Transverse phase space characterization in an accelerator ...

Transverse phase space characterization in an accelerator test facility

A. Wolski * and D. C. ChristieUniversity of Liverpool, Liverpool L69 3BX, United Kingdom, and the Cockcroft Institute,

Daresbury WA4 4AD, United Kingdom

B. L. Militsyn, D. J. Scott , and H. KockelberghSTFC/ASTeC, Daresbury Laboratory, Daresbury WA4 4AD, United Kingdom

(Received 23 October 2019; accepted 27 February 2020; published 10 March 2020)

The transverse phase space of a beam in an accelerator can be characterized using well-establishedmethods based on observation of changes in the beam profile between screens at different locations alongthe beamline, or on observation of changes on a single screen for different strengths of upstreamquadrupoles. Studies on CLARA FE (the Compact Linear Accelerator for Research and Applications FrontEnd, at Daresbury Laboratory, UK) show that where the beam has a complicated (nonelliptical) distributionin transverse phase space, conventional analysis techniques aimed at characterizing the beam in terms of theemittances and Courant–Snyder parameters fail to provide a good description of the beam behavior. Phasespace tomography, however, allows the construction of a detailed representation of the phase spacedistribution that provides a better model for the beam. We compare the results from three measurement andanalysis techniques applied on CLARA FE: emittance and optics measurements from observations on threescreens; emittance and optics measurements from quadrupole scans on a single screen; and phase spacetomography. The results show the advantages of phase space tomography in providing a detailed model ofthe beam distribution in phase space. We present the first experimental results from four-dimensional phasespace tomography, which gives an insight into beam properties (such as the transverse coupling) that are ofimportance for optimizing machine performance.

DOI: 10.1103/PhysRevAccelBeams.23.032804

I. INTRODUCTION

Knowledge of transverse beam emittance and opticalproperties are essential for the commissioning and perfor-mance optimization of many accelerator facilities. Thereare well-established techniques for emittance and opticsmeasurements, often based on observation of changes inbeam size in response to changes in strength of focusing(quadrupole) magnets, or observation of the beam size atdifferent locations along a beam line [1,2]. Beam phasespace tomography is also an established method forproviding detailed information about the phase spacedistribution [3–10]. In this paper, we report the resultsof studies on CLARA FE (Compact Linear Accelerator forResearch and Applications, Front End) at DaresburyLaboratory [11,12], aimed at characterizing the transversephase space of the electron beam. The results of threedifferent measurement techniques are compared, namely:

beam-size measurements at three different locations alongthe beamline (“three-screen analysis”); measurement in thechange of the beam size in response to the change inquadrupole strengths (“quadrupole scan”); and beam phasespace tomography. At the time that the studies were carriedout, the beam in CLARA FE had significant detailedstructure in the phase space distribution (i.e., the beamcould not be described by a simple elliptical phase spacedistribution). We find that in these circumstances, phasespace tomography provides important insights into thetransverse beam properties that cannot be obtained from theother techniques. Quadrupole scans can provide someuseful information, but the results from three-screen analy-sis can vary widely, depending on the precise measurementconditions. Our studies of phase space tomography includethe first experimental demonstration of beam tomographyin four-dimensional phase space [13,14]. We find thatthis technique can provide information on coupling inthe beam, which can be of value for optimizing machineperformance [15].This paper is organized as follows. In Sec. II we briefly

review the definitions that we use for the emittancesand optics functions in coupled beams, and the methodsthat we use for calculating these quantities. In Sec. IIIwe describe the measurement procedures in CLARA FE.

*[email protected]

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PHYSICAL REVIEW ACCELERATORS AND BEAMS 23, 032804 (2020)

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The three-screen analysis method is discussed in Sec. III A,where experimental and simulation results are presented.The results show some limitations of the technique, and wediscuss in particular why it does not produce reliable resultswhen the beam has a complicated structure in phase space(i.e., the beam cannot be described by a simple ellipticaldistribution). In Sec. III B we describe the quadrupole scananalysis method, including application to measurement ofthe full covariance matrix in two (transverse) degrees offreedom. The quadrupole scan technique has some advan-tages over the three-screen analysis, but neither method candetermine the detailed structure of the beam distribution inphase space. Such information can be provided by the finalanalysis technique, phase space tomography, which isconsidered in Sec. III C. We describe the implementationof phase space tomography on CLARA FE, including theuse of normalized phase space [16], and show howtomography can be applied to determine the beam distri-bution in four-dimensional phase space [14]. Simulationsare used to validate the technique, and experimental resultsare again presented. In Sec. IV we show the application ofphase space tomography to provide a detailed characteri-zation of the beam in CLARA FE under a range of machineconditions, looking at the dependence of the phase spacedistribution (including the coupling characteristics) onparameters of the electron source, including the strengthsof the focusing solenoid and bucking coil, and the bunchcharge. Given the complicated structure generally presentin the phase space distribution of the beam in CLARA FE,phase space tomography provides important insights intothe beam properties and behavior that would not beobtained from the three-screen or quadrupole scan analysistechniques. Tomography in four-dimensional phase spaceprovides, in particular, information on transverse couplingthat is of value for optimizing machine performance.Finally, in Sec. V, we summarize the key results, discussthe main conclusions, and consider appropriate directionsfor further work.

II. NORMAL MODE EMITTANCES ANDOPTICAL FUNCTIONS

Since various definitions of beam emittance are used indifferent contexts, we briefly review the definition we usefor the studies presented here, considering in particular thecase where there is coupling in the beam. For clarity,however, we begin with the case of a single degree offreedom. Considering, for example, the transverse hori-zontal direction, the covariance matrix at a specified pointin a beamline can be written:

Σ ¼� hx2i hxpxihxpxi hp2

xi

�; ð1Þ

where x represents the transverse horizontal coordinate of asingle particle at the specified location, px is the horizontal

momentum Px (at the same location) divided by a chosenreference momentum P0, and the brackets hi indicate anaverage over all particles in the beam. Note that we assumethere is no dispersion in the beamline, so that the trajectoryof the beam (nominally passing through the center of eachquadrupole) is independent of its energy: for the presentstudies in CLARA FE, since the layout from the electronsource to the end of the section where the emittancemeasurements are performed is a straight line, this willbe a good approximation.The horizontal (geometric) emittance ϵx and optical

functions (Courant–Snyder parameters βx and αx and γx)can be calculated from:

ϵx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihx2ihp2

xi − hxpxi2q

; ð2Þ

βx ¼hx2iϵx

; ð3Þ

αx ¼ −hxpxiϵx

; ð4Þ

γx ¼hp2

xiϵx

: ð5Þ

These relations imply that:

γxhx2i þ 2αxhxpxi þ βxhp2xi ¼ 2ϵx; ð6Þ

and:

βxγx − α2x ¼ 1: ð7Þ

Equation (6) defines an “emittance ellipse” in phase space.It is straightforward to extend these results to the verticaldirection, to find the vertical emittance and Courant–Snyder parameters. If the beam distribution in phase spacehas elliptical symmetry, then the Courant–Snyder param-eters and emittance are sufficient to describe the overall sizeand shape of the distribution. In general, by an “elliptical”distributionwe refer to one forwhich the phase space densityρ is a function only of the betatron action, defined (in onedegree of freedom, e.g., the transverse horizontal) by:

2Jx ¼ γxx2 þ 2αxxpx þ βxp2x: ð8Þ

For example, a Gaussian elliptical distribution can bedescribed by:

ρðJxÞ ¼N0

ϵxe−

Jxϵx ; ð9Þ

whereN0 is the total number of particles in the beam and, inthis case, using Eqs. (6) and (8), the emittance ϵx is equal tothe mean action, i.e., ϵx ¼ hJxi.

A. WOLSKI et al. PHYS. REV. ACCEL. BEAMS 23, 032804 (2020)

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Note that even for nonelliptical distributions, Eqs. (2)–(5) can be used to calculate values for the emittance andCourant–Snyder parameters. However, whether these val-ues are meaningful or useful depends on the extent to whichthe phase space distribution can be characterized purely interms of its second-order moments: the emittance andCourant–Snyder parameters are essentially a convenientway to parametrize the covariance matrix.In principle, for an elliptical phase space distribution in

one degree of freedom, the Courant–Snyder parameters andemittance at a particular location can be determined fromjust three appropriate measurements of the beam properties,for example, the rms beam size at three different locationsalong the beamline (corresponding to three differentphase angles). This is the principle behind the three-screenmeasurement technique, discussed in Sec. III A. However,if the beam distribution in phase space has a morecomplicated structure, then the distribution cannot bedescribed by just three parameters, and a larger numberof measurements will be needed to determine the distri-bution. The results presented in Sec. III show that this is thecase in CLARA FE. In such situations, a technique such asphase space tomography, discussed in Sec. III C, is neededto provide a good understanding of the beam properties andbehavior.In considering only a single degree of freedom, we

assume that there is no transverse coupling in the beamor in the beamline, so that the transverse horizontaland vertical motions may be treated independently. The4 × 4 covariance matrix for the transverse motion in thiscase takes block-diagonal form:

Σ ¼

0BBBBB@

hx2i hxpxi 0 0

hxpxi hp2xi 0 0

0 0 hy2i hypyi0 0 hypyi hp2

yi

1CCCCCA: ð10Þ

As a beam passes along a beamline, coupling in the beamcan be introduced and modified by skew components in thequadrupoles (for example, from some alignment error in theform of a tilt of the magnet around the beam axis) or from asolenoid field either at the particle source or further downthe beamline. Then, the general covariance matrix for thetransverse phase space distribution takes the form:

Σ ¼

0BBBBB@

hx2i hxpxi hxyi hxpyihxpxi hp2

xi hpxyi hpxpyihxyi hpxyi hy2i hypyihxpyi hpxpyi hypyi hp2

yi

1CCCCCA: ð11Þ

The coupling can be characterized by the values of thecross-plane elements (hxyi, hxpyi, hpxyi, and hpxpyi) in

the 4 × 4 covariance matrix, or by a generalization of theCourant–Snyder parameters (for example, using the meth-ods in [17–19]). In the absence of coupling, the elements ofthe covariance matrix (10) can be determined from sixmeasurements, such as the horizontal and vertical beamsizes at three different locations along the beamline.However, when coupling is present, the covariance matrix(11) has ten independent, nonzero elements, and tenmeasurements are therefore needed to determine all theelements. Screens at three different locations would providejust nine measurements (the beam sizes hx2i, hy2i and tilthxyi at each screen); measurement of the full 4 × 4covariance matrix can be accomplished using screens atadditional locations (for example, [10,20]). Alternatively,the beam sizes and tilt on a single screen can be measuredfor a number of different strengths of the upstream quadru-poles (see, for example, [21,22]). This is the technique usedin quadrupole scan and tomography measurements, dis-cussed in Secs. III B and III C.With coupling present, the emittance calculated using

Eq. (2) will not be the most useful quantity, since it will notbe constant as the beam travels along the beamline. Theconserved quantities where coupling is present are thenormal mode emittances (or eigenemittances) ϵI and ϵII,where �iϵI;II are the eigenvalues of ΣS, with Σ the 4 × 4

covariance matrix (11), and S the antisymmetric matrix:

