0 and x0 arefunctions of x and y, not vice versa.For more general correlations, coupling occurs.

Consider a reasonably general family of two-dimensionalcorrelation given by x0 being a function of y, y0 being afunction of x, and either x or y being a function of the otherdimension (it is hard to imagine a practical system whenany variable depends on either x0 or y0, especially one ofthese on the other). These three correlations lead to asignificantly more complex beam matrix. We will considerboth the cases x depends on y and the reverse.

B. y, x0, and y0 depend on x, y, and x, respectively

We shall transform an initial particle vector ðxi; x0i; yi; y0iÞto a final particle vector ðxf; x0f; yf; y0fÞ such that yf ¼ yi þdxi, y

0f ¼ y0i þ bxi, and x0f ¼ x0i þ eyf. After combining

the first two, adding the third yields

IþCtotal ¼ Iþ

0 0 0 0

0 0 e 0

0 0 0 0

b 0 0 0

0BBBBB@

1CCCCCA

2666664

3777775 Iþ

0 0 0 0

0 0 0 0

d 0 0 0

0 0 0 0

0BBBBB@

1CCCCCA

2666664

3777775

¼ Iþ

0 0 0 0

ed 0 e 0

d 0 0 0

b 0 0 0

0BBBBB@

1CCCCCA

2666664

3777775: (17)

An initially diagonal 4� 4 beam matrix

�0 ¼

�2x 0 0 0

0 �2x0 0 0

0 0 �2y 0

0 0 0 �2y0

0BBBBBB@

1CCCCCCA (18)

becomes

�C: yðxÞ ¼�2

x ed�2x d�2

x b�2x

ed�2x �2

x0 þ e2d2�2x þ e2�2

y ed2�2x þ e�2

y edb�2x

d�2x ed2�2

x þ e�2y �2

y þ d2�2x db�2

x

b�2x edb�2

x db�2x �2

y0 þ b2�2x

0BBB@

1CCCA: (19)

C. x, x0, and y0 depend on y, y, and x, respectively

We repeat the same procedure, but change the first correlation to xf ¼ xi þ dyi. The final beam matrix now becomes

ARBITRARY EMITTANCE PARTITIONING BETWEEN ANY . . . Phys. Rev. ST Accel. Beams 14, 050706 (2011)

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�C: xðyÞ ¼�2

x þ d2�2y ed�2

y d�2y bd2�2

y þ b�2x

ed�2y �2

x0 þ e2�2y e�2

y edb�2y

d�2y e�2

y �2y db�2

y

bd2�2y þ b�2

x edb�2y db�2

y �2y0 þ b2d2�2

y þ b2�2x

0BBB@

1CCCA (20)

which is also quite complicated. More interestingly, everyterm in �C: xðyÞ is different than the equivalent term in�C: yðxÞ, even for the special case of d ¼ 1. In general,we use correlations between x and z too, so in the followingwe generalize the notation where �x, �x0 , �y, and �y0 arereplaced by �1, �2, �3, and �4, respectively.

None of the correlated beam matrices are irreducible interms of symplectic transformations; they can always betransformed to a diagonal beam matrix with the eigene-mittances appearing on the diagonal (and likewise back toany form with an appropriate grouping of correlations).The form of the beam matrices in Eqs. (19) and (20) can beconsidered completely general and irreducible using un-coupled transfer matrices (where the XY and YX subma-trices are zero).

A collection of symplectic transfer matrices needed forrecovering the beam eigenemittances are supplied in theAppendix and which will be used in the followingthree sections. The key observations in the Appendix arethat the x-z counterparts of x-y optics elements are achicane for a drift, an rf cavity for a quadrupole, and atransversely deflecting rf cavity for a skew quadrupole.

IV. EIGENEMITTANCES

The power of the eigenemittance concept is the follow-ing: by establishing a desirable set of eigenemittanceswhen the beam is generated (by incorporating cleverbeam correlations), they will be the observed emittancesafter all correlations are removed. A corollary to this state-ment is that the eigenemittances are unaffected by thespecific transport before these correlations are removed,including acceleration (assuming irresolvable higher-orderissues remain small). The demonstration of the FBT trans-form at the A0 photoinjector at Fermilab has shown that, atleast in certain cases, these higher-order concerns can bekept small [10].

For any beam matrix, the eigenemittances are solvablethrough several different techniques. Three different meth-ods are presented here: the first is the most general andprovides an explicit representation of the required transfermatrix, while the last is the most straightforward and usesmatrix multiplication and the quadratic formula to deter-mine the eigenemittances.

A. Dragt’s method

The eigenemittances and the corresponding transfer ma-trixM which diagonalizes the beam matrix � can be foundusing the following Dragt’s algorithm implemented inthe MARYLIE code [16]. First, we evaluate the matrix

A ¼ e�J6� using its Taylor series, where � is some smallscalar making the norm of the matrix A close to unity. Thenthe matrix A is symplectic having all the eigenvalues on theunit circle. We construct the transfer matrix M fromthe eigenvalues of A which transforms A into the normalform, N ¼ M�1AM [17]. The resulting matrix M trans-forms the original beam matrix into diagonal form �diag ¼M�MT [17]. The matrix M represents the transfer matrixof the entire ‘‘unraveling’’ process, and, in principle, canbe decomposed into a series of required beam-opticscomponents.