S ¼

0BBB@

0 1 0 0

−1 0 0 0

0 0 0 1

0 0 −1 0

1CCCA: ð12Þ

As already mentioned, various formalisms have beendeveloped for generalizing the emittance and Courant–Snyder parameters from one to two (or more) coupleddegrees of freedom. Here, we use the method presented in[19], in which the ði; jÞ element of a covariance matrix Σ isrelated to the normal mode emittances ϵI, ϵII and corre-sponding optical functions βIij, β

IIij by:

Σij ¼Xk¼I;II

ϵkβkij: ð13Þ

The optical functions can be obtained from the eigenvectorsof ΣS. If U is a matrix constructed from the eigenvectors(arranged in columns) of ΣS, then:

ΣS ¼ UΛU−1; ð14Þ

where Λ is a diagonal matrix with diagonal elementscorresponding to the eigenvalues of ΣS. If the eigenvectorsand eigenvalues are arranged so that, in two degrees offreedom:

TRANSVERSE PHASE SPACE CHARACTERIZATION … PHYS. REV. ACCEL. BEAMS 23, 032804 (2020)

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Λ ¼

0BBB@

−iϵI 0 0 0

0 iϵI 0 0

0 0 −iϵII 0

0 0 0 iϵII

1CCCA; ð15Þ

then the optical functions are given by:

βk ¼ UEkU−1S; ð16Þ

where k ¼ I; II, and:

EI ¼

0BBB@

i 0 0 0

0 −i 0 0

0 0 0 0

0 0 0 0

1CCCA; EII ¼

0BBB@

0 0 0 0

0 0 0 0

0 0 i 0

0 0 0 −i

1CCCA: ð17Þ

The covariancematrixΣ can then be expressed in terms of thenormal mode emittances and optical functions using (13).When there is no coupling, the normal mode emittances

and optical functions correspond to the usual quantitiesdefined for independent degrees of freedom. For example,where the transverse horizontal motion is independent ofthe vertical and longitudinal motion, then:

ϵI ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihx2ihp2

xi − hxpxi2q

¼ ϵx; ð18Þ

and:

βI11 ¼ βx; βI12 ¼ −αx: ð19Þ

Since the normal mode emittances and optical functionsdepend on the eigenvalues and eigenvectors of ΣS, whereasthe “uncoupled” emittances and Courant–Snyder parame-ters are calculated from the 2 × 2 submatrices along thediagonal of the covariance matrix Σ, there is no simplerelationship between the coupled and uncoupled quantitiesin the general case.Finally, we note that if the optical functions βkij at a given

point s1 in a beamline are known, the optical functions atany other point s2 are readily computed using:

βkijðs2Þ ¼ M21βkijðs1ÞMT

21; ð20Þ

where M21 is the transfer matrix from s1 to s2 (calculated,for example, from a computational model of the beamline).The normal mode emittances and optical functions definedas described here, therefore provide convenient quantitiesfor describing the variation of the beam sizes hx2i, hy2i (andother elements of the covariance matrix) along a givenbeamline. However, as discussed earlier in this section,the elements of the covariance matrix only provide a usefuldescription of the phase space distribution in simplecases (e.g., for elliptical distributions). For complicated

distributions with significant structure, alternative ways ofdescribing the distribution may be needed.

III. MEASUREMENTS IN CLARA FE

Ultimately, CLARA is planned as a facility that willprovide a high-quality electron beam with energy up to250 MeV for scientific and medical research, and for thedevelopment of new accelerator technologies including(with the addition of an undulator section) the testing ofadvanced techniques and novel modes of FEL operation.So far, only the front end (CLARA FE) has been con-structed: this consists of a low-emittance rf photocathodeelectron source and a linac reaching 35.5 MeV=c beammomentum. The layout of CLARA FE is shown in Fig. 1.The electron source [23] consists of a 2.5 cell S-band rfcavity with copper photocathode, and can deliver short (oforder a few ps) bunches at 10 Hz repetition rate with chargein excess of 250 pC and with beam momentum up to5.0 MeV=c. The source is driven with the third harmonicof a short (2 ps full-width at half maximum) pulsed Ti:Sapphire laser with a pulse energy of up to 100 μJ. Thetypical size of the laser spot on the photocathode is of order600 μm. The source is immersed in the field of a solenoidmagnet which provides emittance compensation and focus-ing of the beam in the initial section of the beamline.A bucking coil located beside the source cancels the fieldfrom the solenoid on the photocathode in the region of thelaser spot.The studies reported in this paper are based on mea-

surements made in the section of CLARA FE followingthe linac, at a nominal beam momentum of 30 MeV=c.Measurements were made under a range of conditionsincluding various bunch charges, and different strengths ofthe solenoid and bucking coil at the electron source. Threetechniques were used, to allow a comparison of the resultsand evaluation of the benefits and limitations of thedifferent methods. The first technique, the three-screenmeasurement and analysis method (described in more detailin Sec. III A, below), is based on observations of thetransverse beam profile at three scintillating (YAG) screens,shown as SCR-01, SCR-02 and SCR-03 in Fig. 1. Thequadrupole scan (Sec. III B) and tomography (Sec. III C)methods use only observations of the beam on SCR-03,though observations on SCR-02 were also made, in order tovalidate the results. For each of the three methods, twoquadrupoles (QUAD-01 and QUAD-02) between the endof the linac and SCR-01 were used for setting the opticalfunctions of the beam on SCR-01, and were kept at fixedstrengths during data collection. As discussed later in thissection, the strengths of these quadrupoles were chosen toproduce a beam waist at SCR-02, based on a model usingthe design parameters for the upstream components(including the electron source and linac). A collimator islocated between SCR-01 and SCR-02, but this was notused during the measurements. For all three measurement


032804-4

techniques, the strengths of three quadrupoles (QUAD-03,QUAD-04, and QUAD-05 in Fig. 1) located between SCR-02 and SCR-03 were varied. For the three-screen analysis,only the beam sizes for one set of magnet strengths arestrictly needed to calculate the emittances and opticalfunctions; however, as described in Sec. III A, measure-ments with different sets of quadrupole strengths can beused to validate the results by showing the consistency ofemittance and optics values obtained for different strengths.In the case of the quadrupole scan and tomography

methods, SCR-03 provides the necessary data, and isreferred to as the “observation point.” For ease of compari-son, for all three techniques we construct the covariancematrix at SCR-02, which is referred to as the “reconstructionpoint.” Initial values for the quadrupole strengths to be usedduring a scan were chosen to provide a wide coverage (from0 to 2π) of the horizontal and vertical phase advances fromSCR-02 to SCR-03: in principle, this provides good con-ditions for reconstruction of the phase space using thequadrupole scan and tomography analysis techniques, sincethe projection of the phase space onto the coordinate axes isthen observed over a wide range of phase space rotationangles. The phase advances are calculated for given quadru-pole strengths using the nominal Courant–Snyder functionsat SCR-02, based on the design parameters of the machine.Simulations were then performed to optimize the number ofquadrupole strengths used in a scan, and the quadrupolestrengths themselves, to allow accurate reconstruction of thephase space using theminimum number of different settingsfor the quadrupoles. Reducing the number of points in a scanreduces the time needed to collect data, but has an adverseimpact on the accuracy of phase space measurements.

For the experiments reported here, each quadrupole scanconsisted of 38 sets of quadrupole strengths. The quadrupolestrengths and corresponding phase advances (from SCR-02to SCR-03) are shown in Fig. 2.

FIG. 1. Layout of CLARA FE (not to scale), showing the electron source, magnetic elements, linac, and diagnostics. For the emittancemeasurements, quadrupoles QUAD-01 and QUAD-02 were used to set the required optics at screen SCR-01. For each measurement,beam images on SCR-03 (the observation point) were recorded for different strengths of quadupoles QUAD-03, QUAD-04 and QUAD-05. The optical functions and phase space distribution were computed at SCR-02 (the reconstruction point). Screens SCR-01, SCR-02and SCR-03 are 0.61 m, 1.53 m and 3.20 m from the entrance of QUAD-01, respectively.

5 10 15 20 25 30 35Scan point index

-3

-2

-1

0

1

2

3

4

5

Foc

usin

g st

reng

th k

1L

(m-1

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pha

se a

dvan

ce

/

QUAD-03

QUAD-04

QUAD-05

x

y

FIG. 2. Strengths of quadrupoles QUAD-03, QUAD-04 andQUAD-05 used during quadrupole scans. The quadrupolestrengths are chosen to provide a wide range of phase advancesfrom SCR-02 (the reconstruction point) to SCR-03 (the obser-vation point): the horizontal and vertical phase advances areshown on the plot as solid and broken lines (respectively).


032804-5

At each point in a quadrupole scan on CLARA FE, tenscreen images were recorded on successive machine pulses(at a rate of 10 Hz, with a single bunch per pulse): thisallows an estimate to be made of random errors arisingfrom pulse-to-pulse variations in beam properties. A back-ground image was recorded without beam (i.e., with thephotocathode laser blocked), so that any constant artefactsin the beam images, for example from dark current, couldbe subtracted. The rms beam sizes were calculated byprojecting the image onto either the x or y axis, withcoordinates measured with respect to a centroid such that:

hxi ¼ hyi ¼ 0: ð21ÞAverage quantities are calculated from a beam image byintegration of the image intensity with an appropriateweighting, for example:

hxi ¼RR

xIðx; yÞdxdyRRIðx; yÞdxdy ; ð22Þ

where Iðx; yÞ is the image intensity at a given point on thescreen. The screens and cameras are specified to provideimages with a linear relation between beam intensity andimage intensity (i.e., without saturation of the images) up tobunch charges of 250 pC, with beam sizes in the rangeachievable on CLARA FE. For the measurements reportedhere, where bunch charges of up to 50 pC were used (withmost measurements at 10 pC) the beam intensity wassignificantly below the point at which saturation occurs.Between each quadrupole scan, the quadrupole magnets

were cycled over a set range of strengths to minimizesystematic errors from hysteresis. Remaining sources ofsystematic errors include calibration factors for the magnets(when converting from coil currents to field gradients),magnet fringe fields, calibration factors for the screens, andaccelerating gradient in the linac. It was found that betteragreement between the analysis results and direct obser-vations (used to validate the results) could be obtained if thebeam momentum in the model used in the analysis wasreduced slightly from the nominal 30 MeV=c. In the resultspresented here, a momentum of 29.5 MeV=c is used. Itshould also be noted that some variation in machineparameters (including rf phase and amplitude in theelectron source and the linac) is likely to have occurredduring data collection, and because of the time required tore-tune the machine it was not always possible to confirmall the parameter values between quadrupole scans.In principle, for each of the three measurement and

analysis methods, the strengths of the quadrupoles betweenSCR-02 and SCR-03 can be chosen randomly. However, ifthe profile of the beam on any of the screens becomes toolarge, too small, or very asymmetric (with large aspectratio) then there can be a large error in the calculation of therms beam size. Before collecting data, therefore, simula-tions were performed to find sets of magnet strengths, with