B. Eigenvalue equation

Alternatively, the eigenemittances "eig of the beam ma-

trix � can be found through solving the eigenvalueproblem [17],

detðJ6�� i"eigIÞ ¼ 0; (21)

where I is the unit matrix, although the correspondingtransfer matrix which diagonalizes the beam matrix re-mains unknown. Equations (5) and (6) in Ref. [4] are aspecial transformation where the correlations are of theform

& ¼ ðI þ CÞ&0; C ¼ 0 B

A 0

" #; (22)

where A and B are 2� 2 matrices and we are now consid-ering either the x-y or x-z phase spaces. Note this is not ofthe most general form, Eqs. (19) and (20), but in the formof what we define as a nonsymmetric cross-diagonal FBT.The transform in Eq. (22) results in the subsequent beammatrix � related to the intrinsic beam matrix �0 as inEq. (9). Reference [4] shows that the constraint detðABÞ ¼TrðABÞ ¼ 0 must hold, and that the eigenemittancesof the modified beam matrix � can be found by solvingthe 4� 4 characteristic equation detðJ4�� i"eigIÞ ¼ 0.

The specific form of C yields the following characteristicequation:

"4eig � "2eig½"212 þ "234 þQþ ð"212 detB2 þ "234 detA2Þ�

þ "212"234 ¼ 0; (23)

whereQ¼�21�

23ða221þb221Þþ�2

1�24ða211þb222Þþ�2

2�23ða222þ

b211Þþ�22�

24ða212þb212Þ, "212 ¼ �2

1�22 and "234 ¼ �2

3�24 are

the uncorrelated beam emittances (since �0 has no corre-lations), and aij and bij are the elements of the matrices A

and B, respectively.

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The biquadratic equation (23) defines two eigenemittan-ces which differ from the intrinsic emittances "12 and "34of the initially diagonal beam matrix �0. There are severalimportant properties caused by the cross-correlation matrixC. First, the product of two eigenemittances remains thesame, which directly follows from the preservation ofthe phase-space density, detðIþ CÞ ¼ 1. Also, the sumof the two eigenemittances grows with the magnitude ofany correlation since Q> 0 and detA2, detB2 � 0. Thisproperty states that the maximum-to-minimum eigenemit-tance ratio can only grow when cross correlations in thebeam matrix are introduced. As a consequence, theseschemes cannot be used for creating a bunch with equalemittances. At the same time, these schemes can be usedfor reducing the smallest eigenemittance which can bebeneficial for producing electron bunches with ultrahightransverse brightness.

C. Application of conserved moments

Using the two conservation properties of four-dimensional symplectic systems, the conservation of thedeterminant and of the trace invariant � 1

2 TrðJ�J�Þ [18],Kim [8] found for the special case where �1 ¼ �3 and theinitial uncoupled emittances are equal "12 ¼ "34 ¼ "0, andthere is an axial magnetic field at the cathode as in Eq. (16),that the eigenemittances are given by an expression equiva-

lent to "2eig;� ¼ "20 þ 2a2�41 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4a2�4

1"20 þ 4a4�8

1

q, which

can be derived from Eq. (23) and is used in the next section.The more general cases, defined in Eqs. (19) and (20),

are studied next. The nonzero hxx0i and hyy0i correlationsare inconvenient, so we can eliminate them by first trans-forming through focusing elements defined by

Rdiag ¼

1 0 0 0

��C;12=�C;11 1 0 0

0 0 1 0

0 0 ��C;34=�C;33 1

0BBBBB@

1CCCCCA; (24)

where the matrix element values are from the correlatedbeam matrices in Eqs. (19) and (20). Those correlatedbeam matrices are transformed to a matrix of the form

�beam ¼

��21 0 D B

0 ��22 E F

D E ��23 0

B F 0 ��24

0BBBBB@

1CCCCCA: (25)

Conservation of the determinant and the trace invariantleads to eigenemittances satisfying

"2eig;� ¼ U� V; (26)

where

U ¼ 12ð ��2

1 ��22 þ ��2

3 ��24 � 2BEþ 2FDÞ (27)

and

V2 ¼ 14ð ��2

1 ��22 þ ��2

3 ��24 � 2BEþ 2FDÞ2

� ð ��21 ��

22 ��

23 ��

24 � F2 ��2

1 ��23 � E2 ��2

1 ��24 �D2 ��2

2 ��24

� B2 ��22 ��

23 þD2F2 þ E2B2 � 2EBDFÞ: (28)

It is convenient to use these equations to solve for theeigenemittances numerically given correlations of theforms shown in Eqs. (19) and (20). Several nominal nu-merical results are plotted in Fig. 2. In these plots, thecorrelations b, e, and f are the most general set of corre-lations, defined by Eq. (17). The correlations b and ecorrespond to the cross-diagonal correlations as in a stan-dard FBT, and are equal in magnitude but opposite in signfor the case of an axial magnetic field and a symmetricFBT. If they are equal in magnitude and sign, they form asymplectic correlation instead of a nonsymplectic correla-tion (for example, a skew quadrupole if in x-y or a trans-versely deflecting rf cavity in x-z). The correlation d refersto a direct correlation between the coordinates (a rotatedellipse either in x-y or in x-z).In Figs. 2(a) and 2(b), the original uncorrelated diagonal

beam-matrix elements�1, �2, �3, �4 [as used inEqs. (19)and (20)] are varied, keeping the product of the emittances�1�2�3�4 ¼ 1, and for b ¼ 1, e ¼ �1 [Fig. 2(a)] andb ¼ 1, e ¼ 1 [Fig. 2(b)]. The four cases in eitherplot correspond to different initial conditions:

(i) �1 ¼ �2 ¼ �3 ¼ �4 ¼ 1 (solid line); (ii) �1 ¼ffiffiffiffiffiffiffi0:5p

,

�3 ¼ffiffiffi2p

, �2 ¼ �4 ¼ 1 (dot-dashed line); (iii) �1 ¼�2 ¼

ffiffiffiffiffiffiffi0:5p

, �3 ¼ �4 ¼ffiffiffi2p

(dashed line); and (iv) �1 ¼0:5, �3 ¼ 2, �2 ¼ �4 ¼ 1 (short dashed line). Thesecombinations were picked to consider an initially symmet-ric beam (i), a beam with an initial emittance ratio of 2 (ii),and two beamwith initial emittance ratios of 4 (iii) and (iv).These plots show that the sign of d is not important, a factproven by expandingB,D, E, and F in terms ofd. Increasingthe magnitude of d increases the eigenemittance ratio, aspreviously discussed. Although the b ¼ e ¼ 0 case is notplotted, it is identical to Fig. 2(b), showing that symplecticcorrelations like b ¼ 1, e ¼ 1 can be ignored. The effect ofthe nonsymplectic correlation b ¼ 1, e ¼ �1 by compar-ing the two plots at d ¼ 0 is very apparent. For large d, theinitial b and e do not matter much. Also, a small �1 [case(ii)] or large �2 [case (iv)] suppresses the effect of thecorrelation d, as one would expect. A final note is thatFig. 2(b), where the only effective correlation is d, isequivalent to simply rotating the coordinate system.Figure 2(c) represents the asymmetric FBT case, where

b ¼ 1, e ¼ �1, d ¼ 0, and the initial beam shape isvaried. For both curves, the divergences �2 ¼ �4 ¼ 1.For the solid line, �3 ¼ 1 and �1 is decreased, causingthe change in the initial emittance ratio. For the dot-dashedline, �1 is decreased while �3 is increased, keeping theproduct �1�3 ¼ 1. In other words, the solid line

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corresponds to the case that a round initial beam is trun-cated to an ellipse, losing charge and the product of theinitial emittance decreases. The dot-dashed line corre-sponds to an initial beam shape that is squished in onedimension but grows in the other, keeping the total chargeand product of the initial emittances constant. For thepurposes of maximizing the x-ray flux from an XFEL,

the dot-dashed curve is more relevant, but the solid curveshows an unexpected decrease in the ratio of final emittan-ces as the ratio of initial emittances is increased.At this point, we have accomplished the first of the main

results of this paper—to provide a simple algorithm tocalculate the eigenemittances from arbitrary correlationsbetween any two dimensions.

FIG. 2. Numerical modeling of the final emittance ratio for a variety of beam-matrix elements: (a) varying d while setting b ¼ 1,e ¼ �1; (b) varying d while setting b ¼ 1, e ¼ 1; (c) final emittance ratio for a nonsymmetric FBT, varying the initial emittanceratio.

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V. CONVENTIONAL x-y FBT

In a FBT, an axial magnetic field is applied at thelocation of the beam cathode, giving the beam canonicalangular momentum and generating the previously dis-cussed eigenemittances. When the beam exits the solenoid,the beam begins to rotate (shear), with b and e typecorrelations. This rotation leads to a large apparent emit-tance growth, added in quadrature to the beam’s intrinsictransverse emittances. Three skew quadrupoles then elimi-nate the rotation, thereby removing the apparent emittancecontribution. However, the b and e correlations success-fully split the eigenemittances as described in the previoussection, so once all correlations are removed with threeskew quadrupoles, one emittance is lower than anequivalent-but-unmagnetized beam would be, while theother emittance is larger (making the product the same).A schematic of this approach is shown in Fig. 3.

After the beam exits the magnetic pole piece, the beammatrix is (assuming the beam is at a waist), from Eq. (16),

�FBT¼

�2x 0 0 �a�2

x

0 �2x0 þa2�2

x a�2x 0

0 a�2x �2

x 0

�a�2x 0 0 �2

x0 þa2�2x

0BBBBB@

1CCCCCA; (29)

where, for a standard FBT, we assume �x ¼ �y and �x0 ¼�y0 . The ‘‘intrinsic emittance,’’ defined as "0 ¼ �x�x0 , is

that emittance that the beam would possess if it weregenerated without any correlations, that is, without anyaxial magnetic field. With the field, the observed emittance

at the exit of the solenoid is "beam ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"20 þL2

q, where as

before, L ¼ ðejBcathjR2cathÞ=ð8��cmÞ ¼ jaj�2

x ¼ 12 jhxy0 �

yx0ij. After the skew quadrupoles, the eigenemittances

"eig;� ¼ "202L

"eig;þ ¼ 2L (30)

are recovered in the limit L2 � "20.

Note that since the �XY ¼ hxyi hxy0ihx0yi hx0y0i

� �submatrix is

constrained by the conservation of canonical angular mo-mentum and hxyi ¼ hx0y0i ¼ 0 for an axisymmetric beam,the matrix �XY cannot change until the first skew quadru-pole. Any effect on the beam matrix due to axisymmetricnonlinearities in the electron diode (or photoinjector for amore practical case) can only occur in the �XX and �YY

submatrices. Because of this, the beam emittances beforethe first skew quadrupole can evolve arbitrarily, but aneffective intrinsic emittance "0 consistent with Eq. (30)

can always be defined by "0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"2beam �L2

q. Thus, the

FBT formulas always apply. If there is no emittance growthin the electron diode, the intrinsic emittance refers to theactual intrinsic thermal emittance during the production ofthe electron beam. If nonlinearities are present, the intrin-sic emittance more closely represents the emittance thatgrows under the presence of the nonlinear fields, for thecase of no applied axial magnetic field. However, the axialfield will modify the emittance growth, and, althoughEq. (30) is valid, it may be hard to predict what "0 actuallyis for practical designs. Despite that complexity, in [10] thelower measured eigenemittance value was smaller thaneven the beam’s thermal emittance from the cathode, asignificant verification of this technique.Earlier work [7] provided the needed symplectic transfer