fixed QUAD-01 and QUAD-02 and variable QUAD-03,QUAD-04 and QUAD-05, for which the transverse beamprofiles on each of the three screens would remainapproximately circular, and with a convenient size. It isalso worth noting that, from (2), a large value of αx at agiven location can indicate a large value for hxpxi at thatlocation: calculation of the emittance then involves takingthe difference between quantities that may be of similarmagnitude, leading to a large uncertainty in the result.A further constraint, therefore, was to find strengths forQUAD-01 and QUAD-02 that would provide a beam waistin x and y (i.e., with αx and αy close to zero) at SCR-02 (theReconstruction Point). Finally, quadrupole strengths werechosen to provide a wide range of horizontal and verticalphase advances from SCR-02 to SCR-03: this is a con-sideration for the tomographic analysis, and is discussedfurther in Sec. III C. Simulations to find sets of suitablestrengths for all five quadrupoles were carried out inGPT [24], tracking particles from the photocathode (withnominal laser spot size and pulse length) to SCR-03,using machine conditions matching those planned for theexperiments.Space charge effects were included in the tracking

simulations in GPT [25,26], though these effects are onlyreally significant at low momentum, upstream of the linac.Space charge was not included in the analysis (using any ofthe three methods discussed in this section). To justify theassumption that space charge can be neglected in theanalysis, we can compare the perveance with the beamemittance. In the case of a beam with an ellipticaldistribution in phase space, the evolution of the rms beamsize σx with distance s along a beamline is described by theenvelope equation:

d2σxds2

þ k1σx −ϵ2xσ3x

−K

2ðσx þ σyÞ¼ 0; ð23Þ

where k1 is the local quadrupole focusing, and theperveance K is:

K ¼ Iβ30γ

30IA

: ð24Þ

Here, β0 is the particle velocity (scaled by the speed oflight), γ0 ¼ 1=

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 − β20

pis the relativistic gamma factor, and

IA ≈ 17.045 kA is the Alfven current. For CLARA FE, weassume representative values of 30 MeV for the beamenergy, 2 A peak current (corresponding, for example, to10 pC bunch charge in a Gaussian longitudinal distributionwith standard deviation 2 ps), beam sizes σx≈σy≈0.3mm,and normalized emittance 1 μm. With these values we findK ≈ 1.2 × 10−9, and 4ϵ2x=σ2x ≈ 1.2 × 10−8. We thereforeexpect to see space charge effects become significant onlyat bunch charges larger by some factor above the nominal


032804-6

10 pC at which most of the measurements reported herewere made.Additional nonlinear and intensity-dependent effects

may come from wakefields. Although wakefields in thelinac may be significant, in the section of CLARA FE usedfor the phase space measurements, the vacuum chamber isessentially smooth and wakefields are small. With thebunch charges used in the experiments reported here (up to50 pC) wakefield effects in the beamline following the linacare not expected to have any observable impact on thebeam.

A. Three-screen method

The three-screen analysis method is based on theprinciple that, given the rms beam size (in either thetransverse horizontal or vertical direction) at three separatelocations, and knowing the transfer matrices between thoselocations, it is possible to calculate the covariance matrixcharacterizing the phase space beam distribution. Asdiscussed in Sec. II above, this technique is appropriatewhen the beam has an elliptical distribution (or a distri-bution that is close to elliptical) in phase space, and there isno coupling. In such cases, the 2 × 2 covariance matricesof the second-order moments of the beam distribution in thetransverse horizontal and vertical phase spaces can betreated independently, and provide a good representation ofthe beam properties.The three-screen measurement and analysis technique

has the advantage over other methods (such as tomography,discussed in Sec. III C) that data collection and analysis canbe performed rapidly. CLARA FE includes a number ofscreens in appropriate locations following the linac,allowing this technique to be readily applied. In principle,the beam in CLARA FE should have a simple (close toelliptical) distribution in transverse phase space, with littleor no coupling. The distribution should be well charac-terized by the 2 × 2 transverse horizontal and verticalcovariance matrices, and the three-screen analysis tech-nique was therefore used during machine commissioning.However, it was found that the results did not provide agood description of the beam behavior, and that valuesfound for the emittances and optical functions could varysignificantly, depending on the strengths of the quadrupolesbetween the screens. Results from phase space tomographyshow that at the time the measurements were made, thebeam distribution had a complicated structure in phasespace. This is also evident from the images observedon SCR-02, which consistently show a lack of ellipticalsymmetry in the coordinate space projection at this screenfor any of the machine configurations that we studied(some examples are given, and compared with predictionsfrom the tomography analysis to be described later, inFig. 9). Here, we report the results of the three-screenmeasurements mainly to illustrate the limits of the tech-nique in this case.

Let us consider just the transverse horizontal motion: theapplication to the vertical motion is straightforward. Ifthe transfer matrix from one location in the beamline s1 toanother location s2 is M21 (with transpose MT

21), then:

Σ2 ¼ M21Σ1MT21; ð25Þ

where Σ1 is the covariance matrix at s1 and Σ2 is thecovariance matrix at s2. For the present case where weconsider motion in just one degree of freedom, thecovariance matrices and transfer matrices are all 2 × 2matrices. At a third location s3:

Σ3 ¼ M32Σ2MT32; ð26Þ

where Σ3 is the covariance matrix at s3, and M32 is thetransfer matrix from s2 to s3. The elements of the transfermatrices can be calculated using a linear model of thebeamline, with known quadrupole strengths.Given measured values of hx21i, hx22i and hx23i (from

observation of the beam images on the three screens), using(25) and (26) we can find hx2px2i and hp2

x2i from:

hx2px2i ¼ −b232hx21i þ CþC−hx22i − b221hx23i

2C−b21b32; ð27Þ

hp2x2i ¼

a32b32hx21i þ C−a21a32hx22i − a21b21hx23iC−b21b32

; ð28Þ

where a21 and b21 are (respectively) the (1, 1) and (1, 2)elements of M−1

21 , a32 and b32 are (respectively) the (1, 1)and (1, 2) elements of M32, and:

C� ¼ a32b21 � a21b32: ð29Þ

The same formulas can be applied to calculate the elementsof the covariance matrix for the vertical motion, and (asdiscussed in Sec. II) the technique can be extended usingadditional screens or by changing the strengths of selectedquadrupoles, to find the elements of the 4 × 4 covariancematrix for motion in two degrees of freedom.In applying the formulas (25)–(28) to CLARA FE, s1, s2,

and s3 correspond to the locations of screens SCR-01,SCR-02, and SCR-03, respectively (see Fig. 1). If there isno coupling in the beam, and if the distribution in phasespace has elliptical symmetry, then the results may bevalidated by repeating the measurements for differentstrengths of the three quadrupoles between screens SCR-02 and SCR-03: since the magnets upstream of SCR-02remain at constant strength, measurements made withdifferent strengths of downstream magnets should all yieldthe same values for the emittances and Courant–Snyderparameters at this screen.Figure 3 shows a typical set of results from the three-

screen analysis applied to CLARA FE, for the transverse


032804-7

horizontal and vertical directions, respectively. Elements ofthe covariance matrix [each scaled by an appropriateCourant–Snyder parameter, calculated from the covariancematrix elements using (3)–(5)] are plotted as functions ofthe phase advance from SCR-02 to SCR-03 in the respec-tive plane (corresponding to different strengths of thequadrupoles QUAD-03, QUAD-04, and QUAD-05). Inthe case of a simple elliptical distribution with no coupling,from Eqs. (3)–(5) we see that scaling the covariance matrixelements by the Courant–Snyder parameters should givevalues that are independent of the phase advance, and equalto the geometric emittance. However, the results in Fig. 3show significant variation in each of the scaled elements of

the covariance matrix over the range of the quadrupolescan: this is particularly evident in the horizontal direction,and is reflected in the values calculated for the emittanceand optics functions. In both the horizontal and the verticalplanes, a number of points in the quadrupole scan lead toimaginary values for the emittance (the covariance matrixhas negative determinant), or nonphysical negative valuesfor the covariance matrix element hp2

xi. These points (21points in the horizontal plane, and 7 in the vertical plane)are omitted from the plots in Fig. 3 and from the calculationof mean values of emittances and optics functions. Theomitted points all have phase advance greater than 1.4 radin the respective plane.As discussed in Sec. II, if the beam has a complicated

structure in phase space, then its properties and behaviorcannot be well characterized in terms of just the elements ofthe 2 × 2 (transverse horizontal and vertical) covariancematrices. To support the argument that the failure to obtainconsistent results using the three-screen analysis techniquein CLARA FE was due to the structure of the beamdistribution in phase space, we performed tracking simu-lations using an initial distribution based on that obtainedfrom a tomography analysis (presented in Sec. III C). In thesimulations, particles were tracked in a computer model ofthe beamline from SCR-02 (the reconstruction point)backward to SCR-01, and forward to SCR-03. At eachscreen, the horizontal and vertical rms beam sizes arecalculated, and the same procedure that was applied to theexperimental data was used to calculate the covariancematrix at SCR-02. The elements of the covariance matrixwere then used to calculate the emittance and opticalparameters at this point. The tracking and optical calcu-lations were repeated for different strengths of the quadru-poles, corresponding to those used in the experiment.Results for the transverse vertical plane are shown inFig. 4. For a Gaussian elliptical distribution in phase space,there are only very small variations in the calculatedcovariance matrix at SCR-02 and in the optical functions,for different quadrupole strengths (the small variations arisefrom statistical variation in the distribution, resulting fromtracking a finite number of particles). The simulation can berepeated, but using a phase space distribution withoutelliptical symmetry, instead of a Gaussian elliptical dis-tribution. Since an appropriate distribution is provided bythe tomography analysis that we present later, in Sec. III C,we use this distribution (illustrated in Fig. 7) in thesimulation for the nonelliptical case. However, we empha-size that the purpose at this point is only to illustrate theimpact of the lack of elliptical symmetry on the three-screen analysis, rather than to demonstrate any specificaspects of the tomography analysis. Using the nonellipticaldistribution in the simulations, we see much larger varia-tions in the covariance matrix at SCR-02 and in theemittance and optical functions at this point, dependingon the strengths of the quadrupoles between SCR-02 and

0.6 0.8 1 1.2

x (rad)

0.20.40.60.8

11.2

x2/

x (m

)

SCR-03

0.6 0.8 1 1.2

x (rad)

20

30

40

50

x (m

)

x = 41.42 2.40 m

0.6 0.8 1 1.2

x (rad)

2

4

6

x (m

)

SCR-02 x = 1.726 0.149 m

0.6 0.8 1 1.2

x (rad)

-0.3

-0.2

-0.1

0

x

SCR-02 x = -0.246 0.021

0.4 0.6 0.8

y (rad)

0.06

0.07

0.08

0.09

y2/

y (m

)

SCR-03

0.4 0.6 0.8

y (rad)

4.7

4.8

4.9

5

y (m

)

y = 4.85 0.10 m

0.4 0.6 0.8

y (rad)

1.22

1.24

1.26

1.28

1.3

y (m

)

SCR-02 y = 1.251 0.027 m

0.4 0.6 0.8

y (rad)

-1.8

-1.6

-1.4

-1.2

y

SCR-02 y = -1.537 0.210

FIG. 3. Emittance and optical functions measured using thethree-screen analysis technique for (left) horizontal and (right)vertical planes. The horizontal axis on each plot shows the phaseadvance between SCR-02 and SCR-03; each point represents theresults for a single set of quadrupole strengths between these twoscreens. The plots show (from top to bottom): the mean-squarebeam size observed at SCR-03 scaled by the beta function; thenormalized emittance; the Courant–Snyder beta function atSCR-02; the Courant–Snyder alpha function at SCR-02. Theemittance, beta function, and alpha function are calculated fromthe covariance matrix at SCR-02 using (27) and (28). The betafunction at SCR-03 (used for scaling the beam size) is calculatedby propagating the Courant–Snyder functions from SCR-02using the appropriate transfer matrix. Points leading to imaginaryvalues for the emittance are omitted. Error ranges show thestandard deviation across the measurements, omitting points withlarge deviation from the mean.


032804-8

SCR-03. For some quadrupole strengths, the calculatedcovariance matrix is unphysical, and it is not possible tofind real values for the emittance or optical functions. Theoverall behavior is qualitatively similar in some respectswith that seen in the experiment, Fig. 3(b). Results ofsimulations for the horizontal plane show the same behav-ior as the vertical plane, with almost no variation in theemittance or optical functions as a function of quadrupolestrength for a Gaussian elliptical phase space distribution,but large variations in the case of a more realistic phasespace distribution based on the results of the tomographyanalysis.

B. Quadrupole scan method

One of the limitations of the three-screen analysismethod described in Sec. III A is the inability to provideinformation on beam coupling. This can be overcome,

however, by combining observations of the transversebeam size at different screens for various strengths ofthe quadrupoles between the screens. If a sufficient numberof quadrupole strengths are used, then beam size measure-ments at a single screen provide sufficient data to calculatethe 4 × 4 transverse beam covariance matrix at a pointupstream of the quadrupoles. The 4 × 4 covariance matrixhas ten independent elements: in principle, just four sets ofquadrupole strengths provide twelve beam size measure-ments (values for hx2i, hy2i and hxyi for each set ofquadrupole strengths), and are more than sufficient todetermine the covariance matrix. In practice, it is desirableto use a greater number of quadrupole strength settings, tooverconstrain the covariance matrix. Although it is againassumed (implicitly) that the beam can be described by asimple elliptical phase space distribution, the quadrupolescan technique differs from the three-screen measurementtechnique in using larger number of measurements, whichallows account to be taken of coupling. Also, depending onthe details of the beam distribution in phase space, byoverconstraining the covariance matrix, it may be possibleto construct a phase space ellipse that better represents thebeam behavior.The quadrupole scan technique that we use is similar to

that presented by Prat and Aiba [22]. The theory can bedeveloped as follows. The covariance matrix Σ3 at alocation s3 in the beamline (SCR-03 in the case ofCLARA FE) is related to the covariance matrix Σ2 at alocation s2 (SCR-02 in CLARA FE) through Eq. (26),where all matrices are now 4 × 4. The relationship betweenthe observable quantities at s3 (assuming a YAG screen atthat location) and the independent elements of Σ2 can bewritten: 0

BBBBBBBBBBBBBBBBBBBBBBBB@

hx23ið1Þhx3y3ið1Þhy23ið1Þhx23ið2Þhx3y3ið2Þhy23ið2Þhx23ið3Þhx3y3ið3Þhy23ið3Þ

..

.

1CCCCCCCCCCCCCCCCCCCCCCCCA

¼ D

0BBBBBBBBBBBBBBBBBBBBB@

hx22ihx2px2ihx2y2ihx2py2ihp2

x2ihpx2y2ihpx2py2ihy22i

hy2py2ihp2

y2i

1CCCCCCCCCCCCCCCCCCCCCA

; ð30Þ

where hx23iðnÞ represents the mean square horizontal trans-verse beam size measured at s3 for a particular set ofquadrupole strengths, and similarly for hy23iðnÞ andhx3y3iðnÞ. With measurements of the beam distribution in

0.4 0.6 0.8

y (rad)

0.0846

0.0848

0.085y2

/y (

m)

SCR-03

0.4 0.6 0.8

y (rad)

4.855

4.86

4.865

4.87

4.875

y (m

)

y = 4.87 0.01 m

0.4 0.6 0.8

y (rad)

1.244

1.246

1.248

y (m

)

SCR-02 y = 1.244 0.002 m

0.4 0.6 0.8

y (rad)

-1.57

-1.56

-1.55

-1.54

y

SCR-02 y = -1.549 0.011

0.4 0.6 0.8

y (rad)

0.06

0.08

0.1

y2/

y (m

)

SCR-03

0.4 0.6 0.8

y (rad)

4.4

4.6

4.8

y (m

)

y = 4.83 0.15 m

0.4 0.6 0.8

y (rad)

1.45

1.5

1.55

1.6

y (m

)

SCR-02 y = 1.465 0.049 m

0.4 0.6 0.8

y (rad)

-2.2

-2

-1.8

-1.6

-1.4

y

SCR-02 y = -1.487 0.215

FIG. 4. Simulation of three-screen analysis technique formeasurement of vertical emittance and optical functions, using(left) a Gaussian elliptical phase space distribution, and (right) arealistic phase space distribution constructed from the results ofthe tomography analysis presented in Sec. III C. The plots showthe same quantities as the corresponding plots in Fig. 3. Note thatfor the Gaussian elliptical distribution, there is only a very smallvariation in each quantity with change in the quadrupolestrengths, but the realistic phase space distribution shows asimilar variation as the experimental case (Fig. 3). The Gaussianelliptical distribution was constructed with nominal parametervalues γϵy ¼ 4.85 μm, βy ¼ 1.25 m and αy ¼ −1.54 at SCR-02.


032804-9

coordinate space at s3 for N different sets of quadrupolestrengths,D is a 3N × 10matrix. The elements ofD can beexpressed, using Eq. (26), in terms of elements of thetransfer matrices M23 from s2 to s3 (with each set of threerows in D corresponding to a single set of quadrupolestrengths). Explicit expressions for the elements of D (for agiven transfer matrix) are as follows:

DT ¼

0BBBBBBBBBBBBBBBBBBB@

m211 m11m31 m2

31

2m11m12 m12m31 þm11m32 2m31m32

2m11m13 m13m31 þm11m33 2m31m33

2m11m14 m14m31 þm11m34 2m31m34

m212 m12m32 m2

32

2m12m13 m13m32 þm12m33 2m32m33

2m12m14 m14m32 þm12m34 2m32m34

m213 m13m33 m2

33

2m13m14 m14m33 þm13m34 2m33m34

m214 m14m34 m2

34

1CCCCCCCCCCCCCCCCCCCA

; ð31Þ

where mij is the ði; jÞ element of the transfer matrix M23

(for a given set of quadrupole strengths). Given observa-tions of the beam profile at s3 for a number of different setsof quadrupole strengths, and the corresponding values forthe elements of D, the elements of the covariance matrix ats2 may be found by inverting Eq. (30). Since D is not asquare matrix, the pseudoinverse of D (found, for example,using singular value decomposition) must be used insteadof the strict inverse.It is worth noting that whereas in one degree of freedom

it is possible to obtain the elements of the covariance matrixat the reconstruction point by varying the strength of asingle quadrupole between the Reconstruction Point andthe observation point, this is not the case in two degrees offreedom. To understand the reason for this, consider thecase of a single thin quadrupole with the reconstructionpoint s2 at the upstream (entrance) face of the quadrupole,and the observation point s3 some distance downstreamfrom the quadrupole. The elements of the covariance matrixhx23i, hx3y3i, and hy23i each have a quadratic dependence onthe quadrupole strength, with coefficients determined bythe elements of the covariance matrix at the reconstructionpoint. By fitting the quadratic curves obtained from aquadrupole scan we therefore obtain nine constraints (threefor each of the observed elements of the covariance matrixat s2); however, the covariance matrix at s2 has tenindependent elements (in two degrees of freedom). Theproblem is therefore underconstrained: in the context ofEq. (30) this is manifest as the matrix D having fewernonzero singular values than are required to determineuniquely the elements of the covariance matrix at theobservation point. Although it is always possible to “invert”D using singular value decomposition, the procedure in this

case would yield a solution for the covariance matrix thatminimizes the sum of the squares of the matrix elements:there is no reason to suppose that this least-squares matrixis near the correct solution. To address this problem,however, it is only necessary to collect data from a scanof two quadrupoles at different locations between theobservation point and the reconstruction point. This breaksthe degeneracy in the system, and (if the system is properlydesigned) more than ten singular values of D will benonzero: in other words, the system becomes overcon-strained, rather than underconstrained.The same data collected for the three-screen method can

be used in the analysis using the quadrupole scan method,and the same practical considerations (concerning, forexample, the desirability of a beam waist at the recon-struction point, and maintaining an approximately roundbeam at the observation point) apply. However, it should benoted that for the three-screen method, the observed beamsizes at all three screens are used to reconstruct the covariancematrix: an independent reconstruction is obtained for eachpoint in the quadrupole scan. For the quadrupole scananalysis method, on the other hand, we use only the observedbeam size at a single screen (SCR-03 in this case) andcombine all the measurements for different quadrupolestrengths to calculate the elements of the covariance matrix.In effect, we calculate the size and shape of the distribution inphase space based on the widths of projections at manydifferent phase angles: this leads to amore reliable result thanis obtained using the three-screen analysismethod, for whichonly three different phase angles are used. Nevertheless,even for a large number of phase angles, the quadrupolescan method does not provide the same detailed informationon the phase space distribution that is provided by thetomography method (discussed in Sec. III C). Rather, itattempts to fit a phase space distribution that may havesignificant detailed structurewith a simpleGaussian ellipticaldistribution.Figure 5 shows the residuals from a fit based on data

from a quadrupole scan in CLARA FE made with nominalmachine settings. Each point indicates the observed andfitted beam size (hx2i, hy2i or hxyi) at the observation pointfor a different set of strengths of the quadrupoles betweenSCR-02 (the reconstruction point) and SCR-03 (the obser-vation point). The results may also be validated bycomparing the beam size predicted at the reconstructionpoint with the actual beam size observed at this point. Forthe case shown, the agreement is within about 15%.To estimate the uncertainty in the elements of the

reconstructed covariance matrix, we use the residualsbetween beam sizes observed on SCR-03 (for a givenset of quadrupole strengths) and the beam sizes predictedfrom the model, using the reconstructed phase spacedistribution at SCR-02: we treat the residuals as an erroron the measured beam sizes. We then construct anensemble consisting of sets of beam sizes at SCR-03


032804-10

produced by simulating the quadrupole scan procedure.Within each set, the beam size for given quadrupolestrengths has a value chosen randomly from a normaldistribution with mean equal to the actual measured beamsize for those quadrupole strengths, and standard deviation

equal to the corresponding residual. From each simulatedquadrupole scan (i.e., for each member of the ensemble) wefind the corresponding emittance and optics functions.Finally, we assume that the uncertainty on these valuescan be found from the standard deviation of the valuesacross the ensemble. The uncertainties found in this wayare shown as the range of variation in the results from thequadrupole scans (in one and two degrees of freedom) forthe emittances and optics functions in Table I.