matrices to recover the eigenemittances. A specific class ofoptics solutions for the symplectic flat-beam transforma-tion was found in [9] based on [6], for the condition thebeam is at a waist at the location of the first skew quadru-pole. In [9], the coordinates of a particle with zero intrinsictransverse divergence is followed through the skew quad-rupoles and drifts, and the quadrupole strengths and driftlengths are found by setting the final horizontal particleposition and divergence to both vanish at the end of thefinal skew quadrupole, which, although do not exactlyrecover the eigenemittances (similar to the solution in[6]), lead to a nearly exact recovery, where the increasein the product of the final emittances goes as "20=L

2.

FIG. 3. Conventional x-y FBT configuration—initial beam correlations are established at the cathode with an applied axial magneticfield, altering the beam’s eigenemittances, and these new eigenemittances are recovered through three skew quadrupoles.

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Let us assume that the first two skew quadrupoles areseparated by a distance M and the third quadrupole is adistance L after the second one. We use a normalizedmagnetic field parameter a ¼ e

2��mc Bcathðr0;edge=redgeÞ2and normalized quadrupole strengths of

b ¼ e

��mc

ZB0quad;1dl c ¼ � e

��mc

ZB0quad;2dl

d ¼ e

��mc

ZB0quad;3dl; (31)

for each of the three quadrupoles (and where b, d, and e nolonger refer to correlations). Using e ¼ b� a and f ¼bþ a we find three unique constraints on the quadrupoleand magnetic field strengths [9]:

1 ¼ MLfc Mc ¼ eðLþMÞc ¼ dþ eð1�MLdcÞ (32)

and six parameters we can chose to satisfy them (L,M, a, b,c, and d). Following [9], a convenient simplification can bemade if we assume the quadrupole separations are equal(M ¼ L) and if we constrain the focal length of the middlequadrupole to be twice the quadrupole separation (Lc ¼1=2). These constraints determine all the other parameters,which are now c ¼ 1

2L , d ¼ 47 c, e ¼ 1

2 c, and f ¼ 4c, which

in turn also give a ¼ � 74 c and b ¼ 9

4 c. These solutions are

the actual field and field gradient values for the axial field atthe cathode and in the quadrupoles.

Written this way, these results are trivial to generalizefor any arbitrary initial beam as long as it is both athorizontal and vertical waists and the correlations areonly between x-y0 and y-x0, as for an asymmetric FBT. Inthat case, if we let hx0yi ¼ u�2

y and hxy0i ¼ v�2x, the same

equations hold, but now with e ¼ bþ u and f ¼ bþ v. Ingeneral, it may not be possible to pick the correlations uand v, so they need to be independent variables. Becausethere are only three constraints, b can also be picked assome convenient value, such that both e and f are positive,and we can let M ¼ L again for convenience. In that case,L, c, and d are found to be

L¼ 1ffiffiffiffiffiffiffiffi2fep ; c¼ 2e; and d¼ e

1� 2L2e2: (33)

It is important to note that the standard FBT optics canbe used for a beam if u and v are negatives of each other(where u ¼ �v ¼ a as before), or consequently, for abeam of any size or ellipticity if there is an axial magneticfield at the cathode. Additionally, using Eqs. (26)–(28), wefind that

"2eig;� ¼ "20

�1þ R2

2

�þ 2a2R2�4

x ��"40

�1� R2

2

�2

þ 2a2R2�4x"

20ð1þ R2Þ þ 4a4R4�8

x

�1=2

(34)

are the eigenemittances of an initially elliptical beam withaxial field on the cathode, and where the uncorrelatedemittances are "0 and R"0, the horizontal beam size atthe cathode is �x, and the vertical beam size at the cathodeis �y ¼ R�x. Note in the limit that the angular momentum

term dominates ðjajR�2x � "0Þ, the eigenemittances are

then "eig;þ ¼ 2jajR�2x and "eig;� ¼ "20=2jaj�2

x, which re-

duce to the usual FBT case, Eq. (30), when R ¼ 1. It isinteresting to note, in this limit and for the case of a beamwhere the horizontal size is increased and the vertical size

is decreased (such that R � 1, �x ¼ �t=ffiffiffiffiRp

, �y ¼ffiffiffiffiRp

�t

and the product of the uncorrelated emittances stays con-

stant, "x ¼ "t=ffiffiffiffiRp

and "y ¼ffiffiffiffiRp

"t), that the eigenemit-

tances are "eig;þ ¼ 2jaj�2t and "eig;� ¼ "2t =ð2jaj�2

t Þ,which are independent of the aspect ratio R. Note thatthe ratio of the eigenemittances is given by "eig;þ="eig;� ¼4a2�4

t ="2t .

Here we have accomplished the second goal of thispaper—we have shown that the optics scheme used for astandard FBT can be used to recover the eigenemittancesfor any cross-diagonal beam matrix, where an initial beamhas arbitrary nonzero moments hx0yi and hxy0i, but hxyi ¼hx0y0i ¼ 0, and the beam is at a horizontal and a verticalwaist, only possibly requiring retuning the first skew quad-rupole. We have also provided an exact expression of theeigenemittances of an elliptical beam in an axial magneticfield.

VI. XZFBT DESIGNS

In this section we provide an algorithm to find the opticsneeded for nearly exactly recovering the eigenemittancesfrom any general two-dimensional correlations, in a moregeneral sense than the algorithm in the previous section.We also provide some examples of photoinjector and othergeometries that can generate these types of correlations.