C. Phase space tomography

Finally, it is possible to use phase space tomography toconstruct a more detailed representation of the beamproperties than is provided by just the emittance and opticalfunctions. In principle, the tomography method is similar tothe quadrupole scan, in that by observing the beam imageon a screen for different strengths of a set of upstreamquadrupoles, it is possible to reconstruct the phase spacedistribution at a point upstream of the quadrupoles. Thedifference is that for the quadrupole scan, only the rmsbeam sizes are used in the analysis: tomography uses all theinformation from the (observed) beam distribution toproduce a more detailed reconstruction of the phase spacedistribution of the beam. When tomography is carried outin coordinate space, two-dimensional images (projections)on a screen for different orientations of an object are usedto reconstruct a three-dimensional representation of theobject. In phase space tomography, different “orientations”correspond to rotations in phase space, which are achievedby changing the horizontal or vertical phase advancesbetween the reconstruction point and the observationpoint. Mathematically, the analysis is essentially thesame as in coordinate space, and standard algorithms deve-loped for tomography in coordinate space (such as filtered

-1 0 1 2 3 4 5 6

Measured beam size (mm2)

-1

0

1

2

3

4

5

6

Fitt

ed b

eam

siz

e (m

m2)

x2

xy

y2

FIG. 5. Beam size at SCR-03 reconstructed by quadrupole scananalysis compared with the beam size observed directly on thescreen. Each point corresponds to a different set of quadrupolestrengths. Circles indicate the horizontal beam size hx2i; boxesindicate the vertical beam size hy2i; crosses indicate the hori-zontal-vertical correlation hxyi. The distances of the points fromthe dashed line (which passes through the origin with unitgradient) indicates the residuals to the fit, which are dominatedby systematic errors. The observed images are highly reproduc-ible over a number of successive machine pulses, and the randomvariation in the measured beam sizes is correspondingly verysmall on the scale of the plot.

TABLE I. Comparison of values for normalized emittances and Courant–Snyder parameters determined from different analysistechniques. The results from analysis in four-dimensional phase space are for the normal mode emittances and coupled optical functions(see Sec. II). The values for the three-screen analysis technique show the mean and standard deviation of the results of the analysis overthe range of points in the quadrupole scan, omitting points that do not return a real value for the emittance. The three-screen analysismethod neglects any detailed structure in the beam distribution in phase space, and in this case leads to unreliable results. As explained inSec. III B, the results from the quadrupole scan analysis show the mean and standard deviation of emittance and optics values obtainedfrom an ensemble of simulated measurements based on the actual measured beam sizes at SCR-03, and the residuals between thepredicted beam sizes (from the reconstructed phase space) and the actual observed beam sizes. The uncertainties in the tomographyresults are based on the standard errors on the parameters of a four-dimensional Gaussian fitted to the reconstructed phase space.

Two-dimensional phase space Four-dimensional phase space

Three-screen Quad scan Tomography Quad scanTomographymeasurement

Tomographysimulation

ϵN;x (μm) 41.4� 2.4 10.6� 1.9 2.34� 0.07 ϵN;I (μm) 12.3� 2.4 8.83� 0.03 8.26� 0.03βx (m) 1.73� 0.15 6.65� 3.44 32.9� 0.8 βI11 (m) 5.49� 2.57 10.24� 0.03 9.46� 0.05αx −0.246� 0.021 −1.36� 0.62 −2.30� 0.06 −βI12 −1.04� 0.43 −1.05� 0.01 −0.950� 0.006ϵN;y (μm) 4.85� 0.10 6.37� 2.30 4.14� 0.08 ϵN;II (μm) 4.55� 1.94 3.93� 0.01 3.85� 0.01βy (m) 1.25� 0.03 1.19� 1.77 1.94� 0.03 βII33 (m) 1.04� 1.69 5.89� 0.03 7.30� 0.03αy −1.54� 0.21 −1.40� 3.06 −2.27� 0.04 −βII34 −1.26� 2.65 −0.692� 0.003 −0.797� 0.003


032804-11

back-projection, or maximum entropy [27–29]) can beapplied to phase space tomography.For analysis of data from CLARA FE, we have used a

form of algebraic reconstruction. The procedure may beoutlined as follows. For simplicity we consider just a singledegree of freedom: the generalization to two (or more)degrees of freedom is straightforward. Let ψ be a vector inwhich each element ψ j represents the beam densityat a particular point ðxj; pxjÞ in phase space at thereconstruction point. Assuming that the points are evenlydistributed on a grid in phase space, then the projected

beam density at the observation point, ρð1Þi at a point x ¼ xiin coordinate space, can be written as a matrix multipli-cation:

ρð1Þi ¼Xj

Pð1Þij ψ j; ð32Þ

where the matrix Pð1Þ has elements:

Pð1Þij ¼

�1 if xi ¼ mð1Þ

11 xj þmð1Þ12 pxj;

0 otherwise:ð33Þ

mð1Þ11 and mð1Þ

12 are elements of the transfer matrix from thereconstruction point to the observation point, for a given setof quadrupole strengths. If the vector ρ has N elements ρi,and there are N 2 points ðxj; pxjÞ in phase space, then Pð1Þ

is an N ×N 2 matrix. If the transfer matrix from thereconstruction point to the observation point is changed(e.g., by changing the strengths of the quadrupoles betweenthe two points), then we construct a new vector ρð2Þfrom the new image at the observation point, correspondingto the new transfer matrix. In general, for the nth transfermatrix, we have:

ρðnÞi ¼Xj

PðnÞij ψ j: ð34Þ

Note that the phase space density ψ is constant, because ψrefers to a point upstream of any quadrupoles whosestrength is changed during the measurements. We cancombine the observations simply by stacking the vectorsρðnÞ and the matrices PðnÞ:

ρ ¼

0BBBBB@

ρð1Þ

ρð2Þ

..

.

ρðnÞ

1CCCCCA; and Pij ¼

0BBBBB@

Pð1Þ

Pð2Þ

..

.

PðnÞ

1CCCCCA: ð35Þ

ρ is a vector with nN elements, and P is an nN ×N 2

matrix. In terms of the pseudoinverse P† of P, we have thefollowing formula for the phase space density at thereconstruction point:

ψ j ¼Xi

P†jiρi: ð36Þ

We perform the analysis in normalized phase space [16],in which the phase space variables ðxN; pxNÞ are defined by:

�xNpxN

�¼

0B@

1ffiffiffiffiβx

p 0

αxffiffiffiffiβx

p ffiffiffiffiffiβx

p

1CA

x

px

!; ð37Þ

where αx and βx are the Courant–Snyder functions at thegiven point in the beam line. The transfer matrix innormalized phase space between any two points in thebeam line is represented by a pure rotation matrix, withrotation angle given by the phase advance. This simplifiesthe implementation of the algebraic tomography methoddescribed above. A further advantage of working in nor-malized phase space is that if the Courant–Snyder functionsat the reconstruction point are chosen to match the beamdistribution, then the beam distribution in phase space atthis point will be perfectly circular: this improves theaccuracy with which parameters such as the emittance maybe calculated. Note that, since we do not know in advancethe actual Courant–Snyder parameters describing the beamdistribution at the reconstruction point, we need to makesome estimate based on (for example) simulations or aquadrupole scan analysis. In practice, it is not essential forthe estimated parameters to match exactly the actual beamparameters: any discrepancy will simply lead to an ellipticaldistortion of the beam distribution in normalized phasespace. To transform experimental observations into nor-malized co-ordinates is straightforward: all that is necessaryis to scale the coordinate axis for the observed beamprojection by a factor 1=

ffiffiffiffiffiffiffiβOPx

p, where βOPx is the Courant–

Snyder beta function at the observation point calculatedfrom the estimated (fixed) Courant–Snyder functions at thereconstruction point and the transfer matrix from thereconstruction point to the observation point.Rather than compute the pseudoinverse of P, we solve

Eq. (34) iteratively, using a least-squares method. For thecomputation of the phase space in a single degree offreedom, we apply a constraint that the particle densitymust be positive at all points in phase space. However,applying this constraint carries a large computationaloverhead, and for computation of the phase space in twodegrees of freedom, which has considerably greater com-putational cost than the case of a single degree of freedom,we do not constrain the least-squares solver in this way.This can result in negative (unphysical) values for theparticle density at some points in phase space; however,when a good fit is achieved, the negative values make arelatively small contribution to the overall phase spacedistribution.Although there is no need for the phase advances

between the reconstruction point and observation point


032804-12

to be evenly distributed over the set of observations fordifferent quadrupole strengths, it generally improves theaccuracy of the tomography analysis to use as wide a rangeof phase advances as possible, with roughly uniformspacing: this maximizes the overall constraints on thephase space distribution for a given number of observa-tions. The sets of quadrupole strengths identified in thepreparatory simulations (described above) were specificallychosen to provide a wide range of phase advances. Thesame data (screen images at the observation point, for arange of different quadrupole strengths) can be used for thethree-screen analysis (described in Sec. III A), the quadru-pole scan analysis (described in Sec. III B) and thetomography analysis described here. Figure 6 shows aset of results from tomography analysis for the nominalmachine settings, and in which the horizontal and verticalphase spaces are treated independently. As was the case forthe quadrupole scan method, the results may be validatedby comparing the predicted beam size at the reconstructionpoint with the beam observed directly at this point (SCR-02): the results of the comparison are shown in the lowerplots in Fig. 6. There is good agreement, and it can beclearly seen that the tomography analysis reveals some

features of the charge distribution in phase space that arenot obtained from the quadrupole scan analysis. Forexample, in the horizontal phase space, we see threedistinct “peaks” in the charge density, which are associatedwith peaks in the variation of the intensity observed onSCR-02, projected onto the horizontal axis. The quadrupolescan analysis provides only the parameters of a Gaussianelliptical distribution in phase space, which would lead to asimple Gaussian variation in the intensity projected onto thehorizontal axis.Although the phase space distributions are not perfectly

elliptical, we can obtain indicative values for the emittancesand Courant–Snyder parameters using a number of differ-ent methods. For example, given a phase space distribution,it is possible to calculate the second-order moments, andthen to find the emittance and Courant–Snyder parametersusing (in one degree of freedom) Eqs. (2)–(5). However,this approach may not give useful results (in terms ofpredicting beam behavior) if the density does not fall tozero rapidly with distance from the center of the beam.Although this may be addressed by imposing a “cutoff”,where the density beyond some amplitude is set to zero, it isnot always clear where such a cutoff should be imposed.Also, it is not clear how to estimate uncertainties (or errors)on the values obtained.An alternative approach is to fit an elliptical distribution

function to the phase space density. For example, anelliptical Gaussian of the form (9) could be used. Usinga standard algorithm, such as nonlinear least-squaresregression, it is then also possible to estimate the uncer-tainty on values obtained for the emittance and Courant–Snyder parameters. This is the approach used to obtainthe values presented in Sec. IVA (Table I). A Gaussianelliptical distribution in horizontal phase space can bewritten:

ρðxÞ ¼ ρ0 exp

�−1

2xTΣ−1x

�; ð38Þ

where ρ0 is the peak density, xT ¼ ðx; pxÞ is a phase-spacevector, and the 2 × 2 symmetric matrix Σ−1 is the inverse ofthe covariance matrix. The peak density ρ0 and the threeindependent components of Σ−1 are used as variables infitting the Gaussian elliptical function to the phase spacedistribution. We use the FITNLM function in MATLAB [30],which performs nonlinear least squares regression using theLevenberg–Marquardt algorithm [31]. This provides valuesfor the variables in the fit that give the best match (in termsof minimizing the squares of the residuals) to the givendata, and the standard error on each variable. The fittingprocedure also provides an indication of the quality of thefit in terms of the coefficient of determination, or r2 value(in the range 0 to 1, with a value of 1 indicating a perfectmatch of the model to the data). Values for the emittanceand optics functions can be obtained from the elements of

-0.5 0 0.5xN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6p

xN (

mm

/m

)

-0.5 0 0.5xN (mm/ m)

0.2

0.4

0.6

0.8

1

Inte

nsity

(ar

b. u

nits

)

-0.5 0 0.5yN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

pyN

(m

m/

m)

-0.5 0 0.5yN (mm/ m)

0.2

0.4

0.6

0.8

1

Inte

nsity

(ar

b. u

nits

)

FIG. 6. Results from phase space tomography in CLARA FE,treating horizontal and vertical phase spaces separately. The topplots show the charge density in the beam at SCR-02 in thehorizontal (left) and vertical (right) phase spaces, in normalizedvariables (co-ordinates scaled by the square root of the betafunction). The black ellipses show the emittance ellipses obtainedby fitting a two-dimensional Gaussian to each phase spacedistribution. The bottom plots show the density projected ontothe horizontal (left) or vertical (right) axes: black lines are fromthe tomographic reconstruction, red lines are from direct obser-vation of the beam image on SCR-02.


032804-13

the covariance matrix Σ describing the fitted Gaussian. Inpractice, we perform the fit in normalized phase space, andapply the appropriate transformation to find the covariancematrix for the Gaussian elliptical distribution that matchesthe given distribution in ordinary (not normalized) phasespace.To estimate the uncertainty in the values obtained for the

emittances and optics functions, we construct an ensembleof matrices fΣ−1

n g, where, for each member Σ−1n of the

ensemble, the value of each element is chosen randomlyfrom a normal distribution with mean equal to the corre-sponding fitted value of Σ−1, and standard deviation equal

to the corresponding standard error. From the ensemblefΣ−1

n g we construct ensembles of values for the emittancesand optics functions: in Table I, the values obtained usingthe tomography techniques are the mean of each ensemble,with uncertainty given by the standard deviation across theensemble.Treating the horizontal and vertical phase spaces sepa-

rately in the analysis means that no information is providedon coupling in the beam, which may arise (for example)from incorrect setting of the bucking coil at the electronsource. It is possible to extend the tomography analysisfrom a single degree of freedom, to treating two degrees of

-0.5 0 0.5xN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6p

xN (

mm

/m

)

-0.5 0 0.5yN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

pyN

(m

m/

m)

-0.5 0 0.5xN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

y N (

mm

/m

)

-0.5 0 0.5pxN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6p

yN (

mm

/m

)

-0.5 0 0.5xN (mm/ m)

-0.6

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0

0.2

0.4

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pyN

(m

m/

m)

-0.5 0 0.5yN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

pxN

(m

m/

m)

-0.5 0 0.5xN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

pxN

(m

m/

m)

-0.5 0 0.5xN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6p

yN (

mm

/m

)

-0.5 0 0.5yN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

pyN

(m

m/

m)

-0.5 0 0.5yN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

pxN

(m

m/

m)

-0.5 0 0.5xN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

y N (

mm

/m

)

-0.5 0 0.5pxN (mm/ m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

pyN

(m

m/

m)

(a) (b)

FIG. 7. Projections of beam density in normalized phase space, found from phase space tomography in two degrees of freedom inCLARA FE. Each plot shows a different projection of the charge density from four-dimensional phase space, using normalized phasespace variables. The black ellipses show projections of the four-dimensional emittance ellipse obtained from a Gaussian fitted to thefour-dimensional (normalized) phase space distribution. Coupling in the beam is evident in the tilt of the charge distribution in the casesthat the axes refer to different degrees of freedom. The left-hand set of plots (a) shows the phase space distribution reconstructed fromexperimental data; the right-hand set of plots (b) shows the results of the tomography analysis applied to simulated data based on themeasured phase space distribution, to validate the technique. Although there are some differences between the analysis results from theexperimental data and the results from the simulated data, there is overall very good agreement in the phase space distribution found ineach case, and in the emittances and optical functions corresponding to fitted emittance ellipses (see Table I).


032804-14

freedom simultaneously [14]. Applying this technique tothe case considered here, the resulting four-dimensionalphase space reconstruction includes information about thebetatron coupling in the beam. Some results from exper-imental data (screen images) are shown in Fig. 7(a).Generally, the fit using Eq. (36) of the phase space beamdensity to the observed images is very good: the residualsfrom a typical example are shown in Fig. 8.The beam emittances and optics functions in two degrees

of freedom can be found from the reconstructed phasespace using a generalization of the method described abovefor a single degree of freedom, based on fitting a Gaussianto the reconstructed phase space distribution. In twodegrees of freedom, the Gaussian function is still givenby Eq. (38), but the phase space vector is nowxT ¼ ðx; px; y; pyÞ, and the matrix Σ−1 is now a 4 × 4

symmetric matrix with ten independent elements that areused as variables in the fit. Values for the normal modeemittances and optics parameters can be found from theeigenvalues and eigenvectors (respectively) of the covari-ance matrix Σ, as described in [19]. Uncertainties in thesevalues can again be obtained from an ensemble of matricesfΣ−1g, constructed using the standard errors on thevariables used in the fit.One drawback of applying phase space tomography in

two degrees of freedom is that the matrix P in Eq. (32)becomes very large: in one degree of freedom, to recon-struct the phase space distribution with resolution N ineach dimension using n observations, P will be an nN ×N 2 matrix. In two degrees of freedom (four-dimensionalphase space), P will be an nN 2 ×N 4 matrix: even for arelatively coarse reconstruction, with N of order 50,computing P and applying its inverse can require significantcomputational resources. The situation is eased somewhat

by the fact that in practice, P is a sparse matrix, and thisallows a significant reduction in the computer memory thatwould otherwise be required; nevertheless, the requiredcomputational resources can be a limit on the resolutionwithwhich the phase space in two degrees of freedommaybereconstructed. The results shownhere use a four-dimensionalphase space resolution N ¼ 69.Projections from the four-dimensional phase space

density found from experimental data in CLARA FE areshown in Fig. 7(a). To validate the technique, we take thefour-dimensional phase space distribution, and use it ina simulation to construct a set of images on SCR-03 cor-responding to different quadrupole strengths. We then takethe simulated images, and again apply the tomographyanalysis: the results are shown in Fig. 7(b). Although thereare some differences between the original and recon-structed distributions they are sufficiently close to indicatethat the technique potentially has good accuracy. We alsofind that there is good agreement between the emittancesand optics functions obtained by fitting ellipses to theprojections of the phase space into the horizontal andvertical planes (see Table I).We can further validate the results by reconstructing the

two-dimensional distribution in coordinate space at thereconstruction point (by projecting the four-dimensionalphase space distribution onto the coordinate axes), andcomparing this with the image that is observed directly.Some examples for such comparisons are shown in Fig. 9.In general, we find that the images reconstructed fromphase space tomography reproduce reasonably well thegeneral shape and some of the more detailed features of theimages that are observed directly. However, the tomogra-phy does not reveal the same level of detail as can be seen inthe observed image. This may be due in part to the limitedresolution of the tomography analysis: as already men-tioned, for the analysis presented here, we used a phasespace resolution of 69 points on each of the four axes(which was at the upper limit set by the available computermemory). However, it is also likely that measurement errorsalso play a role. We note that the residuals of the fits to theimages at the observation point are typically very small (sothat there is no discernible difference between the directlyobserved images at this point and the images reconstructedfrom phase space tomography: see the example in Fig. 8).However, there are systematic differences between thereconstructed images at SCR-02 (the reconstruction point),and the images observed directly on that screen. Inparticular, the vertical size of the reconstructed beam(projecting the phase space distribution onto the verticalaxis) is generally of order 10% larger than the vertical sizeof the image observed directly. Work is in progress tounderstand and correct the systematic errors: possiblesources include calibration errors in the quadrupoles andin the diagnostics used for collecting beam images. It isimportant to have accurate values for the quadrupole

-0.5 0 0.5

x (mm)

-1.5

-1

-0.5

0

0.5

1

1.5

y (m

m)

0.2

0.4

0.6

0.8

1

-0.5 0 0.5

x (mm)

-1.5

-1

-0.5

0

0.5

1

1.5

y (m

m)

-8

-6

-4

-2

0

2

4

6

10-8

FIG. 8. Left: typical beam image at SCR-03 (the observationpoint) for one point in a quadrupole scan. The image recon-structed from the four-dimensional phase space density foundusing Eq. (36) extended to two degrees of freedom (projecting thephase space density ψ into co-ordinate space) is visually indis-tinguishable from the observed image. Right: residuals of the fit,representing the difference between the intensity of each pixel inthe observed image, and the intensity of the corresponding pixelin the image reconstructed from the four-dimensional phase spacedensity. The beam image (left) is scaled so that the intensity variesbetween 0 (dark blue) and 1 (yellow); on this scale, the largestresiduals (right) are of order 10−7.