A. Optics scheme for recovering eigenemittancesfor general two-dimensional correlations

We find it convenient to have the beam correlations inthe form

Rideal correl ¼

1 0 0 0

0 1 0 0

� 0 1 0

� 0 0 1

0BBBBB@

1CCCCCA (35)

(or its equivalent in terms of x as a function of y or zinstead of y or z as a function of x) to simplify thealgebra for diagonalizing the beam matrix. We will usethe same procedure employed in the previous section todetermine parameters for three additional skew quadru-poles (or transverse rf cavities) to nearly exactly diago-nalize the beam matrix. There are an infinite number ofactual optics schemes that can do this; this will be only

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one representation. For significant initial correlations, theincrease in the product of the emittances will be tiny, asin [6,9].

It is trivial to generate the correlation matrix in Eq. (35)starting with one of the form shown in Eq. (19), by using acoupling transfer matrix of the form shown in Eq. (A8) orEq. (A11), where a in those matrices equals �e fromEq. (19). Alternatively, the correlation matrix might havethe slightly different form

Rcorrel ¼

1 0 0 0

� 1 0

� 0 1 0

� 0 0 1

0BBBBB@

1CCCCCA (36)

which is equivalent to the form in Eq. (17) under simple,uncoupled, symplectic transformations. This correlationmatrix transforms into the one shown in Eq. (35) after firsttransforming the beam through a coupling matrix of theform shown in Eq. (A8) or Eq. (A11) where now a ¼ �and then through a horizontal focusing element where theinverse focal length is f ¼ ��þ �.

Now we consider a particle with initial vectorðx; x0; z; z0Þ ¼ ðx0; 0; z0 þ �x0; �x0Þ using the beam corre-lation shown in Eq. (35). Again, we are assuming a particlewith no intrinsic ‘‘divergence’’ because the FBT constraintwill require that this particle is mapped to a final horizontalposition and divergence that are zero. If we let this particlepass through three transverse rf cavities of normalizedstrengths b, c, and d in that order, with a drift of lengthL between the first two andM between the last two (to keepall the notation the same as in the previous optics cases),the final horizontal position and divergence are

xf¼x0f1þLb�þMb�þMc½�þLðbþ�Þ�gþz0½bLþMðbþcÞ�

x0f¼x0fb�þc½�þLðbþ�Þ�þd½�þðLþMÞðbþ�Þ�þcMð1þbL�Þgþz0½bþcþdð1þbcLMÞ�: (37)

Setting the four coefficients to zero leads to only threeunique constraints (as before, one can be derived from theother four), which can be written as

0 ¼ 1þ cLMðbþ �Þ 0 ¼ bLþ bMþ cM

0 ¼ ðbþ cþ dÞ þ bcdLM: (38)

Surprisingly, � has dropped out from these equations, sothe optics are independent of the x-z coupling (or x-ycoupling for that case). We have five parameters to solvethese three equations, with � given. As before, it is conve-nient to separate the rf cavities equally (L ¼ M), whichreduces the three constraints to

L2 ¼ 1

2bðbþ �Þ c ¼ �2b d ¼ bðbþ �Þ�

: (39)

Picking b ¼ � again for convenience (our last free pa-rameter), the optics for a general XZFBT are fully definedby b ¼ � , c ¼ �2� , d ¼ 2� , and L ¼ M ¼ 1=2� , wherethe rf field strength is found from these normalizedcoefficients using Eq. (A11).In the special case � ¼ 0, L cannot equal M, but any

other relation between L and M works. For example, wecan chooseM ¼ 2L, in which case the solution is given by

c ¼ �3b=2, L ¼ 1=bffiffiffi3p

, and d ¼ b=4, where b can bechosen as any convenient value.

B. Examples of correlations for XZFBTs

First, it is useful to consider what an x-z correlationmeans. If the coordinates were x and y, the eigenemittancesare trivial—it is just the beam emittances for a nonrotatedbeam in a rotated coordinate system, ð~x1; ~x2Þ. But this is nottrue for an initially correlated x-z beam because the x and zvelocities are not isotropic—we cannot just rotate anx-z correlated beam to identify and recover theeigenemittances.In [4], we considered creating the needed x-z correla-

tions with a drive laser pulse hitting a vertical cathodesurface at some angle or if the laser pulse acquires a timedelay depending on its transverse coordinate. In order todetermine the nonsymplectic correlations introduced in thebeam matrix we consider two laser pulses having the sametotal spot size and the same duration at a given transverseposition as illustrated in Fig. 4. The coordinates inside thepulses are related as

x ¼ x0; c�t ¼ c�t0 þ x0 tan; (40)

where x0 and �t0 correspond to the coordinates of thenontilted pulse and is the tilt angle of the laser pulse.To condition the beam matrix to recover the beam

eigenemittances, we assume either an identity transformthrough the photoinjector or one of the form shown inEq. (A15), and use the procedure described at the start ofthis section. Note for this special case (just a correlationbetween x and z), the optics do not depend on the ampli-tude of the correlations at all. The optics configuration for

FIG. 4. The correspondence of the photon positions betweenthe normal and tilted incident laser pulses.

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this case would look very similar to Fig. 3, where thecathode is illuminated by a titled cathode and there wouldbe four transversely deflecting rf cavities uniformly spaced(instead of three skew quadrupoles as in Fig. 3).

Alternative initial beam correlations that can be used foran XZFBT include recessing the cathode at an angle withan applied vector potential from an external wiggler fieldand using a photocathode with a work function thatchanges linearly across the cathode surface.