032804-15

-2 0 2

x (mm)

-1

-0.5

0

0.5

1

y (m

m)

-2 0 2

x (mm)

-1

-0.5

0

0.5

1

y (m

m)

-2 0 2

x (mm)

0

0.2

0.4

0.6

0.8

1

Inte

nsity

(ar

b. u

nits

)

0 0.5 1

Intensity (arb. units)

-1

-0.5

0

0.5

1y

(mm

)

-5 0 5

x (mm)

-1.5

-1

-0.5

0

0.5

1

1.5

y (m

m)

-5 0 5

x (mm)

-1.5

-1

-0.5

0

0.5

1

1.5

y (m

m)

-5 0 5

x (mm)

0

0.2

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0.8

1

Inte

nsity

(ar

b. u

nits

)

0 0.5 1


-1.5

-1

-0.5

0

0.5

1

1.5

y (m

m)

-2 -1 0 1 2

x (mm)

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-0.2

-0.1

0

0.1

0.2

0.3

y (m

m)

-2 -1 0 1 2

x (mm)

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-0.2

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0

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0.2

0.3

y (m

m)

-2 -1 0 1 2

x (mm)

0

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1

Inte

nsity

(ar

b. u

nits

)

0 0.5 1


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

y (m

m)

-1 0 1

x (mm)

-0.3

-0.2

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0

0.1

0.2

0.3

y (m

m)

-1 0 1

x (mm)

-0.3

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-0.1

0

0.1

0.2

0.3

y (m

m)

-1 0 1

x (mm)

0

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0.4

0.6

0.8

1

Inte

nsity

(ar

b. u

nits

)

0 0.5 1


-0.3

-0.2

-0.1

0

0.1

0.2

0.3y

(mm

)

(a) (b)

(c) (d)

FIG. 9. Comparison between beam images observed directly at SCR-02 and the reconstruction of the images from phase spacetomography in two degrees of freedom for a range of solenoid and bucking coil currents in the electron source and different bunchcharges in CLARA FE. Within each subfigure, the plot at the top right shows the beam image observed directly on SCR-02; the plot atbottom left shows the charge distribution in coordinate space reconstructed from phase space tomography (a projection of the four-dimensional phase space onto the coordinate plane) at the same location. The plots at top left and bottom right in each subfigure show thecharge density projected onto the horizontal and vertical axes, respectively: black lines are from the tomography analysis and red linesare from the directly observed image. (a) Main solenoid −125 A, bucking coil −2.2 A, bunch charge 10 pC. (b) Main solenoid −125 A,bucking coil −2.2 A, bunch charge 20 pC. (c) Main solenoid −150 A, bucking coil −1.0 A, bunch charge 10 pC bunch charge 10 pC.(d) Main solenoid −150 A, bucking coil −5.0 A, bunch charge 10 pC.


032804-16

strengths, since the tomographic analysis depends onknowing the betatron phase advance between thereconstruction point and the observation point, as wellas the optical functions at the observation point for givenvalues of these functions at the reconstruction point.Similarly, the analysis depends on accurate knowledgeof the calibration factors of the diagnostic screens.Hysteresis in the quadrupole magnets used to change theoptics between the reconstruction point and the observationpoint may also lead to errors in the analysis: to try tominimize hysteresis effects, the quadrupoles were routinelydegaussed (cycled) between scans, but the time taken forthis procedure made it impractical to degauss the quadru-poles at each point in a single scan.

IV. EMITTANCE AND OPTICS MEASUREMENTSUNDER VARIOUS MACHINE CONDITIONS

A. Nominal machine settings

Table I shows the emittance and optics parametersobtained under nominal machine settings using the threedifferent techniques discussed in the previous sections:three-screen measurements, quadrupole scans, and phasespace tomography. For the quadrupole scan and tomogra-phy analysis in four-dimensional phase space, the emit-tance and optics values in the table are those for the normalmode quantities as described in Sec. II. With the nominalmachine settings, the electron source and linac operate withthe beam on-crest of the rf (i.e., to give maximum beamacceleration for a given rf amplitude), with amplitudesproducing beam momentum 5 MeV=c and 30 MeV=crespectively. The current in the bucking coil is set tocancel the solenoid field on the photocathode, and the laserintensity is set to give a bunch charge of 10 pC. The resultsin Table I are based on the same data set (i.e., the same setof beam images) in each case; the only difference betweenthe different methods is in the way that the data areanalysed. In principle, therefore, for a beam with ellipticalsymmetry in the phase space distribution, we would expectto see close agreement between the values obtained usingdifferent techniques. However, while there is reasonableagreement in some of the cases for the data from CLARAFE, there is also wide variation in the values for someparameters (e.g., the horizontal normalized emittance intwo-dimensional phase space). As discussed in Sec. III A,this is likely the result of the complicated structure of thebeam distribution in phase space.For the three-screen and quadrupole scan techniques, the

uncertainties in the values shown in Table I provide anindication of the extent to which the given values describethe beam behavior: in effect, the uncertainties in these casesindicate the quality of a fit of a phase space distributionwith elliptical symmetry to the actual phase space distri-bution. For the tomography technique, however, theuncertainties are based on the standard error on the

parameters describing a Gaussian fitted to the reconstructeddistribution. The error on the parameters may be small,even if the quality of the fit is poor. A better indication ofthe quality of the fit in this case is given by the coefficientof determination, or r2. For the tomography analysis in onedegree of freedom, we find for the case shown in Table Ithat r2 ¼ 0.60 for the horizontal phase space, and r2 ¼0.80 for the vertical phase spaces. In two degrees offreedom, we find r2 ¼ 0.10. This shows a poor fit to aGaussian elliptical distribution in the four-dimensionalphase space, but rather better fits to the distributionsprojected onto the horizontal and vertical (two-dimen-sional) phase spaces.

B. Effect of varying bucking coil strength

The electron source in CLARA FE is constructed so thatthe field from the solenoid can be cancelled at the cathodeby the field from a bucking coil. If the current in thebucking coil is changed from the value needed to achievecancellation, electrons are emitted from the surface of thecathode in a nonzero solenoid field: the effect is tointroduce some coupling into the beam (as a result ofnoncompensated azimuthal momentum), which can appearas changes in the beam emittances. In particular, theindividual normal mode emittances will vary, though theirproduct should remain constant as a function of thesolenoid field strength on the cathode [32,33]. The differ-ence between the normal mode emittances is expected to beminimized when there is zero solenoid field on the cathode:with increasing field strength (parallel to the longitudinalaxis, in either direction) one emittance will increase whilethe other will decrease. Tuning the machine for optimumperformance generally involves minimizing the coupling,to achieve the smallest possible emittance ratio [15], andcharacterizing and understanding the coupling as a functionof the strength of the bucking coil is thus an important stepin machine commissioning. Four-dimensional phase spacetomography offers a powerful tool for providing insightinto coupling in the machine, and was used to study thedependence of the phase space distribution on the current inthe bucking coil.The normal mode emittances as a function of current in

the bucking coil, found from four-dimensional phase spacetomography (as described in Sec. III B) are shown inFig. 10. Although there is some variation in the productof the emittances with changes in the current in the buckingcoil, over a wide range the variation is small. There is alsosome indication of the expected behavior of the individualemittances. The difference between the emittances isminimized for a bucking coil current of approximately−3.5 A: this is different from the nominal value of −2.2 Afor cancelling the field on the cathode. The reason for thediscrepancy is being investigated. Note that before collect-ing data over the range of bucking coil currents, thebucking coil was degaussed with the intention of improving


032804-17

the agreement between the cathode field calculated from acomputer model of the electron source and the field thatwas actually produced for a given current. It is also worthnoting that the time taken for data collection over the fullrange of bucking coil currents took several hours, and it islikely that some variation in machine parameters (such as rfphase and amplitude in the electron source and linac)occurred over this time.Also shown in Fig. 10 are results from a GPT simulation

and from a simple theoretical model: these are included inthe figure to illustrate the expected behavior of the normalmode emittances as a function of the solenoid field on thecathode, and are not intended to show results from anaccurate machine model (although we see that they matchthe results from the GPT simulations very well). For thesimulations, we use parameters for the electron sourcecorresponding to those in CLARA FE, but with the fieldfrom the bucking coil scaled to cancel the solenoid field onthe cathode for a current of−3.5 A in the bucking coil (ratherthan the nominal −2.2 A). Also, the initial distributionof particles in phase space is chosen to give emittances(with zero solenoid field on the cathode) corresponding tothe experimental measurements. This requires the beam

divergence at the cathode to be scaled to exceed significantlythe values believed to be appropriate for CLARA FE;however, it should be remembered that in the simulation,the emittances are calculated immediately after the electronsource, whereas the measurements are made in a section ofbeamline downstream of the linac and numerous othercomponents. Effects (that are not yet well characterized)between the electron source and themeasurement section arelikely to lead to some increase in emittance. The GPT andtheoretical results are therefore included in Fig. 10 purely toillustrate the expected behavior of the emittances as functionsof the strengthof the solenoid field on the cathode, rather thanas a direct comparison of an accurate computational modelwith the experimental results.Also shown in Fig. 10 are results from a simplified

theoretical (analytical) model. This is based on an assumedbeam phase space distribution at the cathode, i.e., immedi-ately after photoemission. If there is no magnetic field onthe cathode, then the covariance matrix is characterized byan emittance and beta function in each transverse direction:

Σ ¼

0BBBBB@

βxϵx 0 0 0

0 ϵxβx

0 0

0 0 βyϵy 0

0 0 0ϵyβy

1CCCCCA: ð39Þ

A solenoid field of strength B0 on the cathode can berepresented by a vector potential:

A ¼�−1

2B0y;

1

2B0x; 0

�; ð40Þ

so that the canonical conjugate momenta px and py

become:

px ¼γmvx þ 1

2eB0y

P0

; ð41Þ

py ¼γmvy − 1

2eB0x

P0

; ð42Þ

wherem and e are the mass and magnitude of the charge ofthe electron, vx and vy are the transverse horizontal andvertical components of the velocity, and P0 ¼ β0γ0mc isthe reference momentum (which can be chosen arbitrarily).The covariance matrix then becomes:

Σ¼

0BBBBB@

βxϵx 0 0 ηβxϵx

0 ϵxβxþ η2βyϵy −ηβyϵy 0

0 −ηβyϵy βyϵy 0

ηβxϵx 0 0ϵyβyþ η2βxϵx

1CCCCCA; ð43Þ

-5 -4 -3 -2 -1 0Bucking coil current (A)

1

1.5

2

2.5

3

3.5

4N

orm

alis

ed e

mitt

ance

(m

)MeasurementSimulationTheoretical

FIG. 10. Normalized normal mode emittances as a function ofcurrent in the bucking coil at the electron source, with fixedbunch charge and main solenoid current. The markers on solidlines show experimental (measured) values determined fromfour-dimensional phase space tomography (as described inSec. III C), the dashed lines show values from GPT simulations,and the dotted lines show values from a simple theoretical model.The upper and lower sets of lines (blue and red, respectively)show the normalized normal mode emittances; the middle set oflines (black) show the geometric mean of the emittances.Parameters in the simulations and theoretical model are chosento fit the experimental results.


032804-18

where:

η ¼ eB0

2P0

: ð44Þ

Finally, from the covariance matrix (43), we find (usingthe methods described in Sec. II) that the normal modeemittances are given by:

ϵI;II ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiχ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiχ2 − ϵ2xϵ

2y

qr; ð45Þ

where:

χ ¼ ϵ2x þ ϵ2y2

þ 2η2βxϵxβyϵy: ð46Þ

The normalized emittances (ϵN;I ¼ β0γ0ϵI, and similarly forϵN;II) remain constant during acceleration of particles in therf field of the electron source (and in the linac). To applythis model to CLARA FE, giving the results shown inFig. 10, the initial beam size and divergence are chosen tofit the emittances at their closest approach: the values usedare close to those used in the GPT simulation. We alsoassume that η ¼ 0 for a bucking coil current of −3.5 A, andscale the dependence of η on the field in the bucking coil soas to match the experimental curves. However, we againemphasize that the results from the theoretical model and

the GPT simulation are included only to give an illustrationof the expected behavior, and are not directly comparablewith the experimental results.Direct inspection of the phase space distribution pro-

vides a further indication of how the coupling changes withthe current in the bucking coil. For example, Fig. 11 showsthe projection onto the x–py plane of the four-dimensionalphase space (reconstructed from the tomography measure-ments) for different values of the current in the buckingcoil. The “tilt” on the distribution corresponds to a cor-relation between the horizontal coordinate and verticalmomentum, and indicates the coupling: we see that thischanges sign as the bucking coil current is varied from−5 A to −1.5 A. The tilt (and hence the coupling) vanishesfor a current of approximately −3.5 A, which is consistentwith the current required to minimize the differencebetween the normal mode emittances.A more complete characterization of the coupling is

given in Fig. 12, which shows the elements of thecovariance matrix at SCR-02 (the reconstruction point)as functions of current in the bucking coil. Couplingbetween motion in the horizontal and vertical directionsis indicated by nonzero values of the elements in the 2 × 2top-right block diagonal. All these elements vanish forbucking coil currents close to −3.5 A. We do not expectthis to correspond exactly to the bucking coil current thatminimizes the separation between the normal mode

-0.5 0 0.5xN (mm/ m)

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/m

)

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pyN

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m/

m)

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pyN

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m)

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pyN

(m

m/

m)

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0

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pyN

(m

m/

m)

-0.5 0 0.5xN (mm/ m)

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0

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0.4

pyN

(m

m/

m)

-0.5 0 0.5xN (mm/ m)

-0.4

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0

0.2

0.4

pyN

(m

m/

m)

-0.5 0 0.5xN (mm/ m)

-0.4

-0.2

0

0.2

0.4

pyN

(m

m/

m)

FIG. 11. Projection of the beam distribution in normalized phase space onto the xN–pyN plane, reconstructed from four-dimensionalphase space tomography, showing variation of coupling with bucking coil current (given above each figure). The correlation (tilt)between horizontal coordinate xN and the vertical momentum pyN indicates the strength of the coupling.


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emittances, since after leaving the cathode (in zero longi-tudinal magnetic field) the particles then pass through asection of main solenoid field, not cancelled by the buckingcoil. The main solenoid field introduces some coupling inthe beam, characterized by nonzero elements off the 2 × 2

block diagonals in the covariance matrix. However,tracking simulations in GPT suggest that in the case ofCLARA FE, the coupling in the covariance matrix intro-duced by the part of the main solenoid not cancelled by thebucking coil is small: the coupling in the covariance matrixis minimized at a current within about 0.1 A of the currentthat gives the closest approach of the normal modeemittances (see Fig. 10).

C. Effects of varying main solenoidstrength and bunch charge

Although space-charge effects in CLARA FE are neg-ligible in the section of beamline where the emittance andoptics measurements are made (with beam momentumaround 30 MeV=c), space-charge forces can play a sig-nificant role in the electron source, depending on the bunchlength and the total bunch charge. In the studies reportedhere, the photocathode laser was operated with pulse length

of 2 ps: as discussed in Sec. III (where we considered thebeam perveance) space-charge effects are expected to beweak up to bunch charges of around 50 pC. However,screen images suggested some significant variation in beamparameters even at lower bunch charges. It is planned in thefuture to use phase space tomography for rigorous studiesof the impact of bunch charge (and other parameters) onbeam properties; but so far, the limited time available forcollecting quadrupole scan data, together with some vari-ability in the machine conditions, has made it impractical tomake detailed, systematic measurements. Nevertheless, toprovide some information on beam behavior, quadrupolescans were performed for bunch charges of 10 pC, 20 pC,and 50 pC, and for a reduced main solenoid current of125 A, as well as the nominal 150 A. Some of the resultsfrom analysis of these quadrupole scans using phase spacetomography in two degrees of freedom are shown in Fig. 9,which compares the reconstructed beam image at SCR-02with the image observed directly on this screen.The images in Fig. 9 indicate significant detailed

structure in the beam distribution, depending on mainsolenoid current, bucking coil current, and bunch charge.This is apparent both from the image observed directlyat the reconstruction point, and from the phase space

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FIG. 12. Elements of the covariance matrix at SCR-02 (the reconstruction point) as functions of current in the bucking coil. Blue solidlines show the values determined from four-dimensional phase space tomography; red dashed lines show the values calculated from thedirectly observed image of the beam on SCR-02. Note that in the case of the tomography analysis, the elements describing couplingbetween the transverse degrees of freedom (shown in the top right 2 × 2 block diagonal) vanish for a bucking coil current ofapproximately −3.5 A: this is consistent with the current at which the difference between the normal mode emittances is minimized (asshown in Fig. 10).


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distribution constructed from four-dimensional tomogra-phy. In such cases, the phase space cannot accurately becharacterized simply by the emittances and optical func-tions that describe the covariance matrix. Nevertheless, toallow some comparison, we calculate the normal modeemittances and optical functions, using the methoddescribed in Sec. II: the values of the normal modeemittances and selected optical functions are shown inTable II. Although there are indications of some patterns(for example, an increase in emittance with bunch charge)no firm conclusions can be drawn because machineconditions between different quadrupole scans were notaccurately reproducible. Nevertheless, the measurementsthat have been made demonstrate the potential value offour-dimensional phase space tomography for developingan understanding of the beam physics in a machine such asCLARA FE, and for tuning the machine for optimumperformance.

V. SUMMARY AND CONCLUSIONS

We have presented the first experimental results fromfour-dimensional phase space tomography in an acceler-ator. The beam emittance and optical properties obtainedfrom phase space tomography have been compared withresults obtained using more commonly employed tech-niques, such as three-screen analysis and quadrupole scans.We considered the suitability of each method for situationswhere the beam contains detailed structures in phase spaceand cannot be described by a simple elliptical distribution.In this case, the three-screen method can give inconsistentand often unphysical results. By contrast, the quadrupolescan method provided approximate values for the emittan-ces which appear broadly to agree with those obtainedusing tomography (see Table I). However, a properdescription of a nonelliptical phase space distributioncannot be given just in terms of a small number ofparameters (emittance and Courant-Snyder parameters).Phase space tomography overcomes this limitation byproviding the beam density at a number of points in phasespace.

Our results for the phase space tomography analysis andthe comparisons with other methods are supported bysimulation studies. The results of the tomography analysishave been validated by comparing (for example) the beamimage at the entrance of the measurement section of thebeamline (the reconstruction point) obtained from a pro-jection of the measured four-dimensional phase space, withthe beam image observed directly on a screen at this point.In general, the agreement suggests that four-dimensionalphase space tomography is providing a useful representationof the beam properties, though the image reconstructed fromtomography lacks the same resolution as the image observeddirectly. There is also evidence for systematic errors in themeasurement that need to be properly understood.A benefit of four-dimensional phase space tomography

(compared to tomography in two-dimensional phase space)is that the technique provides detailed information oncoupling in the beam. This can be important for tuning amachine such as CLARA FE, for example, where solenoidsare used to provide focusing for thebeam, but it is desirable tominimize the coupling that can be introduced by thosesolenoids. Information on coupling can be obtained byapplying the quadrupole scan method in two (transverse)degrees of freedom; but information obtained in this waymay not be accurate or reliable if there is detailed structure inthe beam distribution.The main drawback of the tomography analysis is that

collection of the data may be a time-consuming procedure.In cases where the beam distribution in phase space issmooth and without significant detailed structure (so that itcan be well characterized by the emittance and opticalfunctions) then the three-screen or quadrupole scan tech-niques, using a limited set of observations, may providesufficient information for machine tuning and optimizationrelatively quickly. Phase space tomography generallyrequires data from a larger number of observations, butdepending on the level of detail or accuracy required, itmay be possible to minimize the number of points in thequadrupole scan used to provide the data: the limits of thetechnique have still to be rigorously explored, and willlikely depend on the specific machine to which it is applied.

TABLE II. Beam emittances and optics parameters with different bunch charges and main solenoid strengths, withbucking coil current at the nominal −2.2 A. Note that measurements for some settings of the bunch charge and mainsolenoid strength were repeated (in particular: 10 pC bunch charge and −125 A main solenoid current; and 50 pCbunch charge and −150 A main solenoid current), to indicate the reproducibility of the tomography analysis forgiven machine settings.

Bunch charge (pC) Main solenoid current (A) ϵN;I (μm) ϵN;II (μm) βI11 (m) βII33 (m) βI33 (m) βII11 (m)

10 −125 6.02 3.09 12.5 1.92 0.394 2.2010 −125 10.2 6.33 11.8 1.45 0.368 3.3910 −150 3.75 1.37 8.92 0.417 0.157 8.3720 −125 13.7 5.93 15.8 3.52 0.287 3.9720 −150 8.88 3.33 14.3 1.10 0.065 1.5050 −150 8.98 4.47 18.8 0.981 0.121 3.2850 −150 9.25 4.59 18.8 1.00 0.114 2.88


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Regarding practical application of phase space tomog-raphy, it is worth mentioning that the requirements in termsof beamline design and diagnostics capability are notdemanding. In CLARA FE, the diagnostics section consistsof a short (1.661 m) section of beamline between twotransverse beam profile monitors, and containing three(adjustable strength) quadrupoles. The design of thissection was developed before detailed plans were preparedfor phase space tomography studies, and there is limitedflexibility in optimizing the phase advances and opticalfunctions over the length of the diagnostics section.Nevertheless, it was possible to identify sets of quadrupolestrengths to provide the observations necessary for theanalysis and results presented here.So far, we have used an algebraic reconstruction tech-

nique for the phase space tomography. This technique hasthe advantage (compared to other tomography algorithms)of ease of implementation and flexibility in terms of theinput data. However, it is possible that different algorithmsmay provide better (more accurate, or more detailed)results, and we hope to explore the possible benefits andlimitations of alternative tomography methods. A particularissue with tomography in four-dimensional phase space isthe demand on computer memory for processing the dataand storing the results, especially at high resolution inphase space. However, because of the nature of theproblem, the memory requirements will almost inevitablyscale with the fourth power of the phase space resolution,and it seems unlikely that other tomography methodswould provide significant benefit in this respect. It ispossible that more sophisticated computational techniquesmay allow some reduction in the memory requirements fora given resolution, e.g., [34].While improvements and refinements in the technique

are planned, the results so far show that four-dimensionalphase space tomography is a useful technique for detailedbeam characterization and for machine tuning and opti-mization. It is hoped that further studies will includeinvestigation of space-charge effects in the electron sourceand beam dynamics effects (such as wake fields) in thelinac.

ACKNOWLEDGMENTS

We would like to thank our colleagues in STFC/ASTeCat Daresbury Laboratory for help and support with variousaspects of the simulation and experimental studies ofCLARA FE. This work was supported by the Scienceand Technology Facilities Council, UK, through a grant tothe Cockcroft Institute.

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