It is also possible to consider a two-stage FBT/ZXFBThybrid, where correlations are imposed on the beam athigher energy, using a nonsymplectic beam line element.A round beam at the cathode with a standard FBT can beused to transfer emittance from one transverse dimensionto the other. Having the beam generate an energy correla-tion with transverse position in a downstream elementcould impose an x-z0 coupling that in turns moves theemittance from that second transverse plane with the largersubsequent emittance to the longitudinal plane. The non-symplectic element may also increase the beam’s energyspread and emittance, leading to a complex performancetradeoff. If the initial transverse emittance and energyspread are labeled by initial, the induced transverse emit-tance growth (added in quadrature) and energy spread arelabeled by ind, and the rms induced energy slew correlated

with transverse position is ð��� Þslew, the eigenemittances

after such an element become [19]

"� ¼½ð��� Þ2ind þ ð��� Þ2initial�1=2

ð��� Þslewð"2x;ind þ "2x;initialÞ1=2

"þ ¼ �

���

�

�slew

�z:

(41)

Too large an induced energy spread or emittance growthlimits the ability to perform this kind of two-stageFBT/ZXFBT hybrid. Using collisional ionization orBremstrahlung radiation to generate the correlated energyslew in a wedge-shaped foil was first discussed in 1970[20] and with something similar to eigenemittances calcu-lated in 1983 [21]. Although at first glance it appears thatthe induced energy spread is too large to lead to a signifi-cant improvement in emittances, a small percentage ofparticles dominates the induced energy spread, and if oneis willing to sacrifice a significant portion of the beam (saystart with 1 nC and end with 250 pC by using interceptingoptics after the foil), significant improvements are possible[19]. An alternative approach is to use a transverselytapered wiggler field where the electrons’ incoherentsynchrotron radiation leads to the needed transverseenergy-spread slew. Nominally, a T-scale field is neededfor a 1-m long element at a beam energy of 1 GeV.

At this point, we have completed the third main point ofthis paper. We have described geometries that can lead tocorrelations between the horizontal and longitudinal

directions and we have provided a prescription for using atmost four transversely deflecting cavities to recover theeigenemittances, and, specifically, to minimize the hori-zontal emittance at the expense of the longitudinalemittance.

VII. DISCUSSION

The previous results provide options to provide ultrahighbright transverse emittances in addition to Kim’s originalsuggestion to use a FBT and an EEX with a very shortround bunch [14]. We can use the results in Sec. V todesign arbitrary x-y FBTs where the initial beam is notround or may have unequal transverse emittances, knowingthat initial nonsymplectic correlations can never bring theemittances closer. Equivalently, we can also design anequivalent FBT between one of the transverse dimensionsand the longitudinal dimension. In that case, consider ahighly elliptical photoinjector geometry, where there isabout a 22:1 ratio between the horizontal and vertical sizes(which ensure the vertical emittance already meets thenominal tiny transverse emittance constraint). With initialemittances "x;n="y;n="z;n of 3:3=0:15=1:4 �m, an XZFBT

can be designed to reach a final emittance of0:15=0:15=30 �m, using the notional correlations in theprevious section. Because the eigenemittances are con-served, we only need to start with the right correlationsat the cathode. Typical initial normalized values would be�x ¼ 10 mm and ���x0 ¼ 0:35 mrad for the horizontaldirection, and �y ¼ 1 mm and ���y0 ¼ 1:4 mrad for the

longitudinal direction. Using Eqs. (17) and (26)–(28), thisXZFBTwould require d ¼ 2:5 for an initial x-z correlationand e ¼ 3:5 �m�1 for an initial x0-z correlation, both ofwhich are achievable. There is a concern that nonlinearitiesin a photoinjector may be problematic with such a largeaspect ratio, particularly since an XZFBT does not have theconservation properties of an x-y FBT. Recent FEL simu-lations [22] indicate that final transverse emittance asym-metries as high as 4:1 may only decrease the x-ray flux byas little as 15%, which will reduce the needed initial beamasymmetry. In this case, with a cathode aspect ratio of 5:3:1(where the initial x emittance is increased from 0.7 to1:61 �m and the initial y emittance is decreasedfrom 0.7 to 0:3 �m), a laser tilt can provide the neededx-z coupling to produce an emittance split of0:3=0:075=30 �m. If a cathode with a 1-mm radius canproduce the 0:7 �m transverse emittances, with a 3.3 pslaser pulse, then with a cathode of 2.3-mm rms horizontalsize and 0.43-mm rms vertical size and with a drive laserangle 83 off normal incidence will establish these eige-nemittances. The ability to recover the eigenemittancesestablished in the manner described above depends onthe ability to minimize nonlinear effects from dilutingthe linear correlations in phase space. Emittance compen-sation has demonstrated that this can likely be done if thephotoinjector is designed carefully. The resulting bunch

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length from tilting the drive laser would be about 20 ps, andmay require a lower frequency or DC photoinjector tomaintain linearity.

For a second example, let us consider what emittancepartitioning can be done at a conventional DC photoinjec-tor, such as the gun test stand at the Jefferson NationalAccelerator Laboratory. Nominal parameters are a 5 mmradius cathode producing a beam with a normalized trans-verse emittance of 5 �m and a bunch length of 15 ps with a15 �m longitudinal emittance. The drive laser illuminatesthe cathode at a 45 angle and the phase front is typicallycorrected to remove any horizontal-longitudinal coupling.However, if the drive laser tilt is not corrected, d ¼ 1 fromEq. (17), and the horizontal and longitudinal emittances areconverted to about 3.5 and 21:5 �m, respectively. Withstronger coupling (d ¼ 2 which corresponds to an illumi-nation angle of 63.4 degrees), the converted emittances are2.2 and 33:8 �m respectively, which again indicates thatimprovements of factors of least 2 in emittance seem to bestraightforward.

As a final example, let us consider a two-stage FBT/ZXFBT hybrid as described in the previous section, withthe foil nominally at 100 MeV. This configuration has theadvantage that it is less susceptible to nonlinear fields in aphotoinjector due to the conservation properties supportedby the conventional FBT. A round cathode with anapplied axial magnetic field can be used to modify theintrinsic emittances "x;n="y;n="z;n of 0:7=0:7=1:4 �m to

3:3=0:15=1:4 �m with a conventional FBT. A wedge-shaped foil could provide a ramped energy attenuation asa function of horizontal position. If the foil is designed sothat it provides 50 keV more attenuation at one horizontalend of the bunch compared to the other (for a 1-ps longbeam with 100-�m horizontal size), an XZFBT opticssection would transfer the excess horizontal emittance tothe longitudinal plane, with a final emittance partitioningof 0:15=0:15=30 �m, again using Eqs. (17) and (26)–(28),where b ¼ 5. This technique fails if the foil generates tooexcessive an energy spread, but simulations indicate that aslong as the induced rms ‘‘transverse slice’’ energy spread isno more than 10%–20% of the energy slew, transverseemittances as low as 0:25 �m can be achieved with thisconfiguration.

More generally to minimize the effects of photoinjectornonlinearities, three-dimensional correlations may beneeded, where round cathodes can be conveniently used[23]. Just as one correlation between different dimensionscan reduce one eigenemittance at the expense of another,two correlations, if sufficiently unrelated, can reduce twoeigenemittances at the expense of the third. The two needto be ‘‘related’’ according to the table shown in Fig. 5(a)[23], where the variable in the row is correlated as afunction of the variable in the column. In this table, twocorrelations with the same color must be chosen (but notimmediately next to each other). For example, an x-y

correlation (row x, column y0, where x is a function of y)and a px-z0 correlation (where px is a function of z) areboth yellow; thus we expect the two eigenemittances to belower than the equivalent uncorrelated beam emittances.The plot in Fig. 5(b) shows the effect of these correla-

tions on the eigenemittance values. The origin, hidden inthe back, represents a totally uncorrelated beam (and thethree eigenemittances are degenerate with eigenvalues�i ¼ 1). As the x-y and px-z correlations are increased toa normalized value of 10, at the closest corner, one eige-nemittance has increased, while the other two have de-creased substantially. Not all pairs of correlations reducetwo lower eigenemittances, and, as seen in Fig. 5(a), thetwo final correlated values must be conjugates of eachother [23].Some improvement can be trivially made—keeping a

round cathode and using a slight x-z correlation with anXZFBT to just drop one of the transverse dimensions, as inthe DC gun example above, would certainly improve

FIG. 5. (a) Table of acceptable correlations coupling all threedimensions that lead to two tiny eigenemittances and one largeeigenemittance. Acceptable pairs of correlations are one fromeach of the two groups of the same color, with the black areas asexcluded, where the row index is a function of the column index.(b) Eigenemittance evolution as a function of x-y and x0-zcorrelations [using the yellow sections from part (a)]. Whenboth correlations vanish, the eigenemittances are the initialuncorrelated emittances of 1=1=1 �m.

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XFEL performance. Detailed nonlinear simulations of thedynamics in an actual photoinjector need to be made nextto evaluate the performance of these different options,including likely field errors and misalignments. It is likelythat some two-dimensional FBT configurations or fullthree-dimensional correlations will be more resistant tononlinearities and provide a path to significantly lowertransverse emittances than others.

APPENDIX: SYMPLECTIC BEAM TRANSFORMS

Using the appropriate beam units for proper treatment ofsymplectic transformations that were found in Sec. II, thefollowing is a list of common optics elements and theirtransfer matrices needed for arbitrary FBT optics configu-rations. Of particular importance, we identify the x-zequivalents of standard x-y optics elements. In particular,a chicane acts like a longitudinal drift, an rf cavity provideslongitudinal focusing when the bunch passes the cavity atthe zero phase crossing, and a transversely deflecting rfcavity provides x-z coupling in a manner analogous to thex-y coupling from a skew quadrupole.

1. Drifts in x, y, and z

The full six-dimensional transfer matrix for a drift isgiven by

Rdrift ¼

1 D 0 0 0 0

0 1 0 0 0 0

0 0 1 D 0 0

0 0 0 1 0 0

0 0 0 0 1 Dð��Þ2

0 0 0 0 0 1

0BBBBBBBBBBB@

1CCCCCCCCCCCA: (A1)

Note that there is very little longitudinal ‘‘drift,’’ and itvanishes at high energy. The longitudinal equivalent for adrift is a chicane, with transfer matrix

Rchicane ¼

1 2L 0 0 0 0

0 1 0 0 0 0

0 0 1 2L 0 0

0 0 0 1 0 0

0 0 0 0 1 2"

0 0 0 0 0 1

0BBBBBBBBBBB@

1CCCCCCCCCCCA; (A2)

where

L ¼ S11

cos30þ 2

D

cos0þ S2

2; (A3)

" ¼ S1sin20cos30

þ 2D

sin0

�sin0cos0

� 0

�; (A4)

the dipoles have width D, S1 is the distance betweendipoles in each dogleg, S2 is the distance between thedoglegs, and the dipoles have nominal bend angle 0.Combinations of quadrupoles, drifts, and chicanes canproduce arbitrary drifts in all three dimensionssimultaneously.

2. Focusing in x, y, and z

For the x-y dimension, an ideal (thin lens) quadrupole is

Rquad ¼

1 0 0 0

f 1 0 0

0 0 1 0

0 0 �f 1

0BBBBB@

1CCCCCA; (A5)

where f is the inverse focal length. Combinations of quad-rupoles and drifts can be made to provide this idealizedmatrix:

Rfocus ¼

1 0 0 0

f 1 0 0

0 0 1 0

0 0 g 1

0BBBBB@

1CCCCCA; (A6)

where f and g are arbitrary. The z analogy of a quadrupoleis an rf cavity passed at zero crossing (known as a slewcavity), with transfer matrix

Rrf cav ¼

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 b 1

0BBBBBBBBBBB@

1CCCCCCCCCCCA; (A7)

where b ¼ 2eVgap sinð!d=2�cÞ=ð�mc2dÞ, and Vgap, !,

and d are the cavity’s gap voltage, frequency, and length,respectively.

3. Coupling between two dimensions

A quadrupole rotated 45 degrees has horizontal focusingproportional to a particle’s vertical position and verticalfocusing proportional to a particle’s horizontal position.For a skew quadrupole with field gradient B0, the transfermatrix is given by

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Rskew ¼

1 0 0 0 0 0

0 1 a 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

a 0 0 0 1 0

0 0 0 0 0 1

0BBBBBBBBBBB@

1CCCCCCCCCCCA; (A8)

where a ¼ e��mc

RB0dl and the integral is taken over the

particle’s path in the quadrupole.Note that a skew-quadrupole transfer matrix transforms

the original beam matrix into

�2 ¼ Rskew�0RTskew

¼

�2x 0 0 a�2

x 0 0

0 �2x0 þ a2�2

y a�2y 0 0 0

0 a�2y �2

y 0 0 0

a�2x 0 0 �2

y0 þ a2�2x 0 0

0 0 0 0 �2z 0

0 0 0 0 0 �2z0

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA:

(A9)

Note that, although this matrix looks similar to the beammatrix originating from an axial field on the cathode[Eq. (16)], the signs are different.

A transversely deflecting rf cavity is the x-z analog of askew quadrupole. Consider an rf cavity with transversecenter ðx; yÞ ¼ ð0; 0Þ of horizontal width 2w and verticalwidth 2b with fields

Ez ¼ A sin

�x�

b

�cos

�y�

2w

�ej!t

By ¼ jA�

b!cos

�x�

b

�cos

�y�

2w

�ej!t

Bx ¼ jA�

2w!sin

�x�

b

�sin

�y�

2w

�ej!t:

(A10)

These fields lead to a transfer matrix of

Rrf ¼

1 0 0 0 0 0

0 1 0 0 a 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

a 0 0 0 0 1

0BBBBBBBBBBB@

1CCCCCCCCCCCA; (A11)

where a particle at time t ¼ 0 passes the center of thecavity when the magnetic field vanishes, and a ¼eAd �

b1

m��c .

4. Modifying correlations withsymplectic transformations

In the goal of being able to construct arbitrary FBTs, twouseful transformations are presented. The first has theability to ‘‘fix’’ antidiagonal terms in the beam matrix,and the second can fix diagonal terms. Applicable to anytwo dimensions, we start with the beam matrix:

�C0 ¼

�21 0 0 b�2

1

0 �21 a�2

3 0

0 a�23 �2

3 0

b�21 0 0 �2

4

0BBBBB@

1CCCCCA: (A12)

A symplectic coupling matrix of the form (skew quad-rupole or transversely deflecting rf cavity)

Rcoupling ¼

1 0 0 0

0 1 c 0

0 0 1 0

c 0 0 1

0BBBBB@

1CCCCCA (A13)

transforms the initial correlated beam matrix into

�C: coupled ¼

�21 0 0 ðbþ cÞ�2

1

0 �22 þ 2ac�2

3 þ c2�23 ðaþ cÞ�2

3 0

0 ðaþ cÞ�23 �2

3 0

ðbþ cÞ�21 0 0 �2

4 þ 2abc�21 þ c2�2

1

0BBBBB@

1CCCCCA (A14)

and we see we can somewhat arbitrarily adjust the anti-diagonal elements (e.g., make certain elements on the crossdiagonal vanish or become negative values of each other).Also, a transfer matrix of the form

Rmagnify ¼

M 0 0 0

0 1=M 0 0

0 0 N 0

0 0 0 1=N

0BBBBB@

1CCCCCA (A15)

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magnifies/demagnifies the 12 and 34 submatrices, withonly equivalent changes in the antidiagonal elements:

�magnified¼Rmagnify

�21 0 0 b�2

1

0 �21 a�2

3 0

0 a�23 �2

3 0

b�21 0 0 �2

4

0BBBBB@

1CCCCCART

magnify

¼

M2�21 0 0 Mb�2

1=N

0 �21=M

2 Na�23=M 0

0 Na�23=M N2�2

3 0

Mb�21=N 0 0 �2

4=N2

0BBBBB@

1CCCCCA:

(A16)

These tools may make some optics cases easier to realizeand can help visualize the effect of the optics used in themain sections of this paper.

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[20] C. Bovet, LBL Internal Report No. ERAN 89,1970.

[21] J.M. Peterson, IEEE Trans. Nucl. Sci. 30, 2403(1983).

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Marksteiner, and S. J. Russell, ‘‘Exploring MinimalCorrelation Scenarios to Produce Transversely BrightElectron Beams using the Eigenemittance Concept’’(unpublished).

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